Research ArticleChinese Stock Index Futures Price FluctuationAnalysis and Prediction Based on ComplementaryEnsemble Empirical Mode Decomposition

Ruoyang Chen1 and Bin Pan2

1School of Economics amp Trade Hunan University Changsha Hunan 410082 China2Financial Research Institute Wenzhou University Wenzhou Zhejiang 325035 China

Correspondence should be addressed to Bin Pan panbinwzueducn

Received 8 January 2016 Accepted 24 May 2016

Academic Editor Xiaodong Lin

Copyright copy 2016 R Chen and B PanThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Since the CSI 300 index futures officially began trading on April 15 2010 analysis and predictions of the price fluctuations ofChinese stock index futures prices have become a popular area of active research In this paper the Complementary EnsembleEmpirical Mode Decomposition (CEEMD) method is used to decompose the sequences of Chinese stock index futures pricesinto residue terms low-frequency terms and high-frequency terms to reveal the fluctuation characteristics over different timescales of the sequences Then the CEEMD method is combined with the Particle Swarm Optimization (PSO) algorithm-basedSupport Vector Machine (SVM) model to forecast Chinese stock index futures prices The empirical results show that the residueterm determines the long-term trend of stock index futures prices The low-frequency term which represents medium-term pricefluctuations ismainly affected by policy regulations under the analysis of the IteratedCumulative Sums of Squares (ICSS) algorithmwhereas short-termmarket disequilibrium which is represented by the high-frequency term plays an important local role in stockindex futures price fluctuations In addition in forecasting the daily or even intraday price data of Chinese stock index futures thecombination prediction model is superior to the single SVM model which implies that the accuracy of predicting Chinese stockindex futures prices will be improved by considering fluctuation characteristics in different time scales

1 Introduction

As a powerful financial tool stock index futures can helpcurb abnormal stockmarket fluctuations and effectively avoidinvestment risk by virtue of the primary functions of pricediscovery hedging and arbitrage moreover this tool iswidely used in both developed and developing countriesThe CSI 300 index futures officially began trading on April15 2010 which was a significant milepost in the Chinesecapital markets However the Chinese stock index futuresmarket remains in an emergent state lacking mature marketmanagement mechanisms thus shocks (such as governmentpolicies excessivemarket speculation) often inducewild fluc-tuations in stock index futures prices Simultaneously theseprice fluctuations can spread rapidly and be amplified bycertain trading mechanisms including margin trades two-way trades andmandatory liquidations which can easily lead

tomarket riskTherefore effectively analyzing and accuratelyforecasting fluctuations in Chinese stock index futures pricesis of great theoretical and practical significance for promotingthe healthy development of the capital markets

Many analyses of stock index futures prices frequentlyuse traditional statistical and econometric techniques Forexample Meneu and Torro [1] studied volatility spilloversbetween the spot and futures markets in Spain using asym-metric multivariate GARCH structures Zhong et al [2]investigated the relationship between Mexican stock indexfutures and the spot market using a modified EGARCHmodel and cointegration test Xiong et al [3] analyzedthe long-term and short-term price discovery function ofSGX FTSEXinhua China A50 Index Futures and A-ShareMarket using a cointegration test error correction modeland the impulse response function Moreover using high-frequency CSI 300 stock index and futures data in China

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 3791504 13 pageshttpdxdoiorg10115520163791504

2 Mathematical Problems in Engineering

Zuo et al [4] investigated the Granger causality and long-run equilibrium between daily realized variance and realizedbipower variation and jump variation Based on five-minutehigh-frequency data Chen and Zhang [5] researched theeffects of the CSI 300 index futures trading on the jump riskof the spot market by employing a nonparametric methodand the Granger causality test However there are some keydrawbacks with respect to these traditional models On onehand many factors (including economic fundamentals theeffects of shocks from significant economic and financialevents and investorsrsquo risk preferences [6]) stir up pricefluctuations in stock index futures at different time scalesPeters [7] suggested that traders in the financialmarket woulddisperse and reflect various time scales due to differences ininvestment philosophy ultimately engaging in transactionsin different investment time scales short-term medium-term and long-term As a new product in an emergingmarket Chinese stock index futures have attracted a varietyof investor types and their different investment strategies alsohave a great impact on stock index futures price fluctuationsin different time scales However traditional models conducttheir analyses based solely on the entire price series and theycannot explain the inner driving forces of price fluctuations[8] On the other hand stock index futures price seriesare nonlinear and nonstationary financial time series whiletraditional models are based on the assumption of lineardata and obey a normal distribution thus these models areunable to adequately capture the nonlinear patterns in theprice series

The Empirical Mode Decomposition (EMD) method [9]and the Ensemble EMD method [10] are effective tools toaddress nonlinear nonstationary data and these methodscan analyze the fluctuation characteristics of a time seriesindifferent time scales by decomposing them into IntrinsicMode Functions (IMFs) with independent information Inparticular the EEMD method is a substantial improvementover the EMDmethod and overcomes the drawback of modemixing in the EMD method by adding white noise Thusthe EEMD method is popular and has been widely used andapplied in the financial sector For example Zhang et al[8] used the EEMD method to decompose crude oil pricesinto a trend with a slowly varying part and a fluctuatingprocess and then analyzed the long-term medium-termand short-term fluctuation characteristics of crude oil pricesRuan and Bao [11] applied the EEMD method to decomposehousing prices into a trend term a low-frequency term and ahigh-frequency term to reveal the intrinsic characteristics offluctuations in housing prices Li and Feng [12] employed theEEMD method to separately decompose investor sentimentand the stock index price series into a short-term fluctuationterm amedium-term significant event term and a long-termtrend term integrating econometric models to expand theirresearch However the EEMD method cannot effectivelyoffset the residue noise from the addedwhite noiseThereforeYeh et al [13] further proposed the Complementary Empiri-cal Mode Decomposition (CEEMD) method This methodwhich has been used increasingly in recent years [14 15]provides a fast and accurate method for processing data thatnot only can address the mode mixing in the EMD method

but also can completely offset the added residual white noisein the decomposition process that are generated by the EEMDmethod However this latest data processing method has notyet been applied to the stock index futures market

In addition by combining with other prediction modelsEMD methods can also improve the predictive precision ofa single prediction model For example Yu et al [16] builtan EMD-Fuzzy Neural Network- (FNN-) Adaptive LinearNeural Network (ALNN) model to forecast crude oil pricesthis model yielded superior predictive results in comparisonwith either a single FNN model or a single ALNN modelIn addition Yang et al [17] built an EMD-SVM-SVM modelto forecast crude oil prices and also improved the predictiveresults of a single SVM model Meanwhile The SupportVector Machine (SVM) model [18] is now widely applied inthe stock and stock index futures markets Huang et al [19]used the SVM model a Back Propagation Neural Network(BPNN) Linear Discriminant Analysis (LDA) and othermethods to make predictions regarding the Nikkei 225 Indexand found that the SVMmodel exhibited the highest predic-tive performance among these prediction models Ince andTrafalis [20] demonstrated that the SVM model has greaterprecision in predicting stock prices than the MultilayerPerception (MLP) and Autoregressive Integrated MovingAverage (ARIMA)methods Using five real futures contractssuch as the Standard amp Poor 500 stock index futures Tayand Cao [21] examined the feasibility of SVM modeling bycomparing it with a BP neural network these authors showedthat SVMs are better for forecasting futures series Sai etal [22] optimized four SVM models with different kernelfunctions based on the Genetic Algorithm (GA) and theParticle SwarmOptimization (PSO) algorithm these authorsbuilt eight different programs to forecast Chinese stock indexfutures prices and discovered that the linear kernel functionSVM model based on the PSO algorithm yielded superiorpredictive results for Chinese stock index futures pricesThe above papers on the SVM prediction model bolsterthe conclusion that the SVM model has been used in thestock index futures market although mainly as a single SVMmodel

To improve the analysis of Chinese stock index futuresprices and the ability to forecast them this paper first drewideas from Zhang et al [8] and other scholars Then we usedthe CEEMD method to decompose the Chinese stock indexfutures price series into a residue term a low-frequency termand a high-frequency term Finally we analyzed the long-term medium-term and short-term intrinsic fluctuationcharacteristics of Chinese stock index futures prices Inaddition substantial fluctuations in the low-frequency termare related to the impact of significant events To define theevents that play a role in the low-frequency term we useIterated Cumulative Sums of Squares (ICSS) [23ndash25] to detectthe breakpoints in the low-frequency term Then we focuson the problem of forecasting Chinese stock index futuresprices by referring to Yu et al [16] and Yang et al [17] toconstruct a combination prediction model and to forecastChinese stock index futures prices by building a PSO-based[26ndash28]CEEMDand SVMcombination predictionmodel Inaddition although previous studies on stock and stock index

Mathematical Problems in Engineering 3

futures markets are based on low-frequency daily weeklyand monthly price data [1ndash3 19ndash22 29] current researchregarding intraday high-frequency data gradually becamea focus [4 5 30] Therefore we also use this model toforecast the daily data and the intraday high-frequency dataof Chinese stock index futures prices The main innovationof this paper is the use of the CEEMD method to reveal theintrinsic fluctuation characteristics of Chinese stock indexfutures prices from various time scales which provides a newresearch perspective on the fluctuation analysis of stock indexfutures pricesThefluctuation characteristics of various scalesare introduced into predicting stock index futures prices thusimproving the predictive ability of a single SVM model withrespect to the stock index futures market

The remainder of this paper is structured as followsSection 2 introduces themethods used in this paper Section 3analyzes the Chinese stock index futures prices at varioustime scales Section 4 forecasts Chinese stock index futuresprice Finally Section 5 provides a summary

2 Introduction of Methodology

21 The CEEMD Method EMD methods such as the EMDmethod the EEMD method and the CEEMD method areself-adaptive data processing methods that can decompose atime series into a series of Intrinsic Mode Functions (IMFs)with independent informationThese IMFs are effective toolsto address nonlinear nonstationary data In this paper weemploy the latest extension that is the CEEMD methodto analyze Chinese stock index futures prices The detailedintroduction of the CEEMDmethod is stated as follows

The EMD method proposed by Huang et al [9] is thefirst method to extract series of IMFs and residue terms fromthe original time series In particular IMFs must satisfy thefollowing two conditions (1) the number of extrema andzero-crossing pointsmust be equal or differ by amaximumofone and (2) the mean value of the upper and lower envelopesmust be zero at all times Accordingly decomposition of theEMD method ends when the last residue term (abbreviatedas res) 119903(119905) is a monotonic function or a constant then theoriginal series 119909(119905) can be expressed as

119909 (119905)=119899

sum119894=1

119909119894(119905) + 119903 (119905) (1)

where IMF119894is recorded as 119909

119894(119905) Because mode mixing

can easily emerge in the EMD method Wu and Huang[10] further proposed the EEMD method to address thisdrawback of the EMD method The EEMD method assumesthat the observed data are amalgamations of true time seriesand noise and that the ensemble means of data with differentnoises are closer to true time series Hence adding whitenoise as an additional step may help extract the true IMFsin the time series and the noise can be offset by ensembleaveraging However residue noise continues to be generatedby the added white noise Therefore Yeh et al [13] thenproposed theCEEMDmethod by adding pairs of positive andnegative white noise to generate complementary sequences

Specifically the complementary sequences are obtained bythe following equation

[119909+

119894(119905)

119909minus119894(119905)

] = [1 1

1 minus1] [

119909 (119905)

120576119894(119905)

] (2)

In (2) 120576119894represents 119894th added white noise and 119909+

119894(119905)

and 119909minus119894(119905) represent the sequences after positive and negative

white noise is added for 119894th time to the original 119909(119905) seriesWe can generate complementary IMFs via the EMDmethodand these are expressed as follows

119909+

119895(119905) =

1

119899

119899

sum119894=1

119909+

119894119895(119905)

119909minus

119895(119905) =

1

119899

119899

sum119894=1

119909minus

119894119895(119905)

(3)

Then the final IMFs with no residue noise are extractedby averaging the two complementary IMFs

119909119895=

(119909+119895+ 119909minus119895)

2 (4)

22 ICSS Algorithm The ICSS algorithm proposed by Inclanand Tiao [23] is a relatively mature approach to detectingbreakpoints in sequences The algorithm assumes that in thestable time series 119884

119894119905= 120583119894+ 120576119894119905 120583119894denotes the mean return

for time series 119894 and 120576119894119905 is the residual series with a mean

of 0 and a variance of 1205902119894119905 Letting 119862

119896= sum119896

119894=11205762119905 119896 = 1 119879

be the cumulated sum squares of the residual series 120576119905 the

statistic 119863119896can be defined as follows

119863119896= (

119862119896

119862119879

) minus (119896

119879) 119896 = 1 119879 119863

0= 119863119879

= 0 (5)

If breakpoints do not occur in the variance in the iterationprocess the value of the 119863

119896statistic will vary around 0 By

contrast if there are one or more breakpoints in a sequencethe value of the statistic 119863

119896will deviate significantly upward

or downward from the 0 Let 119896lowast be the value for 119870 atmax119896|119863119896| If max

119896radic(1198792)|119863

119896| exceeds the preset credible

interval 119896lowast will be taken as an estimated break point whereradic(1198792) is a normalizing factor

23 SVM Prediction Model The SVM model employs akernel-based method in which the basic idea is to construct ahyperplane with low risk in high-dimensional feature spaceThis method has the advantage of generalization capabilitiesand good function approximation Among the various typesof SVM models the 120576-SVM model proposed by Vapnik [18]is mainly used to resolve prediction problems Given a set oftraining patterns (119909

1 1199101) (119909

119894 119910119894)119899119894=1

where 119909119894isin 119877 is the

input variable and119910119894isin 119877 is the output variable the prediction

model is learned from these patterns and is used to forecast

4 Mathematical Problems in Engineering

the output variables of unseen input variables Specifically theconstraints and objective function for the 120576-SVMmodel are

min120596119887120591120591

lowast

120596119879120596

2+ 119888

119899

sum119894=1

120591119894+ 119888

119899

sum119894=1

120591lowast

119894

st 120596119879

120601 (119909119894) + 119887 minus 119910

119894le 120576 + 120591

119894

120596119879

120601 (119909119894) + 119887 minus 119910

119894ge 120576 + 120591

lowast

119894

120591119894 120591lowast

119894ge 0 119894 = 1 2 119899

(6)

In (6) 120591119894and 120591lowast

119894are the slack variables that represent

the allowable upper limit and lower limit of training errorsrespectively under the constraint of insensitive loss 120576 119888 is thepenalty factor controlling the degree of penalty on sampleswhose errors exceed 120576

The Lagrange function must be introduced to resolve(6) According to the Duality Theory and the saddle pointcondition the dual form of (6) can be obtained as follows

min120597lowast120597

12

119899

sum119894=1

119899

sum119895=1

(120597lowast

119894minus 120597119894) (120597lowast

119895minus 120597119895) 119896 (119909

119894 119909119895)

+ 120576

119899

sum119894=1

(120597lowast

119894+ 120597119894) minus

119899

sum119894=1

119910119894(120597lowast

119894minus 120597119894)

st119899

sum119894=1

(120597lowast

119894minus 120597119894) = 0 0 le 120597

lowast

119894 120597119894le 119888 119894 = 1 2 119899

120596 =

119899

sum119894=1

(120597lowast

119894minus 120597119894)

(7)

In (7) 120597119894is the nonnegative Lagrangian multiplier of 120597lowast

119894

and 119896(119909119894 119909119895) represents the kernel function Any function

that satisfies Mercerrsquos condition can be a kernel functionThe common kernel functions include the linear kernelfunction 119896(119909 119909

119894) = 119909119879

119909119894and the Radial Basis Function (RBF)

kernel function 119896(119909 119909119894) = exp(minus120574119909 minus 119909

1198942) Ultimately the

hyperplane for the underlying prediction problem is given bythe following

119891 (119909) =

119899

sum119894=1

(120597lowast

119894minus 120597119894) 119896 (119909119894 119909) + 119887 (8)

24 PSO Algorithm The PSO algorithm proposed byKennedy [26] is an effective algorithm to address func-tion parameter optimization Each particle in the algorithmdenoting a potential solution in the solution space has threeindicators including speed position and fitness value Thefitness value is determined by the fitness function whichcan estimate the merit of the particles The velocity of theparticle determines the movement direction and distanceand the velocityrsquos dynamic adjustment follows the movementexperience of the particle itself and other particles thusenabling the optimal selection of an individual in the solutionspace The details are as follows

Suppose that in a 119863-dimensional solution space thepopulation comprising n particles is 119883 = (119883

1 1198832 119883

119899)

20104 20114 20124 20134 201442000

2500

3000

3500

4000

Figure 1 Daily closing price of Chinese stock index futures

the position of 119894th particle is 119883119894

= (1199091198941 1199091198942 119909

119894119863)119879 and

the fitness value of each particle can be calculated basedon the objective function and position of each particleThe velocity of 119894th particle is 119881

119894= (1198811198941 1198811198942 119881

119894119863)119879 its

individual extremum is 119875119894

= (1198751198941 1198751198942 119875

119894119863)119879 and the

group extremum is119875119892= (1198751198921

1198751198922

119875119892119863

)119879 In each iterativeprocess the particle updates its velocity and position throughthe individual extremum and the group extremum Theupdating equation is as follows

119881119896+1

119894119889= 120596119881119896

119894119889+ 11988811199031(119875119896

119894119889minus 119883119896

119894119889) + 11988821199032(119875119896

119892119889minus 119883119896

119894119889)

119883119896+1

119894119889= 119883119896

119894119889+ 119881119896+1

119894119889

(9)

where120596 represents the inertia weight 119889 = 1 2 119863 119896 is thecurrent iteration number119881

119894119889is the velocity component of the

particle 1198881and 1198882are the acceleration factors and nonnegative

constants and 1199031and 1199032are random numbers in the range

[0 1] Because the SVM prediction model is sensitive to themodel parameters and because the literature [22 27 28]indicates that the PSO algorithm performs very well in theparameter optimization process of the SVMmodel this paperuses the PSO algorithm to optimize the parameters of theSVM prediction model

3 Analysis of Chinese Stock IndexFutures Prices in Different Time Scales

31 Data Selection andDescription TheCSI 300 index futuresare the earliest stock index futures listed in China and it hasbeen widely used in many studies [4 5 22] Therefore thispaper selects the daily closing price of the stock index futuresas this sectionrsquos research objectThe time span runs fromApril16 2010 toNovember 20 2014 the sample data total 1115 andthe data are from theWind database See Figure 1 and Table 1for the sample data distribution and the descriptive statisticsrespectively

Nonstationary nonlinear tests are performed on Chinesestock index futures price series based on Ruan and Bao [11]Based on the data in Table 1 the null hypothesis that theChinese stock index future price series are nonstationarycannot be rejected under the condition of less than 10 signif-icance Because the postdifferential price series is a stationaryseries an autocorrelation analysis is further performed on theseries The test results for 119876 statistics demonstrate that thenull hypothesis for this differential series cannot be rejectedunder the condition of less than 10 significance However

Mathematical Problems in Engineering 5

Table 1 Descriptive statistics for Chinese stock index futures prices

Mean Std Dev JB ADF 119876(6) 1198762(6)

Price series 2600691 351503 97269lowastlowastlowast minus2539 mdash mdash119863(price series) minus0783 37611 858432lowastlowastlowast minus35531lowastlowastlowast 7737 26507lowastlowastlowast

Note Augmented Dickey-Fuller (ADF) statistics represent the nonstationary test of the price series 119863(price series) represents the price series after the firstdifference119876(6) statistics represent the autocorrelation test of the time series1198762(6) statistics represent the autocorrelation test of the square of the time seriesand ldquolowast lowast lowastrdquo indicates significance at the 1 significance level

Table 2 Correlations and variance ratios of IMFs and the residue term with the Chinese stock index futures price series

Mean value Pearson correlation coefficient Variance ratio (original series)IMF1 0131 (0238) 0041 (1363) 0273IMF2 minus0096 (minus0151) 0066 (2218)lowastlowast 0372IMF3 minus0431 (minus0523) 0097 (3273)lowastlowastlowast 0614IMF4 minus2103 (minus1277) 0163 (5530)lowastlowastlowast 2446IMF5 minus4652 (minus2413)lowastlowast 0218 (7454)lowastlowastlowast 3354IMF6 46989 (8007)lowastlowastlowast 0660 (29329)lowastlowastlowast 31076IMF7 minus64873 (minus18973)lowastlowastlowast minus0264 (minus9151)lowastlowastlowast 10550Res 2625727 0762 (39214)lowastlowastlowast 70278NoteThe values in parentheses are the 119905 statistics ldquolowastlowastrdquo represents significance at the 5 level and ldquolowast lowast lowastrdquo represents significance at the 1 level

the square of the 119876 statistics rejects the null hypothesis atless than 1 significance in other words autocorrelationis present Thus nonlinear models are more appropriatefor studies of Chinese stock index futures prices based onnonstationary nonlinear tests

32 CEEMD Decomposition Yeh et al [13] indicated thatthe CEEMD method retained the decomposition ability ofthe EEMD method while effectively offsetting the addedwhite noise To demonstrate that the advantages of thisdecomposition remain valid in Chinarsquos stock index futuresprices this paper uses both the CEEMD and EEMDmethodsto decompose the China stock index futures price series andthen compared the differences between the original seriesand the sum of the decomposition sequences obtained by thetwomethods Depending on the general setting the standarddeviation of the added white noise is 02 times the standarddeviation of the daily closing price See Figure 2 for thedifferences between the results

As shown in Figure 2 thewhite noise residue after decom-position by the CEEMD method is nearly zero whereas agreat deal of white noise residue remains after decompositionby the EEMD method indicating that the CEEMD methodcan better maintain the completeness of the original seriesthan the EEMD method Therefore this paper used theCEEMD method to decompose Chinese stock index futuresprices and the IMFs and residue term are presented inFigure 3

As shown in Figure 3 the frequency is the highest forIMF1 and then decreased in order among the remainingIMFs IMF1 IMF2 and IMF3 exhibit dense fluctuationswith amplitudes between positive and negative 50 pointsapproximatelyThe amplitudes of IMF4 and IMF5 increase tobetween positive and negative 100 points and the fluctuationfrequency of IMF5 decreases significantly compared with

the fluctuation frequency of the previous IMFs with afluctuation period of approximately 120 days (4 months)Theamplitude of IMF6 increases to between positive and negative200 pointsmdasheven reaching 400 points once during theinitial stock index futures listing periodmdashwith a fluctuatingperiod that increases to approximately 400 days (13 months)The upward fluctuation frequency of IMF7 decreases toapproximately 100 points whereas the downward fluctuationfrequency is maintained at approximately 200 points itsfrequency also decreases to the lowest among the seven IMFsbut the fluctuating period increases to approximately 600days (20 months) The residue term (res) exhibits a trend ofan initial slow decline and subsequent gradual increase butthere is no clear cyclical fluctuation

As the frequencies decrease the correlation and varianceratios of the IMFs with the Chinese stock index futures priceseries increase indicating that a lower frequency enablesgreater fluctuation information content to be obtained fromthe original series In the low-frequency series the correlationand variance ratio of IMF6 with the original series is signif-icantly larger than that of IMF5 and IMF7 indicating thatmedium-term fluctuations in the original series primarilyoriginated from IMF6 and that the correlations and varianceratios of the residue term with the original series are fargreater than those of the IMFs indicating that this seriescontains the most important information from the originalseries (Table 2)

33 Analysis of Different Time Scales The analysis describedabove established that the frequencies of IMF and theresidue term differ The fluctuations of the high-frequencyseries exhibit ambiguity and disorderliness and the cyclicalfluctuations of the low-frequency series are stronger Theresidue term contains most of the information of the orig-inal series but does not exhibit a clear fluctuating period

6 Mathematical Problems in Engineering

Table 3 Correlations and variance ratios of the component terms with the Chinese stock index futures price series

Pearson correlation coefficient Variance ratio (original series)Residue term 0762 (39214)lowastlowastlowast 70278Low-frequency term 0526 (206229)lowastlowastlowast 37521High-frequency term 0196 (6661)lowastlowastlowast 4051Note ldquolowast lowast lowastrdquo represents significance at the 1 significance level

indicating that the price fluctuation information reflected bythe high-frequency series low-frequency series and residueterm feature more significant differences To more accuratelyreveal the fluctuation characteristics of Chinese stock indexfutures prices at various time scales we analyze the sum ofthe IMFs with similar fluctuation characteristics using a 119905

statistics test of the mean from zero [8] Based on the data inTable 2 the high-frequency IMFsmdashIMF1 IMF2 IMF3 andIMF4mdashare summed as the high-frequency term The low-frequency IMFmdashIMF5 IMF6 and IMF7mdashare summed asthe low-frequency term The residue term alone is retainedThe correlations and variance ratios of the three componentswith the original series are presented in Table 3 and theirdistribution is presented in Figure 4

Consistent with the literature such as [8 9] the residueterm includes most of the information value of the originalseries and reflects the long-term trend of the original seriesIn fact the correlation coefficient of the residue term withthe Chinese stock index futures price series in this paper isas high as 076 and explains 70 of the fluctuations in theoriginal price series (see Table 3)Therefore the residue termcan represent the long-term trend of the Chinese stock indexfutures prices As shown in Figure 4 between the officiallisting of stock index futures inChina (2010) and 2014 there isa slow downward trend in stock index futures prices Becausethe stock index is the ldquobarometerrdquo of a countryrsquos macroe-conomic situation and because stock index futures serve aprice discovery function for stock indexes the futuresmarketprices reflect newly revealed information more quickly thancurrent market prices Therefore macroeconomic changesalso determine the long-term trend of stock index futuresprices As Chinarsquos Gross Domestic Product (GDP) growthrate gradually slows after 2010 (the GDP growth rates from2010 to 2014 are 1045 93 765 767 and 74 resp)(data for GDP growth rates in China from 2010 to 2014 weredownloaded from theNational Bureau of Statistics of the Peo-plersquos Republic of China website (httpwwwstatsgovcn))and macroeconomic growth shows signs of fatigue stockindex futures prices also decrease year by year decreasingfrom 3376 points in the beginning of 2010 to 2332 points inthe beginning of 2014 with an average annual growth rateof approximately minus11192 However this downward trendslows after 2014 and reverses slightly in the second half of 2014mainly because in that year certain important and significantpolitical actions including administrative simplification andanticorruption legislation and enforcement create a healthypolitical environment for the economy In addition a reduc-tion in the reserve requirement ratio and an interest rate cutwill free up cash for investment andChinarsquos economic growth

turns from incremental expansion to an optimized structurewhich results in good prospects for investors Accordinglymore money is flowing into the stock market which drivesstock index prices higher and higher stock index futures priceresult simultaneously

The low-frequency term mainly reflects the medium-term effect of significant events on stock index futures pricesAs shown in Table 3 the correlation between the low-frequency term and the Chinese stock index futures priceseries is 056 and explains up to 375 of the original priceseries indicating that significant events in recent years play animportant role in the fluctuations in the Chinese stock indexfutures prices Figure 4 illustrates the plummet that occursduring the launch of the Chinese stock index futures whichhas a maximum amplitude of 500 points This plummet ismainly due to the strengthening of the countryrsquos efforts tosuppress real estate market speculation in 2010This initiativeshould lead to a stock market decline because real estateinvestment is an important economic driver at the time butthe stock market does not make the appropriate responseTheChinese stock index futures unleash their price discoveryfunction after launch which leads to adjustments in stockindex prices In addition many investors use the hedgingfunction of stock index futures to avoid the risk of a stockmarket crash However after the substantial fluctuationsduring launch the amplitude of the low-frequency termdecreases slightly reflecting in part the stabilization effectof stock index futures on the stock market To furtherinvestigate the effects of significant events on Chinese stockindex futures prices we use the ICSS algorithm to performbreakpoint tests on the rate of change in the low-frequencyterm (first-difference processing on the low-frequency termto identify the breakpoints) Table 4 presents the results ofthe breakpoint test on the low-frequency term The low-frequency term exhibits 11 breakpoints or in other words12 different volatility mechanisms The significant eventsthat cause breakpoints are primarily administrative macro-control policies which are mainly addressed by adjustingmonetary policy perfecting the capital market mechanismand promoting market liberalization The execution of thesepolicies brings volatility to Chinese stock index futures inturn creating a new volatility mechanism These breakpointsare mainly concentrated in 2012 and 2013 During these twoyears the government frequently issues regulatory policiesto bolster financial deepening and reform and investorsadopt a medium-term investment strategy revolving aroundthese policies As an emerging market the Chinese stockindex futures market is easily affected However as thecapital market mechanism gradually adjusts trading in the

Mathematical Problems in Engineering 7

20104 20114 20124 20134 20144

005

1

minus05

minus1

times10minus11

(a)20104 20114 20124 20134 20144

02040

minus20

minus40

(b)

Figure 2 (a) The differences between the stock index futures series and the decomposition sequence obtained by the CEEMD method (b)The differences between the stock index futures series and the decomposition sequence obtained by the EEMDmethod

20104 20114 20124 20134 20144

0

100

200

300

IMF5

20104 20114 20124 20134 20144

0

200

400

600

IMF6

20104 20114 20124 20134 20144

0

100

IMF7

20104 20114 20124 20134 201442000

2500

3000

3500

Res

20104 20114 20124 20134 20144

IMF1 0

50

100

minus50

minus100

20104 20114 20124 20134 20144

IMF3 0

50

100

minus50

minus100

20104 20114 20124 20134 20144

0

100

200

IMF4

minus100

minus200

minus100

minus200

minus200

minus200

minus100

minus300

minus400

20104 20114 20124 20134 20144

0

50

100

IMF2

minus50

minus100

Figure 3 IMFs and residue term

8 Mathematical Problems in Engineering

Table 4 Breakpoint tests on the low-frequency term and corresponding significant events

Breakpointnumber Corresponding time Corresponding significant event The response of stock index futures prices

1 December 13 2010

On December 10 2010 the CentralEconomic Working Conference convenesand indicates that the moderately loosemonetary policy has become robust

This event implies that the supply ofliquidity to the capital market would bereduced which was bad news for the stockindex futures market Hence the stockfutures prices slump

2 September 28 2011

In October 2011 the credit ratings of a dozenEuropean financial institutions aredowngraded and the European debt crisisfurther escalates

The Chinese financial market was affectedby this event and in response the stockindex futures prices maintained thedownward trend

3 January 31 2012

From January 6 to 7 2012 the NationalConference on Financial Work convenes inBeijing to set the tone of the next series ofmajor financial reform issues

This event makes the stock market a positiveprospect for investors and then pushes upthe stock index futures prices

4 May 17 2012

On May of 2012 the stock index futuresprices are affected by a combination ofevents including the US stock market andinternational commodities market crashingand the Chinese PMI index does notcontinue early rising trend

The series of significant events poses aserious negative impact on investorsrsquoexpectations Therefore the stock indexfutures prices stop rising and turn tocollapse

5 June 28 2012

On June 18 2010 the Shanghai StockExchange and the Shenzhen Stock Exchangeseparately publish programs related to thedelisting of listed companies and revise thecorresponding Stock Market Rulesaccordingly further perfecting the initialpublic offering (IPO) and delisting systems

This event would improve investorsrsquoconfidence in the stock market Hence thedownward trend of stock index futures priceis slowed

6 August 28 2012

On August 16 2012 the New York branch ofthe Agricultural Bank of China openscontinuing to accelerate the pace ofinternationalization of the Chinese bankingsector

The event promotes the process of RMBinternationalization which is good news forthe financial market and stock index futuresprices inch up

7 October 18 2012

On October 17 2012 the Chinese StateCouncil meeting made an importantdecision that government would continue toimplement the proactive fiscal policy andprudent monetary policy

The prudent monetary policy implyinglimited funds in the capital market was badnews for stock index futures market Hencethe stock index futures prices tumble over

8 January 31 2013 At the end of January 2013 broad moneygrew 159 and narrow money grew 153

This event prompted a rise in stock indexfutures prices

9 May 21 2013

In May of 2013 the Chinese macroeconomicsituation does not perform well the FederalReserve gradually winds up the quantitativeeasing policy and many banks in China lacksufficient capital at the time

The cumulative effects of shocks of theseevents occurring in May have an enormousnegative influence on the stock index futuresmarket Hence the stock index futuresprices collapse

10 September 10 2013

On September 6 2013 repackaged treasurynote futures debut after a gap of 18 yearswhich is conducive to the establishingmarket-oriented pricing benchmarks andadvances market-oriented reform of theinterest rate

This event rebuilding investorsrsquo confidencein the capital market then pushes up thestock index futures prices

11 February 25 2014

On March 1 2014 the Central Bank releasesthe upper limit on interest rates on smallamounts of foreign currency deposits in theShanghai Free-Trade Zone taking anotherstep toward market-oriented reform of theinterest rate

The rise of deposit interest rate indicates thatmarket as risk-free rate would increasewhich is negative news for the stock indexfutures market Hence the stock indexfutures prices plunge

Mathematical Problems in Engineering 9

20104 20114 20124 20134 20144

0

1000

2000

3000

4000

Original seriesResidue term

Low-frequency termHigh-frequency term

minus1000

Figure 4 Chinese stock index futures prices and the three compo-nents

Chinese stock index futures market further standardizes andthe ability to withstand external impact thus also increasesTherefore the number of breakpoints decreases after 2014

The high-frequency term mainly reflects the short-termdisequilibrium in the Chinese stock index futures marketAs shown in Table 3 the correlation of the high-frequencyterm with the Chinese stock index futures prices is merely0196 and the ability to explain fluctuations in the originalprice series is only 4 indicating that the overall effectof the high-frequency term on Chinese stock index futuresprices is limited However as Figure 4 shows the fluctuationsof the high-frequency term fit well with the local fluctu-ations of the Chinese stock index futures prices in otherwords the high-frequency term plays a primary role in localchanges in Chinese stock index futures prices In additionas opposed to the low-frequency term the high-frequencyterm fluctuates frequently but with smaller amplitudes Thisdifference occurs primarily because the high-frequency termis particularly affected by investorsrsquo speculative arbitrageactivities In particular implementation of a T+0 transactionsystem in the Chinese stock index futures market means thatinvestors buying and selling stock index futures can completetheir transactions in the same day which suits short-termspeculators Based on this transaction system there will bemore investment chances for investors which will enhanceinvestorsrsquo speculative opportunities Hence there are manyshort-term fluctuations in Chinese stock index futures pricesMeanwhile it is possible and convenient for investors toaccess the latest hot investment tip because the role of socialinteractions is partly substituted for by online information inthe Internet age [31] In reality this type of new informationchanges continuously and investors must frequently adjusttheir investment strategies to gainmore additional benefits byspeculating in the spot-futures marketsTherefore the strongeffect of the shock of investorsrsquo speculation on Chinese stockindex futures prices in the short-term is difficult to sustainAlthough fluctuations in the high-frequency term are notimportant to forecast the long-term trends of Chinese stockindex futures their predictive effect of minor fluctuations onstock index futures cannot be ignored

4 Predictions for Chinese StockIndex Futures Prices

In this section this paper further investigates the predictionsof the Chinese stock index futures prices In addition tothe previous dayrsquos price data we also select the five-minutehigh-frequency price data of Chinese stock index futuresfrom April 16 2010 to November 20 2014 as a researchsubject which yields a total of 61325 data points (the data areobtained from theWind database) To forecast the daily pricethe first 745 (approximately two-thirds) pieces of data areused as the training data and the last 370 (approximately one-third) pieces of data are used as the test data To forecast thefive-minute high-frequency price the first 40883 (approxi-mately two-thirds) data points are used as the training dataand the last 20442 (approximately one-third) data points areused as the test data

As our analysis of the price fluctuations of the Chinesestock index futures uses the CEEMDmethod the fluctuationcharacteristics in different time scales have their uniqueeconomic implications we suggest that there may be betterpredictions of Chinese stock index futures prices when thesefluctuation characteristics are considered in the predictionmodel Hence based on the literature [16 17] a combinationprediction model that is the CEEMD-SVM-SVM model isproposed and the PSO algorithm is used to optimize theparameters of the SVM model In our empirical researchwe perform a one-step-ahead prediction of Chinese stockindex futures prices The specific steps are as follows (1) theCEEMD method is used to decompose the Chinese stockindex futures price series into several IMFs and a residueterm (2) the IMFs are then composed into a low-frequencyterm and a high-frequency term based on a fine-to-coarsereconstruction and the residue term remains the same (3)SVM models are established separately for each of the threecomponents obtained in step (2) and the PSO algorithm isused to optimize the parameters and (4) the predictive valueof the three components obtained in step (3) is used as theinput with the real value at the same moment as the outputand then a SVM combination model is built to obtain thefinal predictive value The overall process of the CEEMD-SVM-SVMmodel is shown in Figure 5

The prediction evaluation indexes are the root meansquare error (RMSE) mean absolute prediction error(MAPE) and directional symmetry (DS) the DS predictionevaluation index is used to indicate the modelrsquos ability topredict the directionality trend of Chinese stock index futuresprices The prediction evaluation indexes are expressed asfollows

RMSE = radic1

119899

119899

sum119905=1

[119909 (119905) minus (119905)]2

MAPE =1

119899

119899

sum119905=1

10038161003816100381610038161003816100381610038161003816

119909 (119905) minus (119905)

119909 (119905)

10038161003816100381610038161003816100381610038161003816

DS =1

119899

119899

sum119894=1

119889119894

10 Mathematical Problems in Engineering

Original series

CEEMD

Residue term Low-frequency term High-frequency term

SVM1 SVM2 SVM3

Final predictive value

SVM combination

Figure 5 The overall process of the CEEMD-SVM-SVMmodel

119889119894=

1 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] ge 0

0 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] lt 0

(10)

Among the three evaluation indexes smaller RMSE andMAPE values indicate that the predictionmodel is experienc-ing better performance which is also true of greater DS valueand better performance of the prediction model

The prediction accuracy based on the SVM model issensitive to the SVM modelrsquos kernel function As manyprevious studies (such as [17 22]) indicate that the linearkernel function and the RBF kernel function are appropriatekernel functions for the prediction of SVMmodel the linearkernel function and the RBF kernel function of the SVMare thus both applied to our prediction of Chinese stockindex futures prices Based on the process of the CEEMD-SVM-SVM model we first predict the residue term thenthe low-frequency term and then the high-frequency termof the daily price series and the five-minute price seriesNotably the five-minute price series of Chinese stock indexfutures is decomposed into fourteen IMFs and one residueterm where IMFs from IMF1 to IMF9 are recomposed intothe high-frequency term and IMFs from IMF10 to IMF14are recomposed into the low-frequency termThe predictionerrors of these components are shown in Table 5

As shown in Table 5 in using the SVM model to makepredictions on each component the prediction precision ofthe linear kernel function SVM model is generally higherthan the RBF kernel function SVM model In additionwhether for predicting the daily price series or the five-minute price series the prediction precisions of the residueterm and the low-frequency term are significantly higherthan those of the high-frequency termThis difference occursbecause these effects of shocks on the residue term andthe low-frequency term are fundamental economically theyare significant events that have an impact on Chinese stockindex futures prices that is sustained over a long duration

Therefore the fluctuations of the residue term and the low-frequency term are relatively steady and easier to forecastusing the SVM model By contrast the high-frequency termrepresenting the short-term disequilibrium of Chinese stockindex futures price is affected mainly by daily market activ-ities such as speculation arbitrage and the fluctuation ofthe high-frequency term that is random and changes rapidlyHence the SVMmodel experiences difficulty in learning thepattern of high-frequency term fluctuation resulting in poorpredictive results

Then according to prediction step (4) we use the optimalpredictive value of each component obtained in the previousstep to perform a combination prediction and the predictionerrors of our proposed model and the single SVM modelbased on the same data sample are simultaneously comparedThese results of the prediction errors are listed in Table 6

Table 6 shows that as opposed to the predictive resultsof the three components the RBF kernel function SVMmodel has higher predicted precision than the linear kernelfunction SVM model when performing combination pre-diction which indicates that the RBF kernel function SVMmodel is more appropriate for combination prediction Inaddition we find that the predictive results of the finalpredictive value obtained under the RBF kernel function aresignificantly better than those obtained in the single SVMmodel when predicting the daily price but slightly better thanthe single SVM model when making predictions regardingthe high-frequency price Meanwhile we should considerthat the high-frequency price embodies far more complexinformation than the daily price during the same time spanwhich may restrict the learning ability and generalizationability of the SVM model Hence the CEEMD-SVM-SVMmodel performs better in terms of predicting the daily priceIn any event the CEEMD-SVM-SVM model proposed bythis paper performs better than the single SVM model interms of both forecasting the daily price series and the five-minute price series Thus the introduction of Chinese stockindex futures price characteristics in different time scales into

Mathematical Problems in Engineering 11

Table 5 The prediction errors of different components

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

Residue termRMSE 0007 0008 0006 0009MAPE 0001 0001 0001 0002DS 69837 58424 21678 27241

Low-frequency termRMSE 0007 0008 0004 0008MAPE 0087 0092 0042 0258DS 60326 55434 54613 55254

High-frequency termRMSE 0086 0098 0020 0021MAPE 4806 4725 1146 1149DS 63315 62500 51463 51854

Table 6 Comparison of different modelsrsquo prediction errors

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

SVMRMSE 0053 0054 0008 0008MAPE 0009 0009 0001 0001DS 54891 52445 48171 48161

CEEMD-SVM-SVMRMSE 0051 0009 0008 0007MAPE 0008 0002 0001 0001DS 62228 88043 51565 52715

the SVM model improves the ability of the SVM model tomine the variations in the stock index futures price seriesfurther enhancing its prediction precision Simultaneouslysome useful suggestions for investors can be found in thepredictions of Chinese stock index futures prices generatedby our proposedmodel First investors should be accustomedto analyzing the price series from different time scales tounderstand the fluctuation characteristics of Chinese stockindex futures prices Second investors must actively identifywhat time scale the effects of external shocks on stock indexfutures prices belong to andmust devise the appropriate strat-egy accordingly Finally investors should pay more attentionto a hedging strategy involving Chinese stock index futuresover the long andor medium term and should be cautiousabout short-term speculation

Figures 6 and 7 present the real and predictive values ofthe daily price series andfive-minute price series respectivelyThe final predictive values obtained using the CEEMD-SVM-SVM model appeared to coincide with the real values indi-cating that the predictive values obtained using this modelfit well with the real values and intuitively demonstratingthat the CEEMD-SVM-SVMmodel is suitable for forecastingChinese stock index futures prices

20135 20138 201311 20142 20145 20148 2014112000

2200

2400

2600

2800

Real valuePredictive value

Figure 6 Real value and predictive value of the daily price series

5 Conclusion

The Chinese stock index futures market is a complex systemand fluctuations in the stock index futures prices resultfrom the collective effects of many factors at various levelsThe CEEMD method a method suitable in the analysis ofnonlinear nonstationary data is applied to analyze Chinesestock index futures prices in different time scales and to

12 Mathematical Problems in Engineering

Real valuePredictive value

20135 20138 201311 20142 20145 20148 2014111800

2000

2200

2400

2600

2800

Figure 7 Real value and predictive value of the five-minute priceseries

reveal the intrinsic fluctuation characteristics of Chinesestock index futures prices Then we focus on whether thesecharacteristics improve the predictive precision of Chinesestock index futures prices and propose a CEEMD-SVM-SVMmodel based on the PSO algorithm The empirical resultsdemonstrate that the intrinsic fluctuation characteristics ofChinese stock index futures prices can be revealed in threecomponents including the residue term the low-frequencyterm and the high-frequency term with independent eco-nomic significance The residue term determines the long-term trend of stock index futures prices and its fluctuationsare correlated with macroeconomics and the long-termexpectations of investorsThe low-frequency term reveals theeffect of significant events on medium-term fluctuations instock index futures prices the ICSS algorithm breakpointtests reveal that policy regulation events are the main causeof fluctuations during the low-frequency term The high-frequency term reflects short-term disequilibrium in thestock index futures market although the effect of the high-frequency term on the overall fluctuation in the stock indexfutures prices is limited it nonetheless plays an important rolelocally

In addition in forecasting Chinese stock index futuresprices first we find that the linear kernel function SVMmodel is more suitable for predicting these components thanthe RBF kernel function SVM model The residue term andthe low-frequency term have stronger regular fluctuationsthan the high-frequency term and therefore yield superiorpredictive results Second the RBF kernel function SVMmodel shows higher predictive precision for the final combi-nation prediction than the linear kernel function SVMmodelSimultaneously the predictive results of the combinationprediction model are better than those of a single SVMmodel which indicates that the integration of fluctuationcharacteristics in different time scales increases the abilityof the SVM model to identify regularity in Chinese stockindex futures price fluctuations thus improving predictionprecision Finally the predictive results of the CEEMD-SVM-SVM model on daily prices and intraday prices are basicallyconsistent which indicates that our proposed model hasstability and universal applicability in forecasting Chinesestock index futures prices

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by the National SocialScience Foundation of China (14ZDA045)

References

[1] VMeneu andH Torro ldquoAsymmetric covariance in spot-futuresmarketsrdquo Journal of Futures Markets vol 23 no 11 pp 1019ndash1046 2003

[2] M Zhong A F Darrat and R Otero ldquoPrice discovery andvolatility spillovers in index futures markets some evidencefrom Mexicordquo Journal of Banking and Finance vol 28 no 12pp 3037ndash3054 2004

[3] X Xiong F Wang W Zhang and Y Sun ldquoPrice discoveryand volatility spillovers between SGX FTSEXinhua ChinaA50 index futures and A-share marketrdquo Chinese Journal ofManagement vol 11 no 1535 pp 1507ndash1512 2009

[4] H M Zuo Z T Liu and H W Zeng ldquoResearch of volatilityspillover and information transmission between stock indexfutures and spot market based on the high frequency datardquoJournal of Financial Research vol 4 pp 140ndash154 2012

[5] H Q Chen and C H Zhang ldquoDoes index futures tradingreduce stock market jump riskrdquo Economic Research Journalvol 1 pp 153ndash167 2015

[6] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 48 no 3 pp235ndash254 2015

[7] E E Peters Fractal Market Analysis Applying Chaos Theory toInvestment and Economics John Wiley amp Sons New York NYUSA 1994

[8] X Zhang K K Lai and S-YWang ldquoA new approach for crudeoil price analysis based on empirical mode decompositionrdquoEnergy Economics vol 30 no 3 pp 905ndash918 2008

[9] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnonstationary time series analysisrdquo Proceedings of the RoyalSociety of London A vol 454 no 1971 pp 903ndash995 1998

[10] Z Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[11] L F Ruan and H J Bao ldquoAn empirical analysis on periodicfluctuations of real estate price based on EMDrdquo Chinese Journalof Management Science no 3 pp 41ndash46 2012

[12] H L Li and C E Feng ldquoRelationship between investor senti-ment and stock indices fluctuation based on EEMDrdquo SystemsEngineering-Theory amp Practice vol 10 pp 2495ndash2503 2014

[13] J-R Yeh J-S Shieh andN EHuang ldquoComplementary ensem-ble empiricalmode decomposition a novel noise enhanced dataanalysismethodrdquoAdvances in Adaptive DataAnalysis vol 2 no2 pp 135ndash156 2010

[14] L Zhao W Yu and R Yan ldquoRolling bearing fault diagnosisbased on CEEMD and time series modelingrdquo MathematicalProblems in Engineering vol 2014 Article ID 101867 13 pages2014

Mathematical Problems in Engineering 13

[15] L Tang W Dai L Yu and S Wang ldquoA novel CEEMD-based eelm ensemble learning paradigm for crude oil priceforecastingrdquo International Journal of Information Technologyand Decision Making vol 14 no 1 pp 141ndash169 2015

[16] L Yu S Wang and K K Lai ldquoForecasting crude oil price withan EMD-based neural network ensemble learning paradigmrdquoEnergy Economics vol 30 no 5 pp 2623ndash2635 2008

[17] Y F Yang Y K Bao Z Y Hu and R Zhang ldquoCrude oilprice prediction based on empirical mode decomposition andsupport vector machinesrdquo Chinese Journal of Management vol12 pp 1884ndash1889 2010

[18] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[19] W Huang Y Nakamori and S-Y Wang ldquoForecasting stockmarket movement direction with support vector machinerdquoComputers and Operations Research vol 32 no 10 pp 2513ndash2522 2005

[20] H Ince and T B Trafalis ldquoShort term forecasting with supportvector machines and application to stock price predictionrdquoInternational Journal of General Systems vol 37 no 6 pp 677ndash687 2008

[21] F E H Tay and L Cao ldquoApplication of support vectormachinesin financial time series forecastingrdquo Omega vol 29 no 4 pp309ndash317 2001

[22] Y Sai F T Zhang and T Zhang ldquoResearch of Chinese stockindex futures regression prediction based on support vectormachinesrdquo Chinese Journal of Management Science vol 3 pp35ndash39 2013

[23] C Inclan andG C Tiao ldquoUse of cumulative sums of squares forretrospective detection of changes of variancerdquo Journal of theAmerican Statistical Association vol 89 no 427 pp 913ndash9231994

[24] S H Kang C Cheong and S-M Yoon ldquoStructural changesand volatility transmission in crude oil marketsrdquo Physica AStatisticalMechanics and Its Applications vol 390 no 23-24 pp4317ndash4324 2011

[25] J P Yang X X Chen and C H Zhang ldquoPolicy impact factorson the volatility of Shanghai and Shenzhen stock market inChinardquo Chinese Journal of Management Science vol 20 no 6pp 43ndash51 2012

[26] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer New York NY USA2010

[27] C-L Huang and J-F Dun ldquoA distributed PSOndashSVM hybridsystem with feature selection and parameter optimizationrdquoApplied Soft Computing vol 8 no 4 pp 1381ndash1391 2008

[28] C Zhou K Yin Y Cao and B Ahmed ldquoApplication of timeseries analysis and PSOndashSVMmodel in predicting the Bazimenlandslide in the Three Gorges Reservoir Chinardquo EngineeringGeology vol 204 pp 108ndash120 2016

[29] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis amp Applications vol 16 no 1 pp 93ndash103 2014

[30] X Gong Z He P Li and N Zhu ldquoForecasting return volatilityof the CSI 300 index using the stochastic volatility model withcontinuous volatility and jumpsrdquo Discrete Dynamics in Natureand Society vol 2014 Article ID 964654 10 pages 2014

[31] S Q Guo and P H Liang ldquoSocial Interactions the informationchannel and household stock market participation a empiricalstudy based on a Chinese household finance surveyrdquo EconomicResearch Journal vol S1 pp 116ndash131 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Zuo et al [4] investigated the Granger causality and long-run equilibrium between daily realized variance and realizedbipower variation and jump variation Based on five-minutehigh-frequency data Chen and Zhang [5] researched theeffects of the CSI 300 index futures trading on the jump riskof the spot market by employing a nonparametric methodand the Granger causality test However there are some keydrawbacks with respect to these traditional models On onehand many factors (including economic fundamentals theeffects of shocks from significant economic and financialevents and investorsrsquo risk preferences [6]) stir up pricefluctuations in stock index futures at different time scalesPeters [7] suggested that traders in the financialmarket woulddisperse and reflect various time scales due to differences ininvestment philosophy ultimately engaging in transactionsin different investment time scales short-term medium-term and long-term As a new product in an emergingmarket Chinese stock index futures have attracted a varietyof investor types and their different investment strategies alsohave a great impact on stock index futures price fluctuationsin different time scales However traditional models conducttheir analyses based solely on the entire price series and theycannot explain the inner driving forces of price fluctuations[8] On the other hand stock index futures price seriesare nonlinear and nonstationary financial time series whiletraditional models are based on the assumption of lineardata and obey a normal distribution thus these models areunable to adequately capture the nonlinear patterns in theprice series

The Empirical Mode Decomposition (EMD) method [9]and the Ensemble EMD method [10] are effective tools toaddress nonlinear nonstationary data and these methodscan analyze the fluctuation characteristics of a time seriesindifferent time scales by decomposing them into IntrinsicMode Functions (IMFs) with independent information Inparticular the EEMD method is a substantial improvementover the EMDmethod and overcomes the drawback of modemixing in the EMD method by adding white noise Thusthe EEMD method is popular and has been widely used andapplied in the financial sector For example Zhang et al[8] used the EEMD method to decompose crude oil pricesinto a trend with a slowly varying part and a fluctuatingprocess and then analyzed the long-term medium-termand short-term fluctuation characteristics of crude oil pricesRuan and Bao [11] applied the EEMD method to decomposehousing prices into a trend term a low-frequency term and ahigh-frequency term to reveal the intrinsic characteristics offluctuations in housing prices Li and Feng [12] employed theEEMD method to separately decompose investor sentimentand the stock index price series into a short-term fluctuationterm amedium-term significant event term and a long-termtrend term integrating econometric models to expand theirresearch However the EEMD method cannot effectivelyoffset the residue noise from the addedwhite noiseThereforeYeh et al [13] further proposed the Complementary Empiri-cal Mode Decomposition (CEEMD) method This methodwhich has been used increasingly in recent years [14 15]provides a fast and accurate method for processing data thatnot only can address the mode mixing in the EMD method

but also can completely offset the added residual white noisein the decomposition process that are generated by the EEMDmethod However this latest data processing method has notyet been applied to the stock index futures market

In addition by combining with other prediction modelsEMD methods can also improve the predictive precision ofa single prediction model For example Yu et al [16] builtan EMD-Fuzzy Neural Network- (FNN-) Adaptive LinearNeural Network (ALNN) model to forecast crude oil pricesthis model yielded superior predictive results in comparisonwith either a single FNN model or a single ALNN modelIn addition Yang et al [17] built an EMD-SVM-SVM modelto forecast crude oil prices and also improved the predictiveresults of a single SVM model Meanwhile The SupportVector Machine (SVM) model [18] is now widely applied inthe stock and stock index futures markets Huang et al [19]used the SVM model a Back Propagation Neural Network(BPNN) Linear Discriminant Analysis (LDA) and othermethods to make predictions regarding the Nikkei 225 Indexand found that the SVMmodel exhibited the highest predic-tive performance among these prediction models Ince andTrafalis [20] demonstrated that the SVM model has greaterprecision in predicting stock prices than the MultilayerPerception (MLP) and Autoregressive Integrated MovingAverage (ARIMA)methods Using five real futures contractssuch as the Standard amp Poor 500 stock index futures Tayand Cao [21] examined the feasibility of SVM modeling bycomparing it with a BP neural network these authors showedthat SVMs are better for forecasting futures series Sai etal [22] optimized four SVM models with different kernelfunctions based on the Genetic Algorithm (GA) and theParticle SwarmOptimization (PSO) algorithm these authorsbuilt eight different programs to forecast Chinese stock indexfutures prices and discovered that the linear kernel functionSVM model based on the PSO algorithm yielded superiorpredictive results for Chinese stock index futures pricesThe above papers on the SVM prediction model bolsterthe conclusion that the SVM model has been used in thestock index futures market although mainly as a single SVMmodel

To improve the analysis of Chinese stock index futuresprices and the ability to forecast them this paper first drewideas from Zhang et al [8] and other scholars Then we usedthe CEEMD method to decompose the Chinese stock indexfutures price series into a residue term a low-frequency termand a high-frequency term Finally we analyzed the long-term medium-term and short-term intrinsic fluctuationcharacteristics of Chinese stock index futures prices Inaddition substantial fluctuations in the low-frequency termare related to the impact of significant events To define theevents that play a role in the low-frequency term we useIterated Cumulative Sums of Squares (ICSS) [23ndash25] to detectthe breakpoints in the low-frequency term Then we focuson the problem of forecasting Chinese stock index futuresprices by referring to Yu et al [16] and Yang et al [17] toconstruct a combination prediction model and to forecastChinese stock index futures prices by building a PSO-based[26ndash28]CEEMDand SVMcombination predictionmodel Inaddition although previous studies on stock and stock index

Mathematical Problems in Engineering 3

futures markets are based on low-frequency daily weeklyand monthly price data [1ndash3 19ndash22 29] current researchregarding intraday high-frequency data gradually becamea focus [4 5 30] Therefore we also use this model toforecast the daily data and the intraday high-frequency dataof Chinese stock index futures prices The main innovationof this paper is the use of the CEEMD method to reveal theintrinsic fluctuation characteristics of Chinese stock indexfutures prices from various time scales which provides a newresearch perspective on the fluctuation analysis of stock indexfutures pricesThefluctuation characteristics of various scalesare introduced into predicting stock index futures prices thusimproving the predictive ability of a single SVM model withrespect to the stock index futures market

The remainder of this paper is structured as followsSection 2 introduces themethods used in this paper Section 3analyzes the Chinese stock index futures prices at varioustime scales Section 4 forecasts Chinese stock index futuresprice Finally Section 5 provides a summary

2 Introduction of Methodology

21 The CEEMD Method EMD methods such as the EMDmethod the EEMD method and the CEEMD method areself-adaptive data processing methods that can decompose atime series into a series of Intrinsic Mode Functions (IMFs)with independent informationThese IMFs are effective toolsto address nonlinear nonstationary data In this paper weemploy the latest extension that is the CEEMD methodto analyze Chinese stock index futures prices The detailedintroduction of the CEEMDmethod is stated as follows

The EMD method proposed by Huang et al [9] is thefirst method to extract series of IMFs and residue terms fromthe original time series In particular IMFs must satisfy thefollowing two conditions (1) the number of extrema andzero-crossing pointsmust be equal or differ by amaximumofone and (2) the mean value of the upper and lower envelopesmust be zero at all times Accordingly decomposition of theEMD method ends when the last residue term (abbreviatedas res) 119903(119905) is a monotonic function or a constant then theoriginal series 119909(119905) can be expressed as

119909 (119905)=119899

sum119894=1

119909119894(119905) + 119903 (119905) (1)

where IMF119894is recorded as 119909

119894(119905) Because mode mixing

can easily emerge in the EMD method Wu and Huang[10] further proposed the EEMD method to address thisdrawback of the EMD method The EEMD method assumesthat the observed data are amalgamations of true time seriesand noise and that the ensemble means of data with differentnoises are closer to true time series Hence adding whitenoise as an additional step may help extract the true IMFsin the time series and the noise can be offset by ensembleaveraging However residue noise continues to be generatedby the added white noise Therefore Yeh et al [13] thenproposed theCEEMDmethod by adding pairs of positive andnegative white noise to generate complementary sequences

Specifically the complementary sequences are obtained bythe following equation

[119909+

119894(119905)

119909minus119894(119905)

] = [1 1

1 minus1] [

119909 (119905)

120576119894(119905)

] (2)

In (2) 120576119894represents 119894th added white noise and 119909+

119894(119905)

and 119909minus119894(119905) represent the sequences after positive and negative

white noise is added for 119894th time to the original 119909(119905) seriesWe can generate complementary IMFs via the EMDmethodand these are expressed as follows

119909+

119895(119905) =

1

119899

119899

sum119894=1

119909+

119894119895(119905)

119909minus

119895(119905) =

1

119899

119899

sum119894=1

119909minus

119894119895(119905)

(3)

Then the final IMFs with no residue noise are extractedby averaging the two complementary IMFs

119909119895=

(119909+119895+ 119909minus119895)

2 (4)

22 ICSS Algorithm The ICSS algorithm proposed by Inclanand Tiao [23] is a relatively mature approach to detectingbreakpoints in sequences The algorithm assumes that in thestable time series 119884

119894119905= 120583119894+ 120576119894119905 120583119894denotes the mean return

for time series 119894 and 120576119894119905 is the residual series with a mean

of 0 and a variance of 1205902119894119905 Letting 119862

119896= sum119896

119894=11205762119905 119896 = 1 119879

be the cumulated sum squares of the residual series 120576119905 the

statistic 119863119896can be defined as follows

119863119896= (

119862119896

119862119879

) minus (119896

119879) 119896 = 1 119879 119863

0= 119863119879

= 0 (5)

If breakpoints do not occur in the variance in the iterationprocess the value of the 119863

119896statistic will vary around 0 By

contrast if there are one or more breakpoints in a sequencethe value of the statistic 119863

119896will deviate significantly upward

or downward from the 0 Let 119896lowast be the value for 119870 atmax119896|119863119896| If max

119896radic(1198792)|119863

119896| exceeds the preset credible

interval 119896lowast will be taken as an estimated break point whereradic(1198792) is a normalizing factor

23 SVM Prediction Model The SVM model employs akernel-based method in which the basic idea is to construct ahyperplane with low risk in high-dimensional feature spaceThis method has the advantage of generalization capabilitiesand good function approximation Among the various typesof SVM models the 120576-SVM model proposed by Vapnik [18]is mainly used to resolve prediction problems Given a set oftraining patterns (119909

1 1199101) (119909

119894 119910119894)119899119894=1

where 119909119894isin 119877 is the

input variable and119910119894isin 119877 is the output variable the prediction

model is learned from these patterns and is used to forecast

4 Mathematical Problems in Engineering

the output variables of unseen input variables Specifically theconstraints and objective function for the 120576-SVMmodel are

min120596119887120591120591

lowast

120596119879120596

2+ 119888

119899

sum119894=1

120591119894+ 119888

119899

sum119894=1

120591lowast

119894

st 120596119879

120601 (119909119894) + 119887 minus 119910

119894le 120576 + 120591

119894

120596119879

120601 (119909119894) + 119887 minus 119910

119894ge 120576 + 120591

lowast

119894

120591119894 120591lowast

119894ge 0 119894 = 1 2 119899

(6)

In (6) 120591119894and 120591lowast

119894are the slack variables that represent

the allowable upper limit and lower limit of training errorsrespectively under the constraint of insensitive loss 120576 119888 is thepenalty factor controlling the degree of penalty on sampleswhose errors exceed 120576

The Lagrange function must be introduced to resolve(6) According to the Duality Theory and the saddle pointcondition the dual form of (6) can be obtained as follows

min120597lowast120597

12

119899

sum119894=1

119899

sum119895=1

(120597lowast

119894minus 120597119894) (120597lowast

119895minus 120597119895) 119896 (119909

119894 119909119895)

+ 120576

119899

sum119894=1

(120597lowast

119894+ 120597119894) minus

119899

sum119894=1

119910119894(120597lowast

119894minus 120597119894)

st119899

sum119894=1

(120597lowast

119894minus 120597119894) = 0 0 le 120597

lowast

119894 120597119894le 119888 119894 = 1 2 119899

120596 =

119899

sum119894=1

(120597lowast

119894minus 120597119894)

(7)

In (7) 120597119894is the nonnegative Lagrangian multiplier of 120597lowast

119894

and 119896(119909119894 119909119895) represents the kernel function Any function

that satisfies Mercerrsquos condition can be a kernel functionThe common kernel functions include the linear kernelfunction 119896(119909 119909

119894) = 119909119879

119909119894and the Radial Basis Function (RBF)

kernel function 119896(119909 119909119894) = exp(minus120574119909 minus 119909

1198942) Ultimately the

hyperplane for the underlying prediction problem is given bythe following

119891 (119909) =

119899

sum119894=1

(120597lowast

119894minus 120597119894) 119896 (119909119894 119909) + 119887 (8)

24 PSO Algorithm The PSO algorithm proposed byKennedy [26] is an effective algorithm to address func-tion parameter optimization Each particle in the algorithmdenoting a potential solution in the solution space has threeindicators including speed position and fitness value Thefitness value is determined by the fitness function whichcan estimate the merit of the particles The velocity of theparticle determines the movement direction and distanceand the velocityrsquos dynamic adjustment follows the movementexperience of the particle itself and other particles thusenabling the optimal selection of an individual in the solutionspace The details are as follows

Suppose that in a 119863-dimensional solution space thepopulation comprising n particles is 119883 = (119883

1 1198832 119883

119899)

20104 20114 20124 20134 201442000

2500

3000

3500

4000

Figure 1 Daily closing price of Chinese stock index futures

the position of 119894th particle is 119883119894

= (1199091198941 1199091198942 119909

119894119863)119879 and

the fitness value of each particle can be calculated basedon the objective function and position of each particleThe velocity of 119894th particle is 119881

119894= (1198811198941 1198811198942 119881

119894119863)119879 its

individual extremum is 119875119894

= (1198751198941 1198751198942 119875

119894119863)119879 and the

group extremum is119875119892= (1198751198921

1198751198922

119875119892119863

)119879 In each iterativeprocess the particle updates its velocity and position throughthe individual extremum and the group extremum Theupdating equation is as follows

119881119896+1

119894119889= 120596119881119896

119894119889+ 11988811199031(119875119896

119894119889minus 119883119896

119894119889) + 11988821199032(119875119896

119892119889minus 119883119896

119894119889)

119883119896+1

119894119889= 119883119896

119894119889+ 119881119896+1

119894119889

(9)

where120596 represents the inertia weight 119889 = 1 2 119863 119896 is thecurrent iteration number119881

119894119889is the velocity component of the

particle 1198881and 1198882are the acceleration factors and nonnegative

constants and 1199031and 1199032are random numbers in the range

[0 1] Because the SVM prediction model is sensitive to themodel parameters and because the literature [22 27 28]indicates that the PSO algorithm performs very well in theparameter optimization process of the SVMmodel this paperuses the PSO algorithm to optimize the parameters of theSVM prediction model

3 Analysis of Chinese Stock IndexFutures Prices in Different Time Scales

31 Data Selection andDescription TheCSI 300 index futuresare the earliest stock index futures listed in China and it hasbeen widely used in many studies [4 5 22] Therefore thispaper selects the daily closing price of the stock index futuresas this sectionrsquos research objectThe time span runs fromApril16 2010 toNovember 20 2014 the sample data total 1115 andthe data are from theWind database See Figure 1 and Table 1for the sample data distribution and the descriptive statisticsrespectively

Nonstationary nonlinear tests are performed on Chinesestock index futures price series based on Ruan and Bao [11]Based on the data in Table 1 the null hypothesis that theChinese stock index future price series are nonstationarycannot be rejected under the condition of less than 10 signif-icance Because the postdifferential price series is a stationaryseries an autocorrelation analysis is further performed on theseries The test results for 119876 statistics demonstrate that thenull hypothesis for this differential series cannot be rejectedunder the condition of less than 10 significance However

Mathematical Problems in Engineering 5

Table 1 Descriptive statistics for Chinese stock index futures prices

Mean Std Dev JB ADF 119876(6) 1198762(6)

Price series 2600691 351503 97269lowastlowastlowast minus2539 mdash mdash119863(price series) minus0783 37611 858432lowastlowastlowast minus35531lowastlowastlowast 7737 26507lowastlowastlowast

Note Augmented Dickey-Fuller (ADF) statistics represent the nonstationary test of the price series 119863(price series) represents the price series after the firstdifference119876(6) statistics represent the autocorrelation test of the time series1198762(6) statistics represent the autocorrelation test of the square of the time seriesand ldquolowast lowast lowastrdquo indicates significance at the 1 significance level

Table 2 Correlations and variance ratios of IMFs and the residue term with the Chinese stock index futures price series

Mean value Pearson correlation coefficient Variance ratio (original series)IMF1 0131 (0238) 0041 (1363) 0273IMF2 minus0096 (minus0151) 0066 (2218)lowastlowast 0372IMF3 minus0431 (minus0523) 0097 (3273)lowastlowastlowast 0614IMF4 minus2103 (minus1277) 0163 (5530)lowastlowastlowast 2446IMF5 minus4652 (minus2413)lowastlowast 0218 (7454)lowastlowastlowast 3354IMF6 46989 (8007)lowastlowastlowast 0660 (29329)lowastlowastlowast 31076IMF7 minus64873 (minus18973)lowastlowastlowast minus0264 (minus9151)lowastlowastlowast 10550Res 2625727 0762 (39214)lowastlowastlowast 70278NoteThe values in parentheses are the 119905 statistics ldquolowastlowastrdquo represents significance at the 5 level and ldquolowast lowast lowastrdquo represents significance at the 1 level

the square of the 119876 statistics rejects the null hypothesis atless than 1 significance in other words autocorrelationis present Thus nonlinear models are more appropriatefor studies of Chinese stock index futures prices based onnonstationary nonlinear tests

32 CEEMD Decomposition Yeh et al [13] indicated thatthe CEEMD method retained the decomposition ability ofthe EEMD method while effectively offsetting the addedwhite noise To demonstrate that the advantages of thisdecomposition remain valid in Chinarsquos stock index futuresprices this paper uses both the CEEMD and EEMDmethodsto decompose the China stock index futures price series andthen compared the differences between the original seriesand the sum of the decomposition sequences obtained by thetwomethods Depending on the general setting the standarddeviation of the added white noise is 02 times the standarddeviation of the daily closing price See Figure 2 for thedifferences between the results

As shown in Figure 2 thewhite noise residue after decom-position by the CEEMD method is nearly zero whereas agreat deal of white noise residue remains after decompositionby the EEMD method indicating that the CEEMD methodcan better maintain the completeness of the original seriesthan the EEMD method Therefore this paper used theCEEMD method to decompose Chinese stock index futuresprices and the IMFs and residue term are presented inFigure 3

As shown in Figure 3 the frequency is the highest forIMF1 and then decreased in order among the remainingIMFs IMF1 IMF2 and IMF3 exhibit dense fluctuationswith amplitudes between positive and negative 50 pointsapproximatelyThe amplitudes of IMF4 and IMF5 increase tobetween positive and negative 100 points and the fluctuationfrequency of IMF5 decreases significantly compared with

the fluctuation frequency of the previous IMFs with afluctuation period of approximately 120 days (4 months)Theamplitude of IMF6 increases to between positive and negative200 pointsmdasheven reaching 400 points once during theinitial stock index futures listing periodmdashwith a fluctuatingperiod that increases to approximately 400 days (13 months)The upward fluctuation frequency of IMF7 decreases toapproximately 100 points whereas the downward fluctuationfrequency is maintained at approximately 200 points itsfrequency also decreases to the lowest among the seven IMFsbut the fluctuating period increases to approximately 600days (20 months) The residue term (res) exhibits a trend ofan initial slow decline and subsequent gradual increase butthere is no clear cyclical fluctuation

As the frequencies decrease the correlation and varianceratios of the IMFs with the Chinese stock index futures priceseries increase indicating that a lower frequency enablesgreater fluctuation information content to be obtained fromthe original series In the low-frequency series the correlationand variance ratio of IMF6 with the original series is signif-icantly larger than that of IMF5 and IMF7 indicating thatmedium-term fluctuations in the original series primarilyoriginated from IMF6 and that the correlations and varianceratios of the residue term with the original series are fargreater than those of the IMFs indicating that this seriescontains the most important information from the originalseries (Table 2)

33 Analysis of Different Time Scales The analysis describedabove established that the frequencies of IMF and theresidue term differ The fluctuations of the high-frequencyseries exhibit ambiguity and disorderliness and the cyclicalfluctuations of the low-frequency series are stronger Theresidue term contains most of the information of the orig-inal series but does not exhibit a clear fluctuating period

6 Mathematical Problems in Engineering

Table 3 Correlations and variance ratios of the component terms with the Chinese stock index futures price series

Pearson correlation coefficient Variance ratio (original series)Residue term 0762 (39214)lowastlowastlowast 70278Low-frequency term 0526 (206229)lowastlowastlowast 37521High-frequency term 0196 (6661)lowastlowastlowast 4051Note ldquolowast lowast lowastrdquo represents significance at the 1 significance level

indicating that the price fluctuation information reflected bythe high-frequency series low-frequency series and residueterm feature more significant differences To more accuratelyreveal the fluctuation characteristics of Chinese stock indexfutures prices at various time scales we analyze the sum ofthe IMFs with similar fluctuation characteristics using a 119905

statistics test of the mean from zero [8] Based on the data inTable 2 the high-frequency IMFsmdashIMF1 IMF2 IMF3 andIMF4mdashare summed as the high-frequency term The low-frequency IMFmdashIMF5 IMF6 and IMF7mdashare summed asthe low-frequency term The residue term alone is retainedThe correlations and variance ratios of the three componentswith the original series are presented in Table 3 and theirdistribution is presented in Figure 4

Consistent with the literature such as [8 9] the residueterm includes most of the information value of the originalseries and reflects the long-term trend of the original seriesIn fact the correlation coefficient of the residue term withthe Chinese stock index futures price series in this paper isas high as 076 and explains 70 of the fluctuations in theoriginal price series (see Table 3)Therefore the residue termcan represent the long-term trend of the Chinese stock indexfutures prices As shown in Figure 4 between the officiallisting of stock index futures inChina (2010) and 2014 there isa slow downward trend in stock index futures prices Becausethe stock index is the ldquobarometerrdquo of a countryrsquos macroe-conomic situation and because stock index futures serve aprice discovery function for stock indexes the futuresmarketprices reflect newly revealed information more quickly thancurrent market prices Therefore macroeconomic changesalso determine the long-term trend of stock index futuresprices As Chinarsquos Gross Domestic Product (GDP) growthrate gradually slows after 2010 (the GDP growth rates from2010 to 2014 are 1045 93 765 767 and 74 resp)(data for GDP growth rates in China from 2010 to 2014 weredownloaded from theNational Bureau of Statistics of the Peo-plersquos Republic of China website (httpwwwstatsgovcn))and macroeconomic growth shows signs of fatigue stockindex futures prices also decrease year by year decreasingfrom 3376 points in the beginning of 2010 to 2332 points inthe beginning of 2014 with an average annual growth rateof approximately minus11192 However this downward trendslows after 2014 and reverses slightly in the second half of 2014mainly because in that year certain important and significantpolitical actions including administrative simplification andanticorruption legislation and enforcement create a healthypolitical environment for the economy In addition a reduc-tion in the reserve requirement ratio and an interest rate cutwill free up cash for investment andChinarsquos economic growth

turns from incremental expansion to an optimized structurewhich results in good prospects for investors Accordinglymore money is flowing into the stock market which drivesstock index prices higher and higher stock index futures priceresult simultaneously

The low-frequency term mainly reflects the medium-term effect of significant events on stock index futures pricesAs shown in Table 3 the correlation between the low-frequency term and the Chinese stock index futures priceseries is 056 and explains up to 375 of the original priceseries indicating that significant events in recent years play animportant role in the fluctuations in the Chinese stock indexfutures prices Figure 4 illustrates the plummet that occursduring the launch of the Chinese stock index futures whichhas a maximum amplitude of 500 points This plummet ismainly due to the strengthening of the countryrsquos efforts tosuppress real estate market speculation in 2010This initiativeshould lead to a stock market decline because real estateinvestment is an important economic driver at the time butthe stock market does not make the appropriate responseTheChinese stock index futures unleash their price discoveryfunction after launch which leads to adjustments in stockindex prices In addition many investors use the hedgingfunction of stock index futures to avoid the risk of a stockmarket crash However after the substantial fluctuationsduring launch the amplitude of the low-frequency termdecreases slightly reflecting in part the stabilization effectof stock index futures on the stock market To furtherinvestigate the effects of significant events on Chinese stockindex futures prices we use the ICSS algorithm to performbreakpoint tests on the rate of change in the low-frequencyterm (first-difference processing on the low-frequency termto identify the breakpoints) Table 4 presents the results ofthe breakpoint test on the low-frequency term The low-frequency term exhibits 11 breakpoints or in other words12 different volatility mechanisms The significant eventsthat cause breakpoints are primarily administrative macro-control policies which are mainly addressed by adjustingmonetary policy perfecting the capital market mechanismand promoting market liberalization The execution of thesepolicies brings volatility to Chinese stock index futures inturn creating a new volatility mechanism These breakpointsare mainly concentrated in 2012 and 2013 During these twoyears the government frequently issues regulatory policiesto bolster financial deepening and reform and investorsadopt a medium-term investment strategy revolving aroundthese policies As an emerging market the Chinese stockindex futures market is easily affected However as thecapital market mechanism gradually adjusts trading in the

Mathematical Problems in Engineering 7

20104 20114 20124 20134 20144

005

1

minus05

minus1

times10minus11

(a)20104 20114 20124 20134 20144

02040

minus20

minus40

(b)

Figure 2 (a) The differences between the stock index futures series and the decomposition sequence obtained by the CEEMD method (b)The differences between the stock index futures series and the decomposition sequence obtained by the EEMDmethod

20104 20114 20124 20134 20144

0

100

200

300

IMF5

20104 20114 20124 20134 20144

0

200

400

600

IMF6

20104 20114 20124 20134 20144

0

100

IMF7

20104 20114 20124 20134 201442000

2500

3000

3500

Res

20104 20114 20124 20134 20144

IMF1 0

50

100

minus50

minus100

20104 20114 20124 20134 20144

IMF3 0

50

100

minus50

minus100

20104 20114 20124 20134 20144

0

100

200

IMF4

minus100

minus200

minus100

minus200

minus200

minus200

minus100

minus300

minus400

20104 20114 20124 20134 20144

0

50

100

IMF2

minus50

minus100

Figure 3 IMFs and residue term

8 Mathematical Problems in Engineering

Table 4 Breakpoint tests on the low-frequency term and corresponding significant events

Breakpointnumber Corresponding time Corresponding significant event The response of stock index futures prices

1 December 13 2010

On December 10 2010 the CentralEconomic Working Conference convenesand indicates that the moderately loosemonetary policy has become robust

This event implies that the supply ofliquidity to the capital market would bereduced which was bad news for the stockindex futures market Hence the stockfutures prices slump

2 September 28 2011

In October 2011 the credit ratings of a dozenEuropean financial institutions aredowngraded and the European debt crisisfurther escalates

The Chinese financial market was affectedby this event and in response the stockindex futures prices maintained thedownward trend

3 January 31 2012

From January 6 to 7 2012 the NationalConference on Financial Work convenes inBeijing to set the tone of the next series ofmajor financial reform issues

This event makes the stock market a positiveprospect for investors and then pushes upthe stock index futures prices

4 May 17 2012

On May of 2012 the stock index futuresprices are affected by a combination ofevents including the US stock market andinternational commodities market crashingand the Chinese PMI index does notcontinue early rising trend

The series of significant events poses aserious negative impact on investorsrsquoexpectations Therefore the stock indexfutures prices stop rising and turn tocollapse

5 June 28 2012

On June 18 2010 the Shanghai StockExchange and the Shenzhen Stock Exchangeseparately publish programs related to thedelisting of listed companies and revise thecorresponding Stock Market Rulesaccordingly further perfecting the initialpublic offering (IPO) and delisting systems

This event would improve investorsrsquoconfidence in the stock market Hence thedownward trend of stock index futures priceis slowed

6 August 28 2012

On August 16 2012 the New York branch ofthe Agricultural Bank of China openscontinuing to accelerate the pace ofinternationalization of the Chinese bankingsector

The event promotes the process of RMBinternationalization which is good news forthe financial market and stock index futuresprices inch up

7 October 18 2012

On October 17 2012 the Chinese StateCouncil meeting made an importantdecision that government would continue toimplement the proactive fiscal policy andprudent monetary policy

The prudent monetary policy implyinglimited funds in the capital market was badnews for stock index futures market Hencethe stock index futures prices tumble over

8 January 31 2013 At the end of January 2013 broad moneygrew 159 and narrow money grew 153

This event prompted a rise in stock indexfutures prices

9 May 21 2013

In May of 2013 the Chinese macroeconomicsituation does not perform well the FederalReserve gradually winds up the quantitativeeasing policy and many banks in China lacksufficient capital at the time

The cumulative effects of shocks of theseevents occurring in May have an enormousnegative influence on the stock index futuresmarket Hence the stock index futuresprices collapse

10 September 10 2013

On September 6 2013 repackaged treasurynote futures debut after a gap of 18 yearswhich is conducive to the establishingmarket-oriented pricing benchmarks andadvances market-oriented reform of theinterest rate

This event rebuilding investorsrsquo confidencein the capital market then pushes up thestock index futures prices

11 February 25 2014

On March 1 2014 the Central Bank releasesthe upper limit on interest rates on smallamounts of foreign currency deposits in theShanghai Free-Trade Zone taking anotherstep toward market-oriented reform of theinterest rate

The rise of deposit interest rate indicates thatmarket as risk-free rate would increasewhich is negative news for the stock indexfutures market Hence the stock indexfutures prices plunge

Mathematical Problems in Engineering 9

20104 20114 20124 20134 20144

0

1000

2000

3000

4000

Original seriesResidue term

Low-frequency termHigh-frequency term

minus1000

Figure 4 Chinese stock index futures prices and the three compo-nents

Chinese stock index futures market further standardizes andthe ability to withstand external impact thus also increasesTherefore the number of breakpoints decreases after 2014

The high-frequency term mainly reflects the short-termdisequilibrium in the Chinese stock index futures marketAs shown in Table 3 the correlation of the high-frequencyterm with the Chinese stock index futures prices is merely0196 and the ability to explain fluctuations in the originalprice series is only 4 indicating that the overall effectof the high-frequency term on Chinese stock index futuresprices is limited However as Figure 4 shows the fluctuationsof the high-frequency term fit well with the local fluctu-ations of the Chinese stock index futures prices in otherwords the high-frequency term plays a primary role in localchanges in Chinese stock index futures prices In additionas opposed to the low-frequency term the high-frequencyterm fluctuates frequently but with smaller amplitudes Thisdifference occurs primarily because the high-frequency termis particularly affected by investorsrsquo speculative arbitrageactivities In particular implementation of a T+0 transactionsystem in the Chinese stock index futures market means thatinvestors buying and selling stock index futures can completetheir transactions in the same day which suits short-termspeculators Based on this transaction system there will bemore investment chances for investors which will enhanceinvestorsrsquo speculative opportunities Hence there are manyshort-term fluctuations in Chinese stock index futures pricesMeanwhile it is possible and convenient for investors toaccess the latest hot investment tip because the role of socialinteractions is partly substituted for by online information inthe Internet age [31] In reality this type of new informationchanges continuously and investors must frequently adjusttheir investment strategies to gainmore additional benefits byspeculating in the spot-futures marketsTherefore the strongeffect of the shock of investorsrsquo speculation on Chinese stockindex futures prices in the short-term is difficult to sustainAlthough fluctuations in the high-frequency term are notimportant to forecast the long-term trends of Chinese stockindex futures their predictive effect of minor fluctuations onstock index futures cannot be ignored

4 Predictions for Chinese StockIndex Futures Prices

In this section this paper further investigates the predictionsof the Chinese stock index futures prices In addition tothe previous dayrsquos price data we also select the five-minutehigh-frequency price data of Chinese stock index futuresfrom April 16 2010 to November 20 2014 as a researchsubject which yields a total of 61325 data points (the data areobtained from theWind database) To forecast the daily pricethe first 745 (approximately two-thirds) pieces of data areused as the training data and the last 370 (approximately one-third) pieces of data are used as the test data To forecast thefive-minute high-frequency price the first 40883 (approxi-mately two-thirds) data points are used as the training dataand the last 20442 (approximately one-third) data points areused as the test data

As our analysis of the price fluctuations of the Chinesestock index futures uses the CEEMDmethod the fluctuationcharacteristics in different time scales have their uniqueeconomic implications we suggest that there may be betterpredictions of Chinese stock index futures prices when thesefluctuation characteristics are considered in the predictionmodel Hence based on the literature [16 17] a combinationprediction model that is the CEEMD-SVM-SVM model isproposed and the PSO algorithm is used to optimize theparameters of the SVM model In our empirical researchwe perform a one-step-ahead prediction of Chinese stockindex futures prices The specific steps are as follows (1) theCEEMD method is used to decompose the Chinese stockindex futures price series into several IMFs and a residueterm (2) the IMFs are then composed into a low-frequencyterm and a high-frequency term based on a fine-to-coarsereconstruction and the residue term remains the same (3)SVM models are established separately for each of the threecomponents obtained in step (2) and the PSO algorithm isused to optimize the parameters and (4) the predictive valueof the three components obtained in step (3) is used as theinput with the real value at the same moment as the outputand then a SVM combination model is built to obtain thefinal predictive value The overall process of the CEEMD-SVM-SVMmodel is shown in Figure 5

The prediction evaluation indexes are the root meansquare error (RMSE) mean absolute prediction error(MAPE) and directional symmetry (DS) the DS predictionevaluation index is used to indicate the modelrsquos ability topredict the directionality trend of Chinese stock index futuresprices The prediction evaluation indexes are expressed asfollows

RMSE = radic1

119899

119899

sum119905=1

[119909 (119905) minus (119905)]2

MAPE =1

119899

119899

sum119905=1

10038161003816100381610038161003816100381610038161003816

119909 (119905) minus (119905)

119909 (119905)

10038161003816100381610038161003816100381610038161003816

DS =1

119899

119899

sum119894=1

119889119894

10 Mathematical Problems in Engineering

Original series

CEEMD

Residue term Low-frequency term High-frequency term

SVM1 SVM2 SVM3

Final predictive value

SVM combination

Figure 5 The overall process of the CEEMD-SVM-SVMmodel

119889119894=

1 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] ge 0

0 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] lt 0

(10)

Among the three evaluation indexes smaller RMSE andMAPE values indicate that the predictionmodel is experienc-ing better performance which is also true of greater DS valueand better performance of the prediction model

The prediction accuracy based on the SVM model issensitive to the SVM modelrsquos kernel function As manyprevious studies (such as [17 22]) indicate that the linearkernel function and the RBF kernel function are appropriatekernel functions for the prediction of SVMmodel the linearkernel function and the RBF kernel function of the SVMare thus both applied to our prediction of Chinese stockindex futures prices Based on the process of the CEEMD-SVM-SVM model we first predict the residue term thenthe low-frequency term and then the high-frequency termof the daily price series and the five-minute price seriesNotably the five-minute price series of Chinese stock indexfutures is decomposed into fourteen IMFs and one residueterm where IMFs from IMF1 to IMF9 are recomposed intothe high-frequency term and IMFs from IMF10 to IMF14are recomposed into the low-frequency termThe predictionerrors of these components are shown in Table 5

As shown in Table 5 in using the SVM model to makepredictions on each component the prediction precision ofthe linear kernel function SVM model is generally higherthan the RBF kernel function SVM model In additionwhether for predicting the daily price series or the five-minute price series the prediction precisions of the residueterm and the low-frequency term are significantly higherthan those of the high-frequency termThis difference occursbecause these effects of shocks on the residue term andthe low-frequency term are fundamental economically theyare significant events that have an impact on Chinese stockindex futures prices that is sustained over a long duration

Therefore the fluctuations of the residue term and the low-frequency term are relatively steady and easier to forecastusing the SVM model By contrast the high-frequency termrepresenting the short-term disequilibrium of Chinese stockindex futures price is affected mainly by daily market activ-ities such as speculation arbitrage and the fluctuation ofthe high-frequency term that is random and changes rapidlyHence the SVMmodel experiences difficulty in learning thepattern of high-frequency term fluctuation resulting in poorpredictive results

Then according to prediction step (4) we use the optimalpredictive value of each component obtained in the previousstep to perform a combination prediction and the predictionerrors of our proposed model and the single SVM modelbased on the same data sample are simultaneously comparedThese results of the prediction errors are listed in Table 6

Table 6 shows that as opposed to the predictive resultsof the three components the RBF kernel function SVMmodel has higher predicted precision than the linear kernelfunction SVM model when performing combination pre-diction which indicates that the RBF kernel function SVMmodel is more appropriate for combination prediction Inaddition we find that the predictive results of the finalpredictive value obtained under the RBF kernel function aresignificantly better than those obtained in the single SVMmodel when predicting the daily price but slightly better thanthe single SVM model when making predictions regardingthe high-frequency price Meanwhile we should considerthat the high-frequency price embodies far more complexinformation than the daily price during the same time spanwhich may restrict the learning ability and generalizationability of the SVM model Hence the CEEMD-SVM-SVMmodel performs better in terms of predicting the daily priceIn any event the CEEMD-SVM-SVM model proposed bythis paper performs better than the single SVM model interms of both forecasting the daily price series and the five-minute price series Thus the introduction of Chinese stockindex futures price characteristics in different time scales into

Mathematical Problems in Engineering 11

Table 5 The prediction errors of different components

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

Residue termRMSE 0007 0008 0006 0009MAPE 0001 0001 0001 0002DS 69837 58424 21678 27241

Low-frequency termRMSE 0007 0008 0004 0008MAPE 0087 0092 0042 0258DS 60326 55434 54613 55254

High-frequency termRMSE 0086 0098 0020 0021MAPE 4806 4725 1146 1149DS 63315 62500 51463 51854

Table 6 Comparison of different modelsrsquo prediction errors

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

SVMRMSE 0053 0054 0008 0008MAPE 0009 0009 0001 0001DS 54891 52445 48171 48161

CEEMD-SVM-SVMRMSE 0051 0009 0008 0007MAPE 0008 0002 0001 0001DS 62228 88043 51565 52715

the SVM model improves the ability of the SVM model tomine the variations in the stock index futures price seriesfurther enhancing its prediction precision Simultaneouslysome useful suggestions for investors can be found in thepredictions of Chinese stock index futures prices generatedby our proposedmodel First investors should be accustomedto analyzing the price series from different time scales tounderstand the fluctuation characteristics of Chinese stockindex futures prices Second investors must actively identifywhat time scale the effects of external shocks on stock indexfutures prices belong to andmust devise the appropriate strat-egy accordingly Finally investors should pay more attentionto a hedging strategy involving Chinese stock index futuresover the long andor medium term and should be cautiousabout short-term speculation

Figures 6 and 7 present the real and predictive values ofthe daily price series andfive-minute price series respectivelyThe final predictive values obtained using the CEEMD-SVM-SVM model appeared to coincide with the real values indi-cating that the predictive values obtained using this modelfit well with the real values and intuitively demonstratingthat the CEEMD-SVM-SVMmodel is suitable for forecastingChinese stock index futures prices

20135 20138 201311 20142 20145 20148 2014112000

2200

2400

2600

2800

Real valuePredictive value

Figure 6 Real value and predictive value of the daily price series

5 Conclusion

The Chinese stock index futures market is a complex systemand fluctuations in the stock index futures prices resultfrom the collective effects of many factors at various levelsThe CEEMD method a method suitable in the analysis ofnonlinear nonstationary data is applied to analyze Chinesestock index futures prices in different time scales and to

12 Mathematical Problems in Engineering

Real valuePredictive value

20135 20138 201311 20142 20145 20148 2014111800

2000

2200

2400

2600

2800

Figure 7 Real value and predictive value of the five-minute priceseries

reveal the intrinsic fluctuation characteristics of Chinesestock index futures prices Then we focus on whether thesecharacteristics improve the predictive precision of Chinesestock index futures prices and propose a CEEMD-SVM-SVMmodel based on the PSO algorithm The empirical resultsdemonstrate that the intrinsic fluctuation characteristics ofChinese stock index futures prices can be revealed in threecomponents including the residue term the low-frequencyterm and the high-frequency term with independent eco-nomic significance The residue term determines the long-term trend of stock index futures prices and its fluctuationsare correlated with macroeconomics and the long-termexpectations of investorsThe low-frequency term reveals theeffect of significant events on medium-term fluctuations instock index futures prices the ICSS algorithm breakpointtests reveal that policy regulation events are the main causeof fluctuations during the low-frequency term The high-frequency term reflects short-term disequilibrium in thestock index futures market although the effect of the high-frequency term on the overall fluctuation in the stock indexfutures prices is limited it nonetheless plays an important rolelocally

In addition in forecasting Chinese stock index futuresprices first we find that the linear kernel function SVMmodel is more suitable for predicting these components thanthe RBF kernel function SVM model The residue term andthe low-frequency term have stronger regular fluctuationsthan the high-frequency term and therefore yield superiorpredictive results Second the RBF kernel function SVMmodel shows higher predictive precision for the final combi-nation prediction than the linear kernel function SVMmodelSimultaneously the predictive results of the combinationprediction model are better than those of a single SVMmodel which indicates that the integration of fluctuationcharacteristics in different time scales increases the abilityof the SVM model to identify regularity in Chinese stockindex futures price fluctuations thus improving predictionprecision Finally the predictive results of the CEEMD-SVM-SVM model on daily prices and intraday prices are basicallyconsistent which indicates that our proposed model hasstability and universal applicability in forecasting Chinesestock index futures prices

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by the National SocialScience Foundation of China (14ZDA045)

References

[1] VMeneu andH Torro ldquoAsymmetric covariance in spot-futuresmarketsrdquo Journal of Futures Markets vol 23 no 11 pp 1019ndash1046 2003

[2] M Zhong A F Darrat and R Otero ldquoPrice discovery andvolatility spillovers in index futures markets some evidencefrom Mexicordquo Journal of Banking and Finance vol 28 no 12pp 3037ndash3054 2004

[3] X Xiong F Wang W Zhang and Y Sun ldquoPrice discoveryand volatility spillovers between SGX FTSEXinhua ChinaA50 index futures and A-share marketrdquo Chinese Journal ofManagement vol 11 no 1535 pp 1507ndash1512 2009

[4] H M Zuo Z T Liu and H W Zeng ldquoResearch of volatilityspillover and information transmission between stock indexfutures and spot market based on the high frequency datardquoJournal of Financial Research vol 4 pp 140ndash154 2012

[5] H Q Chen and C H Zhang ldquoDoes index futures tradingreduce stock market jump riskrdquo Economic Research Journalvol 1 pp 153ndash167 2015

[6] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 48 no 3 pp235ndash254 2015

[7] E E Peters Fractal Market Analysis Applying Chaos Theory toInvestment and Economics John Wiley amp Sons New York NYUSA 1994

[8] X Zhang K K Lai and S-YWang ldquoA new approach for crudeoil price analysis based on empirical mode decompositionrdquoEnergy Economics vol 30 no 3 pp 905ndash918 2008

[9] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnonstationary time series analysisrdquo Proceedings of the RoyalSociety of London A vol 454 no 1971 pp 903ndash995 1998

[10] Z Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[11] L F Ruan and H J Bao ldquoAn empirical analysis on periodicfluctuations of real estate price based on EMDrdquo Chinese Journalof Management Science no 3 pp 41ndash46 2012

[12] H L Li and C E Feng ldquoRelationship between investor senti-ment and stock indices fluctuation based on EEMDrdquo SystemsEngineering-Theory amp Practice vol 10 pp 2495ndash2503 2014

[13] J-R Yeh J-S Shieh andN EHuang ldquoComplementary ensem-ble empiricalmode decomposition a novel noise enhanced dataanalysismethodrdquoAdvances in Adaptive DataAnalysis vol 2 no2 pp 135ndash156 2010

[14] L Zhao W Yu and R Yan ldquoRolling bearing fault diagnosisbased on CEEMD and time series modelingrdquo MathematicalProblems in Engineering vol 2014 Article ID 101867 13 pages2014

Mathematical Problems in Engineering 13

[15] L Tang W Dai L Yu and S Wang ldquoA novel CEEMD-based eelm ensemble learning paradigm for crude oil priceforecastingrdquo International Journal of Information Technologyand Decision Making vol 14 no 1 pp 141ndash169 2015

[16] L Yu S Wang and K K Lai ldquoForecasting crude oil price withan EMD-based neural network ensemble learning paradigmrdquoEnergy Economics vol 30 no 5 pp 2623ndash2635 2008

[17] Y F Yang Y K Bao Z Y Hu and R Zhang ldquoCrude oilprice prediction based on empirical mode decomposition andsupport vector machinesrdquo Chinese Journal of Management vol12 pp 1884ndash1889 2010

[18] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[19] W Huang Y Nakamori and S-Y Wang ldquoForecasting stockmarket movement direction with support vector machinerdquoComputers and Operations Research vol 32 no 10 pp 2513ndash2522 2005

[20] H Ince and T B Trafalis ldquoShort term forecasting with supportvector machines and application to stock price predictionrdquoInternational Journal of General Systems vol 37 no 6 pp 677ndash687 2008

[21] F E H Tay and L Cao ldquoApplication of support vectormachinesin financial time series forecastingrdquo Omega vol 29 no 4 pp309ndash317 2001

[22] Y Sai F T Zhang and T Zhang ldquoResearch of Chinese stockindex futures regression prediction based on support vectormachinesrdquo Chinese Journal of Management Science vol 3 pp35ndash39 2013

[23] C Inclan andG C Tiao ldquoUse of cumulative sums of squares forretrospective detection of changes of variancerdquo Journal of theAmerican Statistical Association vol 89 no 427 pp 913ndash9231994

[24] S H Kang C Cheong and S-M Yoon ldquoStructural changesand volatility transmission in crude oil marketsrdquo Physica AStatisticalMechanics and Its Applications vol 390 no 23-24 pp4317ndash4324 2011

[25] J P Yang X X Chen and C H Zhang ldquoPolicy impact factorson the volatility of Shanghai and Shenzhen stock market inChinardquo Chinese Journal of Management Science vol 20 no 6pp 43ndash51 2012

[26] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer New York NY USA2010

[27] C-L Huang and J-F Dun ldquoA distributed PSOndashSVM hybridsystem with feature selection and parameter optimizationrdquoApplied Soft Computing vol 8 no 4 pp 1381ndash1391 2008

[28] C Zhou K Yin Y Cao and B Ahmed ldquoApplication of timeseries analysis and PSOndashSVMmodel in predicting the Bazimenlandslide in the Three Gorges Reservoir Chinardquo EngineeringGeology vol 204 pp 108ndash120 2016

[29] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis amp Applications vol 16 no 1 pp 93ndash103 2014

[30] X Gong Z He P Li and N Zhu ldquoForecasting return volatilityof the CSI 300 index using the stochastic volatility model withcontinuous volatility and jumpsrdquo Discrete Dynamics in Natureand Society vol 2014 Article ID 964654 10 pages 2014

[31] S Q Guo and P H Liang ldquoSocial Interactions the informationchannel and household stock market participation a empiricalstudy based on a Chinese household finance surveyrdquo EconomicResearch Journal vol S1 pp 116ndash131 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

futures markets are based on low-frequency daily weeklyand monthly price data [1ndash3 19ndash22 29] current researchregarding intraday high-frequency data gradually becamea focus [4 5 30] Therefore we also use this model toforecast the daily data and the intraday high-frequency dataof Chinese stock index futures prices The main innovationof this paper is the use of the CEEMD method to reveal theintrinsic fluctuation characteristics of Chinese stock indexfutures prices from various time scales which provides a newresearch perspective on the fluctuation analysis of stock indexfutures pricesThefluctuation characteristics of various scalesare introduced into predicting stock index futures prices thusimproving the predictive ability of a single SVM model withrespect to the stock index futures market

The remainder of this paper is structured as followsSection 2 introduces themethods used in this paper Section 3analyzes the Chinese stock index futures prices at varioustime scales Section 4 forecasts Chinese stock index futuresprice Finally Section 5 provides a summary

2 Introduction of Methodology

21 The CEEMD Method EMD methods such as the EMDmethod the EEMD method and the CEEMD method areself-adaptive data processing methods that can decompose atime series into a series of Intrinsic Mode Functions (IMFs)with independent informationThese IMFs are effective toolsto address nonlinear nonstationary data In this paper weemploy the latest extension that is the CEEMD methodto analyze Chinese stock index futures prices The detailedintroduction of the CEEMDmethod is stated as follows

The EMD method proposed by Huang et al [9] is thefirst method to extract series of IMFs and residue terms fromthe original time series In particular IMFs must satisfy thefollowing two conditions (1) the number of extrema andzero-crossing pointsmust be equal or differ by amaximumofone and (2) the mean value of the upper and lower envelopesmust be zero at all times Accordingly decomposition of theEMD method ends when the last residue term (abbreviatedas res) 119903(119905) is a monotonic function or a constant then theoriginal series 119909(119905) can be expressed as

119909 (119905)=119899

sum119894=1

119909119894(119905) + 119903 (119905) (1)

where IMF119894is recorded as 119909

119894(119905) Because mode mixing

can easily emerge in the EMD method Wu and Huang[10] further proposed the EEMD method to address thisdrawback of the EMD method The EEMD method assumesthat the observed data are amalgamations of true time seriesand noise and that the ensemble means of data with differentnoises are closer to true time series Hence adding whitenoise as an additional step may help extract the true IMFsin the time series and the noise can be offset by ensembleaveraging However residue noise continues to be generatedby the added white noise Therefore Yeh et al [13] thenproposed theCEEMDmethod by adding pairs of positive andnegative white noise to generate complementary sequences

Specifically the complementary sequences are obtained bythe following equation

[119909+

119894(119905)

119909minus119894(119905)

] = [1 1

1 minus1] [

119909 (119905)

120576119894(119905)

] (2)

In (2) 120576119894represents 119894th added white noise and 119909+

119894(119905)

and 119909minus119894(119905) represent the sequences after positive and negative

white noise is added for 119894th time to the original 119909(119905) seriesWe can generate complementary IMFs via the EMDmethodand these are expressed as follows

119909+

119895(119905) =

1

119899

119899

sum119894=1

119909+

119894119895(119905)

119909minus

119895(119905) =

1

119899

119899

sum119894=1

119909minus

119894119895(119905)

(3)

Then the final IMFs with no residue noise are extractedby averaging the two complementary IMFs

119909119895=

(119909+119895+ 119909minus119895)

2 (4)

22 ICSS Algorithm The ICSS algorithm proposed by Inclanand Tiao [23] is a relatively mature approach to detectingbreakpoints in sequences The algorithm assumes that in thestable time series 119884

119894119905= 120583119894+ 120576119894119905 120583119894denotes the mean return

for time series 119894 and 120576119894119905 is the residual series with a mean

of 0 and a variance of 1205902119894119905 Letting 119862

119896= sum119896

119894=11205762119905 119896 = 1 119879

be the cumulated sum squares of the residual series 120576119905 the

statistic 119863119896can be defined as follows

119863119896= (

119862119896

119862119879

) minus (119896

119879) 119896 = 1 119879 119863

0= 119863119879

= 0 (5)

If breakpoints do not occur in the variance in the iterationprocess the value of the 119863

119896statistic will vary around 0 By

contrast if there are one or more breakpoints in a sequencethe value of the statistic 119863

119896will deviate significantly upward

or downward from the 0 Let 119896lowast be the value for 119870 atmax119896|119863119896| If max

119896radic(1198792)|119863

119896| exceeds the preset credible

interval 119896lowast will be taken as an estimated break point whereradic(1198792) is a normalizing factor

23 SVM Prediction Model The SVM model employs akernel-based method in which the basic idea is to construct ahyperplane with low risk in high-dimensional feature spaceThis method has the advantage of generalization capabilitiesand good function approximation Among the various typesof SVM models the 120576-SVM model proposed by Vapnik [18]is mainly used to resolve prediction problems Given a set oftraining patterns (119909

1 1199101) (119909

119894 119910119894)119899119894=1

where 119909119894isin 119877 is the

input variable and119910119894isin 119877 is the output variable the prediction

model is learned from these patterns and is used to forecast

4 Mathematical Problems in Engineering

the output variables of unseen input variables Specifically theconstraints and objective function for the 120576-SVMmodel are

min120596119887120591120591

lowast

120596119879120596

2+ 119888

119899

sum119894=1

120591119894+ 119888

119899

sum119894=1

120591lowast

119894

st 120596119879

120601 (119909119894) + 119887 minus 119910

119894le 120576 + 120591

119894

120596119879

120601 (119909119894) + 119887 minus 119910

119894ge 120576 + 120591

lowast

119894

120591119894 120591lowast

119894ge 0 119894 = 1 2 119899

(6)

In (6) 120591119894and 120591lowast

119894are the slack variables that represent

the allowable upper limit and lower limit of training errorsrespectively under the constraint of insensitive loss 120576 119888 is thepenalty factor controlling the degree of penalty on sampleswhose errors exceed 120576

The Lagrange function must be introduced to resolve(6) According to the Duality Theory and the saddle pointcondition the dual form of (6) can be obtained as follows

min120597lowast120597

12

119899

sum119894=1

119899

sum119895=1

(120597lowast

119894minus 120597119894) (120597lowast

119895minus 120597119895) 119896 (119909

119894 119909119895)

+ 120576

119899

sum119894=1

(120597lowast

119894+ 120597119894) minus

119899

sum119894=1

119910119894(120597lowast

119894minus 120597119894)

st119899

sum119894=1

(120597lowast

119894minus 120597119894) = 0 0 le 120597

lowast

119894 120597119894le 119888 119894 = 1 2 119899

120596 =

119899

sum119894=1

(120597lowast

119894minus 120597119894)

(7)

In (7) 120597119894is the nonnegative Lagrangian multiplier of 120597lowast

119894

and 119896(119909119894 119909119895) represents the kernel function Any function

that satisfies Mercerrsquos condition can be a kernel functionThe common kernel functions include the linear kernelfunction 119896(119909 119909

119894) = 119909119879

119909119894and the Radial Basis Function (RBF)

kernel function 119896(119909 119909119894) = exp(minus120574119909 minus 119909

1198942) Ultimately the

hyperplane for the underlying prediction problem is given bythe following

119891 (119909) =

119899

sum119894=1

(120597lowast

119894minus 120597119894) 119896 (119909119894 119909) + 119887 (8)

24 PSO Algorithm The PSO algorithm proposed byKennedy [26] is an effective algorithm to address func-tion parameter optimization Each particle in the algorithmdenoting a potential solution in the solution space has threeindicators including speed position and fitness value Thefitness value is determined by the fitness function whichcan estimate the merit of the particles The velocity of theparticle determines the movement direction and distanceand the velocityrsquos dynamic adjustment follows the movementexperience of the particle itself and other particles thusenabling the optimal selection of an individual in the solutionspace The details are as follows

Suppose that in a 119863-dimensional solution space thepopulation comprising n particles is 119883 = (119883

1 1198832 119883

119899)

20104 20114 20124 20134 201442000

2500

3000

3500

4000

Figure 1 Daily closing price of Chinese stock index futures

the position of 119894th particle is 119883119894

= (1199091198941 1199091198942 119909

119894119863)119879 and

the fitness value of each particle can be calculated basedon the objective function and position of each particleThe velocity of 119894th particle is 119881

119894= (1198811198941 1198811198942 119881

119894119863)119879 its

individual extremum is 119875119894

= (1198751198941 1198751198942 119875

119894119863)119879 and the

group extremum is119875119892= (1198751198921

1198751198922

119875119892119863

)119879 In each iterativeprocess the particle updates its velocity and position throughthe individual extremum and the group extremum Theupdating equation is as follows

119881119896+1

119894119889= 120596119881119896

119894119889+ 11988811199031(119875119896

119894119889minus 119883119896

119894119889) + 11988821199032(119875119896

119892119889minus 119883119896

119894119889)

119883119896+1

119894119889= 119883119896

119894119889+ 119881119896+1

119894119889

(9)

where120596 represents the inertia weight 119889 = 1 2 119863 119896 is thecurrent iteration number119881

119894119889is the velocity component of the

particle 1198881and 1198882are the acceleration factors and nonnegative

constants and 1199031and 1199032are random numbers in the range

[0 1] Because the SVM prediction model is sensitive to themodel parameters and because the literature [22 27 28]indicates that the PSO algorithm performs very well in theparameter optimization process of the SVMmodel this paperuses the PSO algorithm to optimize the parameters of theSVM prediction model

3 Analysis of Chinese Stock IndexFutures Prices in Different Time Scales

31 Data Selection andDescription TheCSI 300 index futuresare the earliest stock index futures listed in China and it hasbeen widely used in many studies [4 5 22] Therefore thispaper selects the daily closing price of the stock index futuresas this sectionrsquos research objectThe time span runs fromApril16 2010 toNovember 20 2014 the sample data total 1115 andthe data are from theWind database See Figure 1 and Table 1for the sample data distribution and the descriptive statisticsrespectively

Nonstationary nonlinear tests are performed on Chinesestock index futures price series based on Ruan and Bao [11]Based on the data in Table 1 the null hypothesis that theChinese stock index future price series are nonstationarycannot be rejected under the condition of less than 10 signif-icance Because the postdifferential price series is a stationaryseries an autocorrelation analysis is further performed on theseries The test results for 119876 statistics demonstrate that thenull hypothesis for this differential series cannot be rejectedunder the condition of less than 10 significance However

Mathematical Problems in Engineering 5

Table 1 Descriptive statistics for Chinese stock index futures prices

Mean Std Dev JB ADF 119876(6) 1198762(6)

Price series 2600691 351503 97269lowastlowastlowast minus2539 mdash mdash119863(price series) minus0783 37611 858432lowastlowastlowast minus35531lowastlowastlowast 7737 26507lowastlowastlowast

Note Augmented Dickey-Fuller (ADF) statistics represent the nonstationary test of the price series 119863(price series) represents the price series after the firstdifference119876(6) statistics represent the autocorrelation test of the time series1198762(6) statistics represent the autocorrelation test of the square of the time seriesand ldquolowast lowast lowastrdquo indicates significance at the 1 significance level

Table 2 Correlations and variance ratios of IMFs and the residue term with the Chinese stock index futures price series

Mean value Pearson correlation coefficient Variance ratio (original series)IMF1 0131 (0238) 0041 (1363) 0273IMF2 minus0096 (minus0151) 0066 (2218)lowastlowast 0372IMF3 minus0431 (minus0523) 0097 (3273)lowastlowastlowast 0614IMF4 minus2103 (minus1277) 0163 (5530)lowastlowastlowast 2446IMF5 minus4652 (minus2413)lowastlowast 0218 (7454)lowastlowastlowast 3354IMF6 46989 (8007)lowastlowastlowast 0660 (29329)lowastlowastlowast 31076IMF7 minus64873 (minus18973)lowastlowastlowast minus0264 (minus9151)lowastlowastlowast 10550Res 2625727 0762 (39214)lowastlowastlowast 70278NoteThe values in parentheses are the 119905 statistics ldquolowastlowastrdquo represents significance at the 5 level and ldquolowast lowast lowastrdquo represents significance at the 1 level

the square of the 119876 statistics rejects the null hypothesis atless than 1 significance in other words autocorrelationis present Thus nonlinear models are more appropriatefor studies of Chinese stock index futures prices based onnonstationary nonlinear tests

32 CEEMD Decomposition Yeh et al [13] indicated thatthe CEEMD method retained the decomposition ability ofthe EEMD method while effectively offsetting the addedwhite noise To demonstrate that the advantages of thisdecomposition remain valid in Chinarsquos stock index futuresprices this paper uses both the CEEMD and EEMDmethodsto decompose the China stock index futures price series andthen compared the differences between the original seriesand the sum of the decomposition sequences obtained by thetwomethods Depending on the general setting the standarddeviation of the added white noise is 02 times the standarddeviation of the daily closing price See Figure 2 for thedifferences between the results

As shown in Figure 2 thewhite noise residue after decom-position by the CEEMD method is nearly zero whereas agreat deal of white noise residue remains after decompositionby the EEMD method indicating that the CEEMD methodcan better maintain the completeness of the original seriesthan the EEMD method Therefore this paper used theCEEMD method to decompose Chinese stock index futuresprices and the IMFs and residue term are presented inFigure 3

As shown in Figure 3 the frequency is the highest forIMF1 and then decreased in order among the remainingIMFs IMF1 IMF2 and IMF3 exhibit dense fluctuationswith amplitudes between positive and negative 50 pointsapproximatelyThe amplitudes of IMF4 and IMF5 increase tobetween positive and negative 100 points and the fluctuationfrequency of IMF5 decreases significantly compared with

the fluctuation frequency of the previous IMFs with afluctuation period of approximately 120 days (4 months)Theamplitude of IMF6 increases to between positive and negative200 pointsmdasheven reaching 400 points once during theinitial stock index futures listing periodmdashwith a fluctuatingperiod that increases to approximately 400 days (13 months)The upward fluctuation frequency of IMF7 decreases toapproximately 100 points whereas the downward fluctuationfrequency is maintained at approximately 200 points itsfrequency also decreases to the lowest among the seven IMFsbut the fluctuating period increases to approximately 600days (20 months) The residue term (res) exhibits a trend ofan initial slow decline and subsequent gradual increase butthere is no clear cyclical fluctuation

As the frequencies decrease the correlation and varianceratios of the IMFs with the Chinese stock index futures priceseries increase indicating that a lower frequency enablesgreater fluctuation information content to be obtained fromthe original series In the low-frequency series the correlationand variance ratio of IMF6 with the original series is signif-icantly larger than that of IMF5 and IMF7 indicating thatmedium-term fluctuations in the original series primarilyoriginated from IMF6 and that the correlations and varianceratios of the residue term with the original series are fargreater than those of the IMFs indicating that this seriescontains the most important information from the originalseries (Table 2)

33 Analysis of Different Time Scales The analysis describedabove established that the frequencies of IMF and theresidue term differ The fluctuations of the high-frequencyseries exhibit ambiguity and disorderliness and the cyclicalfluctuations of the low-frequency series are stronger Theresidue term contains most of the information of the orig-inal series but does not exhibit a clear fluctuating period

6 Mathematical Problems in Engineering

Table 3 Correlations and variance ratios of the component terms with the Chinese stock index futures price series

Pearson correlation coefficient Variance ratio (original series)Residue term 0762 (39214)lowastlowastlowast 70278Low-frequency term 0526 (206229)lowastlowastlowast 37521High-frequency term 0196 (6661)lowastlowastlowast 4051Note ldquolowast lowast lowastrdquo represents significance at the 1 significance level

indicating that the price fluctuation information reflected bythe high-frequency series low-frequency series and residueterm feature more significant differences To more accuratelyreveal the fluctuation characteristics of Chinese stock indexfutures prices at various time scales we analyze the sum ofthe IMFs with similar fluctuation characteristics using a 119905

statistics test of the mean from zero [8] Based on the data inTable 2 the high-frequency IMFsmdashIMF1 IMF2 IMF3 andIMF4mdashare summed as the high-frequency term The low-frequency IMFmdashIMF5 IMF6 and IMF7mdashare summed asthe low-frequency term The residue term alone is retainedThe correlations and variance ratios of the three componentswith the original series are presented in Table 3 and theirdistribution is presented in Figure 4

Consistent with the literature such as [8 9] the residueterm includes most of the information value of the originalseries and reflects the long-term trend of the original seriesIn fact the correlation coefficient of the residue term withthe Chinese stock index futures price series in this paper isas high as 076 and explains 70 of the fluctuations in theoriginal price series (see Table 3)Therefore the residue termcan represent the long-term trend of the Chinese stock indexfutures prices As shown in Figure 4 between the officiallisting of stock index futures inChina (2010) and 2014 there isa slow downward trend in stock index futures prices Becausethe stock index is the ldquobarometerrdquo of a countryrsquos macroe-conomic situation and because stock index futures serve aprice discovery function for stock indexes the futuresmarketprices reflect newly revealed information more quickly thancurrent market prices Therefore macroeconomic changesalso determine the long-term trend of stock index futuresprices As Chinarsquos Gross Domestic Product (GDP) growthrate gradually slows after 2010 (the GDP growth rates from2010 to 2014 are 1045 93 765 767 and 74 resp)(data for GDP growth rates in China from 2010 to 2014 weredownloaded from theNational Bureau of Statistics of the Peo-plersquos Republic of China website (httpwwwstatsgovcn))and macroeconomic growth shows signs of fatigue stockindex futures prices also decrease year by year decreasingfrom 3376 points in the beginning of 2010 to 2332 points inthe beginning of 2014 with an average annual growth rateof approximately minus11192 However this downward trendslows after 2014 and reverses slightly in the second half of 2014mainly because in that year certain important and significantpolitical actions including administrative simplification andanticorruption legislation and enforcement create a healthypolitical environment for the economy In addition a reduc-tion in the reserve requirement ratio and an interest rate cutwill free up cash for investment andChinarsquos economic growth

turns from incremental expansion to an optimized structurewhich results in good prospects for investors Accordinglymore money is flowing into the stock market which drivesstock index prices higher and higher stock index futures priceresult simultaneously

The low-frequency term mainly reflects the medium-term effect of significant events on stock index futures pricesAs shown in Table 3 the correlation between the low-frequency term and the Chinese stock index futures priceseries is 056 and explains up to 375 of the original priceseries indicating that significant events in recent years play animportant role in the fluctuations in the Chinese stock indexfutures prices Figure 4 illustrates the plummet that occursduring the launch of the Chinese stock index futures whichhas a maximum amplitude of 500 points This plummet ismainly due to the strengthening of the countryrsquos efforts tosuppress real estate market speculation in 2010This initiativeshould lead to a stock market decline because real estateinvestment is an important economic driver at the time butthe stock market does not make the appropriate responseTheChinese stock index futures unleash their price discoveryfunction after launch which leads to adjustments in stockindex prices In addition many investors use the hedgingfunction of stock index futures to avoid the risk of a stockmarket crash However after the substantial fluctuationsduring launch the amplitude of the low-frequency termdecreases slightly reflecting in part the stabilization effectof stock index futures on the stock market To furtherinvestigate the effects of significant events on Chinese stockindex futures prices we use the ICSS algorithm to performbreakpoint tests on the rate of change in the low-frequencyterm (first-difference processing on the low-frequency termto identify the breakpoints) Table 4 presents the results ofthe breakpoint test on the low-frequency term The low-frequency term exhibits 11 breakpoints or in other words12 different volatility mechanisms The significant eventsthat cause breakpoints are primarily administrative macro-control policies which are mainly addressed by adjustingmonetary policy perfecting the capital market mechanismand promoting market liberalization The execution of thesepolicies brings volatility to Chinese stock index futures inturn creating a new volatility mechanism These breakpointsare mainly concentrated in 2012 and 2013 During these twoyears the government frequently issues regulatory policiesto bolster financial deepening and reform and investorsadopt a medium-term investment strategy revolving aroundthese policies As an emerging market the Chinese stockindex futures market is easily affected However as thecapital market mechanism gradually adjusts trading in the

Mathematical Problems in Engineering 7

20104 20114 20124 20134 20144

005

1

minus05

minus1

times10minus11

(a)20104 20114 20124 20134 20144

02040

minus20

minus40

(b)

Figure 2 (a) The differences between the stock index futures series and the decomposition sequence obtained by the CEEMD method (b)The differences between the stock index futures series and the decomposition sequence obtained by the EEMDmethod

20104 20114 20124 20134 20144

0

100

200

300

IMF5

20104 20114 20124 20134 20144

0

200

400

600

IMF6

20104 20114 20124 20134 20144

0

100

IMF7

20104 20114 20124 20134 201442000

2500

3000

3500

Res

20104 20114 20124 20134 20144

IMF1 0

50

100

minus50

minus100

20104 20114 20124 20134 20144

IMF3 0

50

100

minus50

minus100

20104 20114 20124 20134 20144

0

100

200

IMF4

minus100

minus200

minus100

minus200

minus200

minus200

minus100

minus300

minus400

20104 20114 20124 20134 20144

0

50

100

IMF2

minus50

minus100

Figure 3 IMFs and residue term

8 Mathematical Problems in Engineering

Table 4 Breakpoint tests on the low-frequency term and corresponding significant events

Breakpointnumber Corresponding time Corresponding significant event The response of stock index futures prices

1 December 13 2010

On December 10 2010 the CentralEconomic Working Conference convenesand indicates that the moderately loosemonetary policy has become robust

This event implies that the supply ofliquidity to the capital market would bereduced which was bad news for the stockindex futures market Hence the stockfutures prices slump

2 September 28 2011

In October 2011 the credit ratings of a dozenEuropean financial institutions aredowngraded and the European debt crisisfurther escalates

The Chinese financial market was affectedby this event and in response the stockindex futures prices maintained thedownward trend

3 January 31 2012

From January 6 to 7 2012 the NationalConference on Financial Work convenes inBeijing to set the tone of the next series ofmajor financial reform issues

This event makes the stock market a positiveprospect for investors and then pushes upthe stock index futures prices

4 May 17 2012

On May of 2012 the stock index futuresprices are affected by a combination ofevents including the US stock market andinternational commodities market crashingand the Chinese PMI index does notcontinue early rising trend

The series of significant events poses aserious negative impact on investorsrsquoexpectations Therefore the stock indexfutures prices stop rising and turn tocollapse

5 June 28 2012

On June 18 2010 the Shanghai StockExchange and the Shenzhen Stock Exchangeseparately publish programs related to thedelisting of listed companies and revise thecorresponding Stock Market Rulesaccordingly further perfecting the initialpublic offering (IPO) and delisting systems

This event would improve investorsrsquoconfidence in the stock market Hence thedownward trend of stock index futures priceis slowed

6 August 28 2012

On August 16 2012 the New York branch ofthe Agricultural Bank of China openscontinuing to accelerate the pace ofinternationalization of the Chinese bankingsector

The event promotes the process of RMBinternationalization which is good news forthe financial market and stock index futuresprices inch up

7 October 18 2012

On October 17 2012 the Chinese StateCouncil meeting made an importantdecision that government would continue toimplement the proactive fiscal policy andprudent monetary policy

The prudent monetary policy implyinglimited funds in the capital market was badnews for stock index futures market Hencethe stock index futures prices tumble over

8 January 31 2013 At the end of January 2013 broad moneygrew 159 and narrow money grew 153

This event prompted a rise in stock indexfutures prices

9 May 21 2013

In May of 2013 the Chinese macroeconomicsituation does not perform well the FederalReserve gradually winds up the quantitativeeasing policy and many banks in China lacksufficient capital at the time

The cumulative effects of shocks of theseevents occurring in May have an enormousnegative influence on the stock index futuresmarket Hence the stock index futuresprices collapse

10 September 10 2013

On September 6 2013 repackaged treasurynote futures debut after a gap of 18 yearswhich is conducive to the establishingmarket-oriented pricing benchmarks andadvances market-oriented reform of theinterest rate

This event rebuilding investorsrsquo confidencein the capital market then pushes up thestock index futures prices

11 February 25 2014

On March 1 2014 the Central Bank releasesthe upper limit on interest rates on smallamounts of foreign currency deposits in theShanghai Free-Trade Zone taking anotherstep toward market-oriented reform of theinterest rate

The rise of deposit interest rate indicates thatmarket as risk-free rate would increasewhich is negative news for the stock indexfutures market Hence the stock indexfutures prices plunge

Mathematical Problems in Engineering 9

20104 20114 20124 20134 20144

0

1000

2000

3000

4000

Original seriesResidue term

Low-frequency termHigh-frequency term

minus1000

Figure 4 Chinese stock index futures prices and the three compo-nents

Chinese stock index futures market further standardizes andthe ability to withstand external impact thus also increasesTherefore the number of breakpoints decreases after 2014

The high-frequency term mainly reflects the short-termdisequilibrium in the Chinese stock index futures marketAs shown in Table 3 the correlation of the high-frequencyterm with the Chinese stock index futures prices is merely0196 and the ability to explain fluctuations in the originalprice series is only 4 indicating that the overall effectof the high-frequency term on Chinese stock index futuresprices is limited However as Figure 4 shows the fluctuationsof the high-frequency term fit well with the local fluctu-ations of the Chinese stock index futures prices in otherwords the high-frequency term plays a primary role in localchanges in Chinese stock index futures prices In additionas opposed to the low-frequency term the high-frequencyterm fluctuates frequently but with smaller amplitudes Thisdifference occurs primarily because the high-frequency termis particularly affected by investorsrsquo speculative arbitrageactivities In particular implementation of a T+0 transactionsystem in the Chinese stock index futures market means thatinvestors buying and selling stock index futures can completetheir transactions in the same day which suits short-termspeculators Based on this transaction system there will bemore investment chances for investors which will enhanceinvestorsrsquo speculative opportunities Hence there are manyshort-term fluctuations in Chinese stock index futures pricesMeanwhile it is possible and convenient for investors toaccess the latest hot investment tip because the role of socialinteractions is partly substituted for by online information inthe Internet age [31] In reality this type of new informationchanges continuously and investors must frequently adjusttheir investment strategies to gainmore additional benefits byspeculating in the spot-futures marketsTherefore the strongeffect of the shock of investorsrsquo speculation on Chinese stockindex futures prices in the short-term is difficult to sustainAlthough fluctuations in the high-frequency term are notimportant to forecast the long-term trends of Chinese stockindex futures their predictive effect of minor fluctuations onstock index futures cannot be ignored

4 Predictions for Chinese StockIndex Futures Prices

In this section this paper further investigates the predictionsof the Chinese stock index futures prices In addition tothe previous dayrsquos price data we also select the five-minutehigh-frequency price data of Chinese stock index futuresfrom April 16 2010 to November 20 2014 as a researchsubject which yields a total of 61325 data points (the data areobtained from theWind database) To forecast the daily pricethe first 745 (approximately two-thirds) pieces of data areused as the training data and the last 370 (approximately one-third) pieces of data are used as the test data To forecast thefive-minute high-frequency price the first 40883 (approxi-mately two-thirds) data points are used as the training dataand the last 20442 (approximately one-third) data points areused as the test data

As our analysis of the price fluctuations of the Chinesestock index futures uses the CEEMDmethod the fluctuationcharacteristics in different time scales have their uniqueeconomic implications we suggest that there may be betterpredictions of Chinese stock index futures prices when thesefluctuation characteristics are considered in the predictionmodel Hence based on the literature [16 17] a combinationprediction model that is the CEEMD-SVM-SVM model isproposed and the PSO algorithm is used to optimize theparameters of the SVM model In our empirical researchwe perform a one-step-ahead prediction of Chinese stockindex futures prices The specific steps are as follows (1) theCEEMD method is used to decompose the Chinese stockindex futures price series into several IMFs and a residueterm (2) the IMFs are then composed into a low-frequencyterm and a high-frequency term based on a fine-to-coarsereconstruction and the residue term remains the same (3)SVM models are established separately for each of the threecomponents obtained in step (2) and the PSO algorithm isused to optimize the parameters and (4) the predictive valueof the three components obtained in step (3) is used as theinput with the real value at the same moment as the outputand then a SVM combination model is built to obtain thefinal predictive value The overall process of the CEEMD-SVM-SVMmodel is shown in Figure 5

The prediction evaluation indexes are the root meansquare error (RMSE) mean absolute prediction error(MAPE) and directional symmetry (DS) the DS predictionevaluation index is used to indicate the modelrsquos ability topredict the directionality trend of Chinese stock index futuresprices The prediction evaluation indexes are expressed asfollows

RMSE = radic1

119899

119899

sum119905=1

[119909 (119905) minus (119905)]2

MAPE =1

119899

119899

sum119905=1

10038161003816100381610038161003816100381610038161003816

119909 (119905) minus (119905)

119909 (119905)

10038161003816100381610038161003816100381610038161003816

DS =1

119899

119899

sum119894=1

119889119894

10 Mathematical Problems in Engineering

Original series

CEEMD

Residue term Low-frequency term High-frequency term

SVM1 SVM2 SVM3

Final predictive value

SVM combination

Figure 5 The overall process of the CEEMD-SVM-SVMmodel

119889119894=

1 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] ge 0

0 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] lt 0

(10)

Among the three evaluation indexes smaller RMSE andMAPE values indicate that the predictionmodel is experienc-ing better performance which is also true of greater DS valueand better performance of the prediction model

The prediction accuracy based on the SVM model issensitive to the SVM modelrsquos kernel function As manyprevious studies (such as [17 22]) indicate that the linearkernel function and the RBF kernel function are appropriatekernel functions for the prediction of SVMmodel the linearkernel function and the RBF kernel function of the SVMare thus both applied to our prediction of Chinese stockindex futures prices Based on the process of the CEEMD-SVM-SVM model we first predict the residue term thenthe low-frequency term and then the high-frequency termof the daily price series and the five-minute price seriesNotably the five-minute price series of Chinese stock indexfutures is decomposed into fourteen IMFs and one residueterm where IMFs from IMF1 to IMF9 are recomposed intothe high-frequency term and IMFs from IMF10 to IMF14are recomposed into the low-frequency termThe predictionerrors of these components are shown in Table 5

As shown in Table 5 in using the SVM model to makepredictions on each component the prediction precision ofthe linear kernel function SVM model is generally higherthan the RBF kernel function SVM model In additionwhether for predicting the daily price series or the five-minute price series the prediction precisions of the residueterm and the low-frequency term are significantly higherthan those of the high-frequency termThis difference occursbecause these effects of shocks on the residue term andthe low-frequency term are fundamental economically theyare significant events that have an impact on Chinese stockindex futures prices that is sustained over a long duration

Therefore the fluctuations of the residue term and the low-frequency term are relatively steady and easier to forecastusing the SVM model By contrast the high-frequency termrepresenting the short-term disequilibrium of Chinese stockindex futures price is affected mainly by daily market activ-ities such as speculation arbitrage and the fluctuation ofthe high-frequency term that is random and changes rapidlyHence the SVMmodel experiences difficulty in learning thepattern of high-frequency term fluctuation resulting in poorpredictive results

Then according to prediction step (4) we use the optimalpredictive value of each component obtained in the previousstep to perform a combination prediction and the predictionerrors of our proposed model and the single SVM modelbased on the same data sample are simultaneously comparedThese results of the prediction errors are listed in Table 6

Table 6 shows that as opposed to the predictive resultsof the three components the RBF kernel function SVMmodel has higher predicted precision than the linear kernelfunction SVM model when performing combination pre-diction which indicates that the RBF kernel function SVMmodel is more appropriate for combination prediction Inaddition we find that the predictive results of the finalpredictive value obtained under the RBF kernel function aresignificantly better than those obtained in the single SVMmodel when predicting the daily price but slightly better thanthe single SVM model when making predictions regardingthe high-frequency price Meanwhile we should considerthat the high-frequency price embodies far more complexinformation than the daily price during the same time spanwhich may restrict the learning ability and generalizationability of the SVM model Hence the CEEMD-SVM-SVMmodel performs better in terms of predicting the daily priceIn any event the CEEMD-SVM-SVM model proposed bythis paper performs better than the single SVM model interms of both forecasting the daily price series and the five-minute price series Thus the introduction of Chinese stockindex futures price characteristics in different time scales into

Mathematical Problems in Engineering 11

Table 5 The prediction errors of different components

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

Residue termRMSE 0007 0008 0006 0009MAPE 0001 0001 0001 0002DS 69837 58424 21678 27241

Low-frequency termRMSE 0007 0008 0004 0008MAPE 0087 0092 0042 0258DS 60326 55434 54613 55254

High-frequency termRMSE 0086 0098 0020 0021MAPE 4806 4725 1146 1149DS 63315 62500 51463 51854

Table 6 Comparison of different modelsrsquo prediction errors

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

SVMRMSE 0053 0054 0008 0008MAPE 0009 0009 0001 0001DS 54891 52445 48171 48161

CEEMD-SVM-SVMRMSE 0051 0009 0008 0007MAPE 0008 0002 0001 0001DS 62228 88043 51565 52715

the SVM model improves the ability of the SVM model tomine the variations in the stock index futures price seriesfurther enhancing its prediction precision Simultaneouslysome useful suggestions for investors can be found in thepredictions of Chinese stock index futures prices generatedby our proposedmodel First investors should be accustomedto analyzing the price series from different time scales tounderstand the fluctuation characteristics of Chinese stockindex futures prices Second investors must actively identifywhat time scale the effects of external shocks on stock indexfutures prices belong to andmust devise the appropriate strat-egy accordingly Finally investors should pay more attentionto a hedging strategy involving Chinese stock index futuresover the long andor medium term and should be cautiousabout short-term speculation

Figures 6 and 7 present the real and predictive values ofthe daily price series andfive-minute price series respectivelyThe final predictive values obtained using the CEEMD-SVM-SVM model appeared to coincide with the real values indi-cating that the predictive values obtained using this modelfit well with the real values and intuitively demonstratingthat the CEEMD-SVM-SVMmodel is suitable for forecastingChinese stock index futures prices

20135 20138 201311 20142 20145 20148 2014112000

2200

2400

2600

2800

Real valuePredictive value

Figure 6 Real value and predictive value of the daily price series

5 Conclusion

The Chinese stock index futures market is a complex systemand fluctuations in the stock index futures prices resultfrom the collective effects of many factors at various levelsThe CEEMD method a method suitable in the analysis ofnonlinear nonstationary data is applied to analyze Chinesestock index futures prices in different time scales and to

12 Mathematical Problems in Engineering

Real valuePredictive value

20135 20138 201311 20142 20145 20148 2014111800

2000

2200

2400

2600

2800

Figure 7 Real value and predictive value of the five-minute priceseries

reveal the intrinsic fluctuation characteristics of Chinesestock index futures prices Then we focus on whether thesecharacteristics improve the predictive precision of Chinesestock index futures prices and propose a CEEMD-SVM-SVMmodel based on the PSO algorithm The empirical resultsdemonstrate that the intrinsic fluctuation characteristics ofChinese stock index futures prices can be revealed in threecomponents including the residue term the low-frequencyterm and the high-frequency term with independent eco-nomic significance The residue term determines the long-term trend of stock index futures prices and its fluctuationsare correlated with macroeconomics and the long-termexpectations of investorsThe low-frequency term reveals theeffect of significant events on medium-term fluctuations instock index futures prices the ICSS algorithm breakpointtests reveal that policy regulation events are the main causeof fluctuations during the low-frequency term The high-frequency term reflects short-term disequilibrium in thestock index futures market although the effect of the high-frequency term on the overall fluctuation in the stock indexfutures prices is limited it nonetheless plays an important rolelocally

In addition in forecasting Chinese stock index futuresprices first we find that the linear kernel function SVMmodel is more suitable for predicting these components thanthe RBF kernel function SVM model The residue term andthe low-frequency term have stronger regular fluctuationsthan the high-frequency term and therefore yield superiorpredictive results Second the RBF kernel function SVMmodel shows higher predictive precision for the final combi-nation prediction than the linear kernel function SVMmodelSimultaneously the predictive results of the combinationprediction model are better than those of a single SVMmodel which indicates that the integration of fluctuationcharacteristics in different time scales increases the abilityof the SVM model to identify regularity in Chinese stockindex futures price fluctuations thus improving predictionprecision Finally the predictive results of the CEEMD-SVM-SVM model on daily prices and intraday prices are basicallyconsistent which indicates that our proposed model hasstability and universal applicability in forecasting Chinesestock index futures prices

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by the National SocialScience Foundation of China (14ZDA045)

References

[1] VMeneu andH Torro ldquoAsymmetric covariance in spot-futuresmarketsrdquo Journal of Futures Markets vol 23 no 11 pp 1019ndash1046 2003

[2] M Zhong A F Darrat and R Otero ldquoPrice discovery andvolatility spillovers in index futures markets some evidencefrom Mexicordquo Journal of Banking and Finance vol 28 no 12pp 3037ndash3054 2004

[3] X Xiong F Wang W Zhang and Y Sun ldquoPrice discoveryand volatility spillovers between SGX FTSEXinhua ChinaA50 index futures and A-share marketrdquo Chinese Journal ofManagement vol 11 no 1535 pp 1507ndash1512 2009

[4] H M Zuo Z T Liu and H W Zeng ldquoResearch of volatilityspillover and information transmission between stock indexfutures and spot market based on the high frequency datardquoJournal of Financial Research vol 4 pp 140ndash154 2012

[5] H Q Chen and C H Zhang ldquoDoes index futures tradingreduce stock market jump riskrdquo Economic Research Journalvol 1 pp 153ndash167 2015

[6] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 48 no 3 pp235ndash254 2015

[7] E E Peters Fractal Market Analysis Applying Chaos Theory toInvestment and Economics John Wiley amp Sons New York NYUSA 1994

[8] X Zhang K K Lai and S-YWang ldquoA new approach for crudeoil price analysis based on empirical mode decompositionrdquoEnergy Economics vol 30 no 3 pp 905ndash918 2008

[9] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnonstationary time series analysisrdquo Proceedings of the RoyalSociety of London A vol 454 no 1971 pp 903ndash995 1998

[10] Z Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[11] L F Ruan and H J Bao ldquoAn empirical analysis on periodicfluctuations of real estate price based on EMDrdquo Chinese Journalof Management Science no 3 pp 41ndash46 2012

[12] H L Li and C E Feng ldquoRelationship between investor senti-ment and stock indices fluctuation based on EEMDrdquo SystemsEngineering-Theory amp Practice vol 10 pp 2495ndash2503 2014

[13] J-R Yeh J-S Shieh andN EHuang ldquoComplementary ensem-ble empiricalmode decomposition a novel noise enhanced dataanalysismethodrdquoAdvances in Adaptive DataAnalysis vol 2 no2 pp 135ndash156 2010

[14] L Zhao W Yu and R Yan ldquoRolling bearing fault diagnosisbased on CEEMD and time series modelingrdquo MathematicalProblems in Engineering vol 2014 Article ID 101867 13 pages2014

Mathematical Problems in Engineering 13

[15] L Tang W Dai L Yu and S Wang ldquoA novel CEEMD-based eelm ensemble learning paradigm for crude oil priceforecastingrdquo International Journal of Information Technologyand Decision Making vol 14 no 1 pp 141ndash169 2015

[16] L Yu S Wang and K K Lai ldquoForecasting crude oil price withan EMD-based neural network ensemble learning paradigmrdquoEnergy Economics vol 30 no 5 pp 2623ndash2635 2008

[17] Y F Yang Y K Bao Z Y Hu and R Zhang ldquoCrude oilprice prediction based on empirical mode decomposition andsupport vector machinesrdquo Chinese Journal of Management vol12 pp 1884ndash1889 2010

[18] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[19] W Huang Y Nakamori and S-Y Wang ldquoForecasting stockmarket movement direction with support vector machinerdquoComputers and Operations Research vol 32 no 10 pp 2513ndash2522 2005

[20] H Ince and T B Trafalis ldquoShort term forecasting with supportvector machines and application to stock price predictionrdquoInternational Journal of General Systems vol 37 no 6 pp 677ndash687 2008

[21] F E H Tay and L Cao ldquoApplication of support vectormachinesin financial time series forecastingrdquo Omega vol 29 no 4 pp309ndash317 2001

[22] Y Sai F T Zhang and T Zhang ldquoResearch of Chinese stockindex futures regression prediction based on support vectormachinesrdquo Chinese Journal of Management Science vol 3 pp35ndash39 2013

[23] C Inclan andG C Tiao ldquoUse of cumulative sums of squares forretrospective detection of changes of variancerdquo Journal of theAmerican Statistical Association vol 89 no 427 pp 913ndash9231994

[24] S H Kang C Cheong and S-M Yoon ldquoStructural changesand volatility transmission in crude oil marketsrdquo Physica AStatisticalMechanics and Its Applications vol 390 no 23-24 pp4317ndash4324 2011

[25] J P Yang X X Chen and C H Zhang ldquoPolicy impact factorson the volatility of Shanghai and Shenzhen stock market inChinardquo Chinese Journal of Management Science vol 20 no 6pp 43ndash51 2012

[26] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer New York NY USA2010

[27] C-L Huang and J-F Dun ldquoA distributed PSOndashSVM hybridsystem with feature selection and parameter optimizationrdquoApplied Soft Computing vol 8 no 4 pp 1381ndash1391 2008

[28] C Zhou K Yin Y Cao and B Ahmed ldquoApplication of timeseries analysis and PSOndashSVMmodel in predicting the Bazimenlandslide in the Three Gorges Reservoir Chinardquo EngineeringGeology vol 204 pp 108ndash120 2016

[29] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis amp Applications vol 16 no 1 pp 93ndash103 2014

[30] X Gong Z He P Li and N Zhu ldquoForecasting return volatilityof the CSI 300 index using the stochastic volatility model withcontinuous volatility and jumpsrdquo Discrete Dynamics in Natureand Society vol 2014 Article ID 964654 10 pages 2014

[31] S Q Guo and P H Liang ldquoSocial Interactions the informationchannel and household stock market participation a empiricalstudy based on a Chinese household finance surveyrdquo EconomicResearch Journal vol S1 pp 116ndash131 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

the output variables of unseen input variables Specifically theconstraints and objective function for the 120576-SVMmodel are

min120596119887120591120591

lowast

120596119879120596

2+ 119888

119899

sum119894=1

120591119894+ 119888

119899

sum119894=1

120591lowast

119894

st 120596119879

120601 (119909119894) + 119887 minus 119910

119894le 120576 + 120591

119894

120596119879

120601 (119909119894) + 119887 minus 119910

119894ge 120576 + 120591

lowast

119894

120591119894 120591lowast

119894ge 0 119894 = 1 2 119899

(6)

In (6) 120591119894and 120591lowast

119894are the slack variables that represent

the allowable upper limit and lower limit of training errorsrespectively under the constraint of insensitive loss 120576 119888 is thepenalty factor controlling the degree of penalty on sampleswhose errors exceed 120576

The Lagrange function must be introduced to resolve(6) According to the Duality Theory and the saddle pointcondition the dual form of (6) can be obtained as follows

min120597lowast120597

12

119899

sum119894=1

119899

sum119895=1

(120597lowast

119894minus 120597119894) (120597lowast

119895minus 120597119895) 119896 (119909

119894 119909119895)

+ 120576

119899

sum119894=1

(120597lowast

119894+ 120597119894) minus

119899

sum119894=1

119910119894(120597lowast

119894minus 120597119894)

st119899

sum119894=1

(120597lowast

119894minus 120597119894) = 0 0 le 120597

lowast

119894 120597119894le 119888 119894 = 1 2 119899

120596 =

119899

sum119894=1

(120597lowast

119894minus 120597119894)

(7)

In (7) 120597119894is the nonnegative Lagrangian multiplier of 120597lowast

119894

and 119896(119909119894 119909119895) represents the kernel function Any function

that satisfies Mercerrsquos condition can be a kernel functionThe common kernel functions include the linear kernelfunction 119896(119909 119909

119894) = 119909119879

119909119894and the Radial Basis Function (RBF)

kernel function 119896(119909 119909119894) = exp(minus120574119909 minus 119909

1198942) Ultimately the

hyperplane for the underlying prediction problem is given bythe following

119891 (119909) =

119899

sum119894=1

(120597lowast

119894minus 120597119894) 119896 (119909119894 119909) + 119887 (8)

24 PSO Algorithm The PSO algorithm proposed byKennedy [26] is an effective algorithm to address func-tion parameter optimization Each particle in the algorithmdenoting a potential solution in the solution space has threeindicators including speed position and fitness value Thefitness value is determined by the fitness function whichcan estimate the merit of the particles The velocity of theparticle determines the movement direction and distanceand the velocityrsquos dynamic adjustment follows the movementexperience of the particle itself and other particles thusenabling the optimal selection of an individual in the solutionspace The details are as follows

Suppose that in a 119863-dimensional solution space thepopulation comprising n particles is 119883 = (119883

1 1198832 119883

119899)

20104 20114 20124 20134 201442000

2500

3000

3500

4000

Figure 1 Daily closing price of Chinese stock index futures

the position of 119894th particle is 119883119894

= (1199091198941 1199091198942 119909

119894119863)119879 and

the fitness value of each particle can be calculated basedon the objective function and position of each particleThe velocity of 119894th particle is 119881

119894= (1198811198941 1198811198942 119881

119894119863)119879 its

individual extremum is 119875119894

= (1198751198941 1198751198942 119875

119894119863)119879 and the

group extremum is119875119892= (1198751198921

1198751198922

119875119892119863

)119879 In each iterativeprocess the particle updates its velocity and position throughthe individual extremum and the group extremum Theupdating equation is as follows

119881119896+1

119894119889= 120596119881119896

119894119889+ 11988811199031(119875119896

119894119889minus 119883119896

119894119889) + 11988821199032(119875119896

119892119889minus 119883119896

119894119889)

119883119896+1

119894119889= 119883119896

119894119889+ 119881119896+1

119894119889

(9)

where120596 represents the inertia weight 119889 = 1 2 119863 119896 is thecurrent iteration number119881

119894119889is the velocity component of the

particle 1198881and 1198882are the acceleration factors and nonnegative

constants and 1199031and 1199032are random numbers in the range

[0 1] Because the SVM prediction model is sensitive to themodel parameters and because the literature [22 27 28]indicates that the PSO algorithm performs very well in theparameter optimization process of the SVMmodel this paperuses the PSO algorithm to optimize the parameters of theSVM prediction model

3 Analysis of Chinese Stock IndexFutures Prices in Different Time Scales

31 Data Selection andDescription TheCSI 300 index futuresare the earliest stock index futures listed in China and it hasbeen widely used in many studies [4 5 22] Therefore thispaper selects the daily closing price of the stock index futuresas this sectionrsquos research objectThe time span runs fromApril16 2010 toNovember 20 2014 the sample data total 1115 andthe data are from theWind database See Figure 1 and Table 1for the sample data distribution and the descriptive statisticsrespectively

Nonstationary nonlinear tests are performed on Chinesestock index futures price series based on Ruan and Bao [11]Based on the data in Table 1 the null hypothesis that theChinese stock index future price series are nonstationarycannot be rejected under the condition of less than 10 signif-icance Because the postdifferential price series is a stationaryseries an autocorrelation analysis is further performed on theseries The test results for 119876 statistics demonstrate that thenull hypothesis for this differential series cannot be rejectedunder the condition of less than 10 significance However

Mathematical Problems in Engineering 5

Table 1 Descriptive statistics for Chinese stock index futures prices

Mean Std Dev JB ADF 119876(6) 1198762(6)

Price series 2600691 351503 97269lowastlowastlowast minus2539 mdash mdash119863(price series) minus0783 37611 858432lowastlowastlowast minus35531lowastlowastlowast 7737 26507lowastlowastlowast

Note Augmented Dickey-Fuller (ADF) statistics represent the nonstationary test of the price series 119863(price series) represents the price series after the firstdifference119876(6) statistics represent the autocorrelation test of the time series1198762(6) statistics represent the autocorrelation test of the square of the time seriesand ldquolowast lowast lowastrdquo indicates significance at the 1 significance level

Table 2 Correlations and variance ratios of IMFs and the residue term with the Chinese stock index futures price series

Mean value Pearson correlation coefficient Variance ratio (original series)IMF1 0131 (0238) 0041 (1363) 0273IMF2 minus0096 (minus0151) 0066 (2218)lowastlowast 0372IMF3 minus0431 (minus0523) 0097 (3273)lowastlowastlowast 0614IMF4 minus2103 (minus1277) 0163 (5530)lowastlowastlowast 2446IMF5 minus4652 (minus2413)lowastlowast 0218 (7454)lowastlowastlowast 3354IMF6 46989 (8007)lowastlowastlowast 0660 (29329)lowastlowastlowast 31076IMF7 minus64873 (minus18973)lowastlowastlowast minus0264 (minus9151)lowastlowastlowast 10550Res 2625727 0762 (39214)lowastlowastlowast 70278NoteThe values in parentheses are the 119905 statistics ldquolowastlowastrdquo represents significance at the 5 level and ldquolowast lowast lowastrdquo represents significance at the 1 level

the square of the 119876 statistics rejects the null hypothesis atless than 1 significance in other words autocorrelationis present Thus nonlinear models are more appropriatefor studies of Chinese stock index futures prices based onnonstationary nonlinear tests

32 CEEMD Decomposition Yeh et al [13] indicated thatthe CEEMD method retained the decomposition ability ofthe EEMD method while effectively offsetting the addedwhite noise To demonstrate that the advantages of thisdecomposition remain valid in Chinarsquos stock index futuresprices this paper uses both the CEEMD and EEMDmethodsto decompose the China stock index futures price series andthen compared the differences between the original seriesand the sum of the decomposition sequences obtained by thetwomethods Depending on the general setting the standarddeviation of the added white noise is 02 times the standarddeviation of the daily closing price See Figure 2 for thedifferences between the results

As shown in Figure 2 thewhite noise residue after decom-position by the CEEMD method is nearly zero whereas agreat deal of white noise residue remains after decompositionby the EEMD method indicating that the CEEMD methodcan better maintain the completeness of the original seriesthan the EEMD method Therefore this paper used theCEEMD method to decompose Chinese stock index futuresprices and the IMFs and residue term are presented inFigure 3

As shown in Figure 3 the frequency is the highest forIMF1 and then decreased in order among the remainingIMFs IMF1 IMF2 and IMF3 exhibit dense fluctuationswith amplitudes between positive and negative 50 pointsapproximatelyThe amplitudes of IMF4 and IMF5 increase tobetween positive and negative 100 points and the fluctuationfrequency of IMF5 decreases significantly compared with

the fluctuation frequency of the previous IMFs with afluctuation period of approximately 120 days (4 months)Theamplitude of IMF6 increases to between positive and negative200 pointsmdasheven reaching 400 points once during theinitial stock index futures listing periodmdashwith a fluctuatingperiod that increases to approximately 400 days (13 months)The upward fluctuation frequency of IMF7 decreases toapproximately 100 points whereas the downward fluctuationfrequency is maintained at approximately 200 points itsfrequency also decreases to the lowest among the seven IMFsbut the fluctuating period increases to approximately 600days (20 months) The residue term (res) exhibits a trend ofan initial slow decline and subsequent gradual increase butthere is no clear cyclical fluctuation

As the frequencies decrease the correlation and varianceratios of the IMFs with the Chinese stock index futures priceseries increase indicating that a lower frequency enablesgreater fluctuation information content to be obtained fromthe original series In the low-frequency series the correlationand variance ratio of IMF6 with the original series is signif-icantly larger than that of IMF5 and IMF7 indicating thatmedium-term fluctuations in the original series primarilyoriginated from IMF6 and that the correlations and varianceratios of the residue term with the original series are fargreater than those of the IMFs indicating that this seriescontains the most important information from the originalseries (Table 2)

33 Analysis of Different Time Scales The analysis describedabove established that the frequencies of IMF and theresidue term differ The fluctuations of the high-frequencyseries exhibit ambiguity and disorderliness and the cyclicalfluctuations of the low-frequency series are stronger Theresidue term contains most of the information of the orig-inal series but does not exhibit a clear fluctuating period

6 Mathematical Problems in Engineering

Table 3 Correlations and variance ratios of the component terms with the Chinese stock index futures price series

Pearson correlation coefficient Variance ratio (original series)Residue term 0762 (39214)lowastlowastlowast 70278Low-frequency term 0526 (206229)lowastlowastlowast 37521High-frequency term 0196 (6661)lowastlowastlowast 4051Note ldquolowast lowast lowastrdquo represents significance at the 1 significance level

indicating that the price fluctuation information reflected bythe high-frequency series low-frequency series and residueterm feature more significant differences To more accuratelyreveal the fluctuation characteristics of Chinese stock indexfutures prices at various time scales we analyze the sum ofthe IMFs with similar fluctuation characteristics using a 119905

statistics test of the mean from zero [8] Based on the data inTable 2 the high-frequency IMFsmdashIMF1 IMF2 IMF3 andIMF4mdashare summed as the high-frequency term The low-frequency IMFmdashIMF5 IMF6 and IMF7mdashare summed asthe low-frequency term The residue term alone is retainedThe correlations and variance ratios of the three componentswith the original series are presented in Table 3 and theirdistribution is presented in Figure 4

Consistent with the literature such as [8 9] the residueterm includes most of the information value of the originalseries and reflects the long-term trend of the original seriesIn fact the correlation coefficient of the residue term withthe Chinese stock index futures price series in this paper isas high as 076 and explains 70 of the fluctuations in theoriginal price series (see Table 3)Therefore the residue termcan represent the long-term trend of the Chinese stock indexfutures prices As shown in Figure 4 between the officiallisting of stock index futures inChina (2010) and 2014 there isa slow downward trend in stock index futures prices Becausethe stock index is the ldquobarometerrdquo of a countryrsquos macroe-conomic situation and because stock index futures serve aprice discovery function for stock indexes the futuresmarketprices reflect newly revealed information more quickly thancurrent market prices Therefore macroeconomic changesalso determine the long-term trend of stock index futuresprices As Chinarsquos Gross Domestic Product (GDP) growthrate gradually slows after 2010 (the GDP growth rates from2010 to 2014 are 1045 93 765 767 and 74 resp)(data for GDP growth rates in China from 2010 to 2014 weredownloaded from theNational Bureau of Statistics of the Peo-plersquos Republic of China website (httpwwwstatsgovcn))and macroeconomic growth shows signs of fatigue stockindex futures prices also decrease year by year decreasingfrom 3376 points in the beginning of 2010 to 2332 points inthe beginning of 2014 with an average annual growth rateof approximately minus11192 However this downward trendslows after 2014 and reverses slightly in the second half of 2014mainly because in that year certain important and significantpolitical actions including administrative simplification andanticorruption legislation and enforcement create a healthypolitical environment for the economy In addition a reduc-tion in the reserve requirement ratio and an interest rate cutwill free up cash for investment andChinarsquos economic growth

turns from incremental expansion to an optimized structurewhich results in good prospects for investors Accordinglymore money is flowing into the stock market which drivesstock index prices higher and higher stock index futures priceresult simultaneously

The low-frequency term mainly reflects the medium-term effect of significant events on stock index futures pricesAs shown in Table 3 the correlation between the low-frequency term and the Chinese stock index futures priceseries is 056 and explains up to 375 of the original priceseries indicating that significant events in recent years play animportant role in the fluctuations in the Chinese stock indexfutures prices Figure 4 illustrates the plummet that occursduring the launch of the Chinese stock index futures whichhas a maximum amplitude of 500 points This plummet ismainly due to the strengthening of the countryrsquos efforts tosuppress real estate market speculation in 2010This initiativeshould lead to a stock market decline because real estateinvestment is an important economic driver at the time butthe stock market does not make the appropriate responseTheChinese stock index futures unleash their price discoveryfunction after launch which leads to adjustments in stockindex prices In addition many investors use the hedgingfunction of stock index futures to avoid the risk of a stockmarket crash However after the substantial fluctuationsduring launch the amplitude of the low-frequency termdecreases slightly reflecting in part the stabilization effectof stock index futures on the stock market To furtherinvestigate the effects of significant events on Chinese stockindex futures prices we use the ICSS algorithm to performbreakpoint tests on the rate of change in the low-frequencyterm (first-difference processing on the low-frequency termto identify the breakpoints) Table 4 presents the results ofthe breakpoint test on the low-frequency term The low-frequency term exhibits 11 breakpoints or in other words12 different volatility mechanisms The significant eventsthat cause breakpoints are primarily administrative macro-control policies which are mainly addressed by adjustingmonetary policy perfecting the capital market mechanismand promoting market liberalization The execution of thesepolicies brings volatility to Chinese stock index futures inturn creating a new volatility mechanism These breakpointsare mainly concentrated in 2012 and 2013 During these twoyears the government frequently issues regulatory policiesto bolster financial deepening and reform and investorsadopt a medium-term investment strategy revolving aroundthese policies As an emerging market the Chinese stockindex futures market is easily affected However as thecapital market mechanism gradually adjusts trading in the

Mathematical Problems in Engineering 7

20104 20114 20124 20134 20144

005

1

minus05

minus1

times10minus11

(a)20104 20114 20124 20134 20144

02040

minus20

minus40

(b)

Figure 2 (a) The differences between the stock index futures series and the decomposition sequence obtained by the CEEMD method (b)The differences between the stock index futures series and the decomposition sequence obtained by the EEMDmethod

20104 20114 20124 20134 20144

0

100

200

300

IMF5

20104 20114 20124 20134 20144

0

200

400

600

IMF6

20104 20114 20124 20134 20144

0

100

IMF7

20104 20114 20124 20134 201442000

2500

3000

3500

Res

20104 20114 20124 20134 20144

IMF1 0

50

100

minus50

minus100

20104 20114 20124 20134 20144

IMF3 0

50

100

minus50

minus100

20104 20114 20124 20134 20144

0

100

200

IMF4

minus100

minus200

minus100

minus200

minus200

minus200

minus100

minus300

minus400

20104 20114 20124 20134 20144

0

50

100

IMF2

minus50

minus100

Figure 3 IMFs and residue term

8 Mathematical Problems in Engineering

Table 4 Breakpoint tests on the low-frequency term and corresponding significant events

Breakpointnumber Corresponding time Corresponding significant event The response of stock index futures prices

1 December 13 2010

On December 10 2010 the CentralEconomic Working Conference convenesand indicates that the moderately loosemonetary policy has become robust

This event implies that the supply ofliquidity to the capital market would bereduced which was bad news for the stockindex futures market Hence the stockfutures prices slump

2 September 28 2011

In October 2011 the credit ratings of a dozenEuropean financial institutions aredowngraded and the European debt crisisfurther escalates

The Chinese financial market was affectedby this event and in response the stockindex futures prices maintained thedownward trend

3 January 31 2012

From January 6 to 7 2012 the NationalConference on Financial Work convenes inBeijing to set the tone of the next series ofmajor financial reform issues

This event makes the stock market a positiveprospect for investors and then pushes upthe stock index futures prices

4 May 17 2012

On May of 2012 the stock index futuresprices are affected by a combination ofevents including the US stock market andinternational commodities market crashingand the Chinese PMI index does notcontinue early rising trend

The series of significant events poses aserious negative impact on investorsrsquoexpectations Therefore the stock indexfutures prices stop rising and turn tocollapse

5 June 28 2012

On June 18 2010 the Shanghai StockExchange and the Shenzhen Stock Exchangeseparately publish programs related to thedelisting of listed companies and revise thecorresponding Stock Market Rulesaccordingly further perfecting the initialpublic offering (IPO) and delisting systems

This event would improve investorsrsquoconfidence in the stock market Hence thedownward trend of stock index futures priceis slowed

6 August 28 2012

On August 16 2012 the New York branch ofthe Agricultural Bank of China openscontinuing to accelerate the pace ofinternationalization of the Chinese bankingsector

The event promotes the process of RMBinternationalization which is good news forthe financial market and stock index futuresprices inch up

7 October 18 2012

On October 17 2012 the Chinese StateCouncil meeting made an importantdecision that government would continue toimplement the proactive fiscal policy andprudent monetary policy

The prudent monetary policy implyinglimited funds in the capital market was badnews for stock index futures market Hencethe stock index futures prices tumble over

8 January 31 2013 At the end of January 2013 broad moneygrew 159 and narrow money grew 153

This event prompted a rise in stock indexfutures prices

9 May 21 2013

In May of 2013 the Chinese macroeconomicsituation does not perform well the FederalReserve gradually winds up the quantitativeeasing policy and many banks in China lacksufficient capital at the time

The cumulative effects of shocks of theseevents occurring in May have an enormousnegative influence on the stock index futuresmarket Hence the stock index futuresprices collapse

10 September 10 2013

On September 6 2013 repackaged treasurynote futures debut after a gap of 18 yearswhich is conducive to the establishingmarket-oriented pricing benchmarks andadvances market-oriented reform of theinterest rate

This event rebuilding investorsrsquo confidencein the capital market then pushes up thestock index futures prices

11 February 25 2014

On March 1 2014 the Central Bank releasesthe upper limit on interest rates on smallamounts of foreign currency deposits in theShanghai Free-Trade Zone taking anotherstep toward market-oriented reform of theinterest rate

The rise of deposit interest rate indicates thatmarket as risk-free rate would increasewhich is negative news for the stock indexfutures market Hence the stock indexfutures prices plunge

Mathematical Problems in Engineering 9

20104 20114 20124 20134 20144

0

1000

2000

3000

4000

Original seriesResidue term

Low-frequency termHigh-frequency term

minus1000

Figure 4 Chinese stock index futures prices and the three compo-nents

Chinese stock index futures market further standardizes andthe ability to withstand external impact thus also increasesTherefore the number of breakpoints decreases after 2014

The high-frequency term mainly reflects the short-termdisequilibrium in the Chinese stock index futures marketAs shown in Table 3 the correlation of the high-frequencyterm with the Chinese stock index futures prices is merely0196 and the ability to explain fluctuations in the originalprice series is only 4 indicating that the overall effectof the high-frequency term on Chinese stock index futuresprices is limited However as Figure 4 shows the fluctuationsof the high-frequency term fit well with the local fluctu-ations of the Chinese stock index futures prices in otherwords the high-frequency term plays a primary role in localchanges in Chinese stock index futures prices In additionas opposed to the low-frequency term the high-frequencyterm fluctuates frequently but with smaller amplitudes Thisdifference occurs primarily because the high-frequency termis particularly affected by investorsrsquo speculative arbitrageactivities In particular implementation of a T+0 transactionsystem in the Chinese stock index futures market means thatinvestors buying and selling stock index futures can completetheir transactions in the same day which suits short-termspeculators Based on this transaction system there will bemore investment chances for investors which will enhanceinvestorsrsquo speculative opportunities Hence there are manyshort-term fluctuations in Chinese stock index futures pricesMeanwhile it is possible and convenient for investors toaccess the latest hot investment tip because the role of socialinteractions is partly substituted for by online information inthe Internet age [31] In reality this type of new informationchanges continuously and investors must frequently adjusttheir investment strategies to gainmore additional benefits byspeculating in the spot-futures marketsTherefore the strongeffect of the shock of investorsrsquo speculation on Chinese stockindex futures prices in the short-term is difficult to sustainAlthough fluctuations in the high-frequency term are notimportant to forecast the long-term trends of Chinese stockindex futures their predictive effect of minor fluctuations onstock index futures cannot be ignored

4 Predictions for Chinese StockIndex Futures Prices

In this section this paper further investigates the predictionsof the Chinese stock index futures prices In addition tothe previous dayrsquos price data we also select the five-minutehigh-frequency price data of Chinese stock index futuresfrom April 16 2010 to November 20 2014 as a researchsubject which yields a total of 61325 data points (the data areobtained from theWind database) To forecast the daily pricethe first 745 (approximately two-thirds) pieces of data areused as the training data and the last 370 (approximately one-third) pieces of data are used as the test data To forecast thefive-minute high-frequency price the first 40883 (approxi-mately two-thirds) data points are used as the training dataand the last 20442 (approximately one-third) data points areused as the test data

As our analysis of the price fluctuations of the Chinesestock index futures uses the CEEMDmethod the fluctuationcharacteristics in different time scales have their uniqueeconomic implications we suggest that there may be betterpredictions of Chinese stock index futures prices when thesefluctuation characteristics are considered in the predictionmodel Hence based on the literature [16 17] a combinationprediction model that is the CEEMD-SVM-SVM model isproposed and the PSO algorithm is used to optimize theparameters of the SVM model In our empirical researchwe perform a one-step-ahead prediction of Chinese stockindex futures prices The specific steps are as follows (1) theCEEMD method is used to decompose the Chinese stockindex futures price series into several IMFs and a residueterm (2) the IMFs are then composed into a low-frequencyterm and a high-frequency term based on a fine-to-coarsereconstruction and the residue term remains the same (3)SVM models are established separately for each of the threecomponents obtained in step (2) and the PSO algorithm isused to optimize the parameters and (4) the predictive valueof the three components obtained in step (3) is used as theinput with the real value at the same moment as the outputand then a SVM combination model is built to obtain thefinal predictive value The overall process of the CEEMD-SVM-SVMmodel is shown in Figure 5

The prediction evaluation indexes are the root meansquare error (RMSE) mean absolute prediction error(MAPE) and directional symmetry (DS) the DS predictionevaluation index is used to indicate the modelrsquos ability topredict the directionality trend of Chinese stock index futuresprices The prediction evaluation indexes are expressed asfollows

RMSE = radic1

119899

119899

sum119905=1

[119909 (119905) minus (119905)]2

MAPE =1

119899

119899

sum119905=1

10038161003816100381610038161003816100381610038161003816

119909 (119905) minus (119905)

119909 (119905)

10038161003816100381610038161003816100381610038161003816

DS =1

119899

119899

sum119894=1

119889119894

10 Mathematical Problems in Engineering

Original series

CEEMD

Residue term Low-frequency term High-frequency term

SVM1 SVM2 SVM3

Final predictive value

SVM combination

Figure 5 The overall process of the CEEMD-SVM-SVMmodel

119889119894=

1 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] ge 0

0 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] lt 0

(10)

Among the three evaluation indexes smaller RMSE andMAPE values indicate that the predictionmodel is experienc-ing better performance which is also true of greater DS valueand better performance of the prediction model

The prediction accuracy based on the SVM model issensitive to the SVM modelrsquos kernel function As manyprevious studies (such as [17 22]) indicate that the linearkernel function and the RBF kernel function are appropriatekernel functions for the prediction of SVMmodel the linearkernel function and the RBF kernel function of the SVMare thus both applied to our prediction of Chinese stockindex futures prices Based on the process of the CEEMD-SVM-SVM model we first predict the residue term thenthe low-frequency term and then the high-frequency termof the daily price series and the five-minute price seriesNotably the five-minute price series of Chinese stock indexfutures is decomposed into fourteen IMFs and one residueterm where IMFs from IMF1 to IMF9 are recomposed intothe high-frequency term and IMFs from IMF10 to IMF14are recomposed into the low-frequency termThe predictionerrors of these components are shown in Table 5

As shown in Table 5 in using the SVM model to makepredictions on each component the prediction precision ofthe linear kernel function SVM model is generally higherthan the RBF kernel function SVM model In additionwhether for predicting the daily price series or the five-minute price series the prediction precisions of the residueterm and the low-frequency term are significantly higherthan those of the high-frequency termThis difference occursbecause these effects of shocks on the residue term andthe low-frequency term are fundamental economically theyare significant events that have an impact on Chinese stockindex futures prices that is sustained over a long duration

Therefore the fluctuations of the residue term and the low-frequency term are relatively steady and easier to forecastusing the SVM model By contrast the high-frequency termrepresenting the short-term disequilibrium of Chinese stockindex futures price is affected mainly by daily market activ-ities such as speculation arbitrage and the fluctuation ofthe high-frequency term that is random and changes rapidlyHence the SVMmodel experiences difficulty in learning thepattern of high-frequency term fluctuation resulting in poorpredictive results

Then according to prediction step (4) we use the optimalpredictive value of each component obtained in the previousstep to perform a combination prediction and the predictionerrors of our proposed model and the single SVM modelbased on the same data sample are simultaneously comparedThese results of the prediction errors are listed in Table 6

Table 6 shows that as opposed to the predictive resultsof the three components the RBF kernel function SVMmodel has higher predicted precision than the linear kernelfunction SVM model when performing combination pre-diction which indicates that the RBF kernel function SVMmodel is more appropriate for combination prediction Inaddition we find that the predictive results of the finalpredictive value obtained under the RBF kernel function aresignificantly better than those obtained in the single SVMmodel when predicting the daily price but slightly better thanthe single SVM model when making predictions regardingthe high-frequency price Meanwhile we should considerthat the high-frequency price embodies far more complexinformation than the daily price during the same time spanwhich may restrict the learning ability and generalizationability of the SVM model Hence the CEEMD-SVM-SVMmodel performs better in terms of predicting the daily priceIn any event the CEEMD-SVM-SVM model proposed bythis paper performs better than the single SVM model interms of both forecasting the daily price series and the five-minute price series Thus the introduction of Chinese stockindex futures price characteristics in different time scales into

Mathematical Problems in Engineering 11

Table 5 The prediction errors of different components

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

Residue termRMSE 0007 0008 0006 0009MAPE 0001 0001 0001 0002DS 69837 58424 21678 27241

Low-frequency termRMSE 0007 0008 0004 0008MAPE 0087 0092 0042 0258DS 60326 55434 54613 55254

High-frequency termRMSE 0086 0098 0020 0021MAPE 4806 4725 1146 1149DS 63315 62500 51463 51854

Table 6 Comparison of different modelsrsquo prediction errors

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

SVMRMSE 0053 0054 0008 0008MAPE 0009 0009 0001 0001DS 54891 52445 48171 48161

CEEMD-SVM-SVMRMSE 0051 0009 0008 0007MAPE 0008 0002 0001 0001DS 62228 88043 51565 52715

the SVM model improves the ability of the SVM model tomine the variations in the stock index futures price seriesfurther enhancing its prediction precision Simultaneouslysome useful suggestions for investors can be found in thepredictions of Chinese stock index futures prices generatedby our proposedmodel First investors should be accustomedto analyzing the price series from different time scales tounderstand the fluctuation characteristics of Chinese stockindex futures prices Second investors must actively identifywhat time scale the effects of external shocks on stock indexfutures prices belong to andmust devise the appropriate strat-egy accordingly Finally investors should pay more attentionto a hedging strategy involving Chinese stock index futuresover the long andor medium term and should be cautiousabout short-term speculation

Figures 6 and 7 present the real and predictive values ofthe daily price series andfive-minute price series respectivelyThe final predictive values obtained using the CEEMD-SVM-SVM model appeared to coincide with the real values indi-cating that the predictive values obtained using this modelfit well with the real values and intuitively demonstratingthat the CEEMD-SVM-SVMmodel is suitable for forecastingChinese stock index futures prices

20135 20138 201311 20142 20145 20148 2014112000

2200

2400

2600

2800

Real valuePredictive value

Figure 6 Real value and predictive value of the daily price series

5 Conclusion

The Chinese stock index futures market is a complex systemand fluctuations in the stock index futures prices resultfrom the collective effects of many factors at various levelsThe CEEMD method a method suitable in the analysis ofnonlinear nonstationary data is applied to analyze Chinesestock index futures prices in different time scales and to

12 Mathematical Problems in Engineering

Real valuePredictive value

20135 20138 201311 20142 20145 20148 2014111800

2000

2200

2400

2600

2800

Figure 7 Real value and predictive value of the five-minute priceseries

reveal the intrinsic fluctuation characteristics of Chinesestock index futures prices Then we focus on whether thesecharacteristics improve the predictive precision of Chinesestock index futures prices and propose a CEEMD-SVM-SVMmodel based on the PSO algorithm The empirical resultsdemonstrate that the intrinsic fluctuation characteristics ofChinese stock index futures prices can be revealed in threecomponents including the residue term the low-frequencyterm and the high-frequency term with independent eco-nomic significance The residue term determines the long-term trend of stock index futures prices and its fluctuationsare correlated with macroeconomics and the long-termexpectations of investorsThe low-frequency term reveals theeffect of significant events on medium-term fluctuations instock index futures prices the ICSS algorithm breakpointtests reveal that policy regulation events are the main causeof fluctuations during the low-frequency term The high-frequency term reflects short-term disequilibrium in thestock index futures market although the effect of the high-frequency term on the overall fluctuation in the stock indexfutures prices is limited it nonetheless plays an important rolelocally

In addition in forecasting Chinese stock index futuresprices first we find that the linear kernel function SVMmodel is more suitable for predicting these components thanthe RBF kernel function SVM model The residue term andthe low-frequency term have stronger regular fluctuationsthan the high-frequency term and therefore yield superiorpredictive results Second the RBF kernel function SVMmodel shows higher predictive precision for the final combi-nation prediction than the linear kernel function SVMmodelSimultaneously the predictive results of the combinationprediction model are better than those of a single SVMmodel which indicates that the integration of fluctuationcharacteristics in different time scales increases the abilityof the SVM model to identify regularity in Chinese stockindex futures price fluctuations thus improving predictionprecision Finally the predictive results of the CEEMD-SVM-SVM model on daily prices and intraday prices are basicallyconsistent which indicates that our proposed model hasstability and universal applicability in forecasting Chinesestock index futures prices

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by the National SocialScience Foundation of China (14ZDA045)

References

[1] VMeneu andH Torro ldquoAsymmetric covariance in spot-futuresmarketsrdquo Journal of Futures Markets vol 23 no 11 pp 1019ndash1046 2003

[2] M Zhong A F Darrat and R Otero ldquoPrice discovery andvolatility spillovers in index futures markets some evidencefrom Mexicordquo Journal of Banking and Finance vol 28 no 12pp 3037ndash3054 2004

[3] X Xiong F Wang W Zhang and Y Sun ldquoPrice discoveryand volatility spillovers between SGX FTSEXinhua ChinaA50 index futures and A-share marketrdquo Chinese Journal ofManagement vol 11 no 1535 pp 1507ndash1512 2009

[4] H M Zuo Z T Liu and H W Zeng ldquoResearch of volatilityspillover and information transmission between stock indexfutures and spot market based on the high frequency datardquoJournal of Financial Research vol 4 pp 140ndash154 2012

[5] H Q Chen and C H Zhang ldquoDoes index futures tradingreduce stock market jump riskrdquo Economic Research Journalvol 1 pp 153ndash167 2015

[6] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 48 no 3 pp235ndash254 2015

[7] E E Peters Fractal Market Analysis Applying Chaos Theory toInvestment and Economics John Wiley amp Sons New York NYUSA 1994

[8] X Zhang K K Lai and S-YWang ldquoA new approach for crudeoil price analysis based on empirical mode decompositionrdquoEnergy Economics vol 30 no 3 pp 905ndash918 2008

[9] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnonstationary time series analysisrdquo Proceedings of the RoyalSociety of London A vol 454 no 1971 pp 903ndash995 1998

[10] Z Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[11] L F Ruan and H J Bao ldquoAn empirical analysis on periodicfluctuations of real estate price based on EMDrdquo Chinese Journalof Management Science no 3 pp 41ndash46 2012

[12] H L Li and C E Feng ldquoRelationship between investor senti-ment and stock indices fluctuation based on EEMDrdquo SystemsEngineering-Theory amp Practice vol 10 pp 2495ndash2503 2014

[13] J-R Yeh J-S Shieh andN EHuang ldquoComplementary ensem-ble empiricalmode decomposition a novel noise enhanced dataanalysismethodrdquoAdvances in Adaptive DataAnalysis vol 2 no2 pp 135ndash156 2010

[14] L Zhao W Yu and R Yan ldquoRolling bearing fault diagnosisbased on CEEMD and time series modelingrdquo MathematicalProblems in Engineering vol 2014 Article ID 101867 13 pages2014

Mathematical Problems in Engineering 13

[15] L Tang W Dai L Yu and S Wang ldquoA novel CEEMD-based eelm ensemble learning paradigm for crude oil priceforecastingrdquo International Journal of Information Technologyand Decision Making vol 14 no 1 pp 141ndash169 2015

[16] L Yu S Wang and K K Lai ldquoForecasting crude oil price withan EMD-based neural network ensemble learning paradigmrdquoEnergy Economics vol 30 no 5 pp 2623ndash2635 2008

[17] Y F Yang Y K Bao Z Y Hu and R Zhang ldquoCrude oilprice prediction based on empirical mode decomposition andsupport vector machinesrdquo Chinese Journal of Management vol12 pp 1884ndash1889 2010

[18] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[19] W Huang Y Nakamori and S-Y Wang ldquoForecasting stockmarket movement direction with support vector machinerdquoComputers and Operations Research vol 32 no 10 pp 2513ndash2522 2005

[20] H Ince and T B Trafalis ldquoShort term forecasting with supportvector machines and application to stock price predictionrdquoInternational Journal of General Systems vol 37 no 6 pp 677ndash687 2008

[21] F E H Tay and L Cao ldquoApplication of support vectormachinesin financial time series forecastingrdquo Omega vol 29 no 4 pp309ndash317 2001

[22] Y Sai F T Zhang and T Zhang ldquoResearch of Chinese stockindex futures regression prediction based on support vectormachinesrdquo Chinese Journal of Management Science vol 3 pp35ndash39 2013

[23] C Inclan andG C Tiao ldquoUse of cumulative sums of squares forretrospective detection of changes of variancerdquo Journal of theAmerican Statistical Association vol 89 no 427 pp 913ndash9231994

[24] S H Kang C Cheong and S-M Yoon ldquoStructural changesand volatility transmission in crude oil marketsrdquo Physica AStatisticalMechanics and Its Applications vol 390 no 23-24 pp4317ndash4324 2011

[25] J P Yang X X Chen and C H Zhang ldquoPolicy impact factorson the volatility of Shanghai and Shenzhen stock market inChinardquo Chinese Journal of Management Science vol 20 no 6pp 43ndash51 2012

[26] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer New York NY USA2010

[27] C-L Huang and J-F Dun ldquoA distributed PSOndashSVM hybridsystem with feature selection and parameter optimizationrdquoApplied Soft Computing vol 8 no 4 pp 1381ndash1391 2008

[28] C Zhou K Yin Y Cao and B Ahmed ldquoApplication of timeseries analysis and PSOndashSVMmodel in predicting the Bazimenlandslide in the Three Gorges Reservoir Chinardquo EngineeringGeology vol 204 pp 108ndash120 2016

[29] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis amp Applications vol 16 no 1 pp 93ndash103 2014

[30] X Gong Z He P Li and N Zhu ldquoForecasting return volatilityof the CSI 300 index using the stochastic volatility model withcontinuous volatility and jumpsrdquo Discrete Dynamics in Natureand Society vol 2014 Article ID 964654 10 pages 2014

[31] S Q Guo and P H Liang ldquoSocial Interactions the informationchannel and household stock market participation a empiricalstudy based on a Chinese household finance surveyrdquo EconomicResearch Journal vol S1 pp 116ndash131 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Table 1 Descriptive statistics for Chinese stock index futures prices

Mean Std Dev JB ADF 119876(6) 1198762(6)

Price series 2600691 351503 97269lowastlowastlowast minus2539 mdash mdash119863(price series) minus0783 37611 858432lowastlowastlowast minus35531lowastlowastlowast 7737 26507lowastlowastlowast

Note Augmented Dickey-Fuller (ADF) statistics represent the nonstationary test of the price series 119863(price series) represents the price series after the firstdifference119876(6) statistics represent the autocorrelation test of the time series1198762(6) statistics represent the autocorrelation test of the square of the time seriesand ldquolowast lowast lowastrdquo indicates significance at the 1 significance level

Table 2 Correlations and variance ratios of IMFs and the residue term with the Chinese stock index futures price series

Mean value Pearson correlation coefficient Variance ratio (original series)IMF1 0131 (0238) 0041 (1363) 0273IMF2 minus0096 (minus0151) 0066 (2218)lowastlowast 0372IMF3 minus0431 (minus0523) 0097 (3273)lowastlowastlowast 0614IMF4 minus2103 (minus1277) 0163 (5530)lowastlowastlowast 2446IMF5 minus4652 (minus2413)lowastlowast 0218 (7454)lowastlowastlowast 3354IMF6 46989 (8007)lowastlowastlowast 0660 (29329)lowastlowastlowast 31076IMF7 minus64873 (minus18973)lowastlowastlowast minus0264 (minus9151)lowastlowastlowast 10550Res 2625727 0762 (39214)lowastlowastlowast 70278NoteThe values in parentheses are the 119905 statistics ldquolowastlowastrdquo represents significance at the 5 level and ldquolowast lowast lowastrdquo represents significance at the 1 level

the square of the 119876 statistics rejects the null hypothesis atless than 1 significance in other words autocorrelationis present Thus nonlinear models are more appropriatefor studies of Chinese stock index futures prices based onnonstationary nonlinear tests

32 CEEMD Decomposition Yeh et al [13] indicated thatthe CEEMD method retained the decomposition ability ofthe EEMD method while effectively offsetting the addedwhite noise To demonstrate that the advantages of thisdecomposition remain valid in Chinarsquos stock index futuresprices this paper uses both the CEEMD and EEMDmethodsto decompose the China stock index futures price series andthen compared the differences between the original seriesand the sum of the decomposition sequences obtained by thetwomethods Depending on the general setting the standarddeviation of the added white noise is 02 times the standarddeviation of the daily closing price See Figure 2 for thedifferences between the results

As shown in Figure 2 thewhite noise residue after decom-position by the CEEMD method is nearly zero whereas agreat deal of white noise residue remains after decompositionby the EEMD method indicating that the CEEMD methodcan better maintain the completeness of the original seriesthan the EEMD method Therefore this paper used theCEEMD method to decompose Chinese stock index futuresprices and the IMFs and residue term are presented inFigure 3

As shown in Figure 3 the frequency is the highest forIMF1 and then decreased in order among the remainingIMFs IMF1 IMF2 and IMF3 exhibit dense fluctuationswith amplitudes between positive and negative 50 pointsapproximatelyThe amplitudes of IMF4 and IMF5 increase tobetween positive and negative 100 points and the fluctuationfrequency of IMF5 decreases significantly compared with

the fluctuation frequency of the previous IMFs with afluctuation period of approximately 120 days (4 months)Theamplitude of IMF6 increases to between positive and negative200 pointsmdasheven reaching 400 points once during theinitial stock index futures listing periodmdashwith a fluctuatingperiod that increases to approximately 400 days (13 months)The upward fluctuation frequency of IMF7 decreases toapproximately 100 points whereas the downward fluctuationfrequency is maintained at approximately 200 points itsfrequency also decreases to the lowest among the seven IMFsbut the fluctuating period increases to approximately 600days (20 months) The residue term (res) exhibits a trend ofan initial slow decline and subsequent gradual increase butthere is no clear cyclical fluctuation

As the frequencies decrease the correlation and varianceratios of the IMFs with the Chinese stock index futures priceseries increase indicating that a lower frequency enablesgreater fluctuation information content to be obtained fromthe original series In the low-frequency series the correlationand variance ratio of IMF6 with the original series is signif-icantly larger than that of IMF5 and IMF7 indicating thatmedium-term fluctuations in the original series primarilyoriginated from IMF6 and that the correlations and varianceratios of the residue term with the original series are fargreater than those of the IMFs indicating that this seriescontains the most important information from the originalseries (Table 2)

33 Analysis of Different Time Scales The analysis describedabove established that the frequencies of IMF and theresidue term differ The fluctuations of the high-frequencyseries exhibit ambiguity and disorderliness and the cyclicalfluctuations of the low-frequency series are stronger Theresidue term contains most of the information of the orig-inal series but does not exhibit a clear fluctuating period

6 Mathematical Problems in Engineering

Table 3 Correlations and variance ratios of the component terms with the Chinese stock index futures price series

Pearson correlation coefficient Variance ratio (original series)Residue term 0762 (39214)lowastlowastlowast 70278Low-frequency term 0526 (206229)lowastlowastlowast 37521High-frequency term 0196 (6661)lowastlowastlowast 4051Note ldquolowast lowast lowastrdquo represents significance at the 1 significance level

indicating that the price fluctuation information reflected bythe high-frequency series low-frequency series and residueterm feature more significant differences To more accuratelyreveal the fluctuation characteristics of Chinese stock indexfutures prices at various time scales we analyze the sum ofthe IMFs with similar fluctuation characteristics using a 119905

statistics test of the mean from zero [8] Based on the data inTable 2 the high-frequency IMFsmdashIMF1 IMF2 IMF3 andIMF4mdashare summed as the high-frequency term The low-frequency IMFmdashIMF5 IMF6 and IMF7mdashare summed asthe low-frequency term The residue term alone is retainedThe correlations and variance ratios of the three componentswith the original series are presented in Table 3 and theirdistribution is presented in Figure 4

Consistent with the literature such as [8 9] the residueterm includes most of the information value of the originalseries and reflects the long-term trend of the original seriesIn fact the correlation coefficient of the residue term withthe Chinese stock index futures price series in this paper isas high as 076 and explains 70 of the fluctuations in theoriginal price series (see Table 3)Therefore the residue termcan represent the long-term trend of the Chinese stock indexfutures prices As shown in Figure 4 between the officiallisting of stock index futures inChina (2010) and 2014 there isa slow downward trend in stock index futures prices Becausethe stock index is the ldquobarometerrdquo of a countryrsquos macroe-conomic situation and because stock index futures serve aprice discovery function for stock indexes the futuresmarketprices reflect newly revealed information more quickly thancurrent market prices Therefore macroeconomic changesalso determine the long-term trend of stock index futuresprices As Chinarsquos Gross Domestic Product (GDP) growthrate gradually slows after 2010 (the GDP growth rates from2010 to 2014 are 1045 93 765 767 and 74 resp)(data for GDP growth rates in China from 2010 to 2014 weredownloaded from theNational Bureau of Statistics of the Peo-plersquos Republic of China website (httpwwwstatsgovcn))and macroeconomic growth shows signs of fatigue stockindex futures prices also decrease year by year decreasingfrom 3376 points in the beginning of 2010 to 2332 points inthe beginning of 2014 with an average annual growth rateof approximately minus11192 However this downward trendslows after 2014 and reverses slightly in the second half of 2014mainly because in that year certain important and significantpolitical actions including administrative simplification andanticorruption legislation and enforcement create a healthypolitical environment for the economy In addition a reduc-tion in the reserve requirement ratio and an interest rate cutwill free up cash for investment andChinarsquos economic growth

turns from incremental expansion to an optimized structurewhich results in good prospects for investors Accordinglymore money is flowing into the stock market which drivesstock index prices higher and higher stock index futures priceresult simultaneously

The low-frequency term mainly reflects the medium-term effect of significant events on stock index futures pricesAs shown in Table 3 the correlation between the low-frequency term and the Chinese stock index futures priceseries is 056 and explains up to 375 of the original priceseries indicating that significant events in recent years play animportant role in the fluctuations in the Chinese stock indexfutures prices Figure 4 illustrates the plummet that occursduring the launch of the Chinese stock index futures whichhas a maximum amplitude of 500 points This plummet ismainly due to the strengthening of the countryrsquos efforts tosuppress real estate market speculation in 2010This initiativeshould lead to a stock market decline because real estateinvestment is an important economic driver at the time butthe stock market does not make the appropriate responseTheChinese stock index futures unleash their price discoveryfunction after launch which leads to adjustments in stockindex prices In addition many investors use the hedgingfunction of stock index futures to avoid the risk of a stockmarket crash However after the substantial fluctuationsduring launch the amplitude of the low-frequency termdecreases slightly reflecting in part the stabilization effectof stock index futures on the stock market To furtherinvestigate the effects of significant events on Chinese stockindex futures prices we use the ICSS algorithm to performbreakpoint tests on the rate of change in the low-frequencyterm (first-difference processing on the low-frequency termto identify the breakpoints) Table 4 presents the results ofthe breakpoint test on the low-frequency term The low-frequency term exhibits 11 breakpoints or in other words12 different volatility mechanisms The significant eventsthat cause breakpoints are primarily administrative macro-control policies which are mainly addressed by adjustingmonetary policy perfecting the capital market mechanismand promoting market liberalization The execution of thesepolicies brings volatility to Chinese stock index futures inturn creating a new volatility mechanism These breakpointsare mainly concentrated in 2012 and 2013 During these twoyears the government frequently issues regulatory policiesto bolster financial deepening and reform and investorsadopt a medium-term investment strategy revolving aroundthese policies As an emerging market the Chinese stockindex futures market is easily affected However as thecapital market mechanism gradually adjusts trading in the

Mathematical Problems in Engineering 7

20104 20114 20124 20134 20144

005

1

minus05

minus1

times10minus11

(a)20104 20114 20124 20134 20144

02040

minus20

minus40

(b)

Figure 2 (a) The differences between the stock index futures series and the decomposition sequence obtained by the CEEMD method (b)The differences between the stock index futures series and the decomposition sequence obtained by the EEMDmethod

20104 20114 20124 20134 20144

0

100

200

300

IMF5

20104 20114 20124 20134 20144

0

200

400

600

IMF6

20104 20114 20124 20134 20144

0

100

IMF7

20104 20114 20124 20134 201442000

2500

3000

3500

Res

20104 20114 20124 20134 20144

IMF1 0

50

100

minus50

minus100

20104 20114 20124 20134 20144

IMF3 0

50

100

minus50

minus100

20104 20114 20124 20134 20144

0

100

200

IMF4

minus100

minus200

minus100

minus200

minus200

minus200

minus100

minus300

minus400

20104 20114 20124 20134 20144

0

50

100

IMF2

minus50

minus100

Figure 3 IMFs and residue term

8 Mathematical Problems in Engineering

Table 4 Breakpoint tests on the low-frequency term and corresponding significant events

Breakpointnumber Corresponding time Corresponding significant event The response of stock index futures prices

1 December 13 2010

On December 10 2010 the CentralEconomic Working Conference convenesand indicates that the moderately loosemonetary policy has become robust

This event implies that the supply ofliquidity to the capital market would bereduced which was bad news for the stockindex futures market Hence the stockfutures prices slump

2 September 28 2011

In October 2011 the credit ratings of a dozenEuropean financial institutions aredowngraded and the European debt crisisfurther escalates

The Chinese financial market was affectedby this event and in response the stockindex futures prices maintained thedownward trend

3 January 31 2012

From January 6 to 7 2012 the NationalConference on Financial Work convenes inBeijing to set the tone of the next series ofmajor financial reform issues

This event makes the stock market a positiveprospect for investors and then pushes upthe stock index futures prices

4 May 17 2012

On May of 2012 the stock index futuresprices are affected by a combination ofevents including the US stock market andinternational commodities market crashingand the Chinese PMI index does notcontinue early rising trend

The series of significant events poses aserious negative impact on investorsrsquoexpectations Therefore the stock indexfutures prices stop rising and turn tocollapse

5 June 28 2012

On June 18 2010 the Shanghai StockExchange and the Shenzhen Stock Exchangeseparately publish programs related to thedelisting of listed companies and revise thecorresponding Stock Market Rulesaccordingly further perfecting the initialpublic offering (IPO) and delisting systems

This event would improve investorsrsquoconfidence in the stock market Hence thedownward trend of stock index futures priceis slowed

6 August 28 2012

On August 16 2012 the New York branch ofthe Agricultural Bank of China openscontinuing to accelerate the pace ofinternationalization of the Chinese bankingsector

The event promotes the process of RMBinternationalization which is good news forthe financial market and stock index futuresprices inch up

7 October 18 2012

On October 17 2012 the Chinese StateCouncil meeting made an importantdecision that government would continue toimplement the proactive fiscal policy andprudent monetary policy

The prudent monetary policy implyinglimited funds in the capital market was badnews for stock index futures market Hencethe stock index futures prices tumble over

8 January 31 2013 At the end of January 2013 broad moneygrew 159 and narrow money grew 153

This event prompted a rise in stock indexfutures prices

9 May 21 2013

In May of 2013 the Chinese macroeconomicsituation does not perform well the FederalReserve gradually winds up the quantitativeeasing policy and many banks in China lacksufficient capital at the time

The cumulative effects of shocks of theseevents occurring in May have an enormousnegative influence on the stock index futuresmarket Hence the stock index futuresprices collapse

10 September 10 2013

On September 6 2013 repackaged treasurynote futures debut after a gap of 18 yearswhich is conducive to the establishingmarket-oriented pricing benchmarks andadvances market-oriented reform of theinterest rate

This event rebuilding investorsrsquo confidencein the capital market then pushes up thestock index futures prices

11 February 25 2014

On March 1 2014 the Central Bank releasesthe upper limit on interest rates on smallamounts of foreign currency deposits in theShanghai Free-Trade Zone taking anotherstep toward market-oriented reform of theinterest rate

The rise of deposit interest rate indicates thatmarket as risk-free rate would increasewhich is negative news for the stock indexfutures market Hence the stock indexfutures prices plunge

Mathematical Problems in Engineering 9

20104 20114 20124 20134 20144

0

1000

2000

3000

4000

Original seriesResidue term

Low-frequency termHigh-frequency term

minus1000

Figure 4 Chinese stock index futures prices and the three compo-nents

Chinese stock index futures market further standardizes andthe ability to withstand external impact thus also increasesTherefore the number of breakpoints decreases after 2014

The high-frequency term mainly reflects the short-termdisequilibrium in the Chinese stock index futures marketAs shown in Table 3 the correlation of the high-frequencyterm with the Chinese stock index futures prices is merely0196 and the ability to explain fluctuations in the originalprice series is only 4 indicating that the overall effectof the high-frequency term on Chinese stock index futuresprices is limited However as Figure 4 shows the fluctuationsof the high-frequency term fit well with the local fluctu-ations of the Chinese stock index futures prices in otherwords the high-frequency term plays a primary role in localchanges in Chinese stock index futures prices In additionas opposed to the low-frequency term the high-frequencyterm fluctuates frequently but with smaller amplitudes Thisdifference occurs primarily because the high-frequency termis particularly affected by investorsrsquo speculative arbitrageactivities In particular implementation of a T+0 transactionsystem in the Chinese stock index futures market means thatinvestors buying and selling stock index futures can completetheir transactions in the same day which suits short-termspeculators Based on this transaction system there will bemore investment chances for investors which will enhanceinvestorsrsquo speculative opportunities Hence there are manyshort-term fluctuations in Chinese stock index futures pricesMeanwhile it is possible and convenient for investors toaccess the latest hot investment tip because the role of socialinteractions is partly substituted for by online information inthe Internet age [31] In reality this type of new informationchanges continuously and investors must frequently adjusttheir investment strategies to gainmore additional benefits byspeculating in the spot-futures marketsTherefore the strongeffect of the shock of investorsrsquo speculation on Chinese stockindex futures prices in the short-term is difficult to sustainAlthough fluctuations in the high-frequency term are notimportant to forecast the long-term trends of Chinese stockindex futures their predictive effect of minor fluctuations onstock index futures cannot be ignored

4 Predictions for Chinese StockIndex Futures Prices

In this section this paper further investigates the predictionsof the Chinese stock index futures prices In addition tothe previous dayrsquos price data we also select the five-minutehigh-frequency price data of Chinese stock index futuresfrom April 16 2010 to November 20 2014 as a researchsubject which yields a total of 61325 data points (the data areobtained from theWind database) To forecast the daily pricethe first 745 (approximately two-thirds) pieces of data areused as the training data and the last 370 (approximately one-third) pieces of data are used as the test data To forecast thefive-minute high-frequency price the first 40883 (approxi-mately two-thirds) data points are used as the training dataand the last 20442 (approximately one-third) data points areused as the test data

As our analysis of the price fluctuations of the Chinesestock index futures uses the CEEMDmethod the fluctuationcharacteristics in different time scales have their uniqueeconomic implications we suggest that there may be betterpredictions of Chinese stock index futures prices when thesefluctuation characteristics are considered in the predictionmodel Hence based on the literature [16 17] a combinationprediction model that is the CEEMD-SVM-SVM model isproposed and the PSO algorithm is used to optimize theparameters of the SVM model In our empirical researchwe perform a one-step-ahead prediction of Chinese stockindex futures prices The specific steps are as follows (1) theCEEMD method is used to decompose the Chinese stockindex futures price series into several IMFs and a residueterm (2) the IMFs are then composed into a low-frequencyterm and a high-frequency term based on a fine-to-coarsereconstruction and the residue term remains the same (3)SVM models are established separately for each of the threecomponents obtained in step (2) and the PSO algorithm isused to optimize the parameters and (4) the predictive valueof the three components obtained in step (3) is used as theinput with the real value at the same moment as the outputand then a SVM combination model is built to obtain thefinal predictive value The overall process of the CEEMD-SVM-SVMmodel is shown in Figure 5

The prediction evaluation indexes are the root meansquare error (RMSE) mean absolute prediction error(MAPE) and directional symmetry (DS) the DS predictionevaluation index is used to indicate the modelrsquos ability topredict the directionality trend of Chinese stock index futuresprices The prediction evaluation indexes are expressed asfollows

RMSE = radic1

119899

119899

sum119905=1

[119909 (119905) minus (119905)]2

MAPE =1

119899

119899

sum119905=1

10038161003816100381610038161003816100381610038161003816

119909 (119905) minus (119905)

119909 (119905)

10038161003816100381610038161003816100381610038161003816

DS =1

119899

119899

sum119894=1

119889119894

10 Mathematical Problems in Engineering

Original series

CEEMD

Residue term Low-frequency term High-frequency term

SVM1 SVM2 SVM3

Final predictive value

SVM combination

Figure 5 The overall process of the CEEMD-SVM-SVMmodel

119889119894=

1 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] ge 0

0 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] lt 0

(10)

Among the three evaluation indexes smaller RMSE andMAPE values indicate that the predictionmodel is experienc-ing better performance which is also true of greater DS valueand better performance of the prediction model

The prediction accuracy based on the SVM model issensitive to the SVM modelrsquos kernel function As manyprevious studies (such as [17 22]) indicate that the linearkernel function and the RBF kernel function are appropriatekernel functions for the prediction of SVMmodel the linearkernel function and the RBF kernel function of the SVMare thus both applied to our prediction of Chinese stockindex futures prices Based on the process of the CEEMD-SVM-SVM model we first predict the residue term thenthe low-frequency term and then the high-frequency termof the daily price series and the five-minute price seriesNotably the five-minute price series of Chinese stock indexfutures is decomposed into fourteen IMFs and one residueterm where IMFs from IMF1 to IMF9 are recomposed intothe high-frequency term and IMFs from IMF10 to IMF14are recomposed into the low-frequency termThe predictionerrors of these components are shown in Table 5

As shown in Table 5 in using the SVM model to makepredictions on each component the prediction precision ofthe linear kernel function SVM model is generally higherthan the RBF kernel function SVM model In additionwhether for predicting the daily price series or the five-minute price series the prediction precisions of the residueterm and the low-frequency term are significantly higherthan those of the high-frequency termThis difference occursbecause these effects of shocks on the residue term andthe low-frequency term are fundamental economically theyare significant events that have an impact on Chinese stockindex futures prices that is sustained over a long duration

Therefore the fluctuations of the residue term and the low-frequency term are relatively steady and easier to forecastusing the SVM model By contrast the high-frequency termrepresenting the short-term disequilibrium of Chinese stockindex futures price is affected mainly by daily market activ-ities such as speculation arbitrage and the fluctuation ofthe high-frequency term that is random and changes rapidlyHence the SVMmodel experiences difficulty in learning thepattern of high-frequency term fluctuation resulting in poorpredictive results

Then according to prediction step (4) we use the optimalpredictive value of each component obtained in the previousstep to perform a combination prediction and the predictionerrors of our proposed model and the single SVM modelbased on the same data sample are simultaneously comparedThese results of the prediction errors are listed in Table 6

Table 6 shows that as opposed to the predictive resultsof the three components the RBF kernel function SVMmodel has higher predicted precision than the linear kernelfunction SVM model when performing combination pre-diction which indicates that the RBF kernel function SVMmodel is more appropriate for combination prediction Inaddition we find that the predictive results of the finalpredictive value obtained under the RBF kernel function aresignificantly better than those obtained in the single SVMmodel when predicting the daily price but slightly better thanthe single SVM model when making predictions regardingthe high-frequency price Meanwhile we should considerthat the high-frequency price embodies far more complexinformation than the daily price during the same time spanwhich may restrict the learning ability and generalizationability of the SVM model Hence the CEEMD-SVM-SVMmodel performs better in terms of predicting the daily priceIn any event the CEEMD-SVM-SVM model proposed bythis paper performs better than the single SVM model interms of both forecasting the daily price series and the five-minute price series Thus the introduction of Chinese stockindex futures price characteristics in different time scales into

Mathematical Problems in Engineering 11

Table 5 The prediction errors of different components

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

Residue termRMSE 0007 0008 0006 0009MAPE 0001 0001 0001 0002DS 69837 58424 21678 27241

Low-frequency termRMSE 0007 0008 0004 0008MAPE 0087 0092 0042 0258DS 60326 55434 54613 55254

High-frequency termRMSE 0086 0098 0020 0021MAPE 4806 4725 1146 1149DS 63315 62500 51463 51854

Table 6 Comparison of different modelsrsquo prediction errors

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

SVMRMSE 0053 0054 0008 0008MAPE 0009 0009 0001 0001DS 54891 52445 48171 48161

CEEMD-SVM-SVMRMSE 0051 0009 0008 0007MAPE 0008 0002 0001 0001DS 62228 88043 51565 52715

the SVM model improves the ability of the SVM model tomine the variations in the stock index futures price seriesfurther enhancing its prediction precision Simultaneouslysome useful suggestions for investors can be found in thepredictions of Chinese stock index futures prices generatedby our proposedmodel First investors should be accustomedto analyzing the price series from different time scales tounderstand the fluctuation characteristics of Chinese stockindex futures prices Second investors must actively identifywhat time scale the effects of external shocks on stock indexfutures prices belong to andmust devise the appropriate strat-egy accordingly Finally investors should pay more attentionto a hedging strategy involving Chinese stock index futuresover the long andor medium term and should be cautiousabout short-term speculation

Figures 6 and 7 present the real and predictive values ofthe daily price series andfive-minute price series respectivelyThe final predictive values obtained using the CEEMD-SVM-SVM model appeared to coincide with the real values indi-cating that the predictive values obtained using this modelfit well with the real values and intuitively demonstratingthat the CEEMD-SVM-SVMmodel is suitable for forecastingChinese stock index futures prices

20135 20138 201311 20142 20145 20148 2014112000

2200

2400

2600

2800

Real valuePredictive value

Figure 6 Real value and predictive value of the daily price series

5 Conclusion

The Chinese stock index futures market is a complex systemand fluctuations in the stock index futures prices resultfrom the collective effects of many factors at various levelsThe CEEMD method a method suitable in the analysis ofnonlinear nonstationary data is applied to analyze Chinesestock index futures prices in different time scales and to

12 Mathematical Problems in Engineering

Real valuePredictive value

20135 20138 201311 20142 20145 20148 2014111800

2000

2200

2400

2600

2800

Figure 7 Real value and predictive value of the five-minute priceseries

reveal the intrinsic fluctuation characteristics of Chinesestock index futures prices Then we focus on whether thesecharacteristics improve the predictive precision of Chinesestock index futures prices and propose a CEEMD-SVM-SVMmodel based on the PSO algorithm The empirical resultsdemonstrate that the intrinsic fluctuation characteristics ofChinese stock index futures prices can be revealed in threecomponents including the residue term the low-frequencyterm and the high-frequency term with independent eco-nomic significance The residue term determines the long-term trend of stock index futures prices and its fluctuationsare correlated with macroeconomics and the long-termexpectations of investorsThe low-frequency term reveals theeffect of significant events on medium-term fluctuations instock index futures prices the ICSS algorithm breakpointtests reveal that policy regulation events are the main causeof fluctuations during the low-frequency term The high-frequency term reflects short-term disequilibrium in thestock index futures market although the effect of the high-frequency term on the overall fluctuation in the stock indexfutures prices is limited it nonetheless plays an important rolelocally

In addition in forecasting Chinese stock index futuresprices first we find that the linear kernel function SVMmodel is more suitable for predicting these components thanthe RBF kernel function SVM model The residue term andthe low-frequency term have stronger regular fluctuationsthan the high-frequency term and therefore yield superiorpredictive results Second the RBF kernel function SVMmodel shows higher predictive precision for the final combi-nation prediction than the linear kernel function SVMmodelSimultaneously the predictive results of the combinationprediction model are better than those of a single SVMmodel which indicates that the integration of fluctuationcharacteristics in different time scales increases the abilityof the SVM model to identify regularity in Chinese stockindex futures price fluctuations thus improving predictionprecision Finally the predictive results of the CEEMD-SVM-SVM model on daily prices and intraday prices are basicallyconsistent which indicates that our proposed model hasstability and universal applicability in forecasting Chinesestock index futures prices

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by the National SocialScience Foundation of China (14ZDA045)

References

[1] VMeneu andH Torro ldquoAsymmetric covariance in spot-futuresmarketsrdquo Journal of Futures Markets vol 23 no 11 pp 1019ndash1046 2003

[2] M Zhong A F Darrat and R Otero ldquoPrice discovery andvolatility spillovers in index futures markets some evidencefrom Mexicordquo Journal of Banking and Finance vol 28 no 12pp 3037ndash3054 2004

[3] X Xiong F Wang W Zhang and Y Sun ldquoPrice discoveryand volatility spillovers between SGX FTSEXinhua ChinaA50 index futures and A-share marketrdquo Chinese Journal ofManagement vol 11 no 1535 pp 1507ndash1512 2009

[4] H M Zuo Z T Liu and H W Zeng ldquoResearch of volatilityspillover and information transmission between stock indexfutures and spot market based on the high frequency datardquoJournal of Financial Research vol 4 pp 140ndash154 2012

[5] H Q Chen and C H Zhang ldquoDoes index futures tradingreduce stock market jump riskrdquo Economic Research Journalvol 1 pp 153ndash167 2015

[6] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 48 no 3 pp235ndash254 2015

[7] E E Peters Fractal Market Analysis Applying Chaos Theory toInvestment and Economics John Wiley amp Sons New York NYUSA 1994

[8] X Zhang K K Lai and S-YWang ldquoA new approach for crudeoil price analysis based on empirical mode decompositionrdquoEnergy Economics vol 30 no 3 pp 905ndash918 2008

[9] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnonstationary time series analysisrdquo Proceedings of the RoyalSociety of London A vol 454 no 1971 pp 903ndash995 1998

[10] Z Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[11] L F Ruan and H J Bao ldquoAn empirical analysis on periodicfluctuations of real estate price based on EMDrdquo Chinese Journalof Management Science no 3 pp 41ndash46 2012

[12] H L Li and C E Feng ldquoRelationship between investor senti-ment and stock indices fluctuation based on EEMDrdquo SystemsEngineering-Theory amp Practice vol 10 pp 2495ndash2503 2014

[13] J-R Yeh J-S Shieh andN EHuang ldquoComplementary ensem-ble empiricalmode decomposition a novel noise enhanced dataanalysismethodrdquoAdvances in Adaptive DataAnalysis vol 2 no2 pp 135ndash156 2010

[14] L Zhao W Yu and R Yan ldquoRolling bearing fault diagnosisbased on CEEMD and time series modelingrdquo MathematicalProblems in Engineering vol 2014 Article ID 101867 13 pages2014

Mathematical Problems in Engineering 13

[15] L Tang W Dai L Yu and S Wang ldquoA novel CEEMD-based eelm ensemble learning paradigm for crude oil priceforecastingrdquo International Journal of Information Technologyand Decision Making vol 14 no 1 pp 141ndash169 2015

[16] L Yu S Wang and K K Lai ldquoForecasting crude oil price withan EMD-based neural network ensemble learning paradigmrdquoEnergy Economics vol 30 no 5 pp 2623ndash2635 2008

[17] Y F Yang Y K Bao Z Y Hu and R Zhang ldquoCrude oilprice prediction based on empirical mode decomposition andsupport vector machinesrdquo Chinese Journal of Management vol12 pp 1884ndash1889 2010

[18] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[19] W Huang Y Nakamori and S-Y Wang ldquoForecasting stockmarket movement direction with support vector machinerdquoComputers and Operations Research vol 32 no 10 pp 2513ndash2522 2005

[20] H Ince and T B Trafalis ldquoShort term forecasting with supportvector machines and application to stock price predictionrdquoInternational Journal of General Systems vol 37 no 6 pp 677ndash687 2008

[21] F E H Tay and L Cao ldquoApplication of support vectormachinesin financial time series forecastingrdquo Omega vol 29 no 4 pp309ndash317 2001

[22] Y Sai F T Zhang and T Zhang ldquoResearch of Chinese stockindex futures regression prediction based on support vectormachinesrdquo Chinese Journal of Management Science vol 3 pp35ndash39 2013

[23] C Inclan andG C Tiao ldquoUse of cumulative sums of squares forretrospective detection of changes of variancerdquo Journal of theAmerican Statistical Association vol 89 no 427 pp 913ndash9231994

[24] S H Kang C Cheong and S-M Yoon ldquoStructural changesand volatility transmission in crude oil marketsrdquo Physica AStatisticalMechanics and Its Applications vol 390 no 23-24 pp4317ndash4324 2011

[25] J P Yang X X Chen and C H Zhang ldquoPolicy impact factorson the volatility of Shanghai and Shenzhen stock market inChinardquo Chinese Journal of Management Science vol 20 no 6pp 43ndash51 2012

[26] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer New York NY USA2010

[27] C-L Huang and J-F Dun ldquoA distributed PSOndashSVM hybridsystem with feature selection and parameter optimizationrdquoApplied Soft Computing vol 8 no 4 pp 1381ndash1391 2008

[28] C Zhou K Yin Y Cao and B Ahmed ldquoApplication of timeseries analysis and PSOndashSVMmodel in predicting the Bazimenlandslide in the Three Gorges Reservoir Chinardquo EngineeringGeology vol 204 pp 108ndash120 2016

[29] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis amp Applications vol 16 no 1 pp 93ndash103 2014

[30] X Gong Z He P Li and N Zhu ldquoForecasting return volatilityof the CSI 300 index using the stochastic volatility model withcontinuous volatility and jumpsrdquo Discrete Dynamics in Natureand Society vol 2014 Article ID 964654 10 pages 2014

[31] S Q Guo and P H Liang ldquoSocial Interactions the informationchannel and household stock market participation a empiricalstudy based on a Chinese household finance surveyrdquo EconomicResearch Journal vol S1 pp 116ndash131 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Table 3 Correlations and variance ratios of the component terms with the Chinese stock index futures price series

Pearson correlation coefficient Variance ratio (original series)Residue term 0762 (39214)lowastlowastlowast 70278Low-frequency term 0526 (206229)lowastlowastlowast 37521High-frequency term 0196 (6661)lowastlowastlowast 4051Note ldquolowast lowast lowastrdquo represents significance at the 1 significance level

indicating that the price fluctuation information reflected bythe high-frequency series low-frequency series and residueterm feature more significant differences To more accuratelyreveal the fluctuation characteristics of Chinese stock indexfutures prices at various time scales we analyze the sum ofthe IMFs with similar fluctuation characteristics using a 119905

statistics test of the mean from zero [8] Based on the data inTable 2 the high-frequency IMFsmdashIMF1 IMF2 IMF3 andIMF4mdashare summed as the high-frequency term The low-frequency IMFmdashIMF5 IMF6 and IMF7mdashare summed asthe low-frequency term The residue term alone is retainedThe correlations and variance ratios of the three componentswith the original series are presented in Table 3 and theirdistribution is presented in Figure 4

Consistent with the literature such as [8 9] the residueterm includes most of the information value of the originalseries and reflects the long-term trend of the original seriesIn fact the correlation coefficient of the residue term withthe Chinese stock index futures price series in this paper isas high as 076 and explains 70 of the fluctuations in theoriginal price series (see Table 3)Therefore the residue termcan represent the long-term trend of the Chinese stock indexfutures prices As shown in Figure 4 between the officiallisting of stock index futures inChina (2010) and 2014 there isa slow downward trend in stock index futures prices Becausethe stock index is the ldquobarometerrdquo of a countryrsquos macroe-conomic situation and because stock index futures serve aprice discovery function for stock indexes the futuresmarketprices reflect newly revealed information more quickly thancurrent market prices Therefore macroeconomic changesalso determine the long-term trend of stock index futuresprices As Chinarsquos Gross Domestic Product (GDP) growthrate gradually slows after 2010 (the GDP growth rates from2010 to 2014 are 1045 93 765 767 and 74 resp)(data for GDP growth rates in China from 2010 to 2014 weredownloaded from theNational Bureau of Statistics of the Peo-plersquos Republic of China website (httpwwwstatsgovcn))and macroeconomic growth shows signs of fatigue stockindex futures prices also decrease year by year decreasingfrom 3376 points in the beginning of 2010 to 2332 points inthe beginning of 2014 with an average annual growth rateof approximately minus11192 However this downward trendslows after 2014 and reverses slightly in the second half of 2014mainly because in that year certain important and significantpolitical actions including administrative simplification andanticorruption legislation and enforcement create a healthypolitical environment for the economy In addition a reduc-tion in the reserve requirement ratio and an interest rate cutwill free up cash for investment andChinarsquos economic growth

turns from incremental expansion to an optimized structurewhich results in good prospects for investors Accordinglymore money is flowing into the stock market which drivesstock index prices higher and higher stock index futures priceresult simultaneously

The low-frequency term mainly reflects the medium-term effect of significant events on stock index futures pricesAs shown in Table 3 the correlation between the low-frequency term and the Chinese stock index futures priceseries is 056 and explains up to 375 of the original priceseries indicating that significant events in recent years play animportant role in the fluctuations in the Chinese stock indexfutures prices Figure 4 illustrates the plummet that occursduring the launch of the Chinese stock index futures whichhas a maximum amplitude of 500 points This plummet ismainly due to the strengthening of the countryrsquos efforts tosuppress real estate market speculation in 2010This initiativeshould lead to a stock market decline because real estateinvestment is an important economic driver at the time butthe stock market does not make the appropriate responseTheChinese stock index futures unleash their price discoveryfunction after launch which leads to adjustments in stockindex prices In addition many investors use the hedgingfunction of stock index futures to avoid the risk of a stockmarket crash However after the substantial fluctuationsduring launch the amplitude of the low-frequency termdecreases slightly reflecting in part the stabilization effectof stock index futures on the stock market To furtherinvestigate the effects of significant events on Chinese stockindex futures prices we use the ICSS algorithm to performbreakpoint tests on the rate of change in the low-frequencyterm (first-difference processing on the low-frequency termto identify the breakpoints) Table 4 presents the results ofthe breakpoint test on the low-frequency term The low-frequency term exhibits 11 breakpoints or in other words12 different volatility mechanisms The significant eventsthat cause breakpoints are primarily administrative macro-control policies which are mainly addressed by adjustingmonetary policy perfecting the capital market mechanismand promoting market liberalization The execution of thesepolicies brings volatility to Chinese stock index futures inturn creating a new volatility mechanism These breakpointsare mainly concentrated in 2012 and 2013 During these twoyears the government frequently issues regulatory policiesto bolster financial deepening and reform and investorsadopt a medium-term investment strategy revolving aroundthese policies As an emerging market the Chinese stockindex futures market is easily affected However as thecapital market mechanism gradually adjusts trading in the

Mathematical Problems in Engineering 7

20104 20114 20124 20134 20144

005

1

minus05

minus1

times10minus11

(a)20104 20114 20124 20134 20144

02040

minus20

minus40

(b)

Figure 2 (a) The differences between the stock index futures series and the decomposition sequence obtained by the CEEMD method (b)The differences between the stock index futures series and the decomposition sequence obtained by the EEMDmethod

20104 20114 20124 20134 20144

0

100

200

300

IMF5

20104 20114 20124 20134 20144

0

200

400

600

IMF6

20104 20114 20124 20134 20144

0

100

IMF7

20104 20114 20124 20134 201442000

2500

3000

3500

Res

20104 20114 20124 20134 20144

IMF1 0

50

100

minus50

minus100

20104 20114 20124 20134 20144

IMF3 0

50

100

minus50

minus100

20104 20114 20124 20134 20144

0

100

200

IMF4

minus100

minus200

minus100

minus200

minus200

minus200

minus100

minus300

minus400

20104 20114 20124 20134 20144

0

50

100

IMF2

minus50

minus100

Figure 3 IMFs and residue term

8 Mathematical Problems in Engineering

Table 4 Breakpoint tests on the low-frequency term and corresponding significant events

Breakpointnumber Corresponding time Corresponding significant event The response of stock index futures prices

1 December 13 2010

On December 10 2010 the CentralEconomic Working Conference convenesand indicates that the moderately loosemonetary policy has become robust

This event implies that the supply ofliquidity to the capital market would bereduced which was bad news for the stockindex futures market Hence the stockfutures prices slump

2 September 28 2011

In October 2011 the credit ratings of a dozenEuropean financial institutions aredowngraded and the European debt crisisfurther escalates

The Chinese financial market was affectedby this event and in response the stockindex futures prices maintained thedownward trend

3 January 31 2012

From January 6 to 7 2012 the NationalConference on Financial Work convenes inBeijing to set the tone of the next series ofmajor financial reform issues

This event makes the stock market a positiveprospect for investors and then pushes upthe stock index futures prices

4 May 17 2012

On May of 2012 the stock index futuresprices are affected by a combination ofevents including the US stock market andinternational commodities market crashingand the Chinese PMI index does notcontinue early rising trend

The series of significant events poses aserious negative impact on investorsrsquoexpectations Therefore the stock indexfutures prices stop rising and turn tocollapse

5 June 28 2012

On June 18 2010 the Shanghai StockExchange and the Shenzhen Stock Exchangeseparately publish programs related to thedelisting of listed companies and revise thecorresponding Stock Market Rulesaccordingly further perfecting the initialpublic offering (IPO) and delisting systems

This event would improve investorsrsquoconfidence in the stock market Hence thedownward trend of stock index futures priceis slowed

6 August 28 2012

On August 16 2012 the New York branch ofthe Agricultural Bank of China openscontinuing to accelerate the pace ofinternationalization of the Chinese bankingsector

The event promotes the process of RMBinternationalization which is good news forthe financial market and stock index futuresprices inch up

7 October 18 2012

On October 17 2012 the Chinese StateCouncil meeting made an importantdecision that government would continue toimplement the proactive fiscal policy andprudent monetary policy

The prudent monetary policy implyinglimited funds in the capital market was badnews for stock index futures market Hencethe stock index futures prices tumble over

8 January 31 2013 At the end of January 2013 broad moneygrew 159 and narrow money grew 153

This event prompted a rise in stock indexfutures prices

9 May 21 2013

In May of 2013 the Chinese macroeconomicsituation does not perform well the FederalReserve gradually winds up the quantitativeeasing policy and many banks in China lacksufficient capital at the time

The cumulative effects of shocks of theseevents occurring in May have an enormousnegative influence on the stock index futuresmarket Hence the stock index futuresprices collapse

10 September 10 2013

On September 6 2013 repackaged treasurynote futures debut after a gap of 18 yearswhich is conducive to the establishingmarket-oriented pricing benchmarks andadvances market-oriented reform of theinterest rate

This event rebuilding investorsrsquo confidencein the capital market then pushes up thestock index futures prices

11 February 25 2014

On March 1 2014 the Central Bank releasesthe upper limit on interest rates on smallamounts of foreign currency deposits in theShanghai Free-Trade Zone taking anotherstep toward market-oriented reform of theinterest rate

The rise of deposit interest rate indicates thatmarket as risk-free rate would increasewhich is negative news for the stock indexfutures market Hence the stock indexfutures prices plunge

Mathematical Problems in Engineering 9

20104 20114 20124 20134 20144

0

1000

2000

3000

4000

Original seriesResidue term

Low-frequency termHigh-frequency term

minus1000

Figure 4 Chinese stock index futures prices and the three compo-nents

Chinese stock index futures market further standardizes andthe ability to withstand external impact thus also increasesTherefore the number of breakpoints decreases after 2014

The high-frequency term mainly reflects the short-termdisequilibrium in the Chinese stock index futures marketAs shown in Table 3 the correlation of the high-frequencyterm with the Chinese stock index futures prices is merely0196 and the ability to explain fluctuations in the originalprice series is only 4 indicating that the overall effectof the high-frequency term on Chinese stock index futuresprices is limited However as Figure 4 shows the fluctuationsof the high-frequency term fit well with the local fluctu-ations of the Chinese stock index futures prices in otherwords the high-frequency term plays a primary role in localchanges in Chinese stock index futures prices In additionas opposed to the low-frequency term the high-frequencyterm fluctuates frequently but with smaller amplitudes Thisdifference occurs primarily because the high-frequency termis particularly affected by investorsrsquo speculative arbitrageactivities In particular implementation of a T+0 transactionsystem in the Chinese stock index futures market means thatinvestors buying and selling stock index futures can completetheir transactions in the same day which suits short-termspeculators Based on this transaction system there will bemore investment chances for investors which will enhanceinvestorsrsquo speculative opportunities Hence there are manyshort-term fluctuations in Chinese stock index futures pricesMeanwhile it is possible and convenient for investors toaccess the latest hot investment tip because the role of socialinteractions is partly substituted for by online information inthe Internet age [31] In reality this type of new informationchanges continuously and investors must frequently adjusttheir investment strategies to gainmore additional benefits byspeculating in the spot-futures marketsTherefore the strongeffect of the shock of investorsrsquo speculation on Chinese stockindex futures prices in the short-term is difficult to sustainAlthough fluctuations in the high-frequency term are notimportant to forecast the long-term trends of Chinese stockindex futures their predictive effect of minor fluctuations onstock index futures cannot be ignored

4 Predictions for Chinese StockIndex Futures Prices

In this section this paper further investigates the predictionsof the Chinese stock index futures prices In addition tothe previous dayrsquos price data we also select the five-minutehigh-frequency price data of Chinese stock index futuresfrom April 16 2010 to November 20 2014 as a researchsubject which yields a total of 61325 data points (the data areobtained from theWind database) To forecast the daily pricethe first 745 (approximately two-thirds) pieces of data areused as the training data and the last 370 (approximately one-third) pieces of data are used as the test data To forecast thefive-minute high-frequency price the first 40883 (approxi-mately two-thirds) data points are used as the training dataand the last 20442 (approximately one-third) data points areused as the test data

As our analysis of the price fluctuations of the Chinesestock index futures uses the CEEMDmethod the fluctuationcharacteristics in different time scales have their uniqueeconomic implications we suggest that there may be betterpredictions of Chinese stock index futures prices when thesefluctuation characteristics are considered in the predictionmodel Hence based on the literature [16 17] a combinationprediction model that is the CEEMD-SVM-SVM model isproposed and the PSO algorithm is used to optimize theparameters of the SVM model In our empirical researchwe perform a one-step-ahead prediction of Chinese stockindex futures prices The specific steps are as follows (1) theCEEMD method is used to decompose the Chinese stockindex futures price series into several IMFs and a residueterm (2) the IMFs are then composed into a low-frequencyterm and a high-frequency term based on a fine-to-coarsereconstruction and the residue term remains the same (3)SVM models are established separately for each of the threecomponents obtained in step (2) and the PSO algorithm isused to optimize the parameters and (4) the predictive valueof the three components obtained in step (3) is used as theinput with the real value at the same moment as the outputand then a SVM combination model is built to obtain thefinal predictive value The overall process of the CEEMD-SVM-SVMmodel is shown in Figure 5

The prediction evaluation indexes are the root meansquare error (RMSE) mean absolute prediction error(MAPE) and directional symmetry (DS) the DS predictionevaluation index is used to indicate the modelrsquos ability topredict the directionality trend of Chinese stock index futuresprices The prediction evaluation indexes are expressed asfollows

RMSE = radic1

119899

119899

sum119905=1

[119909 (119905) minus (119905)]2

MAPE =1

119899

119899

sum119905=1

10038161003816100381610038161003816100381610038161003816

119909 (119905) minus (119905)

119909 (119905)

10038161003816100381610038161003816100381610038161003816

DS =1

119899

119899

sum119894=1

119889119894

10 Mathematical Problems in Engineering

Original series

CEEMD

Residue term Low-frequency term High-frequency term

SVM1 SVM2 SVM3

Final predictive value

SVM combination

Figure 5 The overall process of the CEEMD-SVM-SVMmodel

119889119894=

1 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] ge 0

0 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] lt 0

(10)

Among the three evaluation indexes smaller RMSE andMAPE values indicate that the predictionmodel is experienc-ing better performance which is also true of greater DS valueand better performance of the prediction model

The prediction accuracy based on the SVM model issensitive to the SVM modelrsquos kernel function As manyprevious studies (such as [17 22]) indicate that the linearkernel function and the RBF kernel function are appropriatekernel functions for the prediction of SVMmodel the linearkernel function and the RBF kernel function of the SVMare thus both applied to our prediction of Chinese stockindex futures prices Based on the process of the CEEMD-SVM-SVM model we first predict the residue term thenthe low-frequency term and then the high-frequency termof the daily price series and the five-minute price seriesNotably the five-minute price series of Chinese stock indexfutures is decomposed into fourteen IMFs and one residueterm where IMFs from IMF1 to IMF9 are recomposed intothe high-frequency term and IMFs from IMF10 to IMF14are recomposed into the low-frequency termThe predictionerrors of these components are shown in Table 5

As shown in Table 5 in using the SVM model to makepredictions on each component the prediction precision ofthe linear kernel function SVM model is generally higherthan the RBF kernel function SVM model In additionwhether for predicting the daily price series or the five-minute price series the prediction precisions of the residueterm and the low-frequency term are significantly higherthan those of the high-frequency termThis difference occursbecause these effects of shocks on the residue term andthe low-frequency term are fundamental economically theyare significant events that have an impact on Chinese stockindex futures prices that is sustained over a long duration

Therefore the fluctuations of the residue term and the low-frequency term are relatively steady and easier to forecastusing the SVM model By contrast the high-frequency termrepresenting the short-term disequilibrium of Chinese stockindex futures price is affected mainly by daily market activ-ities such as speculation arbitrage and the fluctuation ofthe high-frequency term that is random and changes rapidlyHence the SVMmodel experiences difficulty in learning thepattern of high-frequency term fluctuation resulting in poorpredictive results

Then according to prediction step (4) we use the optimalpredictive value of each component obtained in the previousstep to perform a combination prediction and the predictionerrors of our proposed model and the single SVM modelbased on the same data sample are simultaneously comparedThese results of the prediction errors are listed in Table 6

Table 6 shows that as opposed to the predictive resultsof the three components the RBF kernel function SVMmodel has higher predicted precision than the linear kernelfunction SVM model when performing combination pre-diction which indicates that the RBF kernel function SVMmodel is more appropriate for combination prediction Inaddition we find that the predictive results of the finalpredictive value obtained under the RBF kernel function aresignificantly better than those obtained in the single SVMmodel when predicting the daily price but slightly better thanthe single SVM model when making predictions regardingthe high-frequency price Meanwhile we should considerthat the high-frequency price embodies far more complexinformation than the daily price during the same time spanwhich may restrict the learning ability and generalizationability of the SVM model Hence the CEEMD-SVM-SVMmodel performs better in terms of predicting the daily priceIn any event the CEEMD-SVM-SVM model proposed bythis paper performs better than the single SVM model interms of both forecasting the daily price series and the five-minute price series Thus the introduction of Chinese stockindex futures price characteristics in different time scales into

Mathematical Problems in Engineering 11

Table 5 The prediction errors of different components

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

Residue termRMSE 0007 0008 0006 0009MAPE 0001 0001 0001 0002DS 69837 58424 21678 27241

Low-frequency termRMSE 0007 0008 0004 0008MAPE 0087 0092 0042 0258DS 60326 55434 54613 55254

High-frequency termRMSE 0086 0098 0020 0021MAPE 4806 4725 1146 1149DS 63315 62500 51463 51854

Table 6 Comparison of different modelsrsquo prediction errors

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

SVMRMSE 0053 0054 0008 0008MAPE 0009 0009 0001 0001DS 54891 52445 48171 48161

CEEMD-SVM-SVMRMSE 0051 0009 0008 0007MAPE 0008 0002 0001 0001DS 62228 88043 51565 52715

the SVM model improves the ability of the SVM model tomine the variations in the stock index futures price seriesfurther enhancing its prediction precision Simultaneouslysome useful suggestions for investors can be found in thepredictions of Chinese stock index futures prices generatedby our proposedmodel First investors should be accustomedto analyzing the price series from different time scales tounderstand the fluctuation characteristics of Chinese stockindex futures prices Second investors must actively identifywhat time scale the effects of external shocks on stock indexfutures prices belong to andmust devise the appropriate strat-egy accordingly Finally investors should pay more attentionto a hedging strategy involving Chinese stock index futuresover the long andor medium term and should be cautiousabout short-term speculation

Figures 6 and 7 present the real and predictive values ofthe daily price series andfive-minute price series respectivelyThe final predictive values obtained using the CEEMD-SVM-SVM model appeared to coincide with the real values indi-cating that the predictive values obtained using this modelfit well with the real values and intuitively demonstratingthat the CEEMD-SVM-SVMmodel is suitable for forecastingChinese stock index futures prices

20135 20138 201311 20142 20145 20148 2014112000

2200

2400

2600

2800

Real valuePredictive value

Figure 6 Real value and predictive value of the daily price series

5 Conclusion

The Chinese stock index futures market is a complex systemand fluctuations in the stock index futures prices resultfrom the collective effects of many factors at various levelsThe CEEMD method a method suitable in the analysis ofnonlinear nonstationary data is applied to analyze Chinesestock index futures prices in different time scales and to

12 Mathematical Problems in Engineering

Real valuePredictive value

20135 20138 201311 20142 20145 20148 2014111800

2000

2200

2400

2600

2800

Figure 7 Real value and predictive value of the five-minute priceseries

reveal the intrinsic fluctuation characteristics of Chinesestock index futures prices Then we focus on whether thesecharacteristics improve the predictive precision of Chinesestock index futures prices and propose a CEEMD-SVM-SVMmodel based on the PSO algorithm The empirical resultsdemonstrate that the intrinsic fluctuation characteristics ofChinese stock index futures prices can be revealed in threecomponents including the residue term the low-frequencyterm and the high-frequency term with independent eco-nomic significance The residue term determines the long-term trend of stock index futures prices and its fluctuationsare correlated with macroeconomics and the long-termexpectations of investorsThe low-frequency term reveals theeffect of significant events on medium-term fluctuations instock index futures prices the ICSS algorithm breakpointtests reveal that policy regulation events are the main causeof fluctuations during the low-frequency term The high-frequency term reflects short-term disequilibrium in thestock index futures market although the effect of the high-frequency term on the overall fluctuation in the stock indexfutures prices is limited it nonetheless plays an important rolelocally

In addition in forecasting Chinese stock index futuresprices first we find that the linear kernel function SVMmodel is more suitable for predicting these components thanthe RBF kernel function SVM model The residue term andthe low-frequency term have stronger regular fluctuationsthan the high-frequency term and therefore yield superiorpredictive results Second the RBF kernel function SVMmodel shows higher predictive precision for the final combi-nation prediction than the linear kernel function SVMmodelSimultaneously the predictive results of the combinationprediction model are better than those of a single SVMmodel which indicates that the integration of fluctuationcharacteristics in different time scales increases the abilityof the SVM model to identify regularity in Chinese stockindex futures price fluctuations thus improving predictionprecision Finally the predictive results of the CEEMD-SVM-SVM model on daily prices and intraday prices are basicallyconsistent which indicates that our proposed model hasstability and universal applicability in forecasting Chinesestock index futures prices

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by the National SocialScience Foundation of China (14ZDA045)

References

[1] VMeneu andH Torro ldquoAsymmetric covariance in spot-futuresmarketsrdquo Journal of Futures Markets vol 23 no 11 pp 1019ndash1046 2003

[2] M Zhong A F Darrat and R Otero ldquoPrice discovery andvolatility spillovers in index futures markets some evidencefrom Mexicordquo Journal of Banking and Finance vol 28 no 12pp 3037ndash3054 2004

[3] X Xiong F Wang W Zhang and Y Sun ldquoPrice discoveryand volatility spillovers between SGX FTSEXinhua ChinaA50 index futures and A-share marketrdquo Chinese Journal ofManagement vol 11 no 1535 pp 1507ndash1512 2009

[4] H M Zuo Z T Liu and H W Zeng ldquoResearch of volatilityspillover and information transmission between stock indexfutures and spot market based on the high frequency datardquoJournal of Financial Research vol 4 pp 140ndash154 2012

[5] H Q Chen and C H Zhang ldquoDoes index futures tradingreduce stock market jump riskrdquo Economic Research Journalvol 1 pp 153ndash167 2015

[6] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 48 no 3 pp235ndash254 2015

[7] E E Peters Fractal Market Analysis Applying Chaos Theory toInvestment and Economics John Wiley amp Sons New York NYUSA 1994

[8] X Zhang K K Lai and S-YWang ldquoA new approach for crudeoil price analysis based on empirical mode decompositionrdquoEnergy Economics vol 30 no 3 pp 905ndash918 2008

[9] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnonstationary time series analysisrdquo Proceedings of the RoyalSociety of London A vol 454 no 1971 pp 903ndash995 1998

[10] Z Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[11] L F Ruan and H J Bao ldquoAn empirical analysis on periodicfluctuations of real estate price based on EMDrdquo Chinese Journalof Management Science no 3 pp 41ndash46 2012

[12] H L Li and C E Feng ldquoRelationship between investor senti-ment and stock indices fluctuation based on EEMDrdquo SystemsEngineering-Theory amp Practice vol 10 pp 2495ndash2503 2014

[13] J-R Yeh J-S Shieh andN EHuang ldquoComplementary ensem-ble empiricalmode decomposition a novel noise enhanced dataanalysismethodrdquoAdvances in Adaptive DataAnalysis vol 2 no2 pp 135ndash156 2010

[14] L Zhao W Yu and R Yan ldquoRolling bearing fault diagnosisbased on CEEMD and time series modelingrdquo MathematicalProblems in Engineering vol 2014 Article ID 101867 13 pages2014

Mathematical Problems in Engineering 13

[15] L Tang W Dai L Yu and S Wang ldquoA novel CEEMD-based eelm ensemble learning paradigm for crude oil priceforecastingrdquo International Journal of Information Technologyand Decision Making vol 14 no 1 pp 141ndash169 2015

[16] L Yu S Wang and K K Lai ldquoForecasting crude oil price withan EMD-based neural network ensemble learning paradigmrdquoEnergy Economics vol 30 no 5 pp 2623ndash2635 2008

[17] Y F Yang Y K Bao Z Y Hu and R Zhang ldquoCrude oilprice prediction based on empirical mode decomposition andsupport vector machinesrdquo Chinese Journal of Management vol12 pp 1884ndash1889 2010

[18] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[19] W Huang Y Nakamori and S-Y Wang ldquoForecasting stockmarket movement direction with support vector machinerdquoComputers and Operations Research vol 32 no 10 pp 2513ndash2522 2005

[20] H Ince and T B Trafalis ldquoShort term forecasting with supportvector machines and application to stock price predictionrdquoInternational Journal of General Systems vol 37 no 6 pp 677ndash687 2008

[21] F E H Tay and L Cao ldquoApplication of support vectormachinesin financial time series forecastingrdquo Omega vol 29 no 4 pp309ndash317 2001

[22] Y Sai F T Zhang and T Zhang ldquoResearch of Chinese stockindex futures regression prediction based on support vectormachinesrdquo Chinese Journal of Management Science vol 3 pp35ndash39 2013

[23] C Inclan andG C Tiao ldquoUse of cumulative sums of squares forretrospective detection of changes of variancerdquo Journal of theAmerican Statistical Association vol 89 no 427 pp 913ndash9231994

[24] S H Kang C Cheong and S-M Yoon ldquoStructural changesand volatility transmission in crude oil marketsrdquo Physica AStatisticalMechanics and Its Applications vol 390 no 23-24 pp4317ndash4324 2011

[25] J P Yang X X Chen and C H Zhang ldquoPolicy impact factorson the volatility of Shanghai and Shenzhen stock market inChinardquo Chinese Journal of Management Science vol 20 no 6pp 43ndash51 2012

[26] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer New York NY USA2010

[27] C-L Huang and J-F Dun ldquoA distributed PSOndashSVM hybridsystem with feature selection and parameter optimizationrdquoApplied Soft Computing vol 8 no 4 pp 1381ndash1391 2008

[28] C Zhou K Yin Y Cao and B Ahmed ldquoApplication of timeseries analysis and PSOndashSVMmodel in predicting the Bazimenlandslide in the Three Gorges Reservoir Chinardquo EngineeringGeology vol 204 pp 108ndash120 2016

[29] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis amp Applications vol 16 no 1 pp 93ndash103 2014

[30] X Gong Z He P Li and N Zhu ldquoForecasting return volatilityof the CSI 300 index using the stochastic volatility model withcontinuous volatility and jumpsrdquo Discrete Dynamics in Natureand Society vol 2014 Article ID 964654 10 pages 2014

[31] S Q Guo and P H Liang ldquoSocial Interactions the informationchannel and household stock market participation a empiricalstudy based on a Chinese household finance surveyrdquo EconomicResearch Journal vol S1 pp 116ndash131 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

20104 20114 20124 20134 20144

005

1

minus05

minus1

times10minus11

(a)20104 20114 20124 20134 20144

02040

minus20

minus40

(b)

Figure 2 (a) The differences between the stock index futures series and the decomposition sequence obtained by the CEEMD method (b)The differences between the stock index futures series and the decomposition sequence obtained by the EEMDmethod

20104 20114 20124 20134 20144

0

100

200

300

IMF5

20104 20114 20124 20134 20144

0

200

400

600

IMF6

20104 20114 20124 20134 20144

0

100

IMF7

20104 20114 20124 20134 201442000

2500

3000

3500

Res

20104 20114 20124 20134 20144

IMF1 0

50

100

minus50

minus100

20104 20114 20124 20134 20144

IMF3 0

50

100

minus50

minus100

20104 20114 20124 20134 20144

0

100

200

IMF4

minus100

minus200

minus100

minus200

minus200

minus200

minus100

minus300

minus400

20104 20114 20124 20134 20144

0

50

100

IMF2

minus50

minus100

Figure 3 IMFs and residue term

8 Mathematical Problems in Engineering

Table 4 Breakpoint tests on the low-frequency term and corresponding significant events

Breakpointnumber Corresponding time Corresponding significant event The response of stock index futures prices

1 December 13 2010

On December 10 2010 the CentralEconomic Working Conference convenesand indicates that the moderately loosemonetary policy has become robust

This event implies that the supply ofliquidity to the capital market would bereduced which was bad news for the stockindex futures market Hence the stockfutures prices slump

2 September 28 2011

In October 2011 the credit ratings of a dozenEuropean financial institutions aredowngraded and the European debt crisisfurther escalates

The Chinese financial market was affectedby this event and in response the stockindex futures prices maintained thedownward trend

3 January 31 2012

From January 6 to 7 2012 the NationalConference on Financial Work convenes inBeijing to set the tone of the next series ofmajor financial reform issues

This event makes the stock market a positiveprospect for investors and then pushes upthe stock index futures prices

4 May 17 2012

On May of 2012 the stock index futuresprices are affected by a combination ofevents including the US stock market andinternational commodities market crashingand the Chinese PMI index does notcontinue early rising trend

The series of significant events poses aserious negative impact on investorsrsquoexpectations Therefore the stock indexfutures prices stop rising and turn tocollapse

5 June 28 2012

On June 18 2010 the Shanghai StockExchange and the Shenzhen Stock Exchangeseparately publish programs related to thedelisting of listed companies and revise thecorresponding Stock Market Rulesaccordingly further perfecting the initialpublic offering (IPO) and delisting systems

This event would improve investorsrsquoconfidence in the stock market Hence thedownward trend of stock index futures priceis slowed

6 August 28 2012

On August 16 2012 the New York branch ofthe Agricultural Bank of China openscontinuing to accelerate the pace ofinternationalization of the Chinese bankingsector

The event promotes the process of RMBinternationalization which is good news forthe financial market and stock index futuresprices inch up

7 October 18 2012

On October 17 2012 the Chinese StateCouncil meeting made an importantdecision that government would continue toimplement the proactive fiscal policy andprudent monetary policy

The prudent monetary policy implyinglimited funds in the capital market was badnews for stock index futures market Hencethe stock index futures prices tumble over

8 January 31 2013 At the end of January 2013 broad moneygrew 159 and narrow money grew 153

This event prompted a rise in stock indexfutures prices

9 May 21 2013

In May of 2013 the Chinese macroeconomicsituation does not perform well the FederalReserve gradually winds up the quantitativeeasing policy and many banks in China lacksufficient capital at the time

The cumulative effects of shocks of theseevents occurring in May have an enormousnegative influence on the stock index futuresmarket Hence the stock index futuresprices collapse

10 September 10 2013

On September 6 2013 repackaged treasurynote futures debut after a gap of 18 yearswhich is conducive to the establishingmarket-oriented pricing benchmarks andadvances market-oriented reform of theinterest rate

This event rebuilding investorsrsquo confidencein the capital market then pushes up thestock index futures prices

11 February 25 2014

On March 1 2014 the Central Bank releasesthe upper limit on interest rates on smallamounts of foreign currency deposits in theShanghai Free-Trade Zone taking anotherstep toward market-oriented reform of theinterest rate

The rise of deposit interest rate indicates thatmarket as risk-free rate would increasewhich is negative news for the stock indexfutures market Hence the stock indexfutures prices plunge

Mathematical Problems in Engineering 9

20104 20114 20124 20134 20144

0

1000

2000

3000

4000

Original seriesResidue term

Low-frequency termHigh-frequency term

minus1000

Figure 4 Chinese stock index futures prices and the three compo-nents

Chinese stock index futures market further standardizes andthe ability to withstand external impact thus also increasesTherefore the number of breakpoints decreases after 2014

The high-frequency term mainly reflects the short-termdisequilibrium in the Chinese stock index futures marketAs shown in Table 3 the correlation of the high-frequencyterm with the Chinese stock index futures prices is merely0196 and the ability to explain fluctuations in the originalprice series is only 4 indicating that the overall effectof the high-frequency term on Chinese stock index futuresprices is limited However as Figure 4 shows the fluctuationsof the high-frequency term fit well with the local fluctu-ations of the Chinese stock index futures prices in otherwords the high-frequency term plays a primary role in localchanges in Chinese stock index futures prices In additionas opposed to the low-frequency term the high-frequencyterm fluctuates frequently but with smaller amplitudes Thisdifference occurs primarily because the high-frequency termis particularly affected by investorsrsquo speculative arbitrageactivities In particular implementation of a T+0 transactionsystem in the Chinese stock index futures market means thatinvestors buying and selling stock index futures can completetheir transactions in the same day which suits short-termspeculators Based on this transaction system there will bemore investment chances for investors which will enhanceinvestorsrsquo speculative opportunities Hence there are manyshort-term fluctuations in Chinese stock index futures pricesMeanwhile it is possible and convenient for investors toaccess the latest hot investment tip because the role of socialinteractions is partly substituted for by online information inthe Internet age [31] In reality this type of new informationchanges continuously and investors must frequently adjusttheir investment strategies to gainmore additional benefits byspeculating in the spot-futures marketsTherefore the strongeffect of the shock of investorsrsquo speculation on Chinese stockindex futures prices in the short-term is difficult to sustainAlthough fluctuations in the high-frequency term are notimportant to forecast the long-term trends of Chinese stockindex futures their predictive effect of minor fluctuations onstock index futures cannot be ignored

4 Predictions for Chinese StockIndex Futures Prices

In this section this paper further investigates the predictionsof the Chinese stock index futures prices In addition tothe previous dayrsquos price data we also select the five-minutehigh-frequency price data of Chinese stock index futuresfrom April 16 2010 to November 20 2014 as a researchsubject which yields a total of 61325 data points (the data areobtained from theWind database) To forecast the daily pricethe first 745 (approximately two-thirds) pieces of data areused as the training data and the last 370 (approximately one-third) pieces of data are used as the test data To forecast thefive-minute high-frequency price the first 40883 (approxi-mately two-thirds) data points are used as the training dataand the last 20442 (approximately one-third) data points areused as the test data

As our analysis of the price fluctuations of the Chinesestock index futures uses the CEEMDmethod the fluctuationcharacteristics in different time scales have their uniqueeconomic implications we suggest that there may be betterpredictions of Chinese stock index futures prices when thesefluctuation characteristics are considered in the predictionmodel Hence based on the literature [16 17] a combinationprediction model that is the CEEMD-SVM-SVM model isproposed and the PSO algorithm is used to optimize theparameters of the SVM model In our empirical researchwe perform a one-step-ahead prediction of Chinese stockindex futures prices The specific steps are as follows (1) theCEEMD method is used to decompose the Chinese stockindex futures price series into several IMFs and a residueterm (2) the IMFs are then composed into a low-frequencyterm and a high-frequency term based on a fine-to-coarsereconstruction and the residue term remains the same (3)SVM models are established separately for each of the threecomponents obtained in step (2) and the PSO algorithm isused to optimize the parameters and (4) the predictive valueof the three components obtained in step (3) is used as theinput with the real value at the same moment as the outputand then a SVM combination model is built to obtain thefinal predictive value The overall process of the CEEMD-SVM-SVMmodel is shown in Figure 5

The prediction evaluation indexes are the root meansquare error (RMSE) mean absolute prediction error(MAPE) and directional symmetry (DS) the DS predictionevaluation index is used to indicate the modelrsquos ability topredict the directionality trend of Chinese stock index futuresprices The prediction evaluation indexes are expressed asfollows

RMSE = radic1

119899

119899

sum119905=1

[119909 (119905) minus (119905)]2

MAPE =1

119899

119899

sum119905=1

10038161003816100381610038161003816100381610038161003816

119909 (119905) minus (119905)

119909 (119905)

10038161003816100381610038161003816100381610038161003816

DS =1

119899

119899

sum119894=1

119889119894

10 Mathematical Problems in Engineering

Original series

CEEMD

Residue term Low-frequency term High-frequency term

SVM1 SVM2 SVM3

Final predictive value

SVM combination

Figure 5 The overall process of the CEEMD-SVM-SVMmodel

119889119894=

1 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] ge 0

0 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] lt 0

(10)

Among the three evaluation indexes smaller RMSE andMAPE values indicate that the predictionmodel is experienc-ing better performance which is also true of greater DS valueand better performance of the prediction model

The prediction accuracy based on the SVM model issensitive to the SVM modelrsquos kernel function As manyprevious studies (such as [17 22]) indicate that the linearkernel function and the RBF kernel function are appropriatekernel functions for the prediction of SVMmodel the linearkernel function and the RBF kernel function of the SVMare thus both applied to our prediction of Chinese stockindex futures prices Based on the process of the CEEMD-SVM-SVM model we first predict the residue term thenthe low-frequency term and then the high-frequency termof the daily price series and the five-minute price seriesNotably the five-minute price series of Chinese stock indexfutures is decomposed into fourteen IMFs and one residueterm where IMFs from IMF1 to IMF9 are recomposed intothe high-frequency term and IMFs from IMF10 to IMF14are recomposed into the low-frequency termThe predictionerrors of these components are shown in Table 5

As shown in Table 5 in using the SVM model to makepredictions on each component the prediction precision ofthe linear kernel function SVM model is generally higherthan the RBF kernel function SVM model In additionwhether for predicting the daily price series or the five-minute price series the prediction precisions of the residueterm and the low-frequency term are significantly higherthan those of the high-frequency termThis difference occursbecause these effects of shocks on the residue term andthe low-frequency term are fundamental economically theyare significant events that have an impact on Chinese stockindex futures prices that is sustained over a long duration

Therefore the fluctuations of the residue term and the low-frequency term are relatively steady and easier to forecastusing the SVM model By contrast the high-frequency termrepresenting the short-term disequilibrium of Chinese stockindex futures price is affected mainly by daily market activ-ities such as speculation arbitrage and the fluctuation ofthe high-frequency term that is random and changes rapidlyHence the SVMmodel experiences difficulty in learning thepattern of high-frequency term fluctuation resulting in poorpredictive results

Then according to prediction step (4) we use the optimalpredictive value of each component obtained in the previousstep to perform a combination prediction and the predictionerrors of our proposed model and the single SVM modelbased on the same data sample are simultaneously comparedThese results of the prediction errors are listed in Table 6

Table 6 shows that as opposed to the predictive resultsof the three components the RBF kernel function SVMmodel has higher predicted precision than the linear kernelfunction SVM model when performing combination pre-diction which indicates that the RBF kernel function SVMmodel is more appropriate for combination prediction Inaddition we find that the predictive results of the finalpredictive value obtained under the RBF kernel function aresignificantly better than those obtained in the single SVMmodel when predicting the daily price but slightly better thanthe single SVM model when making predictions regardingthe high-frequency price Meanwhile we should considerthat the high-frequency price embodies far more complexinformation than the daily price during the same time spanwhich may restrict the learning ability and generalizationability of the SVM model Hence the CEEMD-SVM-SVMmodel performs better in terms of predicting the daily priceIn any event the CEEMD-SVM-SVM model proposed bythis paper performs better than the single SVM model interms of both forecasting the daily price series and the five-minute price series Thus the introduction of Chinese stockindex futures price characteristics in different time scales into

Mathematical Problems in Engineering 11

Table 5 The prediction errors of different components

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

Residue termRMSE 0007 0008 0006 0009MAPE 0001 0001 0001 0002DS 69837 58424 21678 27241

Low-frequency termRMSE 0007 0008 0004 0008MAPE 0087 0092 0042 0258DS 60326 55434 54613 55254

High-frequency termRMSE 0086 0098 0020 0021MAPE 4806 4725 1146 1149DS 63315 62500 51463 51854

Table 6 Comparison of different modelsrsquo prediction errors

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

SVMRMSE 0053 0054 0008 0008MAPE 0009 0009 0001 0001DS 54891 52445 48171 48161

CEEMD-SVM-SVMRMSE 0051 0009 0008 0007MAPE 0008 0002 0001 0001DS 62228 88043 51565 52715

the SVM model improves the ability of the SVM model tomine the variations in the stock index futures price seriesfurther enhancing its prediction precision Simultaneouslysome useful suggestions for investors can be found in thepredictions of Chinese stock index futures prices generatedby our proposedmodel First investors should be accustomedto analyzing the price series from different time scales tounderstand the fluctuation characteristics of Chinese stockindex futures prices Second investors must actively identifywhat time scale the effects of external shocks on stock indexfutures prices belong to andmust devise the appropriate strat-egy accordingly Finally investors should pay more attentionto a hedging strategy involving Chinese stock index futuresover the long andor medium term and should be cautiousabout short-term speculation

Figures 6 and 7 present the real and predictive values ofthe daily price series andfive-minute price series respectivelyThe final predictive values obtained using the CEEMD-SVM-SVM model appeared to coincide with the real values indi-cating that the predictive values obtained using this modelfit well with the real values and intuitively demonstratingthat the CEEMD-SVM-SVMmodel is suitable for forecastingChinese stock index futures prices

20135 20138 201311 20142 20145 20148 2014112000

2200

2400

2600

2800

Real valuePredictive value

Figure 6 Real value and predictive value of the daily price series

5 Conclusion

The Chinese stock index futures market is a complex systemand fluctuations in the stock index futures prices resultfrom the collective effects of many factors at various levelsThe CEEMD method a method suitable in the analysis ofnonlinear nonstationary data is applied to analyze Chinesestock index futures prices in different time scales and to

12 Mathematical Problems in Engineering

Real valuePredictive value

20135 20138 201311 20142 20145 20148 2014111800

2000

2200

2400

2600

2800

Figure 7 Real value and predictive value of the five-minute priceseries

reveal the intrinsic fluctuation characteristics of Chinesestock index futures prices Then we focus on whether thesecharacteristics improve the predictive precision of Chinesestock index futures prices and propose a CEEMD-SVM-SVMmodel based on the PSO algorithm The empirical resultsdemonstrate that the intrinsic fluctuation characteristics ofChinese stock index futures prices can be revealed in threecomponents including the residue term the low-frequencyterm and the high-frequency term with independent eco-nomic significance The residue term determines the long-term trend of stock index futures prices and its fluctuationsare correlated with macroeconomics and the long-termexpectations of investorsThe low-frequency term reveals theeffect of significant events on medium-term fluctuations instock index futures prices the ICSS algorithm breakpointtests reveal that policy regulation events are the main causeof fluctuations during the low-frequency term The high-frequency term reflects short-term disequilibrium in thestock index futures market although the effect of the high-frequency term on the overall fluctuation in the stock indexfutures prices is limited it nonetheless plays an important rolelocally

In addition in forecasting Chinese stock index futuresprices first we find that the linear kernel function SVMmodel is more suitable for predicting these components thanthe RBF kernel function SVM model The residue term andthe low-frequency term have stronger regular fluctuationsthan the high-frequency term and therefore yield superiorpredictive results Second the RBF kernel function SVMmodel shows higher predictive precision for the final combi-nation prediction than the linear kernel function SVMmodelSimultaneously the predictive results of the combinationprediction model are better than those of a single SVMmodel which indicates that the integration of fluctuationcharacteristics in different time scales increases the abilityof the SVM model to identify regularity in Chinese stockindex futures price fluctuations thus improving predictionprecision Finally the predictive results of the CEEMD-SVM-SVM model on daily prices and intraday prices are basicallyconsistent which indicates that our proposed model hasstability and universal applicability in forecasting Chinesestock index futures prices

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by the National SocialScience Foundation of China (14ZDA045)

References

[1] VMeneu andH Torro ldquoAsymmetric covariance in spot-futuresmarketsrdquo Journal of Futures Markets vol 23 no 11 pp 1019ndash1046 2003

[2] M Zhong A F Darrat and R Otero ldquoPrice discovery andvolatility spillovers in index futures markets some evidencefrom Mexicordquo Journal of Banking and Finance vol 28 no 12pp 3037ndash3054 2004

[3] X Xiong F Wang W Zhang and Y Sun ldquoPrice discoveryand volatility spillovers between SGX FTSEXinhua ChinaA50 index futures and A-share marketrdquo Chinese Journal ofManagement vol 11 no 1535 pp 1507ndash1512 2009

[4] H M Zuo Z T Liu and H W Zeng ldquoResearch of volatilityspillover and information transmission between stock indexfutures and spot market based on the high frequency datardquoJournal of Financial Research vol 4 pp 140ndash154 2012

[5] H Q Chen and C H Zhang ldquoDoes index futures tradingreduce stock market jump riskrdquo Economic Research Journalvol 1 pp 153ndash167 2015

[6] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 48 no 3 pp235ndash254 2015

[7] E E Peters Fractal Market Analysis Applying Chaos Theory toInvestment and Economics John Wiley amp Sons New York NYUSA 1994

[8] X Zhang K K Lai and S-YWang ldquoA new approach for crudeoil price analysis based on empirical mode decompositionrdquoEnergy Economics vol 30 no 3 pp 905ndash918 2008

[9] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnonstationary time series analysisrdquo Proceedings of the RoyalSociety of London A vol 454 no 1971 pp 903ndash995 1998

[10] Z Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[11] L F Ruan and H J Bao ldquoAn empirical analysis on periodicfluctuations of real estate price based on EMDrdquo Chinese Journalof Management Science no 3 pp 41ndash46 2012

[12] H L Li and C E Feng ldquoRelationship between investor senti-ment and stock indices fluctuation based on EEMDrdquo SystemsEngineering-Theory amp Practice vol 10 pp 2495ndash2503 2014

[13] J-R Yeh J-S Shieh andN EHuang ldquoComplementary ensem-ble empiricalmode decomposition a novel noise enhanced dataanalysismethodrdquoAdvances in Adaptive DataAnalysis vol 2 no2 pp 135ndash156 2010

[14] L Zhao W Yu and R Yan ldquoRolling bearing fault diagnosisbased on CEEMD and time series modelingrdquo MathematicalProblems in Engineering vol 2014 Article ID 101867 13 pages2014

Mathematical Problems in Engineering 13

[15] L Tang W Dai L Yu and S Wang ldquoA novel CEEMD-based eelm ensemble learning paradigm for crude oil priceforecastingrdquo International Journal of Information Technologyand Decision Making vol 14 no 1 pp 141ndash169 2015

[16] L Yu S Wang and K K Lai ldquoForecasting crude oil price withan EMD-based neural network ensemble learning paradigmrdquoEnergy Economics vol 30 no 5 pp 2623ndash2635 2008

[17] Y F Yang Y K Bao Z Y Hu and R Zhang ldquoCrude oilprice prediction based on empirical mode decomposition andsupport vector machinesrdquo Chinese Journal of Management vol12 pp 1884ndash1889 2010

[18] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[19] W Huang Y Nakamori and S-Y Wang ldquoForecasting stockmarket movement direction with support vector machinerdquoComputers and Operations Research vol 32 no 10 pp 2513ndash2522 2005

[20] H Ince and T B Trafalis ldquoShort term forecasting with supportvector machines and application to stock price predictionrdquoInternational Journal of General Systems vol 37 no 6 pp 677ndash687 2008

[21] F E H Tay and L Cao ldquoApplication of support vectormachinesin financial time series forecastingrdquo Omega vol 29 no 4 pp309ndash317 2001

[22] Y Sai F T Zhang and T Zhang ldquoResearch of Chinese stockindex futures regression prediction based on support vectormachinesrdquo Chinese Journal of Management Science vol 3 pp35ndash39 2013

[23] C Inclan andG C Tiao ldquoUse of cumulative sums of squares forretrospective detection of changes of variancerdquo Journal of theAmerican Statistical Association vol 89 no 427 pp 913ndash9231994

[24] S H Kang C Cheong and S-M Yoon ldquoStructural changesand volatility transmission in crude oil marketsrdquo Physica AStatisticalMechanics and Its Applications vol 390 no 23-24 pp4317ndash4324 2011

[25] J P Yang X X Chen and C H Zhang ldquoPolicy impact factorson the volatility of Shanghai and Shenzhen stock market inChinardquo Chinese Journal of Management Science vol 20 no 6pp 43ndash51 2012

[26] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer New York NY USA2010

[27] C-L Huang and J-F Dun ldquoA distributed PSOndashSVM hybridsystem with feature selection and parameter optimizationrdquoApplied Soft Computing vol 8 no 4 pp 1381ndash1391 2008

[28] C Zhou K Yin Y Cao and B Ahmed ldquoApplication of timeseries analysis and PSOndashSVMmodel in predicting the Bazimenlandslide in the Three Gorges Reservoir Chinardquo EngineeringGeology vol 204 pp 108ndash120 2016

[29] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis amp Applications vol 16 no 1 pp 93ndash103 2014

[30] X Gong Z He P Li and N Zhu ldquoForecasting return volatilityof the CSI 300 index using the stochastic volatility model withcontinuous volatility and jumpsrdquo Discrete Dynamics in Natureand Society vol 2014 Article ID 964654 10 pages 2014

[31] S Q Guo and P H Liang ldquoSocial Interactions the informationchannel and household stock market participation a empiricalstudy based on a Chinese household finance surveyrdquo EconomicResearch Journal vol S1 pp 116ndash131 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 4 Breakpoint tests on the low-frequency term and corresponding significant events

Breakpointnumber Corresponding time Corresponding significant event The response of stock index futures prices

1 December 13 2010

On December 10 2010 the CentralEconomic Working Conference convenesand indicates that the moderately loosemonetary policy has become robust

This event implies that the supply ofliquidity to the capital market would bereduced which was bad news for the stockindex futures market Hence the stockfutures prices slump

2 September 28 2011

In October 2011 the credit ratings of a dozenEuropean financial institutions aredowngraded and the European debt crisisfurther escalates

The Chinese financial market was affectedby this event and in response the stockindex futures prices maintained thedownward trend

3 January 31 2012

From January 6 to 7 2012 the NationalConference on Financial Work convenes inBeijing to set the tone of the next series ofmajor financial reform issues

This event makes the stock market a positiveprospect for investors and then pushes upthe stock index futures prices

4 May 17 2012

On May of 2012 the stock index futuresprices are affected by a combination ofevents including the US stock market andinternational commodities market crashingand the Chinese PMI index does notcontinue early rising trend

The series of significant events poses aserious negative impact on investorsrsquoexpectations Therefore the stock indexfutures prices stop rising and turn tocollapse

5 June 28 2012

On June 18 2010 the Shanghai StockExchange and the Shenzhen Stock Exchangeseparately publish programs related to thedelisting of listed companies and revise thecorresponding Stock Market Rulesaccordingly further perfecting the initialpublic offering (IPO) and delisting systems

This event would improve investorsrsquoconfidence in the stock market Hence thedownward trend of stock index futures priceis slowed

6 August 28 2012

On August 16 2012 the New York branch ofthe Agricultural Bank of China openscontinuing to accelerate the pace ofinternationalization of the Chinese bankingsector

The event promotes the process of RMBinternationalization which is good news forthe financial market and stock index futuresprices inch up

7 October 18 2012

On October 17 2012 the Chinese StateCouncil meeting made an importantdecision that government would continue toimplement the proactive fiscal policy andprudent monetary policy

The prudent monetary policy implyinglimited funds in the capital market was badnews for stock index futures market Hencethe stock index futures prices tumble over

8 January 31 2013 At the end of January 2013 broad moneygrew 159 and narrow money grew 153

This event prompted a rise in stock indexfutures prices

9 May 21 2013

In May of 2013 the Chinese macroeconomicsituation does not perform well the FederalReserve gradually winds up the quantitativeeasing policy and many banks in China lacksufficient capital at the time

The cumulative effects of shocks of theseevents occurring in May have an enormousnegative influence on the stock index futuresmarket Hence the stock index futuresprices collapse

10 September 10 2013

On September 6 2013 repackaged treasurynote futures debut after a gap of 18 yearswhich is conducive to the establishingmarket-oriented pricing benchmarks andadvances market-oriented reform of theinterest rate

This event rebuilding investorsrsquo confidencein the capital market then pushes up thestock index futures prices

11 February 25 2014

On March 1 2014 the Central Bank releasesthe upper limit on interest rates on smallamounts of foreign currency deposits in theShanghai Free-Trade Zone taking anotherstep toward market-oriented reform of theinterest rate

The rise of deposit interest rate indicates thatmarket as risk-free rate would increasewhich is negative news for the stock indexfutures market Hence the stock indexfutures prices plunge

Mathematical Problems in Engineering 9

20104 20114 20124 20134 20144

0

1000

2000

3000

4000

Original seriesResidue term

Low-frequency termHigh-frequency term

minus1000

Figure 4 Chinese stock index futures prices and the three compo-nents

Chinese stock index futures market further standardizes andthe ability to withstand external impact thus also increasesTherefore the number of breakpoints decreases after 2014

The high-frequency term mainly reflects the short-termdisequilibrium in the Chinese stock index futures marketAs shown in Table 3 the correlation of the high-frequencyterm with the Chinese stock index futures prices is merely0196 and the ability to explain fluctuations in the originalprice series is only 4 indicating that the overall effectof the high-frequency term on Chinese stock index futuresprices is limited However as Figure 4 shows the fluctuationsof the high-frequency term fit well with the local fluctu-ations of the Chinese stock index futures prices in otherwords the high-frequency term plays a primary role in localchanges in Chinese stock index futures prices In additionas opposed to the low-frequency term the high-frequencyterm fluctuates frequently but with smaller amplitudes Thisdifference occurs primarily because the high-frequency termis particularly affected by investorsrsquo speculative arbitrageactivities In particular implementation of a T+0 transactionsystem in the Chinese stock index futures market means thatinvestors buying and selling stock index futures can completetheir transactions in the same day which suits short-termspeculators Based on this transaction system there will bemore investment chances for investors which will enhanceinvestorsrsquo speculative opportunities Hence there are manyshort-term fluctuations in Chinese stock index futures pricesMeanwhile it is possible and convenient for investors toaccess the latest hot investment tip because the role of socialinteractions is partly substituted for by online information inthe Internet age [31] In reality this type of new informationchanges continuously and investors must frequently adjusttheir investment strategies to gainmore additional benefits byspeculating in the spot-futures marketsTherefore the strongeffect of the shock of investorsrsquo speculation on Chinese stockindex futures prices in the short-term is difficult to sustainAlthough fluctuations in the high-frequency term are notimportant to forecast the long-term trends of Chinese stockindex futures their predictive effect of minor fluctuations onstock index futures cannot be ignored

4 Predictions for Chinese StockIndex Futures Prices

In this section this paper further investigates the predictionsof the Chinese stock index futures prices In addition tothe previous dayrsquos price data we also select the five-minutehigh-frequency price data of Chinese stock index futuresfrom April 16 2010 to November 20 2014 as a researchsubject which yields a total of 61325 data points (the data areobtained from theWind database) To forecast the daily pricethe first 745 (approximately two-thirds) pieces of data areused as the training data and the last 370 (approximately one-third) pieces of data are used as the test data To forecast thefive-minute high-frequency price the first 40883 (approxi-mately two-thirds) data points are used as the training dataand the last 20442 (approximately one-third) data points areused as the test data

As our analysis of the price fluctuations of the Chinesestock index futures uses the CEEMDmethod the fluctuationcharacteristics in different time scales have their uniqueeconomic implications we suggest that there may be betterpredictions of Chinese stock index futures prices when thesefluctuation characteristics are considered in the predictionmodel Hence based on the literature [16 17] a combinationprediction model that is the CEEMD-SVM-SVM model isproposed and the PSO algorithm is used to optimize theparameters of the SVM model In our empirical researchwe perform a one-step-ahead prediction of Chinese stockindex futures prices The specific steps are as follows (1) theCEEMD method is used to decompose the Chinese stockindex futures price series into several IMFs and a residueterm (2) the IMFs are then composed into a low-frequencyterm and a high-frequency term based on a fine-to-coarsereconstruction and the residue term remains the same (3)SVM models are established separately for each of the threecomponents obtained in step (2) and the PSO algorithm isused to optimize the parameters and (4) the predictive valueof the three components obtained in step (3) is used as theinput with the real value at the same moment as the outputand then a SVM combination model is built to obtain thefinal predictive value The overall process of the CEEMD-SVM-SVMmodel is shown in Figure 5

The prediction evaluation indexes are the root meansquare error (RMSE) mean absolute prediction error(MAPE) and directional symmetry (DS) the DS predictionevaluation index is used to indicate the modelrsquos ability topredict the directionality trend of Chinese stock index futuresprices The prediction evaluation indexes are expressed asfollows

RMSE = radic1

119899

119899

sum119905=1

[119909 (119905) minus (119905)]2

MAPE =1

119899

119899

sum119905=1

10038161003816100381610038161003816100381610038161003816

119909 (119905) minus (119905)

119909 (119905)

10038161003816100381610038161003816100381610038161003816

DS =1

119899

119899

sum119894=1

119889119894

10 Mathematical Problems in Engineering

Original series

CEEMD

Residue term Low-frequency term High-frequency term

SVM1 SVM2 SVM3

Final predictive value

SVM combination

Figure 5 The overall process of the CEEMD-SVM-SVMmodel

119889119894=

1 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] ge 0

0 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] lt 0

(10)

Among the three evaluation indexes smaller RMSE andMAPE values indicate that the predictionmodel is experienc-ing better performance which is also true of greater DS valueand better performance of the prediction model

The prediction accuracy based on the SVM model issensitive to the SVM modelrsquos kernel function As manyprevious studies (such as [17 22]) indicate that the linearkernel function and the RBF kernel function are appropriatekernel functions for the prediction of SVMmodel the linearkernel function and the RBF kernel function of the SVMare thus both applied to our prediction of Chinese stockindex futures prices Based on the process of the CEEMD-SVM-SVM model we first predict the residue term thenthe low-frequency term and then the high-frequency termof the daily price series and the five-minute price seriesNotably the five-minute price series of Chinese stock indexfutures is decomposed into fourteen IMFs and one residueterm where IMFs from IMF1 to IMF9 are recomposed intothe high-frequency term and IMFs from IMF10 to IMF14are recomposed into the low-frequency termThe predictionerrors of these components are shown in Table 5

As shown in Table 5 in using the SVM model to makepredictions on each component the prediction precision ofthe linear kernel function SVM model is generally higherthan the RBF kernel function SVM model In additionwhether for predicting the daily price series or the five-minute price series the prediction precisions of the residueterm and the low-frequency term are significantly higherthan those of the high-frequency termThis difference occursbecause these effects of shocks on the residue term andthe low-frequency term are fundamental economically theyare significant events that have an impact on Chinese stockindex futures prices that is sustained over a long duration

Therefore the fluctuations of the residue term and the low-frequency term are relatively steady and easier to forecastusing the SVM model By contrast the high-frequency termrepresenting the short-term disequilibrium of Chinese stockindex futures price is affected mainly by daily market activ-ities such as speculation arbitrage and the fluctuation ofthe high-frequency term that is random and changes rapidlyHence the SVMmodel experiences difficulty in learning thepattern of high-frequency term fluctuation resulting in poorpredictive results

Then according to prediction step (4) we use the optimalpredictive value of each component obtained in the previousstep to perform a combination prediction and the predictionerrors of our proposed model and the single SVM modelbased on the same data sample are simultaneously comparedThese results of the prediction errors are listed in Table 6

Table 6 shows that as opposed to the predictive resultsof the three components the RBF kernel function SVMmodel has higher predicted precision than the linear kernelfunction SVM model when performing combination pre-diction which indicates that the RBF kernel function SVMmodel is more appropriate for combination prediction Inaddition we find that the predictive results of the finalpredictive value obtained under the RBF kernel function aresignificantly better than those obtained in the single SVMmodel when predicting the daily price but slightly better thanthe single SVM model when making predictions regardingthe high-frequency price Meanwhile we should considerthat the high-frequency price embodies far more complexinformation than the daily price during the same time spanwhich may restrict the learning ability and generalizationability of the SVM model Hence the CEEMD-SVM-SVMmodel performs better in terms of predicting the daily priceIn any event the CEEMD-SVM-SVM model proposed bythis paper performs better than the single SVM model interms of both forecasting the daily price series and the five-minute price series Thus the introduction of Chinese stockindex futures price characteristics in different time scales into

Mathematical Problems in Engineering 11

Table 5 The prediction errors of different components

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

Residue termRMSE 0007 0008 0006 0009MAPE 0001 0001 0001 0002DS 69837 58424 21678 27241

Low-frequency termRMSE 0007 0008 0004 0008MAPE 0087 0092 0042 0258DS 60326 55434 54613 55254

High-frequency termRMSE 0086 0098 0020 0021MAPE 4806 4725 1146 1149DS 63315 62500 51463 51854

Table 6 Comparison of different modelsrsquo prediction errors

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

SVMRMSE 0053 0054 0008 0008MAPE 0009 0009 0001 0001DS 54891 52445 48171 48161

CEEMD-SVM-SVMRMSE 0051 0009 0008 0007MAPE 0008 0002 0001 0001DS 62228 88043 51565 52715

the SVM model improves the ability of the SVM model tomine the variations in the stock index futures price seriesfurther enhancing its prediction precision Simultaneouslysome useful suggestions for investors can be found in thepredictions of Chinese stock index futures prices generatedby our proposedmodel First investors should be accustomedto analyzing the price series from different time scales tounderstand the fluctuation characteristics of Chinese stockindex futures prices Second investors must actively identifywhat time scale the effects of external shocks on stock indexfutures prices belong to andmust devise the appropriate strat-egy accordingly Finally investors should pay more attentionto a hedging strategy involving Chinese stock index futuresover the long andor medium term and should be cautiousabout short-term speculation

Figures 6 and 7 present the real and predictive values ofthe daily price series andfive-minute price series respectivelyThe final predictive values obtained using the CEEMD-SVM-SVM model appeared to coincide with the real values indi-cating that the predictive values obtained using this modelfit well with the real values and intuitively demonstratingthat the CEEMD-SVM-SVMmodel is suitable for forecastingChinese stock index futures prices

20135 20138 201311 20142 20145 20148 2014112000

2200

2400

2600

2800

Real valuePredictive value

Figure 6 Real value and predictive value of the daily price series

5 Conclusion

The Chinese stock index futures market is a complex systemand fluctuations in the stock index futures prices resultfrom the collective effects of many factors at various levelsThe CEEMD method a method suitable in the analysis ofnonlinear nonstationary data is applied to analyze Chinesestock index futures prices in different time scales and to

12 Mathematical Problems in Engineering

Real valuePredictive value

20135 20138 201311 20142 20145 20148 2014111800

2000

2200

2400

2600

2800

Figure 7 Real value and predictive value of the five-minute priceseries

reveal the intrinsic fluctuation characteristics of Chinesestock index futures prices Then we focus on whether thesecharacteristics improve the predictive precision of Chinesestock index futures prices and propose a CEEMD-SVM-SVMmodel based on the PSO algorithm The empirical resultsdemonstrate that the intrinsic fluctuation characteristics ofChinese stock index futures prices can be revealed in threecomponents including the residue term the low-frequencyterm and the high-frequency term with independent eco-nomic significance The residue term determines the long-term trend of stock index futures prices and its fluctuationsare correlated with macroeconomics and the long-termexpectations of investorsThe low-frequency term reveals theeffect of significant events on medium-term fluctuations instock index futures prices the ICSS algorithm breakpointtests reveal that policy regulation events are the main causeof fluctuations during the low-frequency term The high-frequency term reflects short-term disequilibrium in thestock index futures market although the effect of the high-frequency term on the overall fluctuation in the stock indexfutures prices is limited it nonetheless plays an important rolelocally

In addition in forecasting Chinese stock index futuresprices first we find that the linear kernel function SVMmodel is more suitable for predicting these components thanthe RBF kernel function SVM model The residue term andthe low-frequency term have stronger regular fluctuationsthan the high-frequency term and therefore yield superiorpredictive results Second the RBF kernel function SVMmodel shows higher predictive precision for the final combi-nation prediction than the linear kernel function SVMmodelSimultaneously the predictive results of the combinationprediction model are better than those of a single SVMmodel which indicates that the integration of fluctuationcharacteristics in different time scales increases the abilityof the SVM model to identify regularity in Chinese stockindex futures price fluctuations thus improving predictionprecision Finally the predictive results of the CEEMD-SVM-SVM model on daily prices and intraday prices are basicallyconsistent which indicates that our proposed model hasstability and universal applicability in forecasting Chinesestock index futures prices

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by the National SocialScience Foundation of China (14ZDA045)

References

[1] VMeneu andH Torro ldquoAsymmetric covariance in spot-futuresmarketsrdquo Journal of Futures Markets vol 23 no 11 pp 1019ndash1046 2003

[2] M Zhong A F Darrat and R Otero ldquoPrice discovery andvolatility spillovers in index futures markets some evidencefrom Mexicordquo Journal of Banking and Finance vol 28 no 12pp 3037ndash3054 2004

[3] X Xiong F Wang W Zhang and Y Sun ldquoPrice discoveryand volatility spillovers between SGX FTSEXinhua ChinaA50 index futures and A-share marketrdquo Chinese Journal ofManagement vol 11 no 1535 pp 1507ndash1512 2009

[4] H M Zuo Z T Liu and H W Zeng ldquoResearch of volatilityspillover and information transmission between stock indexfutures and spot market based on the high frequency datardquoJournal of Financial Research vol 4 pp 140ndash154 2012

[5] H Q Chen and C H Zhang ldquoDoes index futures tradingreduce stock market jump riskrdquo Economic Research Journalvol 1 pp 153ndash167 2015

[6] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 48 no 3 pp235ndash254 2015

[7] E E Peters Fractal Market Analysis Applying Chaos Theory toInvestment and Economics John Wiley amp Sons New York NYUSA 1994

[8] X Zhang K K Lai and S-YWang ldquoA new approach for crudeoil price analysis based on empirical mode decompositionrdquoEnergy Economics vol 30 no 3 pp 905ndash918 2008

[9] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnonstationary time series analysisrdquo Proceedings of the RoyalSociety of London A vol 454 no 1971 pp 903ndash995 1998

[10] Z Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[11] L F Ruan and H J Bao ldquoAn empirical analysis on periodicfluctuations of real estate price based on EMDrdquo Chinese Journalof Management Science no 3 pp 41ndash46 2012

[12] H L Li and C E Feng ldquoRelationship between investor senti-ment and stock indices fluctuation based on EEMDrdquo SystemsEngineering-Theory amp Practice vol 10 pp 2495ndash2503 2014

[13] J-R Yeh J-S Shieh andN EHuang ldquoComplementary ensem-ble empiricalmode decomposition a novel noise enhanced dataanalysismethodrdquoAdvances in Adaptive DataAnalysis vol 2 no2 pp 135ndash156 2010

[14] L Zhao W Yu and R Yan ldquoRolling bearing fault diagnosisbased on CEEMD and time series modelingrdquo MathematicalProblems in Engineering vol 2014 Article ID 101867 13 pages2014

Mathematical Problems in Engineering 13

[15] L Tang W Dai L Yu and S Wang ldquoA novel CEEMD-based eelm ensemble learning paradigm for crude oil priceforecastingrdquo International Journal of Information Technologyand Decision Making vol 14 no 1 pp 141ndash169 2015

[16] L Yu S Wang and K K Lai ldquoForecasting crude oil price withan EMD-based neural network ensemble learning paradigmrdquoEnergy Economics vol 30 no 5 pp 2623ndash2635 2008

[17] Y F Yang Y K Bao Z Y Hu and R Zhang ldquoCrude oilprice prediction based on empirical mode decomposition andsupport vector machinesrdquo Chinese Journal of Management vol12 pp 1884ndash1889 2010

[18] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[19] W Huang Y Nakamori and S-Y Wang ldquoForecasting stockmarket movement direction with support vector machinerdquoComputers and Operations Research vol 32 no 10 pp 2513ndash2522 2005

[20] H Ince and T B Trafalis ldquoShort term forecasting with supportvector machines and application to stock price predictionrdquoInternational Journal of General Systems vol 37 no 6 pp 677ndash687 2008

[21] F E H Tay and L Cao ldquoApplication of support vectormachinesin financial time series forecastingrdquo Omega vol 29 no 4 pp309ndash317 2001

[22] Y Sai F T Zhang and T Zhang ldquoResearch of Chinese stockindex futures regression prediction based on support vectormachinesrdquo Chinese Journal of Management Science vol 3 pp35ndash39 2013

[23] C Inclan andG C Tiao ldquoUse of cumulative sums of squares forretrospective detection of changes of variancerdquo Journal of theAmerican Statistical Association vol 89 no 427 pp 913ndash9231994

[24] S H Kang C Cheong and S-M Yoon ldquoStructural changesand volatility transmission in crude oil marketsrdquo Physica AStatisticalMechanics and Its Applications vol 390 no 23-24 pp4317ndash4324 2011

[25] J P Yang X X Chen and C H Zhang ldquoPolicy impact factorson the volatility of Shanghai and Shenzhen stock market inChinardquo Chinese Journal of Management Science vol 20 no 6pp 43ndash51 2012

[26] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer New York NY USA2010

[27] C-L Huang and J-F Dun ldquoA distributed PSOndashSVM hybridsystem with feature selection and parameter optimizationrdquoApplied Soft Computing vol 8 no 4 pp 1381ndash1391 2008

[28] C Zhou K Yin Y Cao and B Ahmed ldquoApplication of timeseries analysis and PSOndashSVMmodel in predicting the Bazimenlandslide in the Three Gorges Reservoir Chinardquo EngineeringGeology vol 204 pp 108ndash120 2016

[29] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis amp Applications vol 16 no 1 pp 93ndash103 2014

[30] X Gong Z He P Li and N Zhu ldquoForecasting return volatilityof the CSI 300 index using the stochastic volatility model withcontinuous volatility and jumpsrdquo Discrete Dynamics in Natureand Society vol 2014 Article ID 964654 10 pages 2014

[31] S Q Guo and P H Liang ldquoSocial Interactions the informationchannel and household stock market participation a empiricalstudy based on a Chinese household finance surveyrdquo EconomicResearch Journal vol S1 pp 116ndash131 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

20104 20114 20124 20134 20144

0

1000

2000

3000

4000

Original seriesResidue term

Low-frequency termHigh-frequency term

minus1000

Figure 4 Chinese stock index futures prices and the three compo-nents

Chinese stock index futures market further standardizes andthe ability to withstand external impact thus also increasesTherefore the number of breakpoints decreases after 2014

The high-frequency term mainly reflects the short-termdisequilibrium in the Chinese stock index futures marketAs shown in Table 3 the correlation of the high-frequencyterm with the Chinese stock index futures prices is merely0196 and the ability to explain fluctuations in the originalprice series is only 4 indicating that the overall effectof the high-frequency term on Chinese stock index futuresprices is limited However as Figure 4 shows the fluctuationsof the high-frequency term fit well with the local fluctu-ations of the Chinese stock index futures prices in otherwords the high-frequency term plays a primary role in localchanges in Chinese stock index futures prices In additionas opposed to the low-frequency term the high-frequencyterm fluctuates frequently but with smaller amplitudes Thisdifference occurs primarily because the high-frequency termis particularly affected by investorsrsquo speculative arbitrageactivities In particular implementation of a T+0 transactionsystem in the Chinese stock index futures market means thatinvestors buying and selling stock index futures can completetheir transactions in the same day which suits short-termspeculators Based on this transaction system there will bemore investment chances for investors which will enhanceinvestorsrsquo speculative opportunities Hence there are manyshort-term fluctuations in Chinese stock index futures pricesMeanwhile it is possible and convenient for investors toaccess the latest hot investment tip because the role of socialinteractions is partly substituted for by online information inthe Internet age [31] In reality this type of new informationchanges continuously and investors must frequently adjusttheir investment strategies to gainmore additional benefits byspeculating in the spot-futures marketsTherefore the strongeffect of the shock of investorsrsquo speculation on Chinese stockindex futures prices in the short-term is difficult to sustainAlthough fluctuations in the high-frequency term are notimportant to forecast the long-term trends of Chinese stockindex futures their predictive effect of minor fluctuations onstock index futures cannot be ignored

4 Predictions for Chinese StockIndex Futures Prices

In this section this paper further investigates the predictionsof the Chinese stock index futures prices In addition tothe previous dayrsquos price data we also select the five-minutehigh-frequency price data of Chinese stock index futuresfrom April 16 2010 to November 20 2014 as a researchsubject which yields a total of 61325 data points (the data areobtained from theWind database) To forecast the daily pricethe first 745 (approximately two-thirds) pieces of data areused as the training data and the last 370 (approximately one-third) pieces of data are used as the test data To forecast thefive-minute high-frequency price the first 40883 (approxi-mately two-thirds) data points are used as the training dataand the last 20442 (approximately one-third) data points areused as the test data

As our analysis of the price fluctuations of the Chinesestock index futures uses the CEEMDmethod the fluctuationcharacteristics in different time scales have their uniqueeconomic implications we suggest that there may be betterpredictions of Chinese stock index futures prices when thesefluctuation characteristics are considered in the predictionmodel Hence based on the literature [16 17] a combinationprediction model that is the CEEMD-SVM-SVM model isproposed and the PSO algorithm is used to optimize theparameters of the SVM model In our empirical researchwe perform a one-step-ahead prediction of Chinese stockindex futures prices The specific steps are as follows (1) theCEEMD method is used to decompose the Chinese stockindex futures price series into several IMFs and a residueterm (2) the IMFs are then composed into a low-frequencyterm and a high-frequency term based on a fine-to-coarsereconstruction and the residue term remains the same (3)SVM models are established separately for each of the threecomponents obtained in step (2) and the PSO algorithm isused to optimize the parameters and (4) the predictive valueof the three components obtained in step (3) is used as theinput with the real value at the same moment as the outputand then a SVM combination model is built to obtain thefinal predictive value The overall process of the CEEMD-SVM-SVMmodel is shown in Figure 5

The prediction evaluation indexes are the root meansquare error (RMSE) mean absolute prediction error(MAPE) and directional symmetry (DS) the DS predictionevaluation index is used to indicate the modelrsquos ability topredict the directionality trend of Chinese stock index futuresprices The prediction evaluation indexes are expressed asfollows

RMSE = radic1

119899

119899

sum119905=1

[119909 (119905) minus (119905)]2

MAPE =1

119899

119899

sum119905=1

10038161003816100381610038161003816100381610038161003816

119909 (119905) minus (119905)

119909 (119905)

10038161003816100381610038161003816100381610038161003816

DS =1

119899

119899

sum119894=1

119889119894

10 Mathematical Problems in Engineering

Original series

CEEMD

Residue term Low-frequency term High-frequency term

SVM1 SVM2 SVM3

Final predictive value

SVM combination

Figure 5 The overall process of the CEEMD-SVM-SVMmodel

119889119894=

1 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] ge 0

0 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] lt 0

(10)

Among the three evaluation indexes smaller RMSE andMAPE values indicate that the predictionmodel is experienc-ing better performance which is also true of greater DS valueand better performance of the prediction model

The prediction accuracy based on the SVM model issensitive to the SVM modelrsquos kernel function As manyprevious studies (such as [17 22]) indicate that the linearkernel function and the RBF kernel function are appropriatekernel functions for the prediction of SVMmodel the linearkernel function and the RBF kernel function of the SVMare thus both applied to our prediction of Chinese stockindex futures prices Based on the process of the CEEMD-SVM-SVM model we first predict the residue term thenthe low-frequency term and then the high-frequency termof the daily price series and the five-minute price seriesNotably the five-minute price series of Chinese stock indexfutures is decomposed into fourteen IMFs and one residueterm where IMFs from IMF1 to IMF9 are recomposed intothe high-frequency term and IMFs from IMF10 to IMF14are recomposed into the low-frequency termThe predictionerrors of these components are shown in Table 5

As shown in Table 5 in using the SVM model to makepredictions on each component the prediction precision ofthe linear kernel function SVM model is generally higherthan the RBF kernel function SVM model In additionwhether for predicting the daily price series or the five-minute price series the prediction precisions of the residueterm and the low-frequency term are significantly higherthan those of the high-frequency termThis difference occursbecause these effects of shocks on the residue term andthe low-frequency term are fundamental economically theyare significant events that have an impact on Chinese stockindex futures prices that is sustained over a long duration

Therefore the fluctuations of the residue term and the low-frequency term are relatively steady and easier to forecastusing the SVM model By contrast the high-frequency termrepresenting the short-term disequilibrium of Chinese stockindex futures price is affected mainly by daily market activ-ities such as speculation arbitrage and the fluctuation ofthe high-frequency term that is random and changes rapidlyHence the SVMmodel experiences difficulty in learning thepattern of high-frequency term fluctuation resulting in poorpredictive results

Then according to prediction step (4) we use the optimalpredictive value of each component obtained in the previousstep to perform a combination prediction and the predictionerrors of our proposed model and the single SVM modelbased on the same data sample are simultaneously comparedThese results of the prediction errors are listed in Table 6

Table 6 shows that as opposed to the predictive resultsof the three components the RBF kernel function SVMmodel has higher predicted precision than the linear kernelfunction SVM model when performing combination pre-diction which indicates that the RBF kernel function SVMmodel is more appropriate for combination prediction Inaddition we find that the predictive results of the finalpredictive value obtained under the RBF kernel function aresignificantly better than those obtained in the single SVMmodel when predicting the daily price but slightly better thanthe single SVM model when making predictions regardingthe high-frequency price Meanwhile we should considerthat the high-frequency price embodies far more complexinformation than the daily price during the same time spanwhich may restrict the learning ability and generalizationability of the SVM model Hence the CEEMD-SVM-SVMmodel performs better in terms of predicting the daily priceIn any event the CEEMD-SVM-SVM model proposed bythis paper performs better than the single SVM model interms of both forecasting the daily price series and the five-minute price series Thus the introduction of Chinese stockindex futures price characteristics in different time scales into

Mathematical Problems in Engineering 11

Table 5 The prediction errors of different components

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

Residue termRMSE 0007 0008 0006 0009MAPE 0001 0001 0001 0002DS 69837 58424 21678 27241

Low-frequency termRMSE 0007 0008 0004 0008MAPE 0087 0092 0042 0258DS 60326 55434 54613 55254

High-frequency termRMSE 0086 0098 0020 0021MAPE 4806 4725 1146 1149DS 63315 62500 51463 51854

Table 6 Comparison of different modelsrsquo prediction errors

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

SVMRMSE 0053 0054 0008 0008MAPE 0009 0009 0001 0001DS 54891 52445 48171 48161

CEEMD-SVM-SVMRMSE 0051 0009 0008 0007MAPE 0008 0002 0001 0001DS 62228 88043 51565 52715

the SVM model improves the ability of the SVM model tomine the variations in the stock index futures price seriesfurther enhancing its prediction precision Simultaneouslysome useful suggestions for investors can be found in thepredictions of Chinese stock index futures prices generatedby our proposedmodel First investors should be accustomedto analyzing the price series from different time scales tounderstand the fluctuation characteristics of Chinese stockindex futures prices Second investors must actively identifywhat time scale the effects of external shocks on stock indexfutures prices belong to andmust devise the appropriate strat-egy accordingly Finally investors should pay more attentionto a hedging strategy involving Chinese stock index futuresover the long andor medium term and should be cautiousabout short-term speculation

Figures 6 and 7 present the real and predictive values ofthe daily price series andfive-minute price series respectivelyThe final predictive values obtained using the CEEMD-SVM-SVM model appeared to coincide with the real values indi-cating that the predictive values obtained using this modelfit well with the real values and intuitively demonstratingthat the CEEMD-SVM-SVMmodel is suitable for forecastingChinese stock index futures prices

20135 20138 201311 20142 20145 20148 2014112000

2200

2400

2600

2800

Real valuePredictive value

Figure 6 Real value and predictive value of the daily price series

5 Conclusion

The Chinese stock index futures market is a complex systemand fluctuations in the stock index futures prices resultfrom the collective effects of many factors at various levelsThe CEEMD method a method suitable in the analysis ofnonlinear nonstationary data is applied to analyze Chinesestock index futures prices in different time scales and to

12 Mathematical Problems in Engineering

Real valuePredictive value

20135 20138 201311 20142 20145 20148 2014111800

2000

2200

2400

2600

2800

Figure 7 Real value and predictive value of the five-minute priceseries

reveal the intrinsic fluctuation characteristics of Chinesestock index futures prices Then we focus on whether thesecharacteristics improve the predictive precision of Chinesestock index futures prices and propose a CEEMD-SVM-SVMmodel based on the PSO algorithm The empirical resultsdemonstrate that the intrinsic fluctuation characteristics ofChinese stock index futures prices can be revealed in threecomponents including the residue term the low-frequencyterm and the high-frequency term with independent eco-nomic significance The residue term determines the long-term trend of stock index futures prices and its fluctuationsare correlated with macroeconomics and the long-termexpectations of investorsThe low-frequency term reveals theeffect of significant events on medium-term fluctuations instock index futures prices the ICSS algorithm breakpointtests reveal that policy regulation events are the main causeof fluctuations during the low-frequency term The high-frequency term reflects short-term disequilibrium in thestock index futures market although the effect of the high-frequency term on the overall fluctuation in the stock indexfutures prices is limited it nonetheless plays an important rolelocally

In addition in forecasting Chinese stock index futuresprices first we find that the linear kernel function SVMmodel is more suitable for predicting these components thanthe RBF kernel function SVM model The residue term andthe low-frequency term have stronger regular fluctuationsthan the high-frequency term and therefore yield superiorpredictive results Second the RBF kernel function SVMmodel shows higher predictive precision for the final combi-nation prediction than the linear kernel function SVMmodelSimultaneously the predictive results of the combinationprediction model are better than those of a single SVMmodel which indicates that the integration of fluctuationcharacteristics in different time scales increases the abilityof the SVM model to identify regularity in Chinese stockindex futures price fluctuations thus improving predictionprecision Finally the predictive results of the CEEMD-SVM-SVM model on daily prices and intraday prices are basicallyconsistent which indicates that our proposed model hasstability and universal applicability in forecasting Chinesestock index futures prices

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by the National SocialScience Foundation of China (14ZDA045)

References

[1] VMeneu andH Torro ldquoAsymmetric covariance in spot-futuresmarketsrdquo Journal of Futures Markets vol 23 no 11 pp 1019ndash1046 2003

[2] M Zhong A F Darrat and R Otero ldquoPrice discovery andvolatility spillovers in index futures markets some evidencefrom Mexicordquo Journal of Banking and Finance vol 28 no 12pp 3037ndash3054 2004

[3] X Xiong F Wang W Zhang and Y Sun ldquoPrice discoveryand volatility spillovers between SGX FTSEXinhua ChinaA50 index futures and A-share marketrdquo Chinese Journal ofManagement vol 11 no 1535 pp 1507ndash1512 2009

[4] H M Zuo Z T Liu and H W Zeng ldquoResearch of volatilityspillover and information transmission between stock indexfutures and spot market based on the high frequency datardquoJournal of Financial Research vol 4 pp 140ndash154 2012

[5] H Q Chen and C H Zhang ldquoDoes index futures tradingreduce stock market jump riskrdquo Economic Research Journalvol 1 pp 153ndash167 2015

[6] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 48 no 3 pp235ndash254 2015

[7] E E Peters Fractal Market Analysis Applying Chaos Theory toInvestment and Economics John Wiley amp Sons New York NYUSA 1994

[8] X Zhang K K Lai and S-YWang ldquoA new approach for crudeoil price analysis based on empirical mode decompositionrdquoEnergy Economics vol 30 no 3 pp 905ndash918 2008

[9] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnonstationary time series analysisrdquo Proceedings of the RoyalSociety of London A vol 454 no 1971 pp 903ndash995 1998

[10] Z Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[11] L F Ruan and H J Bao ldquoAn empirical analysis on periodicfluctuations of real estate price based on EMDrdquo Chinese Journalof Management Science no 3 pp 41ndash46 2012

[12] H L Li and C E Feng ldquoRelationship between investor senti-ment and stock indices fluctuation based on EEMDrdquo SystemsEngineering-Theory amp Practice vol 10 pp 2495ndash2503 2014

[13] J-R Yeh J-S Shieh andN EHuang ldquoComplementary ensem-ble empiricalmode decomposition a novel noise enhanced dataanalysismethodrdquoAdvances in Adaptive DataAnalysis vol 2 no2 pp 135ndash156 2010

[14] L Zhao W Yu and R Yan ldquoRolling bearing fault diagnosisbased on CEEMD and time series modelingrdquo MathematicalProblems in Engineering vol 2014 Article ID 101867 13 pages2014

Mathematical Problems in Engineering 13

[15] L Tang W Dai L Yu and S Wang ldquoA novel CEEMD-based eelm ensemble learning paradigm for crude oil priceforecastingrdquo International Journal of Information Technologyand Decision Making vol 14 no 1 pp 141ndash169 2015

[16] L Yu S Wang and K K Lai ldquoForecasting crude oil price withan EMD-based neural network ensemble learning paradigmrdquoEnergy Economics vol 30 no 5 pp 2623ndash2635 2008

[17] Y F Yang Y K Bao Z Y Hu and R Zhang ldquoCrude oilprice prediction based on empirical mode decomposition andsupport vector machinesrdquo Chinese Journal of Management vol12 pp 1884ndash1889 2010

[18] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[19] W Huang Y Nakamori and S-Y Wang ldquoForecasting stockmarket movement direction with support vector machinerdquoComputers and Operations Research vol 32 no 10 pp 2513ndash2522 2005

[20] H Ince and T B Trafalis ldquoShort term forecasting with supportvector machines and application to stock price predictionrdquoInternational Journal of General Systems vol 37 no 6 pp 677ndash687 2008

[21] F E H Tay and L Cao ldquoApplication of support vectormachinesin financial time series forecastingrdquo Omega vol 29 no 4 pp309ndash317 2001

[22] Y Sai F T Zhang and T Zhang ldquoResearch of Chinese stockindex futures regression prediction based on support vectormachinesrdquo Chinese Journal of Management Science vol 3 pp35ndash39 2013

[23] C Inclan andG C Tiao ldquoUse of cumulative sums of squares forretrospective detection of changes of variancerdquo Journal of theAmerican Statistical Association vol 89 no 427 pp 913ndash9231994

[24] S H Kang C Cheong and S-M Yoon ldquoStructural changesand volatility transmission in crude oil marketsrdquo Physica AStatisticalMechanics and Its Applications vol 390 no 23-24 pp4317ndash4324 2011

[25] J P Yang X X Chen and C H Zhang ldquoPolicy impact factorson the volatility of Shanghai and Shenzhen stock market inChinardquo Chinese Journal of Management Science vol 20 no 6pp 43ndash51 2012

[26] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer New York NY USA2010

[27] C-L Huang and J-F Dun ldquoA distributed PSOndashSVM hybridsystem with feature selection and parameter optimizationrdquoApplied Soft Computing vol 8 no 4 pp 1381ndash1391 2008

[28] C Zhou K Yin Y Cao and B Ahmed ldquoApplication of timeseries analysis and PSOndashSVMmodel in predicting the Bazimenlandslide in the Three Gorges Reservoir Chinardquo EngineeringGeology vol 204 pp 108ndash120 2016

[29] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis amp Applications vol 16 no 1 pp 93ndash103 2014

[30] X Gong Z He P Li and N Zhu ldquoForecasting return volatilityof the CSI 300 index using the stochastic volatility model withcontinuous volatility and jumpsrdquo Discrete Dynamics in Natureand Society vol 2014 Article ID 964654 10 pages 2014

[31] S Q Guo and P H Liang ldquoSocial Interactions the informationchannel and household stock market participation a empiricalstudy based on a Chinese household finance surveyrdquo EconomicResearch Journal vol S1 pp 116ndash131 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Original series

CEEMD

Residue term Low-frequency term High-frequency term

SVM1 SVM2 SVM3

Final predictive value

SVM combination

Figure 5 The overall process of the CEEMD-SVM-SVMmodel

119889119894=

1 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] ge 0

0 [119909 (119905) minus 119909 (119905 minus 1)] lowast [ (119905) minus 119909 (119905 minus 1)] lt 0

(10)

Among the three evaluation indexes smaller RMSE andMAPE values indicate that the predictionmodel is experienc-ing better performance which is also true of greater DS valueand better performance of the prediction model

The prediction accuracy based on the SVM model issensitive to the SVM modelrsquos kernel function As manyprevious studies (such as [17 22]) indicate that the linearkernel function and the RBF kernel function are appropriatekernel functions for the prediction of SVMmodel the linearkernel function and the RBF kernel function of the SVMare thus both applied to our prediction of Chinese stockindex futures prices Based on the process of the CEEMD-SVM-SVM model we first predict the residue term thenthe low-frequency term and then the high-frequency termof the daily price series and the five-minute price seriesNotably the five-minute price series of Chinese stock indexfutures is decomposed into fourteen IMFs and one residueterm where IMFs from IMF1 to IMF9 are recomposed intothe high-frequency term and IMFs from IMF10 to IMF14are recomposed into the low-frequency termThe predictionerrors of these components are shown in Table 5

As shown in Table 5 in using the SVM model to makepredictions on each component the prediction precision ofthe linear kernel function SVM model is generally higherthan the RBF kernel function SVM model In additionwhether for predicting the daily price series or the five-minute price series the prediction precisions of the residueterm and the low-frequency term are significantly higherthan those of the high-frequency termThis difference occursbecause these effects of shocks on the residue term andthe low-frequency term are fundamental economically theyare significant events that have an impact on Chinese stockindex futures prices that is sustained over a long duration

Therefore the fluctuations of the residue term and the low-frequency term are relatively steady and easier to forecastusing the SVM model By contrast the high-frequency termrepresenting the short-term disequilibrium of Chinese stockindex futures price is affected mainly by daily market activ-ities such as speculation arbitrage and the fluctuation ofthe high-frequency term that is random and changes rapidlyHence the SVMmodel experiences difficulty in learning thepattern of high-frequency term fluctuation resulting in poorpredictive results

Then according to prediction step (4) we use the optimalpredictive value of each component obtained in the previousstep to perform a combination prediction and the predictionerrors of our proposed model and the single SVM modelbased on the same data sample are simultaneously comparedThese results of the prediction errors are listed in Table 6

Table 6 shows that as opposed to the predictive resultsof the three components the RBF kernel function SVMmodel has higher predicted precision than the linear kernelfunction SVM model when performing combination pre-diction which indicates that the RBF kernel function SVMmodel is more appropriate for combination prediction Inaddition we find that the predictive results of the finalpredictive value obtained under the RBF kernel function aresignificantly better than those obtained in the single SVMmodel when predicting the daily price but slightly better thanthe single SVM model when making predictions regardingthe high-frequency price Meanwhile we should considerthat the high-frequency price embodies far more complexinformation than the daily price during the same time spanwhich may restrict the learning ability and generalizationability of the SVM model Hence the CEEMD-SVM-SVMmodel performs better in terms of predicting the daily priceIn any event the CEEMD-SVM-SVM model proposed bythis paper performs better than the single SVM model interms of both forecasting the daily price series and the five-minute price series Thus the introduction of Chinese stockindex futures price characteristics in different time scales into

Mathematical Problems in Engineering 11

Table 5 The prediction errors of different components

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

Residue termRMSE 0007 0008 0006 0009MAPE 0001 0001 0001 0002DS 69837 58424 21678 27241

Low-frequency termRMSE 0007 0008 0004 0008MAPE 0087 0092 0042 0258DS 60326 55434 54613 55254

High-frequency termRMSE 0086 0098 0020 0021MAPE 4806 4725 1146 1149DS 63315 62500 51463 51854

Table 6 Comparison of different modelsrsquo prediction errors

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

SVMRMSE 0053 0054 0008 0008MAPE 0009 0009 0001 0001DS 54891 52445 48171 48161

CEEMD-SVM-SVMRMSE 0051 0009 0008 0007MAPE 0008 0002 0001 0001DS 62228 88043 51565 52715

the SVM model improves the ability of the SVM model tomine the variations in the stock index futures price seriesfurther enhancing its prediction precision Simultaneouslysome useful suggestions for investors can be found in thepredictions of Chinese stock index futures prices generatedby our proposedmodel First investors should be accustomedto analyzing the price series from different time scales tounderstand the fluctuation characteristics of Chinese stockindex futures prices Second investors must actively identifywhat time scale the effects of external shocks on stock indexfutures prices belong to andmust devise the appropriate strat-egy accordingly Finally investors should pay more attentionto a hedging strategy involving Chinese stock index futuresover the long andor medium term and should be cautiousabout short-term speculation

Figures 6 and 7 present the real and predictive values ofthe daily price series andfive-minute price series respectivelyThe final predictive values obtained using the CEEMD-SVM-SVM model appeared to coincide with the real values indi-cating that the predictive values obtained using this modelfit well with the real values and intuitively demonstratingthat the CEEMD-SVM-SVMmodel is suitable for forecastingChinese stock index futures prices

20135 20138 201311 20142 20145 20148 2014112000

2200

2400

2600

2800

Real valuePredictive value

Figure 6 Real value and predictive value of the daily price series

5 Conclusion

The Chinese stock index futures market is a complex systemand fluctuations in the stock index futures prices resultfrom the collective effects of many factors at various levelsThe CEEMD method a method suitable in the analysis ofnonlinear nonstationary data is applied to analyze Chinesestock index futures prices in different time scales and to

12 Mathematical Problems in Engineering

Real valuePredictive value

20135 20138 201311 20142 20145 20148 2014111800

2000

2200

2400

2600

2800

Figure 7 Real value and predictive value of the five-minute priceseries

reveal the intrinsic fluctuation characteristics of Chinesestock index futures prices Then we focus on whether thesecharacteristics improve the predictive precision of Chinesestock index futures prices and propose a CEEMD-SVM-SVMmodel based on the PSO algorithm The empirical resultsdemonstrate that the intrinsic fluctuation characteristics ofChinese stock index futures prices can be revealed in threecomponents including the residue term the low-frequencyterm and the high-frequency term with independent eco-nomic significance The residue term determines the long-term trend of stock index futures prices and its fluctuationsare correlated with macroeconomics and the long-termexpectations of investorsThe low-frequency term reveals theeffect of significant events on medium-term fluctuations instock index futures prices the ICSS algorithm breakpointtests reveal that policy regulation events are the main causeof fluctuations during the low-frequency term The high-frequency term reflects short-term disequilibrium in thestock index futures market although the effect of the high-frequency term on the overall fluctuation in the stock indexfutures prices is limited it nonetheless plays an important rolelocally

In addition in forecasting Chinese stock index futuresprices first we find that the linear kernel function SVMmodel is more suitable for predicting these components thanthe RBF kernel function SVM model The residue term andthe low-frequency term have stronger regular fluctuationsthan the high-frequency term and therefore yield superiorpredictive results Second the RBF kernel function SVMmodel shows higher predictive precision for the final combi-nation prediction than the linear kernel function SVMmodelSimultaneously the predictive results of the combinationprediction model are better than those of a single SVMmodel which indicates that the integration of fluctuationcharacteristics in different time scales increases the abilityof the SVM model to identify regularity in Chinese stockindex futures price fluctuations thus improving predictionprecision Finally the predictive results of the CEEMD-SVM-SVM model on daily prices and intraday prices are basicallyconsistent which indicates that our proposed model hasstability and universal applicability in forecasting Chinesestock index futures prices

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by the National SocialScience Foundation of China (14ZDA045)

References

[1] VMeneu andH Torro ldquoAsymmetric covariance in spot-futuresmarketsrdquo Journal of Futures Markets vol 23 no 11 pp 1019ndash1046 2003

[2] M Zhong A F Darrat and R Otero ldquoPrice discovery andvolatility spillovers in index futures markets some evidencefrom Mexicordquo Journal of Banking and Finance vol 28 no 12pp 3037ndash3054 2004

[3] X Xiong F Wang W Zhang and Y Sun ldquoPrice discoveryand volatility spillovers between SGX FTSEXinhua ChinaA50 index futures and A-share marketrdquo Chinese Journal ofManagement vol 11 no 1535 pp 1507ndash1512 2009

[4] H M Zuo Z T Liu and H W Zeng ldquoResearch of volatilityspillover and information transmission between stock indexfutures and spot market based on the high frequency datardquoJournal of Financial Research vol 4 pp 140ndash154 2012

[5] H Q Chen and C H Zhang ldquoDoes index futures tradingreduce stock market jump riskrdquo Economic Research Journalvol 1 pp 153ndash167 2015

[6] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 48 no 3 pp235ndash254 2015

[7] E E Peters Fractal Market Analysis Applying Chaos Theory toInvestment and Economics John Wiley amp Sons New York NYUSA 1994

[8] X Zhang K K Lai and S-YWang ldquoA new approach for crudeoil price analysis based on empirical mode decompositionrdquoEnergy Economics vol 30 no 3 pp 905ndash918 2008

[9] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnonstationary time series analysisrdquo Proceedings of the RoyalSociety of London A vol 454 no 1971 pp 903ndash995 1998

[10] Z Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[11] L F Ruan and H J Bao ldquoAn empirical analysis on periodicfluctuations of real estate price based on EMDrdquo Chinese Journalof Management Science no 3 pp 41ndash46 2012

[12] H L Li and C E Feng ldquoRelationship between investor senti-ment and stock indices fluctuation based on EEMDrdquo SystemsEngineering-Theory amp Practice vol 10 pp 2495ndash2503 2014

[13] J-R Yeh J-S Shieh andN EHuang ldquoComplementary ensem-ble empiricalmode decomposition a novel noise enhanced dataanalysismethodrdquoAdvances in Adaptive DataAnalysis vol 2 no2 pp 135ndash156 2010

[14] L Zhao W Yu and R Yan ldquoRolling bearing fault diagnosisbased on CEEMD and time series modelingrdquo MathematicalProblems in Engineering vol 2014 Article ID 101867 13 pages2014

Mathematical Problems in Engineering 13

[15] L Tang W Dai L Yu and S Wang ldquoA novel CEEMD-based eelm ensemble learning paradigm for crude oil priceforecastingrdquo International Journal of Information Technologyand Decision Making vol 14 no 1 pp 141ndash169 2015

[16] L Yu S Wang and K K Lai ldquoForecasting crude oil price withan EMD-based neural network ensemble learning paradigmrdquoEnergy Economics vol 30 no 5 pp 2623ndash2635 2008

[17] Y F Yang Y K Bao Z Y Hu and R Zhang ldquoCrude oilprice prediction based on empirical mode decomposition andsupport vector machinesrdquo Chinese Journal of Management vol12 pp 1884ndash1889 2010

[18] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[19] W Huang Y Nakamori and S-Y Wang ldquoForecasting stockmarket movement direction with support vector machinerdquoComputers and Operations Research vol 32 no 10 pp 2513ndash2522 2005

[20] H Ince and T B Trafalis ldquoShort term forecasting with supportvector machines and application to stock price predictionrdquoInternational Journal of General Systems vol 37 no 6 pp 677ndash687 2008

[21] F E H Tay and L Cao ldquoApplication of support vectormachinesin financial time series forecastingrdquo Omega vol 29 no 4 pp309ndash317 2001

[22] Y Sai F T Zhang and T Zhang ldquoResearch of Chinese stockindex futures regression prediction based on support vectormachinesrdquo Chinese Journal of Management Science vol 3 pp35ndash39 2013

[23] C Inclan andG C Tiao ldquoUse of cumulative sums of squares forretrospective detection of changes of variancerdquo Journal of theAmerican Statistical Association vol 89 no 427 pp 913ndash9231994

[24] S H Kang C Cheong and S-M Yoon ldquoStructural changesand volatility transmission in crude oil marketsrdquo Physica AStatisticalMechanics and Its Applications vol 390 no 23-24 pp4317ndash4324 2011

[25] J P Yang X X Chen and C H Zhang ldquoPolicy impact factorson the volatility of Shanghai and Shenzhen stock market inChinardquo Chinese Journal of Management Science vol 20 no 6pp 43ndash51 2012

[26] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer New York NY USA2010

[27] C-L Huang and J-F Dun ldquoA distributed PSOndashSVM hybridsystem with feature selection and parameter optimizationrdquoApplied Soft Computing vol 8 no 4 pp 1381ndash1391 2008

[28] C Zhou K Yin Y Cao and B Ahmed ldquoApplication of timeseries analysis and PSOndashSVMmodel in predicting the Bazimenlandslide in the Three Gorges Reservoir Chinardquo EngineeringGeology vol 204 pp 108ndash120 2016

[29] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis amp Applications vol 16 no 1 pp 93ndash103 2014

[30] X Gong Z He P Li and N Zhu ldquoForecasting return volatilityof the CSI 300 index using the stochastic volatility model withcontinuous volatility and jumpsrdquo Discrete Dynamics in Natureand Society vol 2014 Article ID 964654 10 pages 2014

[31] S Q Guo and P H Liang ldquoSocial Interactions the informationchannel and household stock market participation a empiricalstudy based on a Chinese household finance surveyrdquo EconomicResearch Journal vol S1 pp 116ndash131 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Table 5 The prediction errors of different components

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

Residue termRMSE 0007 0008 0006 0009MAPE 0001 0001 0001 0002DS 69837 58424 21678 27241

Low-frequency termRMSE 0007 0008 0004 0008MAPE 0087 0092 0042 0258DS 60326 55434 54613 55254

High-frequency termRMSE 0086 0098 0020 0021MAPE 4806 4725 1146 1149DS 63315 62500 51463 51854

Table 6 Comparison of different modelsrsquo prediction errors

Daily price series Five-minute price seriesLinear kernel function RBF kernel function Linear kernel function RBF kernel function

SVMRMSE 0053 0054 0008 0008MAPE 0009 0009 0001 0001DS 54891 52445 48171 48161

CEEMD-SVM-SVMRMSE 0051 0009 0008 0007MAPE 0008 0002 0001 0001DS 62228 88043 51565 52715

the SVM model improves the ability of the SVM model tomine the variations in the stock index futures price seriesfurther enhancing its prediction precision Simultaneouslysome useful suggestions for investors can be found in thepredictions of Chinese stock index futures prices generatedby our proposedmodel First investors should be accustomedto analyzing the price series from different time scales tounderstand the fluctuation characteristics of Chinese stockindex futures prices Second investors must actively identifywhat time scale the effects of external shocks on stock indexfutures prices belong to andmust devise the appropriate strat-egy accordingly Finally investors should pay more attentionto a hedging strategy involving Chinese stock index futuresover the long andor medium term and should be cautiousabout short-term speculation

Figures 6 and 7 present the real and predictive values ofthe daily price series andfive-minute price series respectivelyThe final predictive values obtained using the CEEMD-SVM-SVM model appeared to coincide with the real values indi-cating that the predictive values obtained using this modelfit well with the real values and intuitively demonstratingthat the CEEMD-SVM-SVMmodel is suitable for forecastingChinese stock index futures prices

20135 20138 201311 20142 20145 20148 2014112000

2200

2400

2600

2800

Real valuePredictive value

Figure 6 Real value and predictive value of the daily price series

5 Conclusion

The Chinese stock index futures market is a complex systemand fluctuations in the stock index futures prices resultfrom the collective effects of many factors at various levelsThe CEEMD method a method suitable in the analysis ofnonlinear nonstationary data is applied to analyze Chinesestock index futures prices in different time scales and to

12 Mathematical Problems in Engineering

Real valuePredictive value

20135 20138 201311 20142 20145 20148 2014111800

2000

2200

2400

2600

2800

Figure 7 Real value and predictive value of the five-minute priceseries

reveal the intrinsic fluctuation characteristics of Chinesestock index futures prices Then we focus on whether thesecharacteristics improve the predictive precision of Chinesestock index futures prices and propose a CEEMD-SVM-SVMmodel based on the PSO algorithm The empirical resultsdemonstrate that the intrinsic fluctuation characteristics ofChinese stock index futures prices can be revealed in threecomponents including the residue term the low-frequencyterm and the high-frequency term with independent eco-nomic significance The residue term determines the long-term trend of stock index futures prices and its fluctuationsare correlated with macroeconomics and the long-termexpectations of investorsThe low-frequency term reveals theeffect of significant events on medium-term fluctuations instock index futures prices the ICSS algorithm breakpointtests reveal that policy regulation events are the main causeof fluctuations during the low-frequency term The high-frequency term reflects short-term disequilibrium in thestock index futures market although the effect of the high-frequency term on the overall fluctuation in the stock indexfutures prices is limited it nonetheless plays an important rolelocally

In addition in forecasting Chinese stock index futuresprices first we find that the linear kernel function SVMmodel is more suitable for predicting these components thanthe RBF kernel function SVM model The residue term andthe low-frequency term have stronger regular fluctuationsthan the high-frequency term and therefore yield superiorpredictive results Second the RBF kernel function SVMmodel shows higher predictive precision for the final combi-nation prediction than the linear kernel function SVMmodelSimultaneously the predictive results of the combinationprediction model are better than those of a single SVMmodel which indicates that the integration of fluctuationcharacteristics in different time scales increases the abilityof the SVM model to identify regularity in Chinese stockindex futures price fluctuations thus improving predictionprecision Finally the predictive results of the CEEMD-SVM-SVM model on daily prices and intraday prices are basicallyconsistent which indicates that our proposed model hasstability and universal applicability in forecasting Chinesestock index futures prices

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by the National SocialScience Foundation of China (14ZDA045)

References

[1] VMeneu andH Torro ldquoAsymmetric covariance in spot-futuresmarketsrdquo Journal of Futures Markets vol 23 no 11 pp 1019ndash1046 2003

[2] M Zhong A F Darrat and R Otero ldquoPrice discovery andvolatility spillovers in index futures markets some evidencefrom Mexicordquo Journal of Banking and Finance vol 28 no 12pp 3037ndash3054 2004

[3] X Xiong F Wang W Zhang and Y Sun ldquoPrice discoveryand volatility spillovers between SGX FTSEXinhua ChinaA50 index futures and A-share marketrdquo Chinese Journal ofManagement vol 11 no 1535 pp 1507ndash1512 2009

[4] H M Zuo Z T Liu and H W Zeng ldquoResearch of volatilityspillover and information transmission between stock indexfutures and spot market based on the high frequency datardquoJournal of Financial Research vol 4 pp 140ndash154 2012

[5] H Q Chen and C H Zhang ldquoDoes index futures tradingreduce stock market jump riskrdquo Economic Research Journalvol 1 pp 153ndash167 2015

[6] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 48 no 3 pp235ndash254 2015

[7] E E Peters Fractal Market Analysis Applying Chaos Theory toInvestment and Economics John Wiley amp Sons New York NYUSA 1994

[8] X Zhang K K Lai and S-YWang ldquoA new approach for crudeoil price analysis based on empirical mode decompositionrdquoEnergy Economics vol 30 no 3 pp 905ndash918 2008

[9] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnonstationary time series analysisrdquo Proceedings of the RoyalSociety of London A vol 454 no 1971 pp 903ndash995 1998

[10] Z Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[11] L F Ruan and H J Bao ldquoAn empirical analysis on periodicfluctuations of real estate price based on EMDrdquo Chinese Journalof Management Science no 3 pp 41ndash46 2012

[12] H L Li and C E Feng ldquoRelationship between investor senti-ment and stock indices fluctuation based on EEMDrdquo SystemsEngineering-Theory amp Practice vol 10 pp 2495ndash2503 2014

[13] J-R Yeh J-S Shieh andN EHuang ldquoComplementary ensem-ble empiricalmode decomposition a novel noise enhanced dataanalysismethodrdquoAdvances in Adaptive DataAnalysis vol 2 no2 pp 135ndash156 2010

[14] L Zhao W Yu and R Yan ldquoRolling bearing fault diagnosisbased on CEEMD and time series modelingrdquo MathematicalProblems in Engineering vol 2014 Article ID 101867 13 pages2014

Mathematical Problems in Engineering 13

[15] L Tang W Dai L Yu and S Wang ldquoA novel CEEMD-based eelm ensemble learning paradigm for crude oil priceforecastingrdquo International Journal of Information Technologyand Decision Making vol 14 no 1 pp 141ndash169 2015

[16] L Yu S Wang and K K Lai ldquoForecasting crude oil price withan EMD-based neural network ensemble learning paradigmrdquoEnergy Economics vol 30 no 5 pp 2623ndash2635 2008

[17] Y F Yang Y K Bao Z Y Hu and R Zhang ldquoCrude oilprice prediction based on empirical mode decomposition andsupport vector machinesrdquo Chinese Journal of Management vol12 pp 1884ndash1889 2010

[18] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[19] W Huang Y Nakamori and S-Y Wang ldquoForecasting stockmarket movement direction with support vector machinerdquoComputers and Operations Research vol 32 no 10 pp 2513ndash2522 2005

[20] H Ince and T B Trafalis ldquoShort term forecasting with supportvector machines and application to stock price predictionrdquoInternational Journal of General Systems vol 37 no 6 pp 677ndash687 2008

[21] F E H Tay and L Cao ldquoApplication of support vectormachinesin financial time series forecastingrdquo Omega vol 29 no 4 pp309ndash317 2001

[22] Y Sai F T Zhang and T Zhang ldquoResearch of Chinese stockindex futures regression prediction based on support vectormachinesrdquo Chinese Journal of Management Science vol 3 pp35ndash39 2013

[23] C Inclan andG C Tiao ldquoUse of cumulative sums of squares forretrospective detection of changes of variancerdquo Journal of theAmerican Statistical Association vol 89 no 427 pp 913ndash9231994

[24] S H Kang C Cheong and S-M Yoon ldquoStructural changesand volatility transmission in crude oil marketsrdquo Physica AStatisticalMechanics and Its Applications vol 390 no 23-24 pp4317ndash4324 2011

[25] J P Yang X X Chen and C H Zhang ldquoPolicy impact factorson the volatility of Shanghai and Shenzhen stock market inChinardquo Chinese Journal of Management Science vol 20 no 6pp 43ndash51 2012

[26] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer New York NY USA2010

[27] C-L Huang and J-F Dun ldquoA distributed PSOndashSVM hybridsystem with feature selection and parameter optimizationrdquoApplied Soft Computing vol 8 no 4 pp 1381ndash1391 2008

[28] C Zhou K Yin Y Cao and B Ahmed ldquoApplication of timeseries analysis and PSOndashSVMmodel in predicting the Bazimenlandslide in the Three Gorges Reservoir Chinardquo EngineeringGeology vol 204 pp 108ndash120 2016

[29] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis amp Applications vol 16 no 1 pp 93ndash103 2014

[30] X Gong Z He P Li and N Zhu ldquoForecasting return volatilityof the CSI 300 index using the stochastic volatility model withcontinuous volatility and jumpsrdquo Discrete Dynamics in Natureand Society vol 2014 Article ID 964654 10 pages 2014

[31] S Q Guo and P H Liang ldquoSocial Interactions the informationchannel and household stock market participation a empiricalstudy based on a Chinese household finance surveyrdquo EconomicResearch Journal vol S1 pp 116ndash131 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

Real valuePredictive value

20135 20138 201311 20142 20145 20148 2014111800

2000

2200

2400

2600

2800

Figure 7 Real value and predictive value of the five-minute priceseries

reveal the intrinsic fluctuation characteristics of Chinesestock index futures prices Then we focus on whether thesecharacteristics improve the predictive precision of Chinesestock index futures prices and propose a CEEMD-SVM-SVMmodel based on the PSO algorithm The empirical resultsdemonstrate that the intrinsic fluctuation characteristics ofChinese stock index futures prices can be revealed in threecomponents including the residue term the low-frequencyterm and the high-frequency term with independent eco-nomic significance The residue term determines the long-term trend of stock index futures prices and its fluctuationsare correlated with macroeconomics and the long-termexpectations of investorsThe low-frequency term reveals theeffect of significant events on medium-term fluctuations instock index futures prices the ICSS algorithm breakpointtests reveal that policy regulation events are the main causeof fluctuations during the low-frequency term The high-frequency term reflects short-term disequilibrium in thestock index futures market although the effect of the high-frequency term on the overall fluctuation in the stock indexfutures prices is limited it nonetheless plays an important rolelocally

In addition in forecasting Chinese stock index futuresprices first we find that the linear kernel function SVMmodel is more suitable for predicting these components thanthe RBF kernel function SVM model The residue term andthe low-frequency term have stronger regular fluctuationsthan the high-frequency term and therefore yield superiorpredictive results Second the RBF kernel function SVMmodel shows higher predictive precision for the final combi-nation prediction than the linear kernel function SVMmodelSimultaneously the predictive results of the combinationprediction model are better than those of a single SVMmodel which indicates that the integration of fluctuationcharacteristics in different time scales increases the abilityof the SVM model to identify regularity in Chinese stockindex futures price fluctuations thus improving predictionprecision Finally the predictive results of the CEEMD-SVM-SVM model on daily prices and intraday prices are basicallyconsistent which indicates that our proposed model hasstability and universal applicability in forecasting Chinesestock index futures prices

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by the National SocialScience Foundation of China (14ZDA045)

References

[1] VMeneu andH Torro ldquoAsymmetric covariance in spot-futuresmarketsrdquo Journal of Futures Markets vol 23 no 11 pp 1019ndash1046 2003

[2] M Zhong A F Darrat and R Otero ldquoPrice discovery andvolatility spillovers in index futures markets some evidencefrom Mexicordquo Journal of Banking and Finance vol 28 no 12pp 3037ndash3054 2004

[3] X Xiong F Wang W Zhang and Y Sun ldquoPrice discoveryand volatility spillovers between SGX FTSEXinhua ChinaA50 index futures and A-share marketrdquo Chinese Journal ofManagement vol 11 no 1535 pp 1507ndash1512 2009

[4] H M Zuo Z T Liu and H W Zeng ldquoResearch of volatilityspillover and information transmission between stock indexfutures and spot market based on the high frequency datardquoJournal of Financial Research vol 4 pp 140ndash154 2012

[5] H Q Chen and C H Zhang ldquoDoes index futures tradingreduce stock market jump riskrdquo Economic Research Journalvol 1 pp 153ndash167 2015

[6] FWen Z He Z Dai and X Yang ldquoCharacteristics of investorsrsquorisk preference for stock marketsrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 48 no 3 pp235ndash254 2015

[7] E E Peters Fractal Market Analysis Applying Chaos Theory toInvestment and Economics John Wiley amp Sons New York NYUSA 1994

[8] X Zhang K K Lai and S-YWang ldquoA new approach for crudeoil price analysis based on empirical mode decompositionrdquoEnergy Economics vol 30 no 3 pp 905ndash918 2008

[9] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hilbert spectrum for nonlinear andnonstationary time series analysisrdquo Proceedings of the RoyalSociety of London A vol 454 no 1971 pp 903ndash995 1998

[10] Z Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[11] L F Ruan and H J Bao ldquoAn empirical analysis on periodicfluctuations of real estate price based on EMDrdquo Chinese Journalof Management Science no 3 pp 41ndash46 2012

[12] H L Li and C E Feng ldquoRelationship between investor senti-ment and stock indices fluctuation based on EEMDrdquo SystemsEngineering-Theory amp Practice vol 10 pp 2495ndash2503 2014

[13] J-R Yeh J-S Shieh andN EHuang ldquoComplementary ensem-ble empiricalmode decomposition a novel noise enhanced dataanalysismethodrdquoAdvances in Adaptive DataAnalysis vol 2 no2 pp 135ndash156 2010

[14] L Zhao W Yu and R Yan ldquoRolling bearing fault diagnosisbased on CEEMD and time series modelingrdquo MathematicalProblems in Engineering vol 2014 Article ID 101867 13 pages2014

Mathematical Problems in Engineering 13

[15] L Tang W Dai L Yu and S Wang ldquoA novel CEEMD-based eelm ensemble learning paradigm for crude oil priceforecastingrdquo International Journal of Information Technologyand Decision Making vol 14 no 1 pp 141ndash169 2015

[16] L Yu S Wang and K K Lai ldquoForecasting crude oil price withan EMD-based neural network ensemble learning paradigmrdquoEnergy Economics vol 30 no 5 pp 2623ndash2635 2008

[17] Y F Yang Y K Bao Z Y Hu and R Zhang ldquoCrude oilprice prediction based on empirical mode decomposition andsupport vector machinesrdquo Chinese Journal of Management vol12 pp 1884ndash1889 2010

[18] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[19] W Huang Y Nakamori and S-Y Wang ldquoForecasting stockmarket movement direction with support vector machinerdquoComputers and Operations Research vol 32 no 10 pp 2513ndash2522 2005

[20] H Ince and T B Trafalis ldquoShort term forecasting with supportvector machines and application to stock price predictionrdquoInternational Journal of General Systems vol 37 no 6 pp 677ndash687 2008

[21] F E H Tay and L Cao ldquoApplication of support vectormachinesin financial time series forecastingrdquo Omega vol 29 no 4 pp309ndash317 2001

[22] Y Sai F T Zhang and T Zhang ldquoResearch of Chinese stockindex futures regression prediction based on support vectormachinesrdquo Chinese Journal of Management Science vol 3 pp35ndash39 2013

[23] C Inclan andG C Tiao ldquoUse of cumulative sums of squares forretrospective detection of changes of variancerdquo Journal of theAmerican Statistical Association vol 89 no 427 pp 913ndash9231994

[24] S H Kang C Cheong and S-M Yoon ldquoStructural changesand volatility transmission in crude oil marketsrdquo Physica AStatisticalMechanics and Its Applications vol 390 no 23-24 pp4317ndash4324 2011

[25] J P Yang X X Chen and C H Zhang ldquoPolicy impact factorson the volatility of Shanghai and Shenzhen stock market inChinardquo Chinese Journal of Management Science vol 20 no 6pp 43ndash51 2012

[26] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer New York NY USA2010

[27] C-L Huang and J-F Dun ldquoA distributed PSOndashSVM hybridsystem with feature selection and parameter optimizationrdquoApplied Soft Computing vol 8 no 4 pp 1381ndash1391 2008

[28] C Zhou K Yin Y Cao and B Ahmed ldquoApplication of timeseries analysis and PSOndashSVMmodel in predicting the Bazimenlandslide in the Three Gorges Reservoir Chinardquo EngineeringGeology vol 204 pp 108ndash120 2016

[29] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis amp Applications vol 16 no 1 pp 93ndash103 2014

[30] X Gong Z He P Li and N Zhu ldquoForecasting return volatilityof the CSI 300 index using the stochastic volatility model withcontinuous volatility and jumpsrdquo Discrete Dynamics in Natureand Society vol 2014 Article ID 964654 10 pages 2014

[31] S Q Guo and P H Liang ldquoSocial Interactions the informationchannel and household stock market participation a empiricalstudy based on a Chinese household finance surveyrdquo EconomicResearch Journal vol S1 pp 116ndash131 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

[15] L Tang W Dai L Yu and S Wang ldquoA novel CEEMD-based eelm ensemble learning paradigm for crude oil priceforecastingrdquo International Journal of Information Technologyand Decision Making vol 14 no 1 pp 141ndash169 2015

[16] L Yu S Wang and K K Lai ldquoForecasting crude oil price withan EMD-based neural network ensemble learning paradigmrdquoEnergy Economics vol 30 no 5 pp 2623ndash2635 2008

[17] Y F Yang Y K Bao Z Y Hu and R Zhang ldquoCrude oilprice prediction based on empirical mode decomposition andsupport vector machinesrdquo Chinese Journal of Management vol12 pp 1884ndash1889 2010

[18] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[19] W Huang Y Nakamori and S-Y Wang ldquoForecasting stockmarket movement direction with support vector machinerdquoComputers and Operations Research vol 32 no 10 pp 2513ndash2522 2005

[20] H Ince and T B Trafalis ldquoShort term forecasting with supportvector machines and application to stock price predictionrdquoInternational Journal of General Systems vol 37 no 6 pp 677ndash687 2008

[21] F E H Tay and L Cao ldquoApplication of support vectormachinesin financial time series forecastingrdquo Omega vol 29 no 4 pp309ndash317 2001

[22] Y Sai F T Zhang and T Zhang ldquoResearch of Chinese stockindex futures regression prediction based on support vectormachinesrdquo Chinese Journal of Management Science vol 3 pp35ndash39 2013

[23] C Inclan andG C Tiao ldquoUse of cumulative sums of squares forretrospective detection of changes of variancerdquo Journal of theAmerican Statistical Association vol 89 no 427 pp 913ndash9231994

[24] S H Kang C Cheong and S-M Yoon ldquoStructural changesand volatility transmission in crude oil marketsrdquo Physica AStatisticalMechanics and Its Applications vol 390 no 23-24 pp4317ndash4324 2011

[25] J P Yang X X Chen and C H Zhang ldquoPolicy impact factorson the volatility of Shanghai and Shenzhen stock market inChinardquo Chinese Journal of Management Science vol 20 no 6pp 43ndash51 2012

[26] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer New York NY USA2010

[27] C-L Huang and J-F Dun ldquoA distributed PSOndashSVM hybridsystem with feature selection and parameter optimizationrdquoApplied Soft Computing vol 8 no 4 pp 1381ndash1391 2008

[28] C Zhou K Yin Y Cao and B Ahmed ldquoApplication of timeseries analysis and PSOndashSVMmodel in predicting the Bazimenlandslide in the Three Gorges Reservoir Chinardquo EngineeringGeology vol 204 pp 108ndash120 2016

[29] Z Dai and F Wen ldquoRobust CVaR-based portfolio optimizationunder a genal affine data perturbation uncertainty setrdquo Journalof Computational Analysis amp Applications vol 16 no 1 pp 93ndash103 2014

[30] X Gong Z He P Li and N Zhu ldquoForecasting return volatilityof the CSI 300 index using the stochastic volatility model withcontinuous volatility and jumpsrdquo Discrete Dynamics in Natureand Society vol 2014 Article ID 964654 10 pages 2014

[31] S Q Guo and P H Liang ldquoSocial Interactions the informationchannel and household stock market participation a empiricalstudy based on a Chinese household finance surveyrdquo EconomicResearch Journal vol S1 pp 116ndash131 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of