Research Article A New Least Squares Support Vector ...downloads.hindawi.com/journals/mpe/2016/4615903.pdf · Research Article A New Least Squares Support Vector Machines Ensemble

Research ArticleA New Least Squares Support Vector Machines Ensemble Modelfor Aero Engine Performance Parameter Chaotic Prediction

Dangdang Du Xiaoliang Jia and Chaobo Hao

School of Mechanical Engineering Northwestern Polytechnical University Xirsquoan 710072 China

Correspondence should be addressed to Xiaoliang Jia jiaxlnwpueducn

Received 25 September 2015 Accepted 26 January 2016

Academic Editor Francesco Franco

Copyright copy 2016 Dangdang Du et al This 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

Aiming at the nonlinearity chaos and small-sample of aero engine performance parameters data a new ensemble model namedthe least squares support vector machine (LSSVM) ensemble model with phase space reconstruction (PSR) and particle swarmoptimization (PSO) is presented First to guarantee the diversity of individualmembers different single kernel LSSVMs are selectedas base predictors and they also output the primary prediction results independently Then all the primary prediction results areintegrated to produce the most appropriate prediction results by another particular LSSVMmdasha multiple kernel LSSVM whichreduces the dependence of modeling accuracy on kernel function and parameters Phase space reconstruction theory is applied toextract the chaotic characteristic of input data source and reconstruct the data sample and particle swarm optimization algorithmis used to obtain the best LSSVM individual members A case study is employed to verify the effectiveness of presented model withreal operation data of aero engine The results show that prediction accuracy of the proposed model improves obviously comparedwith other three models

1 Introduction

With increasing demands in the field of operation safetyasset availability and economy the health monitoring of aeroengine has been widely considered as the key prerequisite forthe competition of an airline company One of the main tasksof healthmonitoring is to predict the performance parameterof aero engine By predicting and analyzing the trend of per-formance parameters one can obtain valuable information toavoid future risk and loss due to faults or accidents and reduceassociated maintenance costs [1] Therefore it is necessary todesign a high accurate and robust prediction model for aeroengine performance parameter (AEPP)

A variety of traditional time series prediction approacheshave already been proposed for this problem such as fuzzyrule [2] Kalman filter [3] grey prediction [4] ARMA [5] andmultiple regression [6] These approaches are very maturein theory but the accuracy is not always high and therobustness is not always satisfied in the application [7] Withthe development of artificial intelligence techniques recent

studies for AEPP prediction are mainly focused on artificialneural network (ANN) [8 9] and support vector machine(SVM) [10 11]

Comparedwith standard SVM least squares support vec-tormachine (LSSVM) adopts equality constraints and a linearKarush-Kuhn-Tucker system which has a more powerfulcomputational ability in solving the nonlinear and small-sample problem [12 13] In addition LSSVM eliminates localminima and structure design complexity of ANNThereforeLSSVM is a good choice for AEPP prediction model design-ingHowever themodeling accuracy of a single LSSVM is notonly influenced by the input data source but also affected byits kernel function and regularization parameters [12] Thusseveral main disadvantages are worth to be addressed Firstlyusing a data-driven technique to design an LSSVM modeldata source should be considered as the first factor AEPPdata is different from the pure random system that is thechaotic characteristic of AEPP data should be extracted toreconstruct input data samples before modeling Secondlyas two common parameter optimized methods for LSSVM

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 4615903 8 pageshttpdxdoiorg10115520164615903

2 Mathematical Problems in Engineering

conventional cross-validation and grid search methods haveseveral defects such as high time consuming and a prioriknowledge requirement

In addition although a single LSSVM with optimalparameters and reconstructed input data samples may havean excellent prediction performance under certain circum-stances because its kernel function is fixed it perhaps hassome kinds of inherent bias under other cases In the litera-tures due to the super robustness and generalization ensem-ble model has been proved to be an effective way to reducebiases of single model Ensemble model can make full useof diversity to compensate for disadvantages among the indi-vidual members and the reasonable combination strategy isbelieved to be able to produce better prediction accuracy andgeneralization than singlemodel [13ndash16] By using combiningsubmodels the multilayer networks of LS-SVMs ensemblehave been discussed deeply which is very encouraging andpromising for further research [13] but up to now theapplication of LSSVM ensemble model for AEPP predictionis relatively fresh and untouched in the open literature

For ensemble model design there are two points thatshould be considered One is that the selected individualmembers need to exhibit much diversity (disagreement) andaccuracy The other is the effectiveness of the combinationstrategy [17] For the diversity of individual members itis an easy and common way to build individual membersby using data decomposition [18] However this methodis proved to be effective when the original data sampleis sufficient and it is not suitable for small-sample dataCompared with the existing combination strategies such assimple averaging weighting method mean squared errorweighting method and least squares estimation weightingaveraging method intelligent method based combinationstrategies include ANN combiner and SVM combiner whichhave become the current trend [18] however ANN com-biner cannot avoid falling into local optima and for SVMcombiner it is not easy to select appropriate kernel functionso it is necessary to further improve these ensemble strate-gies

As previously mentioned in this paper a new PSR-PSO-(SK)LSSVM-(MK)LSSVM ensemble (PPLLE) modelforAEPPprediction is proposed Firstly a set of diverse singlekernel LSSVMs are created as base predictors Subsequentlythese individualmember LSSVMsoutput the primary predic-tion results independently Finally all the primary predictionresults are combined to produce themost appropriate predic-tion results by another particular multiple kernel LSSVM Inthe process of modeling phase space reconstruction (PSR)theory is applied to extract the chaotic characteristic ofinput data source and reconstruct the data samples Particleswarm optimization algorithm is used to search the bestparameters for LSSVM members to ensure their predictionaccuracies

The rest of this paper is organized as followsThenext sec-tion provides a brief introduction to the related knowledgeSection 3 formulates the proposed PPLLE model For illus-tration purpose the detailed application on AEPP predictionand model comparisons is proposed in Section 4 Section 5concludes this study

2 An Overview of the Related Knowledge

21 Data Samples Reconstruction Based on PSR TheoryAlthough the nonlinear chaos behavior is the main challengeconfronting the chaotic data series prediction the underlyingdata generating mechanisms can still be explored by PSRtheory [19] By means of the ability of revealing the natureof dynamic system state the PSR theory is useful in systemcharacterization in nonlinear prediction and in estimatingbounds on the size of the system [20 21]

According to Takensrsquo theorem for the nonlinear timeseries 119909

119894119899

119894=1 the current state information can be represented

by an119898-dimensional vector

119909119894+120591= 119891 (119909

119894 119909119894minus120591 119909

119894minus(119898minus1)120591) (1)

where 120591 is the delay time119898 is the embedded dimension theyare two important parameters for phase space constructionand119891 is themapping relation between the inputs and outputs

The autocorrelation function of time series at the firstminimum value is taken as the delay time 120591 of the recon-structed phase space we write

CR =sum119899minus120591

119894=1(119909119894minus 119909) (119909

119894+120591minus 119909)

sum119899minus120591

119894=1(119909119894minus 119909)2

(2)

where 119909 is the mean of 119909119894

To calculate the correlation dimension the correlationintegral 119862(119903) needs to be computed

119862 (119903) =1

1198732

119873

sum119894=1

119873

sum119895=1

119906 (119903 minus10038171003817100381710038171003817119883119894minus 119883119895

10038171003817100381710038171003817) (3)

where 119903 is the selected radius and 119903 gt 0 119906(119909) is the Heavisidefunction

The correlation dimension 119889(119898) is calculated by theformula as below

119889 (119898) =ln119862 (119903)ln 119903

(4)

Suppose119884119894= 119909119894119883119894= (119909119894minus120591 119909119894minus2120591 119909

119894minus119898120591) then we can

reconstruct 119899 minus 119898 data samples 119883119894 119884119894119899

119898+1

22 Least Squares Support Vector Machine LSSVM is theleast squares form of a standard SVM it was firstly proposedby Suykens and Vandewalle [12] LSSVM uses a set of linearequations during the training process and chooses all trainingdata as support vectors so it has excellent generalization andlow computation complexity [12ndash14]

In LSSVM the regression issue can be expressed as thefollowing optimization problem

min 1

21198822+1

2120574

119873

sum119905=1

1198902

119905(120574 gt 0)

st 119910119905= ⟨119882 120593 (X

119905)⟩ + 119887 + 119890

119905

(5)

where 120593(X119905) is a nonlinear function which maps the input

data into a higher dimensional space 119890119905is the error at time

119905 119887 is the bias and 120574 is the regulation constant

Mathematical Problems in Engineering 3

According to the Lagrange function and Karush-Kuhn-Tucker theorem the LSSVM for nonlinear functions can begiven as below

119910119905=

119873

sum119905=1

120572119905119870(119883119883

119905) + 119887 (6)

where 120572119905is the Lagrangemultiplier and119870(119883119883

119905) is the kernel

function which is applied to substitute the mapping processand avoid computing the function 120593(X

119905)

Typical kernel functions include linear kernel functionpolynomial kernel function radial basis kernel functionsigmoid kernel function and multiple kernel function Someof them are listed as follows

(1) linear kernel function (LKF)

119870(119883119883119905) = (119883119883

1015840

119905) (7)

(2) polynomial kernel function (PKF)

119870(119883119883119905) = [(119883119883

119905) + 1]

119902 (8)

(3) Gaussian radial basis kernel function (RBF)

119870(119883119883119905) = exp(minus

1003817100381710038171003817119883 minus 11988311990510038171003817100381710038172

1205902) (9)

(4) sigmoid kernel function (SKF)

119870(119883119883119905) = tanh (] (119883119883

119905) + 119890) (10)

(5) multiple kernel function (MKF)

119870(119883119883119905) = 12058811198701(119883119883

119905) + sdot sdot sdot + 120588

119899119870119899(119883119883

119905)

st119899

sum119894=1

120588119894= 1 (0 le 120588

119894le 1)

(11)

The nonlinear mapping ability of LSSVM is mainlydetermined by its kernel function form and relevant param-eters setting that is various kernel functions or parametershave different influence on the prediction ability of LSSVMpredictor (the parameters setting will be discussed in the nextsection) As to kernel function the LKF is suited to expressingthe linear component of the mapping relation and the RBFpossesses a wider convergence domain and an outstandinglearning ability and high resolution power while the PKFhas a powerful approximation and generalization abilityMeanwhile kernel functions can also be divided into localkernel function and global kernel function For the globalkernel function it has the overall situation characteristic andis commonly good at fitting the sample points which arefar away from the testing points but the fitting effect is notperfect on the sample points which are near the testing pointsand vice versa to the local kernel function [15] Each kindof kernel function has its own advantages and disadvantagesthe prediction performances of LSSVM with different kernelfunctions are not identical

Here we define the LSSVM configured with a multiplekernel function as themultiple kernel LSSVM (MK-LSSVM)otherwise we call it the single kernel LSSVM (SK-LSSVM)

23 Parameters Optimized Based on PSO Particle swarmoptimization (PSO) algorithm is a popular swarm intelligenceevolutionary algorithm used for solving global optimizationproblem [22] It can search the global optimal solution indifferent regions of the solution space in parallel

In PSO the position of each particle represents a solutionto the optimization problem 119909

119896= (1199091198961 1199091198962 119909

119896119889) is the

position vector and V119896= (V1198961 V1198962 V

119896119889) is the velocity

vector of the 119896th particle Similarly 119875119896= (1199011198961 1199011198962 119901

119896119889)

represents the best position of the 119896th particle which has beenachieved and 119875

119892= (1199011198921 1199011198922 119901

119892119889) represents the best

position among the whole particle groupThe values of position and velocity of the particle are

updated as follows

V119905+1119894119889= [120596V119905

119894119889+ 11988811199031(119901119905

119894119889minus 119909119905

119894119889) + 11988821199032(119901119905

119892119889minus 119909119905

119894119889)]

119909119905+1

119894119889= 119909119905

119894119889+ V119905+1119894119889

(12)

where 1198881and 1198882are the acceleration constant 119903

1and 1199032are two

random numbers in the range [0 1] and 120596 is inertia weightfactor To improve the convergence speed of PSO 119888

1 1198882 and

120596 of PSO are adjusted by using the formulas as below

120596 = 120596max minus120596max minus 120596min

119905max119905

1198881= 1198881max minus

1198881max minus 1198881min119905max

119905

1198882= 1198882max minus

1198882max minus 1198882min119905max

119905

(13)

where 119905max expresses the maximum iteration number and 119905 isthe current iteration number

3 Overall Process of Designingthe PPLLE Model

The core idea of the ensemble model lies in that all theindividual members are accurate as much as possible anddiverse enough and it adopts an appropriate ensemblestrategy to combine these outputs of the selected members[13ndash18]

31 Selection of the Appropriate Individual Member PredictorsFor LSSVM predictionmodel several diverse strategies suchas data diversity parameter diversity and kernel diversityhave been proved effectively for the creation of ensemblemembers with much dissimilarity [13] Because kernel func-tion has a crucial and direct effect on the learning andgeneralizing performance of LSSVM various kernel func-tions can be used to create diverse LSSVMs In this study119899 independent SK-LSSVMs such as LKF-LSSVM PKF-LSSVM and RBF-LSSVM are selected as individual memberLSSVM predictors

32 Combination of the Selected Individual Member LSSVMPredictors After the diverse individual member LSSVM


Individual

Individual

Reconstructdata samples

Combiner MK-LSSVM

Data source

PSR

Stage 2Stage 1 Stage 3

Partitiondata samples

PSO

Individual

y

x1

x2

xn

middot middot middot


SK-LSSVM1

SK-LSSVM2

SK-LSSVMn

Figure 1 Basic framework of the PPLLE model

predictors have been selected the other key question ishow to determine the weight coefficient of each individualpredictor that is how to construct the combiner effectivelyAs depicted in previous section the MKF integrates theadvantages of global kernel function and local kernel functionand offsets some shortages of both simultaneously Henceanother specialMK-LSSVM is chosen as the combiner In thispaper theMKF is composed of aRBF and aPKF the former isa typical local kernel function and the latter is a representativeglobe kernel function A similar MK-LSSVMmodel has highprediction accuracy and generalization ability which hasbeen proved with chaotic time series by Tian et al [15]

33 Overall Process of Designing the PPLLE Model The basicframework of the proposed PPLLEmodel is given in Figure 1where 119899 is the number of the individual member LSSVMpredictors

As shown in Figure 1 there are three main stages in thebasic framework which can be summarized as follows

Stage 1 (sample dataset reconstruction and partition) Thedata source is reconstructed as data samples by using PSRthen the reconstructed data samples are divided into twoindispensable subsets training subset and testing subset

Stage 2 (individual member creation and prediction) Basedon kernel function diversity principle 119899 independent SK-LSSVMs are created as the individual member Each SK-LSSVM is trained by using the training subset Accordinglythe computational results

1 2 3

119899of the 119899 SK-

LSSVM predictors can be obtained respectively In theprocess of SK-LSSVM creating PSO is used to optimizeparameters of each member SK-LSSVM

Stage 3 (combiner creation and prediction) When the com-putational results of the individual member predictors inthe second stage are acquired they are aggregated intoan ensemble result by another special MK-LSSVM Sim-ilarly to create the optimal MK-LSSVM PSO is appliedagain

Here 119891(sdot) is the mapping function determined by thespecial combiner MK-LSSVM thus the final predictionoutput of the PPLLE model can be given as below

= 119891 (1 2 3

119899) (14)

4 Case Study

Due to different gas path component degradations such asfouling erosion corrosion and foreign object damage theperformance of an aero engine will decline over the servicetime [23] A lot of gas path performance parameters areoften used in health monitoring of aero engine from differentangles and levels such as exhaust gas temperature (EGT)fuel flow (FF) and low pressure fan speed (N1) Amongthese performance parameters EGT is considered as one ofthe most crucial working performance parameters of aeroengine which is measured to represent outlet temperatureof combustor chamber in practice When other conditionsremain the same the higher the EGT is the more seriousthe performance degradation of aero engine is [4] EGTgradually rises when theworking life of aero engine increasesif the EGT value reaches or exceeds the scheduled thresholdprovided by the original equipment manufacturer then theaero engine needs to be arranged for maintenance timely

In this study we select EGT as the AEPP representativeto predict by using the proposed PPLLE model and it isworth mentioning that other similar parameters can also bepredicted in the same way

41 Data Description and Samples Reconstruction In thisstudy the EGT data come from the real flight recorders ofthe cruise state of a certain type of aero engine and thesampling interval is 5 flight cycles The data series consistsof 148 EGT datasets covering the period from February 2013to September 2014 To increase the quality of the predictionresults some abnormal samples have been discarded fromthe original data series The observed EGT data is shown inFigure 2

For the observed EGT data series EGT119894148

119894=1 according

to (2) (3) and (4) the delay time 120591 is set as 1 and embed-ding dimension 119898 = 5 is obtained by computing Thus(EGT

119894minus5EGT

119894minus4 EGT

119894minus1) is taken as the input vector119883

119894

and119884119894= EGT

119894(119894 = 6 7 148) is used as the corresponding

expected value so we can get the reconstructed data samples119883119894 119884119894148

119894=6 The data samples 119883

119894 119884119894120

119894=6are used as training

subset to train each individual LSSVM of the ensemblemodel and the samples 119883

119894 119884119894148

119894=121are chosen as testing

subset to validate the ensemble model The one-step aheadprediction used in this paper is explained as in Figure 3After the ensemble model has been trained vector 119883

121

is entered into 4 individual predictors (SK-LSSVM predic-tors) to compute their predicted values 1

121 2

121

4

121

respectively Then these predicted values are aggregated intoan ensemble result by using a combination predictor (MK-LSSVM predictor) Hence the final predicted value

121is

obtained In this way from 119894 = 121 to 148 all the finalpredicted values

121to 148

can be got in turn


Table 1 Optimal parameters of LSSVM2simLSSVM

5

LSSVM1

(LKF-LSSVM)LSSVM

2

(PKF-LSSVM)LSSVM

3

(RBF-LSSVM)LSSVM

4

(SKF-LSSVM)LSSVM

5

(MK-LSSVM)119902 = 3

120574 = 9341

1205902= 032

120574 = 17682

V = 1 119890 = 1120574 = 12513

119902 = 2 1205902 = 051120588 = 027 120574 = 1683

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150690

700

710

720

730

740

750

Sample number

Observed data

EGT

(∘C)

Figure 2 The observed EGT data

42 Evaluation Indices Mean absolute percentage error(MAPE) mean absolute error (MAE) mean squared error(MSE) and Theilrsquos Inequality Coefficient (TIC) are used toevaluate the prediction ability of the prediction model

MAPE = 1119896

119896

sum119894=1

10038161003816100381610038161003816100381610038161003816

119910119894minus 119894

119910119894

10038161003816100381610038161003816100381610038161003816times 100

MAE = 1119896

119896

sum119894=1

1003816100381610038161003816119910119894 minus 1198941003816100381610038161003816

MSE = 1119896

119896

sum119894=1

(119910119894minus 119894)2

TIC =radicsum119896

119894=1(119910119894minus 119894)2

radicsum119896

119894=1(119910119894)2+ radicsum

119896

119894=1(119894)2

(15)

where 119910119894and

119894are the observed values and corresponding

prediction values respectively

43 Model Parameters Setting In the modeling process ofLSSVMs the parameters of PSO are set as follows 119888

1min =1198882min = 2 119888

1max = 1198882max = 3 120596min = 1 120596max = 3 119901 =

50 and 119905max = 1000 By using the PSO the correspondingoptimal parameters of LSSVM

2simLSSVM

5are obtained and

listed in Table 1 An appropriate individual member numberof the ensemble model is able to achieve a balance between

Individual

Individual

Combination predictor

Individual

EndYesNo i = i + 1

i = 121

yi

x1

i

x2

i

x4

i



Xi

i gt 148

predictor1

predictor2

predictor4

Figure 3 One-step ahead prediction from 119894 = 121 to 148

120 125 130 135 140 145 150690

700

710

720

730

740

750

Sample number

Observed dataPPLLE predicted data

EGT

(∘C)

Figure 4 Prediction results of the PPLLE model on EGT testingdataset

the prediction efficiency and the prediction ability [17] In thisstudy the member number is set as 5

44 Results and Discussion Figure 4 illustrates the predic-tion results for the EGT testing dataset by PPLLE modeland corresponding observed EGT value The black symbolrepresents the observed value and the red symbol expressesthe prediction value From Figure 4 we can find that therise and fall trends of the two curves are approximatelythe same and only the individual points have some highergaps of the size which means EGT is predicted with good


Table 2 Comparison of different models on EGT testing dataset

Model MAPE () MAE MSE TICPPLLE 051 367 1404 000258PPLLElowast 062 448 2248 000327RBF-chaos 085 616 4970 000485Single LSSVM 110 799 7566 000598

accuracy on the testing data samples as a whole There aretwo causes that may explain the gaps between the observedvalues and prediction values Firstly it is difficult to give athorough consideration to extract the EGT characteristicswhen determining the model input data Secondly due to theinfluence of subjective factors it is impossible to eliminate allthe outliers properly

In contrast the single LSSVM model proposed by Tianet al [15] RBF-chaos model proposed by Zhang et al [24]and PPLLElowast (PSR-PSO-LSSVM-LSSVMlowast ensemble) modelare built The kernel function and parameters of the singleLSSVM model are the same as those of the LSSVM

5model

listed in Table 1 The RBF-chaos model aggregated chaoscharacteristics and RBF neural networks (here the inputlayer hidden layer and output layer of RBF neural networkare set as 5 11 and 1 resp)The difference between the PPLLEmodel and PPLLElowast model lies in that the latter uses an RBF-LSSVM (ie SK-LSSVM) as the combiner

In Table 2 the MAPE MAE MSE and TIC values ofthe PPLLE PPLLElowast RBF-chaos and single LSSVM modelson the testing dataset are listed It shows that the PPLLEmodel performs the best among the four modes with MAPEof 051 compared with those of 062 085 and 110 by thePPLLElowast RBF-chaos and single LSSVMmodels respectivelyTheMAE of PPLLE PPLLElowast RBF-chaos and single LSSVMmodels are 367 448 616 and 799 respectively whichdemonstrates the prediction accuracy of the proposedmodelPPLLE model predicts the EGT with MSE of 1404 betterthan PPLLElowast RBF-chaos and single LSSVM models withthose of 2248 4970 and 7566 respectively Besides itshould be pointed out that the TIC of PPLLE is 000258which is quite acceptable compared with those of the other3 models A strong support is also exhibited by Figure 5where the curve of PPLLE model intuitively shows the goodprediction accuracy and excellent ability in tracking theobserved EGT compared to the other 3 models

Figures 6(a)ndash6(d) show a detailed profile of relativepercentage error (RPE) between the observed values andprediction values of different models on the EGT testingdata samples It illustrates that the PPLLE model has anoutstanding approximation ability with the RPE rangingfrom minus07 to 09 the RPE ranging around [minus14 29]in Figure 6(d) shows that the single LSSVM has the worstperformance RPE distribution range of PPLLElowast model isbetter than that of RBF-chaos model which are exhibitedby Figures 6(c) and 6(d) Comparison results of Figure 6also prove the effectiveness of our proposed approach Someof the main reasons why the PPLLE model is superiorto others can be summarized as follows (1) the PPLLEensemble model based on kernel diverse principle eliminates

120 125 130 135 140 145 150690

700

710

720

730

740

750

Sample number


RBF-chaos predicted dataSingle LSSVM predicted data

PPLLElowast predicted data

EGT

(∘C)

Figure 5 Prediction results of different models on EGT testingdataset

the possible inherent biases of single LSSVM and makes fulluse of the advantages of individual member LSSVMs (2) thePSR extracts the chaotic feature of the original data sourceand reconstructs data samples which elucidates the inputcharacteristic for the PPLLE model (3) the PSO ensure thateach individual LSSVM achieves the best performance (4)the particular ensemble strategy of PPLLE employs an MK-LSSVM and further enhances the prediction ability of theensemble model

5 Conclusions

Designing a high accuracy and robust model for AEPPprediction is quite challenging since AEPP data is nonlinearchaotic and small-sample and the traditional single predic-tion model may have some inherent biases To solve thisproblem and to realize high prediction accuracy level a newLSSVM ensemble model based on PSR and PSO is presentedand applied to AEPP prediction in this paper

For the presented PPLLE prediction model individualmember LSSVMs based on kernel diverse principle eliminatethe inherent biases of single LSSVM and make full use ofthe advantages of them as much as possible PSR is appliedto reconstruct data samples which alleviates the influence ofthe chaotic feature of the original data source to the PPLLEmodel PSO is used to guarantee that each individual LSSVMachieves the best performance The particular ensemblestrategy employs an MK-LSSVM combiner as the MKFintegrates the advantages of global kernel function and localkernel function and it offsets some shortages of both thisensemble strategy further enhances the prediction ability ofthe ensemble model

EGT is selected as the representative health monitoringparameter of aero engine for validating the effectiveness of the


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

PPLLE(a) RPE of PPLLE

120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

PPLLElowast

(b) RPE of PPLLElowast

120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

RBF-chaos(c) RPE of RBF-chaos

120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

Single LSSVM

(d) RPE of single LSSVM

Figure 6 RPE comparison of different models on EGT testing dataset

proposed PPLLE model For comparison the PPLLElowast RBF-chaos and single LSSVMmodels are also developed and eval-uated The PPLLE predicts EGT with MAPE of 051 betterthan the PPLLElowast RBF-chaos and single LSSVMmodels withthose of 062 085 and 110 respectively Similarly thePPLLE predicts EGT with TIC of 000258 better than thePPLLElowast RBF-chaos and single LSSVM models with thoseof 000327 000485 and 000598 respectively In additionMAE andMSE indices also confirm that the presentedmodelgives improved prediction accuracy In a word the above fourevaluation indices consistently demonstrate that the PPLLEmodel is more suitable for AEPP prediction problem and thePPLLE model can meet the actual demand of engineeringapplication Moreover comparing results imply that thisensemble model has a promising application in other similarengineering areas where the data have complex nonlinearchaos relationships

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L-F Yang and I Ioachim ldquoAdaptive estimation of aircraft flightparameters for engine health monitoring systemrdquo Journal ofAircraft vol 39 no 3 pp 404ndash408 2002

[2] J Hong L Han X Miao et al ldquoFuzzy logic inference for pre-dicting aero-engine bearing grad-liferdquo in Proceedings of the 9thInternational Flins Conference on Computational IntelligenceFoundations and Applications vol 4 pp 367ndash373 ChengduChina August 2010

[3] G You and NWang ldquoAero-engine conditionmonitoring basedonKalmanfilter theoryrdquoAdvancedMaterials Research vol 490ndash495 no 4 pp 176ndash181 2012


[4] N-B Zhao J-L Yang S-Y Li and Y-W Sun ldquoA GM (1 1)Markov chain-based aeroengine performance degradation fore-cast approach using exhaust gas temperaturerdquo MathematicalProblems in Engineering vol 2014 Article ID 832851 11 pages2014

[5] Z G Liu Z J Cai and XM Tan ldquoForecasting research of aero-engine rotate speed signal based on ARMA modelrdquo ProcediaEngineering vol 15 pp 115ndash121 2011

[6] Y-X Song K-X Zhang and Y-S Shi ldquoResearch on aeroengineperformance parameters forecast based on multiple linearregression forecastingmethodrdquo Journal of Aerospace Power vol24 no 2 pp 427ndash431 2009 (Chinese)

[7] C Chatfield The Analysis of Time Series An IntroductionChapmanampHallCRC BocaRaton Fla USA 6th edition 2003

[8] S G Luan S S Zhong and Y Li ldquoHybrid recurrent processneural network for aero engine condition monitoringrdquo NeuralNetwork World vol 18 no 2 pp 133ndash145 2008

[9] D Gang and S S Zhong ldquoAircraft engine lubricating oilmonitoring by process neural networkrdquo Neural Network Worldvol 16 no 1 pp 15ndash24 2006

[10] C Zhang and N Wang ldquoAero-engine condition monitoringbased on support vector machinerdquo Physics Procedia vol 24 pp1546ndash1552 2012

[11] X-Y Fu and S-S Zhong ldquoAeroengine turbine exhaust gastemperature prediction using process support vectormachinesrdquoinAdvances in Neural NetworksmdashSNN 2013 vol 7951 of LectureNotes in Computer Science pp 300ndash310 Springer Berlin Ger-many 2013

[12] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999

[13] J A K Suykens T Van Gestel J De Brabanter B DeMoor andJ Vandewalle Least Squares Support Vector Machines WorldScientific Singapore 2002

[14] X Yan and N A Chowdhury ldquoMid-term electricity marketclearing price forecasting a hybrid LSSVM and ARMAXapproachrdquo International Journal of Electrical Power amp EnergySystems vol 53 no 1 pp 20ndash26 2013

[15] Z-D Tian X-W Gao and T Shi ldquoCombination kernel func-tion least squares support vector machine for chaotic timeseries predictionrdquo Acta Physica Sinica vol 63 no 16 Article ID160508 2014 (Chinese)

[16] O Cagcag Yolcu ldquoA hybrid fuzzy time series approach based onfuzzy clustering and artificial neural network with single multi-plicative neuronmodelrdquoMathematical Problems in Engineeringvol 2013 Article ID 560472 9 pages 2013

[17] L Yu W Y Yue S Y Wang and K K Lai ldquoSupport vectormachine based multiagent ensemble learning for credit riskevaluationrdquo Expert Systems with Applications vol 37 no 2 pp1351ndash1360 2010

[18] Y Lv J Liu T Yang andD Zeng ldquoA novel least squares supportvector machine ensemble model for NO

119909emission prediction

of a coal-fired boilerrdquo Energy vol 55 pp 319ndash329 2013[19] Q Zhang P W-T Tse X Wan and G Xu ldquoRemaining useful

life estimation for mechanical systems based on similarity ofphase space trajectoryrdquo Expert Systems with Applications vol42 no 5 pp 2353ndash2360 2015

[20] B F Feeny and G Lin ldquoFractional derivatives applied to phase-space reconstructionsrdquoNonlinear Dynamics vol 38 no 1ndash4 pp85ndash99 2004

[21] L Su and C Li ldquoLocal prediction of chaotic time series basedon polynomial coefficient autoregressive modelrdquoMathematicalProblems in Engineering vol 2015 Article ID 901807 14 pages2015

[22] Y B Yuan and X H Yuan ldquoAn improved PSO approach toshort-term economic dispatch of cascaded hydropower plantsrdquoKybernetes vol 39 no 8 pp 1359ndash1365 2010

[23] L Wang Y G Li M F Abdul Ghafir and A Swingler ldquoARough Set-based gas turbine fault classification approach usingenhanced fault signaturesrdquo Proceedings of the Institution ofMechanical Engineers vol 225 no 8 pp 1052ndash1065 2011

[24] Z Y Zhang T Wang and X G Liu ldquoMelt index prediction byaggregated RBF neural networks trained with chaotic theoryrdquoNeurocomputing vol 131 pp 368ndash376 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


conventional cross-validation and grid search methods haveseveral defects such as high time consuming and a prioriknowledge requirement

In addition although a single LSSVM with optimalparameters and reconstructed input data samples may havean excellent prediction performance under certain circum-stances because its kernel function is fixed it perhaps hassome kinds of inherent bias under other cases In the litera-tures due to the super robustness and generalization ensem-ble model has been proved to be an effective way to reducebiases of single model Ensemble model can make full useof diversity to compensate for disadvantages among the indi-vidual members and the reasonable combination strategy isbelieved to be able to produce better prediction accuracy andgeneralization than singlemodel [13ndash16] By using combiningsubmodels the multilayer networks of LS-SVMs ensemblehave been discussed deeply which is very encouraging andpromising for further research [13] but up to now theapplication of LSSVM ensemble model for AEPP predictionis relatively fresh and untouched in the open literature

For ensemble model design there are two points thatshould be considered One is that the selected individualmembers need to exhibit much diversity (disagreement) andaccuracy The other is the effectiveness of the combinationstrategy [17] For the diversity of individual members itis an easy and common way to build individual membersby using data decomposition [18] However this methodis proved to be effective when the original data sampleis sufficient and it is not suitable for small-sample dataCompared with the existing combination strategies such assimple averaging weighting method mean squared errorweighting method and least squares estimation weightingaveraging method intelligent method based combinationstrategies include ANN combiner and SVM combiner whichhave become the current trend [18] however ANN com-biner cannot avoid falling into local optima and for SVMcombiner it is not easy to select appropriate kernel functionso it is necessary to further improve these ensemble strate-gies

As previously mentioned in this paper a new PSR-PSO-(SK)LSSVM-(MK)LSSVM ensemble (PPLLE) modelforAEPPprediction is proposed Firstly a set of diverse singlekernel LSSVMs are created as base predictors Subsequentlythese individualmember LSSVMsoutput the primary predic-tion results independently Finally all the primary predictionresults are combined to produce themost appropriate predic-tion results by another particular multiple kernel LSSVM Inthe process of modeling phase space reconstruction (PSR)theory is applied to extract the chaotic characteristic ofinput data source and reconstruct the data samples Particleswarm optimization algorithm is used to search the bestparameters for LSSVM members to ensure their predictionaccuracies

The rest of this paper is organized as followsThenext sec-tion provides a brief introduction to the related knowledgeSection 3 formulates the proposed PPLLE model For illus-tration purpose the detailed application on AEPP predictionand model comparisons is proposed in Section 4 Section 5concludes this study

2 An Overview of the Related Knowledge

21 Data Samples Reconstruction Based on PSR TheoryAlthough the nonlinear chaos behavior is the main challengeconfronting the chaotic data series prediction the underlyingdata generating mechanisms can still be explored by PSRtheory [19] By means of the ability of revealing the natureof dynamic system state the PSR theory is useful in systemcharacterization in nonlinear prediction and in estimatingbounds on the size of the system [20 21]

According to Takensrsquo theorem for the nonlinear timeseries 119909

119894119899

119894=1 the current state information can be represented

by an119898-dimensional vector

119909119894+120591= 119891 (119909

119894 119909119894minus120591 119909

119894minus(119898minus1)120591) (1)

where 120591 is the delay time119898 is the embedded dimension theyare two important parameters for phase space constructionand119891 is themapping relation between the inputs and outputs

The autocorrelation function of time series at the firstminimum value is taken as the delay time 120591 of the recon-structed phase space we write

CR =sum119899minus120591

119894=1(119909119894minus 119909) (119909

119894+120591minus 119909)

sum119899minus120591

119894=1(119909119894minus 119909)2

(2)

where 119909 is the mean of 119909119894

To calculate the correlation dimension the correlationintegral 119862(119903) needs to be computed

119862 (119903) =1

1198732

119873

sum119894=1

119873

sum119895=1

119906 (119903 minus10038171003817100381710038171003817119883119894minus 119883119895

10038171003817100381710038171003817) (3)

where 119903 is the selected radius and 119903 gt 0 119906(119909) is the Heavisidefunction

The correlation dimension 119889(119898) is calculated by theformula as below

119889 (119898) =ln119862 (119903)ln 119903

(4)

Suppose119884119894= 119909119894119883119894= (119909119894minus120591 119909119894minus2120591 119909

119894minus119898120591) then we can

reconstruct 119899 minus 119898 data samples 119883119894 119884119894119899

119898+1

22 Least Squares Support Vector Machine LSSVM is theleast squares form of a standard SVM it was firstly proposedby Suykens and Vandewalle [12] LSSVM uses a set of linearequations during the training process and chooses all trainingdata as support vectors so it has excellent generalization andlow computation complexity [12ndash14]

In LSSVM the regression issue can be expressed as thefollowing optimization problem

min 1

21198822+1

2120574

119873

sum119905=1

1198902

119905(120574 gt 0)

st 119910119905= ⟨119882 120593 (X

119905)⟩ + 119887 + 119890

119905

(5)

where 120593(X119905) is a nonlinear function which maps the input

data into a higher dimensional space 119890119905is the error at time

119905 119887 is the bias and 120574 is the regulation constant



119910119905=

119873

sum119905=1

120572119905119870(119883119883

119905) + 119887 (6)




119905)



119870(119883119883119905) = (119883119883

1015840

119905) (7)


119870(119883119883119905) = [(119883119883

119905) + 1]

119902 (8)


119870(119883119883119905) = exp(minus

1003817100381710038171003817119883 minus 11988311990510038171003817100381710038172

1205902) (9)


119870(119883119883119905) = tanh (] (119883119883

119905) + 119890) (10)


119870(119883119883119905) = 12058811198701(119883119883

119905) + sdot sdot sdot + 120588

119899119870119899(119883119883

119905)

st119899

sum119894=1

120588119894= 1 (0 le 120588

119894le 1)

(11)





119896= (1199091198961 1199091198962 119909

119896119889) is the




119896119889)


119892= (1199011198921 1199011198922 119901



updated as follows

V119905+1119894119889= [120596V119905

119894119889+ 11988811199031(119901119905

119894119889minus 119909119905

119894119889) + 11988821199032(119901119905

119892119889minus 119909119905

119894119889)]

119909119905+1

119894119889= 119909119905

119894119889+ V119905+1119894119889

(12)


1and 1199032are two


1 1198882 and



119905max119905

1198881= 1198881max minus

1198881max minus 1198881min119905max

119905

1198882= 1198882max minus

1198882max minus 1198882min119905max

119905

(13)







Individual

Individual


Combiner MK-LSSVM

Data source

PSR



PSO

Individual

y

x1

x2

xn



SK-LSSVM1

SK-LSSVM2

SK-LSSVMn







1 2 3

119899of the 119899 SK-




= 119891 (1 2 3

119899) (14)

4 Case Study





119894=1 according


119894minus5EGT

119894minus4 EGT


119894

and119884119894= EGT




119894 119884119894120



119894 119884119894148



121


121 2

121

4

121


121is


121to 148

can be got in turn



5

LSSVM1

(LKF-LSSVM)LSSVM

2

(PKF-LSSVM)LSSVM

3

(RBF-LSSVM)LSSVM

4

(SKF-LSSVM)LSSVM

5

(MK-LSSVM)119902 = 3

120574 = 9341

1205902= 032

120574 = 17682

V = 1 119890 = 1120574 = 12513

119902 = 2 1205902 = 051120588 = 027 120574 = 1683

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150690

700

710

720

730

740

750

Sample number

Observed data

EGT

(∘C)



MAPE = 1119896

119896

sum119894=1

10038161003816100381610038161003816100381610038161003816

119910119894minus 119894

119910119894

10038161003816100381610038161003816100381610038161003816times 100

MAE = 1119896

119896

sum119894=1

1003816100381610038161003816119910119894 minus 1198941003816100381610038161003816

MSE = 1119896

119896

sum119894=1

(119910119894minus 119894)2

TIC =radicsum119896

119894=1(119910119894minus 119894)2

radicsum119896

119894=1(119910119894)2+ radicsum

119896

119894=1(119894)2

(15)

where 119910119894and




1min =1198882min = 2 119888

1max = 1198882max = 3 120596min = 1 120596max = 3 119901 =


2simLSSVM

5are obtained and


Individual

Individual


Individual

EndYesNo i = i + 1

i = 121

yi

x1

i

x2

i

x4

i



Xi

i gt 148

predictor1

predictor2

predictor4


120 125 130 135 140 145 150690

700

710

720

730

740

750

Sample number


EGT

(∘C)









5model




120 125 130 135 140 145 150690

700

710

720

730

740

750

Sample number




EGT

(∘C)



5 Conclusions





120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

PPLLElowast


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

Single LSSVM






References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119910119905=

119873

sum119905=1

120572119905119870(119883119883

119905) + 119887 (6)




119905)



119870(119883119883119905) = (119883119883

1015840

119905) (7)


119870(119883119883119905) = [(119883119883

119905) + 1]

119902 (8)


119870(119883119883119905) = exp(minus

1003817100381710038171003817119883 minus 11988311990510038171003817100381710038172

1205902) (9)


119870(119883119883119905) = tanh (] (119883119883

119905) + 119890) (10)


119870(119883119883119905) = 12058811198701(119883119883

119905) + sdot sdot sdot + 120588

119899119870119899(119883119883

119905)

st119899

sum119894=1

120588119894= 1 (0 le 120588

119894le 1)

(11)





119896= (1199091198961 1199091198962 119909

119896119889) is the




119896119889)


119892= (1199011198921 1199011198922 119901



updated as follows

V119905+1119894119889= [120596V119905

119894119889+ 11988811199031(119901119905

119894119889minus 119909119905

119894119889) + 11988821199032(119901119905

119892119889minus 119909119905

119894119889)]

119909119905+1

119894119889= 119909119905

119894119889+ V119905+1119894119889

(12)


1and 1199032are two


1 1198882 and



119905max119905

1198881= 1198881max minus

1198881max minus 1198881min119905max

119905

1198882= 1198882max minus

1198882max minus 1198882min119905max

119905

(13)







Individual

Individual


Combiner MK-LSSVM

Data source

PSR



PSO

Individual

y

x1

x2

xn



SK-LSSVM1

SK-LSSVM2

SK-LSSVMn







1 2 3

119899of the 119899 SK-




= 119891 (1 2 3

119899) (14)

4 Case Study





119894=1 according


119894minus5EGT

119894minus4 EGT


119894

and119884119894= EGT




119894 119884119894120



119894 119884119894148



121


121 2

121

4

121


121is


121to 148

can be got in turn



5

LSSVM1

(LKF-LSSVM)LSSVM

2

(PKF-LSSVM)LSSVM

3

(RBF-LSSVM)LSSVM

4

(SKF-LSSVM)LSSVM

5

(MK-LSSVM)119902 = 3

120574 = 9341

1205902= 032

120574 = 17682

V = 1 119890 = 1120574 = 12513

119902 = 2 1205902 = 051120588 = 027 120574 = 1683

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150690

700

710

720

730

740

750

Sample number

Observed data

EGT

(∘C)



MAPE = 1119896

119896

sum119894=1

10038161003816100381610038161003816100381610038161003816

119910119894minus 119894

119910119894

10038161003816100381610038161003816100381610038161003816times 100

MAE = 1119896

119896

sum119894=1

1003816100381610038161003816119910119894 minus 1198941003816100381610038161003816

MSE = 1119896

119896

sum119894=1

(119910119894minus 119894)2

TIC =radicsum119896

119894=1(119910119894minus 119894)2

radicsum119896

119894=1(119910119894)2+ radicsum

119896

119894=1(119894)2

(15)

where 119910119894and




1min =1198882min = 2 119888

1max = 1198882max = 3 120596min = 1 120596max = 3 119901 =


2simLSSVM

5are obtained and


Individual

Individual


Individual

EndYesNo i = i + 1

i = 121

yi

x1

i

x2

i

x4

i



Xi

i gt 148

predictor1

predictor2

predictor4


120 125 130 135 140 145 150690

700

710

720

730

740

750

Sample number


EGT

(∘C)









5model




120 125 130 135 140 145 150690

700

710

720

730

740

750

Sample number




EGT

(∘C)



5 Conclusions





120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

PPLLElowast


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

Single LSSVM






References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Individual

Individual


Combiner MK-LSSVM

Data source

PSR



PSO

Individual

y

x1

x2

xn



SK-LSSVM1

SK-LSSVM2

SK-LSSVMn







1 2 3

119899of the 119899 SK-




= 119891 (1 2 3

119899) (14)

4 Case Study





119894=1 according


119894minus5EGT

119894minus4 EGT


119894

and119884119894= EGT




119894 119884119894120



119894 119884119894148



121


121 2

121

4

121


121is


121to 148

can be got in turn



5

LSSVM1

(LKF-LSSVM)LSSVM

2

(PKF-LSSVM)LSSVM

3

(RBF-LSSVM)LSSVM

4

(SKF-LSSVM)LSSVM

5

(MK-LSSVM)119902 = 3

120574 = 9341

1205902= 032

120574 = 17682

V = 1 119890 = 1120574 = 12513

119902 = 2 1205902 = 051120588 = 027 120574 = 1683

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150690

700

710

720

730

740

750

Sample number

Observed data

EGT

(∘C)



MAPE = 1119896

119896

sum119894=1

10038161003816100381610038161003816100381610038161003816

119910119894minus 119894

119910119894

10038161003816100381610038161003816100381610038161003816times 100

MAE = 1119896

119896

sum119894=1

1003816100381610038161003816119910119894 minus 1198941003816100381610038161003816

MSE = 1119896

119896

sum119894=1

(119910119894minus 119894)2

TIC =radicsum119896

119894=1(119910119894minus 119894)2

radicsum119896

119894=1(119910119894)2+ radicsum

119896

119894=1(119894)2

(15)

where 119910119894and




1min =1198882min = 2 119888

1max = 1198882max = 3 120596min = 1 120596max = 3 119901 =


2simLSSVM

5are obtained and


Individual

Individual


Individual

EndYesNo i = i + 1

i = 121

yi

x1

i

x2

i

x4

i



Xi

i gt 148

predictor1

predictor2

predictor4


120 125 130 135 140 145 150690

700

710

720

730

740

750

Sample number


EGT

(∘C)









5model




120 125 130 135 140 145 150690

700

710

720

730

740

750

Sample number




EGT

(∘C)



5 Conclusions





120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

PPLLElowast


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

Single LSSVM






References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











5

LSSVM1

(LKF-LSSVM)LSSVM

2

(PKF-LSSVM)LSSVM

3

(RBF-LSSVM)LSSVM

4

(SKF-LSSVM)LSSVM

5

(MK-LSSVM)119902 = 3

120574 = 9341

1205902= 032

120574 = 17682

V = 1 119890 = 1120574 = 12513

119902 = 2 1205902 = 051120588 = 027 120574 = 1683

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150690

700

710

720

730

740

750

Sample number

Observed data

EGT

(∘C)



MAPE = 1119896

119896

sum119894=1

10038161003816100381610038161003816100381610038161003816

119910119894minus 119894

119910119894

10038161003816100381610038161003816100381610038161003816times 100

MAE = 1119896

119896

sum119894=1

1003816100381610038161003816119910119894 minus 1198941003816100381610038161003816

MSE = 1119896

119896

sum119894=1

(119910119894minus 119894)2

TIC =radicsum119896

119894=1(119910119894minus 119894)2

radicsum119896

119894=1(119910119894)2+ radicsum

119896

119894=1(119894)2

(15)

where 119910119894and




1min =1198882min = 2 119888

1max = 1198882max = 3 120596min = 1 120596max = 3 119901 =


2simLSSVM

5are obtained and


Individual

Individual


Individual

EndYesNo i = i + 1

i = 121

yi

x1

i

x2

i

x4

i



Xi

i gt 148

predictor1

predictor2

predictor4


120 125 130 135 140 145 150690

700

710

720

730

740

750

Sample number


EGT

(∘C)









5model




120 125 130 135 140 145 150690

700

710

720

730

740

750

Sample number




EGT

(∘C)



5 Conclusions





120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

PPLLElowast


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

Single LSSVM






References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














5model




120 125 130 135 140 145 150690

700

710

720

730

740

750

Sample number




EGT

(∘C)



5 Conclusions





120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

PPLLElowast


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

Single LSSVM






References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

PPLLElowast


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E


120 125 130 135 140 145 150minus3

minus2

minus1

0

1

2

3

Sample number

EG

T RP

E

Single LSSVM






References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A New Least Squares Support Vector ...downloads.hindawi.com/journals/mpe/2016/4615903.pdf · Research Article A New Least Squares Support Vector Machines Ensemble