Research ArticleA Kind of Urban Road Travel Time Forecasting Model withLoop Detectors

Guangyu Zhu123 Li Wang12 Peng Zhang4 and Kang Song12

1MOEKey Laboratory for Transportation Complex SystemsTheory and Technology Beijing Jiaotong University Beijing 100044 China2Center of Cooperative Innovation for Beijing Metropolitan Transportation Beijing 100044 China3Key Laboratory of System Control and Information Processing Ministry of Education Shanghai 200240 China4Beijing Municipality Key Laboratory of Urban Traffic Operation Simulation and Decision SupportBeijing Transportation Research Center Beijing 100073 China

Correspondence should be addressed to Guangyu Zhu gyzhubjtueducn

Received 2 December 2015 Accepted 18 January 2016

Academic Editor Zhipeng Cai

Copyright copy 2016 Guangyu Zhu 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

Urban road travel time is an important parameter to reflect the traffic flow state Besides it is one of the important parameters forthe trafficmanagement department to formulate guidancemeasures provide traffic information service and improve the efficiencyof the detectors group Therefore it is crucial to improve the forecast accuracy of travel time in traffic management practice Basedon the analysis of the change-point and the ARIMA model this paper constructs a model for the massive data collected by loopdetectors to forecast travel time parameters Firstly the preprocessing algorithm for the data of loop detectors is given and thecalculating model of the travel time is studied Secondly a change-point detection algorithm is designed to classify the sequenceof large number of travel time data items into several patterns Then this paper establishes a forecast model to forecast travel timein different patterns using the improved ARIMA model At last the model is verified by simulation and the verification results ofseveral groups of examples show that the model has high accuracy and practicality

1 Introduction

The travel time (TT) refers to the average time of all vehiclesto pass a section of a road as is shown in Figure 1 If119879

119894119895means

the time that vehicle 119894 (119894 = 1 2 119898) travels from detector119897119895to detector 119897

119895+1(119895 = 1 2 119899 minus 1) then the travel time of

section [1198971 119897119899] can be defined as TT

119894= sum119898

119894=1(sum119899minus1

119895=1119879119894119895)119898

Urban road travel time is an important parameter toreflect the state of traffic flow of a road [1 2] Based on theforecast information of travel time the traveler can choosetheir travel route reasonably [3] and the traffic managementdepartment can establish impeccable guiding measures [4]Thus the precise forecast of travel time plays important rolein improving the quality of urban traffic information serviceand the efficiency of detector group on the road [5] whichhas drawn great attention from scholars all the time

Mori et al give a thorough classification of the methodsfor travel time forecasting and they divide the forecasting

model into naive model traffic flow model data modeland hybrid model [6] Vlahogianni et al give a short-termtraffic forecasting method of where we are and where weare going [7] Shao et al give the method of real-timetravel time forecasting based on the improved Kalman filter[8] Chilukuri et al forecast the short travel time of thehighway by using microsimulation technique [9] Yao andZhang give the short-term forecasting algorithm of intervaltravel time for urban freeway by analyzing the floating cardata which provides the basis for the subsection forecastof the travel time [10] Zhao et al propose a forecastingalgorithm based on equal interval interpolation and Sage-Husa adaptive Kalman filtering which effectively improvethe forecast accuracy of travel time [11] Gui and Yu comeup with a new idea for travel time forecast by establishinga forecast model with the selective forgetting ability whichenables the algorithm to adapt to trip conditions changes well[12]

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2016 Article ID 9043835 11 pageshttpdxdoiorg10115520169043835

2 International Journal of Distributed Sensor Networks

O D

l1 l2 li ln

d1 dnminus1

Figure 1 Distribution of the loops on the road

The literatures which have beenmentioned above providesome idea for this paper to forecast the travel time based onmassive data collected from loop detectors First travel timeand the traffic state of the road have a certain correlationBesides most of the travel time forecast models are usinghistorical data for analysis And the characteristics of trafficflow tend to change with the seasons and the environment incertain regularity Thus if the traffic flow can be divided intoseveral state intervals this means that the different intervalsin the same pattern have the similar statistical characteristicsof the mean and variance As a result it is easier to get themore optimized forecast result than to obtain it by using theglobal search

Therefore this paper proposes a travel time forecastingmodel based on change-point detection which uses thechange-point detection to identify different patterns of traveltime series and set up the forecasting model by ARIMA ineach of the patterns

The rest of the content of this paper is summarized asfollows

(1) Preprocessing of the massive data collected by theloop detectors and calculation of travel time param-eter

(2) The pattern partition of travel time series based onchange-point analysis and setting up the forecastingmodel based on ARIMA

(3) Verification of the travel time forecastingmodel basedon actual data

2 Identification and Correction ofthe Loop Detectorrsquos Data

Because of the reasons such as the detector fault the faultof communication system and the environmental factorthe real-time detector data contain some unpredictable datamissing or invalid data Therefore it is necessary to pre-process the data collected by the traffic detectors [13] Sothis paper gives the basic rules to identify and correct loopdetectorrsquos data based on practical experience

Basic Rule 1 When the data of traffic volume speed andoccupancy rate is negative or null it is recorded as the errordata When the data of volume is significantly greater com-pared to the maximum volume of road (119876max = 119891

11988811986211987960)

it is recoded as the error data When the data of speed issignificantly greater compared to the maximum allowablespeed or capacity of the urban road it is recorded as the errordata When the data of occupancy rate is not less than 100it is recorded as error data

Basic Rule 2 When the data of occupancy rate is greater thansome reasonable threshold such as 95 and the data of speedis greater than the normal range such as 5 kmh it is recordedas the error dataWhen the speed is zero and the volume is notzero the data is the error data When the volume is zero andthe occupancy rate is not zero the data is the error dataWhenthe average effective vehicle length (AEVL (m) = (10 times V times

ℎ)119902 V speed ⟨kmh⟩ ℎ occupancy rate ⟨⟩ and 119902 volume⟨vehicle volume(lanehour)minus1⟩) is beyond reasonable limits(such as AEVL isin [15m 30m]) it is the error data

Basic Rule 3 Each data item should be recorded similarly by119899 piece of data in time 119905 before it And the data should bedone in first-order difference If the difference value of first-order does not belong to the reasonable change range madeby 119899 data before it this data can be defined as the abruptlychanging distortion data

The data collected by the detectors can be expressed asfour-tuple structure that is [119905 119902 V ℎ] Based on the basicrules which have been discussed above this paper proposesan algorithm to accomplish the real-time identification andcorrection of loop detectorrsquos data

Algorithm 1Step 1 It is determining 119876max 119881max and 119867max according tothe actual situation of the detectors on the road Test all ofthe data if 119902 gt 119876max or V gt 119881max or ℎ gt 119867max the data willbe defined as error data

Step 2 When the occupancy rate ℎ gt 1198671(1198671

= 95) andV gt 1198811(1198811= 5 kmh) the data is recorded as the error data

Step 3 When V = 0 (kmh) and 119902 = 0 (vehiclenumber(lanehour)minus1) the data is the error data

Step 4 When 119902 = 0 (vehicle number(lanehour)minus1) and V =

0ampampℎ = 0 the data will be defined as error data

Step 5 Calculate the average effective vehicle length (AEVL)according to the current detected data If AVEL notin [15 30]exclude this data

Step 6 It is using the reasonable and nearest data to replacethose error data which have been found in Steps 1ndash5

Step 7 It is using the first-order differential operation toprocess data If the difference value does not belong to thereasonable change range made by the differential mean valueand variance of 119899 piece of data before it (such as [119889minus120576

0lowast120590119889 119889+

1205760lowast 120590119889]) this data can be defined as the abruptly changing

distortion data Then use 119901119899minus1

+ (11205760) lowast 119889119899to replace it

3 Calculation of Travel Time Based onthe Data from Loop Detector

Using the preprocessed data the travel time parameter valuescan be calculated [14]The calculation result of the travel timeparameter is usually related to the speed of the vehicle on theroad

International Journal of Distributed Sensor Networks 3

Assume the speed is conformed to liner change and theupstream and downstream of each section of the road have adetector and each trip chain has multiple parts so that V

119894(119905)

speed of the vehicle between detector 119896 and detector 119896 + 1can be expressed as

V119894 (119905) = 119881 (119896 119889) +

119904119894 (119905) minus 119904

119896

119904119896+1

minus 119904119896

(119881 (119896 + 1 119889) minus 119881 (119896 119889)) (1)

119904(119905) = int V(119905)119889119905 and this equation is a standard differentialequation Due to differential equations it is very difficultto obtain exact solutions generally we need to seek anapproximation to replace it So 119904(119905) can be obtained byformula

119904119894 (119905) = 119904

0

119894+ (

119881 (119896 119889) (119904119896+1 minus 119904119896)

119881 (119896 + 1 119889) minus 119881 (119896 119889)+ 119904119896+1

minus 119904119896)

sdot (119890[(119881(119896+1119889)minus119881(119896119889))(119904119896+1minus119904119896)](119905minus119905

0

119894)minus 1)

(2)

119904119896+1

119904119896stand for the location of detectors 119896 + 1 and 119896

119881(119896 119889) stands for the speed of detector 119896 in time period 119889

and also is the slope of the vehicle motion curve 119904119894(119905) stands

for specific motion trajectories of time period 119889 within roadsection 119896 1199040

119894 1199050

119894 is the initial state of the vehicle entering

119896 119889 range If 119881(119896 + 1 119889) = 119881(119896 119889) formula (2) is

119904119894 (119905) = 119904

0

119894+ 119881 (119896 119889) (119905 minus 119905

0

119894) (3)

According to formulas (1) (2) and (3) the formula fortime calculation of the road section based on the vehiclemoving track can be divided into two situations

(1) When the speed is fast consider the following

1199040

119894+ (

119881 (119896 119889) (119904119896+1 minus 119904119896)

119881 (119896 + 1 119889) minus 119881 (119896 119889)+ 1199040

119894minus 119904119896)

sdot (119890[(119881(119896+1119889)minus119881(119896119889))(119904119896+1minus119904119896)](119905minus119905

0

119894)minus 1) gt 119904

119896+1

119904lowast

119894= 119904119894(119905lowast

119894) = 119904119896+1

(4)

The approximate result for the travel time of the roadsection is

119905lowast

119894asymp 1199050

119894+

(119904119896+1

minus 119904119896)

119881 (119896 + 1 119889) minus 119881 (119896 119889)

sdot ln(119881 (119896 119889) (119904119896+1 minus 119904

119896) (119881 (119896 + 1 119889) minus 119881 (119896 119889)) + 119904

119896+1minus 119904119896

119881 (119896 119889) (119904119896+1 minus 119904119896) (119881 (119896 + 1 119889) minus 119881 (119896 119889)) + 119904

0

119894minus 119904119896

)

(5)

(2) When the speed is slow consider the followingWhen 119904

119894(119905119889+1

) lt 119904119896+1

the travel time obtained by for-mula (2) is

119905lowast

119894= 1199050

119894+

119904119896+1

minus 1199040

119894

119881 (119896 119889) (6)

This algorithm based on the method mentioned abovecan be summarized as follows

Algorithm 2Step 1 Launch vehicles and select sections (119870 = 1)

Step 2 For vehicle in 1199040

119894 1199050

119894 and 119896 119889 use formulas (2) (3)

(5) and (6) to calculate 119905lowast

119894 and 119904

lowast

119894 when it runs out of the

area If 119904lowast119894

= 119904119896+1

set 119896 = 119896 + 1 and move to next sectionOtherwise 119905lowast

119894= 119905119889+1

set 119889 = 119889+1 and move to the next datacollection period

Step 3 If 119896 ge 119898 it means that the vehicle has arrived atthe destination Record the departure time and arrival timeof the vehicle and then stop Otherwise go back to Step 2 torecalculate the travel time

Step 4 Set 119894 = 119894+1Thenumber of 119894+1 vehicles has continuousheadway at 1199040

119894+1 1199050

119894+1+ ℎ If 1199050

119894+1+ ℎ gt 119899 stop That indicates

that the calculation of the specified time period has beencompleted

This algorithm is firstly assuming themotion trajectory ofthe vehicle on the road and then through using the location-time curve gets the time at which a vehicle runs out of thedetector area to obtain the travel time [15] This method oftravel time estimation through time space motion trajectoryhas high accuracy The error between the results of thecalculation in [13 16] and the result of this algorithm is below6 which means that this algorithmrsquos result is acceptable

4 Forecast Model for Short-Term Travel TimeBased on ARIMA

41 Change-Point Searching Because the traffic data hasdifferent numerical characteristics in different time periodsit can be divided into numbers of similar small states byconditional change-point searching which can effectivelyimprove the fitting degree of the model In [17 18] a newalgorithm for state division based on the demand variationof the observation function is introduced The mean andthe variance of the sequence can be expressed by statisticalformula

Whole sequence is

119909119894

1003816100381610038161003816

119899

1=

sum119899

119894=1119909119894

119899

Dev (119909119894

1003816100381610038161003816

119899

1) =

119899

sum

119894=1

(119909119894minus 119909119894

1003816100381610038161003816

119899

1)2

(7)

Convex (concave) wave is

119909119905

1003816100381610038161003816

119897119895

119897119895minus1=

(119909119897119895minus1

+ sdot sdot sdot + 119909119897119895)

(119897119895minus 119897119895minus1

+ 1)

Dev (119909119894

1003816100381610038161003816

119897119895

119897119895minus1) =

119897119895

sum

119905=119897119895minus1

(119909119905minus 119909119905

1003816100381610038161003816

119897119895

119897119895minus1)

2

119895 = 2 119896

(8)

4 International Journal of Distributed Sensor Networks

119897119895 is valley point of peak carve or peak points of the valley

curveThe observation function decides whether to retain the

possible change-point Before that the control parameter andobservation function values of the minimum state variableare required

Observation function is

119879 (119908) = (

119861 (119908 119909119894

1003816100381610038161003816

119899

1)

119861 (119908 + 1 119909119894

1003816100381610038161003816

119899

1)

) 119908 = 0 1 119869 minus 1

119861 (119908 119909119894

1003816100381610038161003816

119899

1)

=

119908minus1

sum

119892=0

Dev(119909119905

1003816100381610038161003816

119887(119892+1)minus1

119887(119892) ) + Dev (119909

119905

1003816100381610038161003816

119899

119887(119908)) 119908 = 1 119869

Dev (119909119894

1003816100381610038161003816

119899

1) 119908 = 0

(9)

119887(119908)

is the index set of all 119869 points on the curve and 119887(0)

=

1199091198971

= 1199091

The algorithm is as follows

Algorithm 3Step 1 Make the travel time series into carve 119906 represents thetime of cycles119908 represents the number of change-points and119890119890is the auxiliary variable Set 119906 = 1 119908 = 1 119890

119890= 0 and

120573119895= 119897119895 119895 = 1 119896

Step 2 Select two convex waves along the axis of time from120573119906on the carve such as (120573

119906minus 120573119906+1

minus 120573119906+2

) Then calculate 119905lowast

119860 (120573119906 120573119906+1

120573119906+2

)

=

120573119906+1

sum

119905=120573119906

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+1

120573119906)

2

+

120573119906+2

sum

119905=120573119906+1+1

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+2

120573119906+1+1)

2

119872 (120573119906 119905lowast

1 120573119906+2

)

= max(

119905

sum

119905=120573119906

(119909119905minus 119909119905

1003816100381610038161003816

119905

120573119906)2

+

120573119906+2

sum

119905=119905+1

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+2

119905+1)

2

)

119898 (120573119906 119905lowast

2 120573119906+2

)

= min(

119905

sum

119905=120573119906

(119909119905minus 119909119905

1003816100381610038161003816

119905

120573119906)2

+

120573119906+2

sum

119905=119905+1

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+2

119905+1)

2

)

(10)

Step 3 Change-point estimation is

119877119906=

2119860 (120573119906 120573119906+1

120573119906+2

)

(119872 (120573119906 119905lowast

1 120573119906+2

) + 119898 (120573119906 119905lowast

2 120573119906+2

))

119888119903119906=

119872(120573119906 119905lowast

1 120573119906+2

)

119898 (120573119906 119905lowast

2 120573119906+2

)

(11)

Consider the following

(1) If 119872(120573119906 119905lowast

1 120573119906+2

) asymp 119898(120573119906 119905lowast

2 120573119906+2

) 1 le 119888119903119906

lt 13and the sequence change is gently you do not need toset change-point in (119897

119906 119897119906+2

)

(2) If 119877119906

le (14) + 32(119888119903119906+ 1) regardless of whether

119905lowast

1= 120573119906+1

there is 119887(119908) = 120573119906+1

(3) If 1 ge 119877119906

ge (14) + 32(119888119903119906+ 1) seek the nearest

peak or valley point of 120573119906+1

and set the left oneas 120573119871

119906+1and right one as 120573

119877

119906+1 If exist119905

119888isin 120573

119871

119906+1

120573119877

119906+1 have Dev(120573

119906 119905119888 120573119906+2

) = min(Dev(120573119906 120573119871

119906+1

120573119906+2

)Dev(120573119906 120573119877

119906+1 120573119906+2

)) Then 119887(119908)

= 119905119888

(4) If 119877119906gt 1 then 119887

(119908)= 119905lowast

2

(5) Make effective judgments for 119887(119908) offered by (2) (3)

and (4) if 119879(119908 minus 1) minus 1 gt 1205760 keep this change-point

otherwise remove it

Step 4 If you have 119887(119908) then 119906 = 119906 + 1 120573

119906= 119887(119908) 120573119906+1

=

120573119906+1

120573119906+2

= 120573119906+2

119908 = 119908 + 1 and 119890119890= 0 Otherwise 119890

119890=

1 120573119906

= 120573119906 120573119906+1

= 120573119906+1+119890119890

and 120573119906+2+119890119890

= 120573119906+2+119890119890

When120573119906+2

= 120573119896 the searching is complete and the algorithm ends

Otherwise go back to Step 2

The paper [18] has provided a complete method on howto improve this kind of algorithm However there has beena crucial control parameter 119890

0for which the algorithm does

not give the specific processing formula That algorithmsets 119890

0as a constant value such as 05 In this paper we

will carry out several experiments with different 1198900 which

provide reference to the parameter of travel time inferredfrom detectorsrsquo data

42 ARIMA Forecasting Model The preprocessing of thetime series short-term forecasting model includes stationarytest and random test If the time series is nonstationary itneeds to be transformed into stationary series by differentialoperation In this circumstance the ARIMA model is con-verted to ARMA model The sequence of 119889 order differencesis expressed as

nabla119889119909119905=

119889

sum

119894=1

(minus1)119889119862119894

119889119909119905minus119894

(12)

ARIMA (119901 119889 119902) model can be expressed as

Φ (119861) nabla119889119909119905= Φ119901 (119861) 120576119905

119864 (120576119905) = 0

Var (120576119905) = 1205902

120576

119864 (120576119905120576119904) = 0 119904 = 119905

119864119909119904120576119905= 0 forall119904 lt 119905

(13)

International Journal of Distributed Sensor Networks 5

The order number 119901 119902 of ARIMA (119901 119889 119902)model is basedon Autocorrelation Coefficient (ACF) and Partial Auto-correlation Coefficient (PACF) of ARMA after differentialoperation And according to the characteristics of PACF andACF coefficients the model identification is carried out

For random inspection the data collected by the detectoris a large density data point so the calculation result of traveltime is also a large sample of high density which needs to testthe hypothesis by using 119876 statistics

119876 = 119899

119898

sum

119896=1

2

119896sim 1205942(119898) (119898 is delay-stage) (14)

When119876 is less than the quintile of 12059421minus120572

(119898) the sequenceis pure random sequence However when the travel time ismodeled by the ARIMA model if the sample space becomessmall the modified LB statistics can be used

LB = 119899 (119899 + 2)

119898

sum

119896=1

(2

119896

119899 minus 119896) (15)

Because there is only a short-term significant correlationin the sequence the test for the hypothesis is only for 119876 andLBwith short-termdelay-stage which is generally119898 less than10

After differential operation the ARIMA model isdegraded to the standard ARMAmodel its standard form is

119909119905= 120583 +

Φ119901 (119861)

Φ119901 (119861)

120576119905 (16)

In the formula

120576119905sim 119882119873(0 120590

2

119905)

Φ119901 (119861) = 1 minus 120579

1119861 minus sdot sdot sdot minus 120579

119901119861119901

Φ119901 (119861) = 1 minus 120593

1119861 minus sdot sdot sdot minus 120593

119902119861119902

(17)

There are 119901 + 119902 + 2 unknown parameters in ARMAmodel120601

1 1206012 120601

119901 1205791 1205792 120579

119902 120583 1205902

120576Matrix estimation is

usually used to obtain the value of 120583 and 1205902

120576

Calculate the expectation and variance of formula (16) oneach side and get

= 119864 (119909) = 119909 =sum119899

1119909119894

119899

1205902

120576=

sum119899

1(119909119894minus 119909)2

119899=

1 + 1205932

1+ sdot sdot sdot + 120593

2

119901

1 + 1205792

1+ sdot sdot sdot + 1205792

119901

1205902

119909

(18)

The parameters of the equation are reduced to thenumber of 119901 + 119902 Least square estimation is used to estimatethe parameters of the ARMA model which is obtained bydifferential operation

In the case of ARMA (119901 119902) the parameter vector is

= (1206011 120601

119901 1205791 120579

119902)1015840

119865119905() = 120601

1119909119905minus1

+ 1206012119909119905minus2

+ sdot sdot sdot + 120601119901119909119905minus119901

+ 120576119905minus 1205791120576119905minus1

minus 1205792120576119905minus2

minus sdot sdot sdot minus 120579119902120576119905minus119902

(19)

Thus overall observed sum of squared residuals of thesample is

119876() =

119899

sum

1

1205762

119905=

119899

sum

1

(119909119905minus 119865119905())2

(20)

Set the objective function for parameter estimation asmin119876() and use the weighted least squares method to solvethe parameters

The essence of the weighted least square method is totransform the original data to obtain the new explanatoryvariables and explained variable Assume that 119909

119894is time series

data and 120596 is the weight of this point so that weighted leastsquare method is

1199091015840

119894= 119909119894sdot 120596119894

(1199091015840 refers to travel time after transformation)

(21)

Then use (11990910158401 119909

1015840

119899) to do the least square parameter

estimation of ordinaryARIMAmodel we can get the optimalparameter vector

1015840

of themodel inweighted transformationIn this way the formula for the weighted forecast formula is

119865119905(1015840

) = 1205931015840

1119909119905minus1

+ 1205931015840

2119909119905minus2

+ sdot sdot sdot + 1205931015840

119901119909119905minus119901

+ 120576119905

minus 1205791015840

1120576119905minus1

minus 1205791015840

2120576119905minus2

minus sdot sdot sdot minus 1205791015840

119902120576119905minus119902

(22)

After removing theweight the final forecast value is obtained

119865119905() =

119865119905(1015840

)

120596119905

(23)

5 Application Example

51 Preprocessing of the Massive Data from Loop DetectorsThis paper takes the actual data of 2nd ring road in a big cityas an example (detector number is 020lowastlowast line number is Lan1ndashLan 6 date is on Mar 3rd 2013 data collection time is 24hours sampling interval is 2 minutes parameters are trafficvolume speed and occupancy rate and the total number ofdata points is 720) to verify the travel time forecasting modelbased on loop detectors which has been mentioned aboveThe actual data of Lan 1 is shown in Figure 2

In Figure 2 mutation points can be observed in the dataseries of all the three parameters In fact the data of trafficconditions cannot change more than 500 times within twominutes So it can be concluded that there are abnormal ordistorted data in the actual data and it is necessary to filterthose data

According toAlgorithm 1 we can finish the data cleaningFirstly according to the definition the control parametersbased on the basic traffic flow principle and the actualphysical meaning are set up in Table 1

6 International Journal of Distributed Sensor Networks

Table 1 Selection of filtering modelrsquos parameters

Parameter name Abbreviation Experimental valuesSection maximum flow 119902max 119902max = 360030Section maximum speed Vmax Vmax = 150Section maximum occupancy rate 119900max 119900max = 100Constraints of occupancy rate and speed 119867

1 1198811

1198671= 95 119881

1= 5

119902 volume ⟨vehicle volume(lanehour)minus1⟩ V speed ⟨kmh⟩ and 119900 occupancy rate ⟨⟩

00000000 0600 1800120000000000 0600 1800120000000000 0600 18001200

0

20

40

60

80

50

100

150

200

0

10

20

30

40

50

Figure 2 Data distribution of Lan 1rsquos volume-speed-occupancy rate in 24 hours

100 200 300 400 500 600 700

minus04

minus02

02

04

Figure 3 Fault data points of filtering intermediate state search

Under the control of parameters listed in Table 1 wecan finish the data cleaning to find out the data beyond themaximum control range or contrary to the theory of trafficflow The result is shown in Figure 3

Use a one-dimensional matrix to record the effectivenessof each record All the initial value of thematrix is 1When theabnormal data is detected the corresponding matrix valueis changed into 0 From Figure 3 there is a series of errordata points at the time 300ndash600 which is consistent with theoriginal graph shown in Figure 4

Because these error data points do not have actual physi-cal meaning they are replaced by the closest normal recordAfter the cleaning the figure of volume-speed-occupancyrate is shown in Figure 5

Data quality has been improved to a certain extentespecially for the speed data But there still has beenmutationin the filtering results Test the first-order differential ofthe data to determine the mutation data The first-orderdifference graph of intermediate state is shown in Figure 6

Table 2 Result of the filter by different parameters

1198900value 119890

0= 3 119890

0= 4 119890

0= 5

Number of change ranges beyond 83 34 21Rate of the data beyond 115 47 21

Control parameter of differential operational is 1198900

Assuming that 1198900

= 3 is the parameter of the reasonablechange region if the actual value of the first-order is exceededthree times of the standard deviation control range changethe corresponding position of the effective matrix into 0

When 1198900

= 3 there are 83 abnormal data points as isshown in Figure 7 which account for 115 of the total recordThis result is too strict for the data of detectors so we canincrease the value of 119890

0to release the strictness for change

range of the data as is shown in Table 2According to the results shown in the table 119890

0= 4 is more

reasonable Figure 8 is comparison chart between the finalresult of the filter and the original data and it can be seenthat the algorithm has basically achieved the requirements ofthe loop detector datarsquos cleaning and preprocessing

FromFigure 3 it can be seen that the algorithmhas a smallcorrection for the traffic data and the data with the occupancyrate but the algorithm has better effect on the speed data Inthe process of predicting travel time the speed of the detectoris often used only which means that the algorithm can besimplified so that it only needs to produce speed data

Because only the speed data is processed we need to setthe upper and lower limit of speed and the limit of first-orderdifferential change range to restrict data As a result of usingthe detector data to calculate and predict travel time we needthree continuous detectorrsquos data points to simulate the traveltime forecast inwhole roadnetworkThe sketchmap is shownin Figure 9

International Journal of Distributed Sensor Networks 7

00000000 0600 18001200

0

20

40

60

80

00000000 0600 18001200

50

100

150

200

00000000 0600 18001200

0

10

20

30

40

50

Figure 4 Comparison between the actual data and the results of the filterrsquos intermediate state

00000000 0600 18001200 00000000 0600 18001200 00000000 0600 18001200

0

20

40

60

80

20

40

60

80

100

120

0

10

20

30

40

50

Figure 5 Result of filterrsquos intermediate state

Figure 6 First-order difference graph of intermediate state

100 200 300 400 500 600 700

01

02

minus01

minus02

Figure 7 Distortion data distribution when 1198900= 3

Under the condition that 1198900= 3 the filter result of speed

after cleaning of the three detectors is as shown in Figures 1011 and 12

As shown in Figures 10sim12 the red correction curvebasically achieved a reasonable correction of the distortiondata

52 Calculation of Travel Time Parameters Use the traveltime conversion model given by formulas (5) and (6) wecan do the travel time conversion according to the speeddata For example when one calculates the travel timedriving from east to west the vehicles pass through sec-tions Detector 1 Detector 2 and Detector 2 Detector 3The result is shown in Figure 13 and the unit of time is sm

53 Pattern Partition of Travel Time Series Based on Change-Point Analysis Because it is not clear how to choose 119890

0up to

the size of the sample we select 7 values to search the change-point The result is in Table 3

According to the characteristics of the travel time serieswe need to select the result that has 5 to 10 change-points So1198900= 007 119890

0= 005 and 119890

0= 002 are all reasonable control

parameters The results of all the reasonable circumstanceare shown in Figure 14 We use different colors to indicatedifferent state

According to the time sequence diagram the results of thealgorithm are basically completed and the travel time seriesis decomposed into a series of time periods with practicalmeaning

8 International Journal of Distributed Sensor Networks

50

100

150

200

0

10

20

30

40

50

00000000 0600 18001200 00000000 0600 18001200 00000000 0600 18001200

0

20

40

60

80

Figure 8 Comparison of the source data and the filter result

Detector 1Detector 2Detector 3

4494m 157m

Figure 9 Distribution of the three serial detectors on the road

0800 1200 1600 20000

20

40

60

80

Figure 10 Filter result of the speed data of Detector 1

0800 1200 1600 20000

10

20

30

40

50

60

Figure 11 Filter result of the speed data of Detector 2

On the situation of 1198900

= 007 and 1198900

= 005 thereare 90 important change-points that were missed So 119890

0=

002 and 1 25 90 224 375 380 384 390 393 414 are idealcontrolling parameter and result of change-point control

0800 1200 1600 20000

10

20

30

40

50

60

Figure 12 Filter result of the speed data of Detector 3

0800 1200 1600 200000

01

02

03

04

Figure 13 Travel time calculation result from Detector 1Detector 2 to Detector 2 Detector 3

And the travel time is divided into 9 divisions 600sim650650sim900 900sim1328 1328sim1830 1830sim1838 1838sim18501850sim1856 1856sim1948 and 1948sim2300

Then if we assume that this moment is 1700 we canpredict the travel time of 1702 and 1704 The historical dataof the forecast model is shown in Figure 15

54 Travel Time Forecasting Based on ARIMAModel To sumup we choose ARIMA model to build up the forecast modeland regress the parameterUsing the data of 221sim330 collectedby Detector 1 at the time 1328sim1700 and testing the time

International Journal of Distributed Sensor Networks 9

Table 3 Result of change-point searching of each section

1198900

Searching result of change-point Number of change-points001 1 25 59 70 77 90 224 319 326 342 357 376 379 390 393 414 415 17002 1 25 90 224 375 380 384 390 393 414 10005 1 225 375 380 390 393 412 7007 1 225 380 393 412 501 1 411 2012 1 416 2015 1 1

Table 4 ARIMA model and forecast result of different weight functions

Weight functions Fitting equations Forecast value on 1702 Forecast value on 1704Square-root 119909

1015840

119905= 0000321317 + 0726226119909

1015840

119905minus1+ 0191767119909

2

119905minus20208528 0206241

Square 1199091015840

119905= 00000607593 + 0798534119909

1015840

119905minus1+ 0204619119909

2

119905minus20213584 0215573

Growth rate curve 1199091015840

119905= 0000340756 + 0717946119909

1015840

119905minus1+ 0193551119909

2

119905minus20209668 0208116

Liner 1199091015840

119905= 0000948595 + 0617164119909

1015840

119905minus1+ 0110425119909

2

119905minus20206278 0202864

Actual value on 1702 020577Actual value on 1704 0196385

Table 5 Error matrix of different model fittings

Weight functions ME MAE MAPE MSESquare-root (global) minus0000361064 001288 710089 0000279502Square (global) minus0000751537 00133415 734701 0000294168Growth rate curve (global) minus0000697771 00131061 722872 0000289045Liner (global) 136255 lowast 10

minus17 00120611 672481 0000258082Square-root (proximal point) minus000357858 00134934 629296 0000200429Square (proximal point) minus000703095 00144531 674308 0000263613Growth rate curve (proximal point) minus000513248 00133818 624778 0000211068Liner (proximal point) 0000443646 00113057 527639 0000140726

sequence we found it is a nonrandom stationary sequenceThe fix order result of the ARIMA model for the time of1328sim1700 is (2 0 0)

After the weighted least squares are transformed intoordinary least squares we use four kinds of weight functionto do the fitting experiment and also have the error analysisto the output results The forecast results of different weightfunctions are as in Table 4

Error analysis result is shown in Table 5 and we can seethat the crucial indexMAPE has a certain degree of reductionat the proximal point

The forecast results of four different weighting functionscanmeet the basic requirements of the accuracy error of 10At the same time we can know that the linear weight functionhas good fitting and forecasting effect on the experimentaldata The linear weighting function is the optimal weightfunction for this forecast according to the statistics in Table 5

6 ConclusionThis paper uses the change-point detection algorithm todivide travel time series into several patterns and set up

forecasting model through ARIMA for different patternsbased on massive data collected by the loop detectors onthe roads Different from traditional forecasting methodsit is easier to get the more optimized forecasting resultthan to obtain it by using the global search because thedifferent intervals in same pattern have similar statisticalcharacteristics of the mean and variance In the process ofdividing the travel time series the calculation of algorithmis complicated and the derivation of control parameters isonly obtained by experiments which still needs research inthe future

Conflict of Interests

The authors declare no conflict of interests

Authorsrsquo Contribution

Guangyu Zhu designed the forecasting model of travel timebased on the change-point detection algorithm and wrote thepaper Li Wang designed the preprocessing algorithm and

10 International Journal of Distributed Sensor Networks

0800 1200 1600 2000

0800 1200 1600 2000

0800 1200 1600 2000

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

e0 = 002 1 25 90 224 375 380 384 390 393 414

e0 = 005 1 225 375 380 390 393 412

e0 = 007 1 225 380 393 412

Figure 14 Result of the global state change-point searching

0600 0900 1200 1500000

005

010

015

020

025

Figure 15 Historical data of the forecast model

performed the data preprocess and revised the paper PengZhang and Kang Song analyzed the data

Acknowledgments

This work is supported by the National Science Founda-tion of China (nos 61572069 and 61503022) the Funda-mental Research Funds for the Central Universities (no2014JBM211) the Open Project Program of Key Laboratoryof System Control and Information Processing Ministry ofEducation Shanghai Jiaotong University (no Scip201507)Beijing Municipality Key Laboratory of Urban Traffic Oper-ation Simulation and Decision Support Beijing Transporta-tion Research Center the project of the Department of Trafficand Transportation of Hebei Province (no A0201-150505)and The National Key Technology Support Program (no2014BAG01B02)

References

[1] Z-P Li H Yu Y-C Liu and F-Q Liu ldquoAn improved adaptiveexponential smoothing model for short-term travel time fore-casting of urban arterial streetrdquo Acta Automatica Sinica vol 34no 11 pp 1404ndash1409 2008

[2] Y Hollander and R Liu ldquoEstimation of the distribution of traveltimes by repeated simulationrdquo Transportation Research Part CEmerging Technologies vol 16 no 2 pp 212ndash231 2008

[3] G-Y Jiang Technologies and Application of the Identification ofRoad Traffic Condition Communications Press Beijing China2004 (Chinese)

[4] M Gramaglia C J Bernardos and M Calderon ldquoVirtualinduction loops based on cooperative vehicular communica-tionsrdquo Sensors vol 13 no 2 pp 1467ndash1476 2013

[5] X Zhang and J A Rice ldquoShort-term travel time predictionrdquoTransportation Research Part C vol 11 no 3-4 pp 187ndash2102003

[6] U Mori A Mendiburu M Alvarez and J A Lozano ldquoA reviewof travel time estimation and forecasting for Advanced TravellerInformation Systemsrdquo Transportmetrica A Transport Sciencevol 11 no 2 pp 119ndash157 2015

[7] E I Vlahogianni M G Karlaftis and J C Golias ldquoShort-term traffic forecasting where we are and where wersquore goingrdquoTransportation Research Part C Emerging Technologies vol 43part 1 pp 3ndash19 2014

[8] C Shao K Zhang and Y Gu ldquoA study of route traveltime forecast method based on real data of urban expresswaynetworkrdquo China Civil Engineering Journal vol 36 no 1 pp 16ndash20 2003

[9] B R Chilukuri J A Laval and A Guin ldquoMicrosimulation-based framework for freeway travel time forecastingrdquo Trans-portation Research Record vol 2470 pp 34ndash45 2014

[10] E Yao and Y Zhang ldquoA new algorithm of shortmdashterm traveltime prediction for urban expresswayrdquo Journal of WuhanUniversity of Technology (Transportation ScienceampEngineering)vol 6 pp 1133ndash1137 2013

[11] J Zhao H Wang and W Liu ldquoPrediction of expressway traveltime based on adaptive interpolation kalman filteringrdquo Journalof South ChinaUniversity of Technology vol 2 pp 109ndash115 2014

[12] Z Gui and H Yu ldquoTrip travel time forecasting based onselective forgetting extreme learning machinerdquo Mathematical

International Journal of Distributed Sensor Networks 11

Problems in Engineering vol 2014 Article ID 829256 7 pages2014

[13] Y-J Jou Y-H Wen T-T Lee and H-J Cho ldquoMissing datatreatment on travel time estimation for ATISrdquo in Proceedingsof the IEEE International Conference on Systems Man andCybernetics (SMC rsquo03) vol 1 pp 102ndash107 IEEE WashingtonDC USA October 2003

[14] LWang and CWang ldquoA newmethod of real-time informationcollection in intelligent transportation systemrdquo Systems Engi-neering vol 23 no 2 pp 86ndash89 2005

[15] Z Cai Y Duan and A G Bourgeois ldquoDelay efficient oppor-tunistic routing in asynchronous multi-channel cognitive radionetworksrdquo Journal of Combinatorial Optimization vol 29 no 4pp 815ndash835 2015

[16] J W C VanLint and N J Zijpp ldquoAn improved travel timeestimation algorithm using dual-loop detectorsrdquo in Proceedingsof the Annual Meeting of the Transportation Research BoardWashington DC USA 2003

[17] Z Cai R Goebel and G Lin ldquoSize-constrained tree parti-tioning approximating the multicast k-tree routing problemrdquoTheoretical Computer Science vol 412 no 3 pp 240ndash245 2011

[18] G Zhu and H Yan ldquoA kind of demand-forecasting modelbased on analysis of demand booming and principle of naiveforecastingrdquo Systems EngineeringTheory amp Practice vol 24 no5 pp 22ndash33 2004

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AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

2 International Journal of Distributed Sensor Networks

O D

l1 l2 li ln

d1 dnminus1

Figure 1 Distribution of the loops on the road

The literatures which have beenmentioned above providesome idea for this paper to forecast the travel time based onmassive data collected from loop detectors First travel timeand the traffic state of the road have a certain correlationBesides most of the travel time forecast models are usinghistorical data for analysis And the characteristics of trafficflow tend to change with the seasons and the environment incertain regularity Thus if the traffic flow can be divided intoseveral state intervals this means that the different intervalsin the same pattern have the similar statistical characteristicsof the mean and variance As a result it is easier to get themore optimized forecast result than to obtain it by using theglobal search

Therefore this paper proposes a travel time forecastingmodel based on change-point detection which uses thechange-point detection to identify different patterns of traveltime series and set up the forecasting model by ARIMA ineach of the patterns

The rest of the content of this paper is summarized asfollows

(1) Preprocessing of the massive data collected by theloop detectors and calculation of travel time param-eter

(2) The pattern partition of travel time series based onchange-point analysis and setting up the forecastingmodel based on ARIMA

(3) Verification of the travel time forecastingmodel basedon actual data

2 Identification and Correction ofthe Loop Detectorrsquos Data

Because of the reasons such as the detector fault the faultof communication system and the environmental factorthe real-time detector data contain some unpredictable datamissing or invalid data Therefore it is necessary to pre-process the data collected by the traffic detectors [13] Sothis paper gives the basic rules to identify and correct loopdetectorrsquos data based on practical experience

Basic Rule 1 When the data of traffic volume speed andoccupancy rate is negative or null it is recorded as the errordata When the data of volume is significantly greater com-pared to the maximum volume of road (119876max = 119891

11988811986211987960)

it is recoded as the error data When the data of speed issignificantly greater compared to the maximum allowablespeed or capacity of the urban road it is recorded as the errordata When the data of occupancy rate is not less than 100it is recorded as error data

Basic Rule 2 When the data of occupancy rate is greater thansome reasonable threshold such as 95 and the data of speedis greater than the normal range such as 5 kmh it is recordedas the error dataWhen the speed is zero and the volume is notzero the data is the error data When the volume is zero andthe occupancy rate is not zero the data is the error dataWhenthe average effective vehicle length (AEVL (m) = (10 times V times

ℎ)119902 V speed ⟨kmh⟩ ℎ occupancy rate ⟨⟩ and 119902 volume⟨vehicle volume(lanehour)minus1⟩) is beyond reasonable limits(such as AEVL isin [15m 30m]) it is the error data

Basic Rule 3 Each data item should be recorded similarly by119899 piece of data in time 119905 before it And the data should bedone in first-order difference If the difference value of first-order does not belong to the reasonable change range madeby 119899 data before it this data can be defined as the abruptlychanging distortion data

The data collected by the detectors can be expressed asfour-tuple structure that is [119905 119902 V ℎ] Based on the basicrules which have been discussed above this paper proposesan algorithm to accomplish the real-time identification andcorrection of loop detectorrsquos data

Algorithm 1Step 1 It is determining 119876max 119881max and 119867max according tothe actual situation of the detectors on the road Test all ofthe data if 119902 gt 119876max or V gt 119881max or ℎ gt 119867max the data willbe defined as error data

Step 2 When the occupancy rate ℎ gt 1198671(1198671

= 95) andV gt 1198811(1198811= 5 kmh) the data is recorded as the error data

Step 3 When V = 0 (kmh) and 119902 = 0 (vehiclenumber(lanehour)minus1) the data is the error data

Step 4 When 119902 = 0 (vehicle number(lanehour)minus1) and V =

0ampampℎ = 0 the data will be defined as error data

Step 5 Calculate the average effective vehicle length (AEVL)according to the current detected data If AVEL notin [15 30]exclude this data

Step 6 It is using the reasonable and nearest data to replacethose error data which have been found in Steps 1ndash5

Step 7 It is using the first-order differential operation toprocess data If the difference value does not belong to thereasonable change range made by the differential mean valueand variance of 119899 piece of data before it (such as [119889minus120576

0lowast120590119889 119889+

1205760lowast 120590119889]) this data can be defined as the abruptly changing

distortion data Then use 119901119899minus1

+ (11205760) lowast 119889119899to replace it

3 Calculation of Travel Time Based onthe Data from Loop Detector

Using the preprocessed data the travel time parameter valuescan be calculated [14]The calculation result of the travel timeparameter is usually related to the speed of the vehicle on theroad

International Journal of Distributed Sensor Networks 3

Assume the speed is conformed to liner change and theupstream and downstream of each section of the road have adetector and each trip chain has multiple parts so that V

119894(119905)

speed of the vehicle between detector 119896 and detector 119896 + 1can be expressed as

V119894 (119905) = 119881 (119896 119889) +

119904119894 (119905) minus 119904

119896

119904119896+1

minus 119904119896

(119881 (119896 + 1 119889) minus 119881 (119896 119889)) (1)

119904(119905) = int V(119905)119889119905 and this equation is a standard differentialequation Due to differential equations it is very difficultto obtain exact solutions generally we need to seek anapproximation to replace it So 119904(119905) can be obtained byformula

119904119894 (119905) = 119904

0

119894+ (

119881 (119896 119889) (119904119896+1 minus 119904119896)

119881 (119896 + 1 119889) minus 119881 (119896 119889)+ 119904119896+1

minus 119904119896)

sdot (119890[(119881(119896+1119889)minus119881(119896119889))(119904119896+1minus119904119896)](119905minus119905

0

119894)minus 1)

(2)

119904119896+1

119904119896stand for the location of detectors 119896 + 1 and 119896

119881(119896 119889) stands for the speed of detector 119896 in time period 119889

and also is the slope of the vehicle motion curve 119904119894(119905) stands

for specific motion trajectories of time period 119889 within roadsection 119896 1199040

119894 1199050

119894 is the initial state of the vehicle entering

119896 119889 range If 119881(119896 + 1 119889) = 119881(119896 119889) formula (2) is

119904119894 (119905) = 119904

0

119894+ 119881 (119896 119889) (119905 minus 119905

0

119894) (3)

According to formulas (1) (2) and (3) the formula fortime calculation of the road section based on the vehiclemoving track can be divided into two situations

(1) When the speed is fast consider the following

1199040

119894+ (

119881 (119896 119889) (119904119896+1 minus 119904119896)

119881 (119896 + 1 119889) minus 119881 (119896 119889)+ 1199040

119894minus 119904119896)

sdot (119890[(119881(119896+1119889)minus119881(119896119889))(119904119896+1minus119904119896)](119905minus119905

0

119894)minus 1) gt 119904

119896+1

119904lowast

119894= 119904119894(119905lowast

119894) = 119904119896+1

(4)

The approximate result for the travel time of the roadsection is

119905lowast

119894asymp 1199050

119894+

(119904119896+1

minus 119904119896)

119881 (119896 + 1 119889) minus 119881 (119896 119889)

sdot ln(119881 (119896 119889) (119904119896+1 minus 119904

119896) (119881 (119896 + 1 119889) minus 119881 (119896 119889)) + 119904

119896+1minus 119904119896

119881 (119896 119889) (119904119896+1 minus 119904119896) (119881 (119896 + 1 119889) minus 119881 (119896 119889)) + 119904

0

119894minus 119904119896

)

(5)

(2) When the speed is slow consider the followingWhen 119904

119894(119905119889+1

) lt 119904119896+1

the travel time obtained by for-mula (2) is

119905lowast

119894= 1199050

119894+

119904119896+1

minus 1199040

119894

119881 (119896 119889) (6)

This algorithm based on the method mentioned abovecan be summarized as follows

Algorithm 2Step 1 Launch vehicles and select sections (119870 = 1)

Step 2 For vehicle in 1199040

119894 1199050

119894 and 119896 119889 use formulas (2) (3)

(5) and (6) to calculate 119905lowast

119894 and 119904

lowast

119894 when it runs out of the

area If 119904lowast119894

= 119904119896+1

set 119896 = 119896 + 1 and move to next sectionOtherwise 119905lowast

119894= 119905119889+1

set 119889 = 119889+1 and move to the next datacollection period

Step 3 If 119896 ge 119898 it means that the vehicle has arrived atthe destination Record the departure time and arrival timeof the vehicle and then stop Otherwise go back to Step 2 torecalculate the travel time

Step 4 Set 119894 = 119894+1Thenumber of 119894+1 vehicles has continuousheadway at 1199040

119894+1 1199050

119894+1+ ℎ If 1199050

119894+1+ ℎ gt 119899 stop That indicates

that the calculation of the specified time period has beencompleted

This algorithm is firstly assuming themotion trajectory ofthe vehicle on the road and then through using the location-time curve gets the time at which a vehicle runs out of thedetector area to obtain the travel time [15] This method oftravel time estimation through time space motion trajectoryhas high accuracy The error between the results of thecalculation in [13 16] and the result of this algorithm is below6 which means that this algorithmrsquos result is acceptable

4 Forecast Model for Short-Term Travel TimeBased on ARIMA

41 Change-Point Searching Because the traffic data hasdifferent numerical characteristics in different time periodsit can be divided into numbers of similar small states byconditional change-point searching which can effectivelyimprove the fitting degree of the model In [17 18] a newalgorithm for state division based on the demand variationof the observation function is introduced The mean andthe variance of the sequence can be expressed by statisticalformula

Whole sequence is

119909119894

1003816100381610038161003816

119899

1=

sum119899

119894=1119909119894

119899

Dev (119909119894

1003816100381610038161003816

119899

1) =

119899

sum

119894=1

(119909119894minus 119909119894

1003816100381610038161003816

119899

1)2

(7)

Convex (concave) wave is

119909119905

1003816100381610038161003816

119897119895

119897119895minus1=

(119909119897119895minus1

+ sdot sdot sdot + 119909119897119895)

(119897119895minus 119897119895minus1

+ 1)

Dev (119909119894

1003816100381610038161003816

119897119895

119897119895minus1) =

119897119895

sum

119905=119897119895minus1

(119909119905minus 119909119905

1003816100381610038161003816

119897119895

119897119895minus1)

2

119895 = 2 119896

(8)

4 International Journal of Distributed Sensor Networks

119897119895 is valley point of peak carve or peak points of the valley

curveThe observation function decides whether to retain the

possible change-point Before that the control parameter andobservation function values of the minimum state variableare required

Observation function is

119879 (119908) = (

119861 (119908 119909119894

1003816100381610038161003816

119899

1)

119861 (119908 + 1 119909119894

1003816100381610038161003816

119899

1)

) 119908 = 0 1 119869 minus 1

119861 (119908 119909119894

1003816100381610038161003816

119899

1)

=

119908minus1

sum

119892=0

Dev(119909119905

1003816100381610038161003816

119887(119892+1)minus1

119887(119892) ) + Dev (119909

119905

1003816100381610038161003816

119899

119887(119908)) 119908 = 1 119869

Dev (119909119894

1003816100381610038161003816

119899

1) 119908 = 0

(9)

119887(119908)

is the index set of all 119869 points on the curve and 119887(0)

=

1199091198971

= 1199091

The algorithm is as follows

Algorithm 3Step 1 Make the travel time series into carve 119906 represents thetime of cycles119908 represents the number of change-points and119890119890is the auxiliary variable Set 119906 = 1 119908 = 1 119890

119890= 0 and

120573119895= 119897119895 119895 = 1 119896

Step 2 Select two convex waves along the axis of time from120573119906on the carve such as (120573

119906minus 120573119906+1

minus 120573119906+2

) Then calculate 119905lowast

119860 (120573119906 120573119906+1

120573119906+2

)

=

120573119906+1

sum

119905=120573119906

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+1

120573119906)

2

+

120573119906+2

sum

119905=120573119906+1+1

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+2

120573119906+1+1)

2

119872 (120573119906 119905lowast

1 120573119906+2

)

= max(

119905

sum

119905=120573119906

(119909119905minus 119909119905

1003816100381610038161003816

119905

120573119906)2

+

120573119906+2

sum

119905=119905+1

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+2

119905+1)

2

)

119898 (120573119906 119905lowast

2 120573119906+2

)

= min(

119905

sum

119905=120573119906

(119909119905minus 119909119905

1003816100381610038161003816

119905

120573119906)2

+

120573119906+2

sum

119905=119905+1

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+2

119905+1)

2

)

(10)

Step 3 Change-point estimation is

119877119906=

2119860 (120573119906 120573119906+1

120573119906+2

)

(119872 (120573119906 119905lowast

1 120573119906+2

) + 119898 (120573119906 119905lowast

2 120573119906+2

))

119888119903119906=

119872(120573119906 119905lowast

1 120573119906+2

)

119898 (120573119906 119905lowast

2 120573119906+2

)

(11)

Consider the following

(1) If 119872(120573119906 119905lowast

1 120573119906+2

) asymp 119898(120573119906 119905lowast

2 120573119906+2

) 1 le 119888119903119906

lt 13and the sequence change is gently you do not need toset change-point in (119897

119906 119897119906+2

)

(2) If 119877119906

le (14) + 32(119888119903119906+ 1) regardless of whether

119905lowast

1= 120573119906+1

there is 119887(119908) = 120573119906+1

(3) If 1 ge 119877119906

ge (14) + 32(119888119903119906+ 1) seek the nearest

peak or valley point of 120573119906+1

and set the left oneas 120573119871

119906+1and right one as 120573

119877

119906+1 If exist119905

119888isin 120573

119871

119906+1

120573119877

119906+1 have Dev(120573

119906 119905119888 120573119906+2

) = min(Dev(120573119906 120573119871

119906+1

120573119906+2

)Dev(120573119906 120573119877

119906+1 120573119906+2

)) Then 119887(119908)

= 119905119888

(4) If 119877119906gt 1 then 119887

(119908)= 119905lowast

2

(5) Make effective judgments for 119887(119908) offered by (2) (3)

and (4) if 119879(119908 minus 1) minus 1 gt 1205760 keep this change-point

otherwise remove it

Step 4 If you have 119887(119908) then 119906 = 119906 + 1 120573

119906= 119887(119908) 120573119906+1

=

120573119906+1

120573119906+2

= 120573119906+2

119908 = 119908 + 1 and 119890119890= 0 Otherwise 119890

119890=

1 120573119906

= 120573119906 120573119906+1

= 120573119906+1+119890119890

and 120573119906+2+119890119890

= 120573119906+2+119890119890

When120573119906+2

= 120573119896 the searching is complete and the algorithm ends

Otherwise go back to Step 2

The paper [18] has provided a complete method on howto improve this kind of algorithm However there has beena crucial control parameter 119890

0for which the algorithm does

not give the specific processing formula That algorithmsets 119890

0as a constant value such as 05 In this paper we

will carry out several experiments with different 1198900 which

provide reference to the parameter of travel time inferredfrom detectorsrsquo data

42 ARIMA Forecasting Model The preprocessing of thetime series short-term forecasting model includes stationarytest and random test If the time series is nonstationary itneeds to be transformed into stationary series by differentialoperation In this circumstance the ARIMA model is con-verted to ARMA model The sequence of 119889 order differencesis expressed as

nabla119889119909119905=

119889

sum

119894=1

(minus1)119889119862119894

119889119909119905minus119894

(12)

ARIMA (119901 119889 119902) model can be expressed as

Φ (119861) nabla119889119909119905= Φ119901 (119861) 120576119905

119864 (120576119905) = 0

Var (120576119905) = 1205902

120576

119864 (120576119905120576119904) = 0 119904 = 119905

119864119909119904120576119905= 0 forall119904 lt 119905

(13)

International Journal of Distributed Sensor Networks 5

The order number 119901 119902 of ARIMA (119901 119889 119902)model is basedon Autocorrelation Coefficient (ACF) and Partial Auto-correlation Coefficient (PACF) of ARMA after differentialoperation And according to the characteristics of PACF andACF coefficients the model identification is carried out

For random inspection the data collected by the detectoris a large density data point so the calculation result of traveltime is also a large sample of high density which needs to testthe hypothesis by using 119876 statistics

119876 = 119899

119898

sum

119896=1

2

119896sim 1205942(119898) (119898 is delay-stage) (14)

When119876 is less than the quintile of 12059421minus120572

(119898) the sequenceis pure random sequence However when the travel time ismodeled by the ARIMA model if the sample space becomessmall the modified LB statistics can be used

LB = 119899 (119899 + 2)

119898

sum

119896=1

(2

119896

119899 minus 119896) (15)

Because there is only a short-term significant correlationin the sequence the test for the hypothesis is only for 119876 andLBwith short-termdelay-stage which is generally119898 less than10

After differential operation the ARIMA model isdegraded to the standard ARMAmodel its standard form is

119909119905= 120583 +

Φ119901 (119861)

Φ119901 (119861)

120576119905 (16)

In the formula

120576119905sim 119882119873(0 120590

2

119905)

Φ119901 (119861) = 1 minus 120579

1119861 minus sdot sdot sdot minus 120579

119901119861119901

Φ119901 (119861) = 1 minus 120593

1119861 minus sdot sdot sdot minus 120593

119902119861119902

(17)

There are 119901 + 119902 + 2 unknown parameters in ARMAmodel120601

1 1206012 120601

119901 1205791 1205792 120579

119902 120583 1205902

120576Matrix estimation is

usually used to obtain the value of 120583 and 1205902

120576

Calculate the expectation and variance of formula (16) oneach side and get

= 119864 (119909) = 119909 =sum119899

1119909119894

119899

1205902

120576=

sum119899

1(119909119894minus 119909)2

119899=

1 + 1205932

1+ sdot sdot sdot + 120593

2

119901

1 + 1205792

1+ sdot sdot sdot + 1205792

119901

1205902

119909

(18)

The parameters of the equation are reduced to thenumber of 119901 + 119902 Least square estimation is used to estimatethe parameters of the ARMA model which is obtained bydifferential operation

In the case of ARMA (119901 119902) the parameter vector is

= (1206011 120601

119901 1205791 120579

119902)1015840

119865119905() = 120601

1119909119905minus1

+ 1206012119909119905minus2

+ sdot sdot sdot + 120601119901119909119905minus119901

+ 120576119905minus 1205791120576119905minus1

minus 1205792120576119905minus2

minus sdot sdot sdot minus 120579119902120576119905minus119902

(19)

Thus overall observed sum of squared residuals of thesample is

119876() =

119899

sum

1

1205762

119905=

119899

sum

1

(119909119905minus 119865119905())2

(20)

Set the objective function for parameter estimation asmin119876() and use the weighted least squares method to solvethe parameters

The essence of the weighted least square method is totransform the original data to obtain the new explanatoryvariables and explained variable Assume that 119909

119894is time series

data and 120596 is the weight of this point so that weighted leastsquare method is

1199091015840

119894= 119909119894sdot 120596119894

(1199091015840 refers to travel time after transformation)

(21)

Then use (11990910158401 119909

1015840

119899) to do the least square parameter

estimation of ordinaryARIMAmodel we can get the optimalparameter vector

1015840

of themodel inweighted transformationIn this way the formula for the weighted forecast formula is

119865119905(1015840

) = 1205931015840

1119909119905minus1

+ 1205931015840

2119909119905minus2

+ sdot sdot sdot + 1205931015840

119901119909119905minus119901

+ 120576119905

minus 1205791015840

1120576119905minus1

minus 1205791015840

2120576119905minus2

minus sdot sdot sdot minus 1205791015840

119902120576119905minus119902

(22)

After removing theweight the final forecast value is obtained

119865119905() =

119865119905(1015840

)

120596119905

(23)

5 Application Example

51 Preprocessing of the Massive Data from Loop DetectorsThis paper takes the actual data of 2nd ring road in a big cityas an example (detector number is 020lowastlowast line number is Lan1ndashLan 6 date is on Mar 3rd 2013 data collection time is 24hours sampling interval is 2 minutes parameters are trafficvolume speed and occupancy rate and the total number ofdata points is 720) to verify the travel time forecasting modelbased on loop detectors which has been mentioned aboveThe actual data of Lan 1 is shown in Figure 2

In Figure 2 mutation points can be observed in the dataseries of all the three parameters In fact the data of trafficconditions cannot change more than 500 times within twominutes So it can be concluded that there are abnormal ordistorted data in the actual data and it is necessary to filterthose data

According toAlgorithm 1 we can finish the data cleaningFirstly according to the definition the control parametersbased on the basic traffic flow principle and the actualphysical meaning are set up in Table 1

6 International Journal of Distributed Sensor Networks

Table 1 Selection of filtering modelrsquos parameters

Parameter name Abbreviation Experimental valuesSection maximum flow 119902max 119902max = 360030Section maximum speed Vmax Vmax = 150Section maximum occupancy rate 119900max 119900max = 100Constraints of occupancy rate and speed 119867

1 1198811

1198671= 95 119881

1= 5

119902 volume ⟨vehicle volume(lanehour)minus1⟩ V speed ⟨kmh⟩ and 119900 occupancy rate ⟨⟩

00000000 0600 1800120000000000 0600 1800120000000000 0600 18001200

0

20

40

60

80

50

100

150

200

0

10

20

30

40

50

Figure 2 Data distribution of Lan 1rsquos volume-speed-occupancy rate in 24 hours

100 200 300 400 500 600 700

minus04

minus02

02

04

Figure 3 Fault data points of filtering intermediate state search

Under the control of parameters listed in Table 1 wecan finish the data cleaning to find out the data beyond themaximum control range or contrary to the theory of trafficflow The result is shown in Figure 3

Use a one-dimensional matrix to record the effectivenessof each record All the initial value of thematrix is 1When theabnormal data is detected the corresponding matrix valueis changed into 0 From Figure 3 there is a series of errordata points at the time 300ndash600 which is consistent with theoriginal graph shown in Figure 4

Because these error data points do not have actual physi-cal meaning they are replaced by the closest normal recordAfter the cleaning the figure of volume-speed-occupancyrate is shown in Figure 5

Data quality has been improved to a certain extentespecially for the speed data But there still has beenmutationin the filtering results Test the first-order differential ofthe data to determine the mutation data The first-orderdifference graph of intermediate state is shown in Figure 6

Table 2 Result of the filter by different parameters

1198900value 119890

0= 3 119890

0= 4 119890

0= 5

Number of change ranges beyond 83 34 21Rate of the data beyond 115 47 21

Control parameter of differential operational is 1198900

Assuming that 1198900

= 3 is the parameter of the reasonablechange region if the actual value of the first-order is exceededthree times of the standard deviation control range changethe corresponding position of the effective matrix into 0

When 1198900

= 3 there are 83 abnormal data points as isshown in Figure 7 which account for 115 of the total recordThis result is too strict for the data of detectors so we canincrease the value of 119890

0to release the strictness for change

range of the data as is shown in Table 2According to the results shown in the table 119890

0= 4 is more

reasonable Figure 8 is comparison chart between the finalresult of the filter and the original data and it can be seenthat the algorithm has basically achieved the requirements ofthe loop detector datarsquos cleaning and preprocessing

FromFigure 3 it can be seen that the algorithmhas a smallcorrection for the traffic data and the data with the occupancyrate but the algorithm has better effect on the speed data Inthe process of predicting travel time the speed of the detectoris often used only which means that the algorithm can besimplified so that it only needs to produce speed data

Because only the speed data is processed we need to setthe upper and lower limit of speed and the limit of first-orderdifferential change range to restrict data As a result of usingthe detector data to calculate and predict travel time we needthree continuous detectorrsquos data points to simulate the traveltime forecast inwhole roadnetworkThe sketchmap is shownin Figure 9

International Journal of Distributed Sensor Networks 7

00000000 0600 18001200

0

20

40

60

80

00000000 0600 18001200

50

100

150

200

00000000 0600 18001200

0

10

20

30

40

50

Figure 4 Comparison between the actual data and the results of the filterrsquos intermediate state

00000000 0600 18001200 00000000 0600 18001200 00000000 0600 18001200

0

20

40

60

80

20

40

60

80

100

120

0

10

20

30

40

50

Figure 5 Result of filterrsquos intermediate state

Figure 6 First-order difference graph of intermediate state

100 200 300 400 500 600 700

01

02

minus01

minus02

Figure 7 Distortion data distribution when 1198900= 3

Under the condition that 1198900= 3 the filter result of speed

after cleaning of the three detectors is as shown in Figures 1011 and 12

As shown in Figures 10sim12 the red correction curvebasically achieved a reasonable correction of the distortiondata

52 Calculation of Travel Time Parameters Use the traveltime conversion model given by formulas (5) and (6) wecan do the travel time conversion according to the speeddata For example when one calculates the travel timedriving from east to west the vehicles pass through sec-tions Detector 1 Detector 2 and Detector 2 Detector 3The result is shown in Figure 13 and the unit of time is sm

53 Pattern Partition of Travel Time Series Based on Change-Point Analysis Because it is not clear how to choose 119890

0up to

the size of the sample we select 7 values to search the change-point The result is in Table 3

According to the characteristics of the travel time serieswe need to select the result that has 5 to 10 change-points So1198900= 007 119890

0= 005 and 119890

0= 002 are all reasonable control

parameters The results of all the reasonable circumstanceare shown in Figure 14 We use different colors to indicatedifferent state

According to the time sequence diagram the results of thealgorithm are basically completed and the travel time seriesis decomposed into a series of time periods with practicalmeaning

8 International Journal of Distributed Sensor Networks

50

100

150

200

0

10

20

30

40

50

00000000 0600 18001200 00000000 0600 18001200 00000000 0600 18001200

0

20

40

60

80

Figure 8 Comparison of the source data and the filter result

Detector 1Detector 2Detector 3

4494m 157m

Figure 9 Distribution of the three serial detectors on the road

0800 1200 1600 20000

20

40

60

80

Figure 10 Filter result of the speed data of Detector 1

0800 1200 1600 20000

10

20

30

40

50

60

Figure 11 Filter result of the speed data of Detector 2

On the situation of 1198900

= 007 and 1198900

= 005 thereare 90 important change-points that were missed So 119890

0=

002 and 1 25 90 224 375 380 384 390 393 414 are idealcontrolling parameter and result of change-point control

0800 1200 1600 20000

10

20

30

40

50

60

Figure 12 Filter result of the speed data of Detector 3

0800 1200 1600 200000

01

02

03

04

Figure 13 Travel time calculation result from Detector 1Detector 2 to Detector 2 Detector 3

And the travel time is divided into 9 divisions 600sim650650sim900 900sim1328 1328sim1830 1830sim1838 1838sim18501850sim1856 1856sim1948 and 1948sim2300

Then if we assume that this moment is 1700 we canpredict the travel time of 1702 and 1704 The historical dataof the forecast model is shown in Figure 15

54 Travel Time Forecasting Based on ARIMAModel To sumup we choose ARIMA model to build up the forecast modeland regress the parameterUsing the data of 221sim330 collectedby Detector 1 at the time 1328sim1700 and testing the time

International Journal of Distributed Sensor Networks 9

Table 3 Result of change-point searching of each section

1198900

Searching result of change-point Number of change-points001 1 25 59 70 77 90 224 319 326 342 357 376 379 390 393 414 415 17002 1 25 90 224 375 380 384 390 393 414 10005 1 225 375 380 390 393 412 7007 1 225 380 393 412 501 1 411 2012 1 416 2015 1 1

Table 4 ARIMA model and forecast result of different weight functions

Weight functions Fitting equations Forecast value on 1702 Forecast value on 1704Square-root 119909

1015840

119905= 0000321317 + 0726226119909

1015840

119905minus1+ 0191767119909

2

119905minus20208528 0206241

Square 1199091015840

119905= 00000607593 + 0798534119909

1015840

119905minus1+ 0204619119909

2

119905minus20213584 0215573

Growth rate curve 1199091015840

119905= 0000340756 + 0717946119909

1015840

119905minus1+ 0193551119909

2

119905minus20209668 0208116

Liner 1199091015840

119905= 0000948595 + 0617164119909

1015840

119905minus1+ 0110425119909

2

119905minus20206278 0202864

Actual value on 1702 020577Actual value on 1704 0196385

Table 5 Error matrix of different model fittings

Weight functions ME MAE MAPE MSESquare-root (global) minus0000361064 001288 710089 0000279502Square (global) minus0000751537 00133415 734701 0000294168Growth rate curve (global) minus0000697771 00131061 722872 0000289045Liner (global) 136255 lowast 10

minus17 00120611 672481 0000258082Square-root (proximal point) minus000357858 00134934 629296 0000200429Square (proximal point) minus000703095 00144531 674308 0000263613Growth rate curve (proximal point) minus000513248 00133818 624778 0000211068Liner (proximal point) 0000443646 00113057 527639 0000140726

sequence we found it is a nonrandom stationary sequenceThe fix order result of the ARIMA model for the time of1328sim1700 is (2 0 0)

After the weighted least squares are transformed intoordinary least squares we use four kinds of weight functionto do the fitting experiment and also have the error analysisto the output results The forecast results of different weightfunctions are as in Table 4

Error analysis result is shown in Table 5 and we can seethat the crucial indexMAPE has a certain degree of reductionat the proximal point

The forecast results of four different weighting functionscanmeet the basic requirements of the accuracy error of 10At the same time we can know that the linear weight functionhas good fitting and forecasting effect on the experimentaldata The linear weighting function is the optimal weightfunction for this forecast according to the statistics in Table 5

6 ConclusionThis paper uses the change-point detection algorithm todivide travel time series into several patterns and set up

forecasting model through ARIMA for different patternsbased on massive data collected by the loop detectors onthe roads Different from traditional forecasting methodsit is easier to get the more optimized forecasting resultthan to obtain it by using the global search because thedifferent intervals in same pattern have similar statisticalcharacteristics of the mean and variance In the process ofdividing the travel time series the calculation of algorithmis complicated and the derivation of control parameters isonly obtained by experiments which still needs research inthe future

Conflict of Interests

The authors declare no conflict of interests

Authorsrsquo Contribution

Guangyu Zhu designed the forecasting model of travel timebased on the change-point detection algorithm and wrote thepaper Li Wang designed the preprocessing algorithm and

10 International Journal of Distributed Sensor Networks

0800 1200 1600 2000

0800 1200 1600 2000

0800 1200 1600 2000

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

e0 = 002 1 25 90 224 375 380 384 390 393 414

e0 = 005 1 225 375 380 390 393 412

e0 = 007 1 225 380 393 412

Figure 14 Result of the global state change-point searching

0600 0900 1200 1500000

005

010

015

020

025

Figure 15 Historical data of the forecast model

performed the data preprocess and revised the paper PengZhang and Kang Song analyzed the data

Acknowledgments

This work is supported by the National Science Founda-tion of China (nos 61572069 and 61503022) the Funda-mental Research Funds for the Central Universities (no2014JBM211) the Open Project Program of Key Laboratoryof System Control and Information Processing Ministry ofEducation Shanghai Jiaotong University (no Scip201507)Beijing Municipality Key Laboratory of Urban Traffic Oper-ation Simulation and Decision Support Beijing Transporta-tion Research Center the project of the Department of Trafficand Transportation of Hebei Province (no A0201-150505)and The National Key Technology Support Program (no2014BAG01B02)

References

[1] Z-P Li H Yu Y-C Liu and F-Q Liu ldquoAn improved adaptiveexponential smoothing model for short-term travel time fore-casting of urban arterial streetrdquo Acta Automatica Sinica vol 34no 11 pp 1404ndash1409 2008

[2] Y Hollander and R Liu ldquoEstimation of the distribution of traveltimes by repeated simulationrdquo Transportation Research Part CEmerging Technologies vol 16 no 2 pp 212ndash231 2008

[3] G-Y Jiang Technologies and Application of the Identification ofRoad Traffic Condition Communications Press Beijing China2004 (Chinese)

[4] M Gramaglia C J Bernardos and M Calderon ldquoVirtualinduction loops based on cooperative vehicular communica-tionsrdquo Sensors vol 13 no 2 pp 1467ndash1476 2013

[5] X Zhang and J A Rice ldquoShort-term travel time predictionrdquoTransportation Research Part C vol 11 no 3-4 pp 187ndash2102003

[6] U Mori A Mendiburu M Alvarez and J A Lozano ldquoA reviewof travel time estimation and forecasting for Advanced TravellerInformation Systemsrdquo Transportmetrica A Transport Sciencevol 11 no 2 pp 119ndash157 2015

[7] E I Vlahogianni M G Karlaftis and J C Golias ldquoShort-term traffic forecasting where we are and where wersquore goingrdquoTransportation Research Part C Emerging Technologies vol 43part 1 pp 3ndash19 2014

[8] C Shao K Zhang and Y Gu ldquoA study of route traveltime forecast method based on real data of urban expresswaynetworkrdquo China Civil Engineering Journal vol 36 no 1 pp 16ndash20 2003

[9] B R Chilukuri J A Laval and A Guin ldquoMicrosimulation-based framework for freeway travel time forecastingrdquo Trans-portation Research Record vol 2470 pp 34ndash45 2014

[10] E Yao and Y Zhang ldquoA new algorithm of shortmdashterm traveltime prediction for urban expresswayrdquo Journal of WuhanUniversity of Technology (Transportation ScienceampEngineering)vol 6 pp 1133ndash1137 2013

[11] J Zhao H Wang and W Liu ldquoPrediction of expressway traveltime based on adaptive interpolation kalman filteringrdquo Journalof South ChinaUniversity of Technology vol 2 pp 109ndash115 2014

[12] Z Gui and H Yu ldquoTrip travel time forecasting based onselective forgetting extreme learning machinerdquo Mathematical

International Journal of Distributed Sensor Networks 11

Problems in Engineering vol 2014 Article ID 829256 7 pages2014

[13] Y-J Jou Y-H Wen T-T Lee and H-J Cho ldquoMissing datatreatment on travel time estimation for ATISrdquo in Proceedingsof the IEEE International Conference on Systems Man andCybernetics (SMC rsquo03) vol 1 pp 102ndash107 IEEE WashingtonDC USA October 2003

[14] LWang and CWang ldquoA newmethod of real-time informationcollection in intelligent transportation systemrdquo Systems Engi-neering vol 23 no 2 pp 86ndash89 2005

[15] Z Cai Y Duan and A G Bourgeois ldquoDelay efficient oppor-tunistic routing in asynchronous multi-channel cognitive radionetworksrdquo Journal of Combinatorial Optimization vol 29 no 4pp 815ndash835 2015

[16] J W C VanLint and N J Zijpp ldquoAn improved travel timeestimation algorithm using dual-loop detectorsrdquo in Proceedingsof the Annual Meeting of the Transportation Research BoardWashington DC USA 2003

[17] Z Cai R Goebel and G Lin ldquoSize-constrained tree parti-tioning approximating the multicast k-tree routing problemrdquoTheoretical Computer Science vol 412 no 3 pp 240ndash245 2011

[18] G Zhu and H Yan ldquoA kind of demand-forecasting modelbased on analysis of demand booming and principle of naiveforecastingrdquo Systems EngineeringTheory amp Practice vol 24 no5 pp 22ndash33 2004

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Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 3

Assume the speed is conformed to liner change and theupstream and downstream of each section of the road have adetector and each trip chain has multiple parts so that V

119894(119905)

speed of the vehicle between detector 119896 and detector 119896 + 1can be expressed as

V119894 (119905) = 119881 (119896 119889) +

119904119894 (119905) minus 119904

119896

119904119896+1

minus 119904119896

(119881 (119896 + 1 119889) minus 119881 (119896 119889)) (1)

119904(119905) = int V(119905)119889119905 and this equation is a standard differentialequation Due to differential equations it is very difficultto obtain exact solutions generally we need to seek anapproximation to replace it So 119904(119905) can be obtained byformula

119904119894 (119905) = 119904

0

119894+ (

119881 (119896 119889) (119904119896+1 minus 119904119896)

119881 (119896 + 1 119889) minus 119881 (119896 119889)+ 119904119896+1

minus 119904119896)

sdot (119890[(119881(119896+1119889)minus119881(119896119889))(119904119896+1minus119904119896)](119905minus119905

0

119894)minus 1)

(2)

119904119896+1

119904119896stand for the location of detectors 119896 + 1 and 119896

119881(119896 119889) stands for the speed of detector 119896 in time period 119889

and also is the slope of the vehicle motion curve 119904119894(119905) stands

for specific motion trajectories of time period 119889 within roadsection 119896 1199040

119894 1199050

119894 is the initial state of the vehicle entering

119896 119889 range If 119881(119896 + 1 119889) = 119881(119896 119889) formula (2) is

119904119894 (119905) = 119904

0

119894+ 119881 (119896 119889) (119905 minus 119905

0

119894) (3)

According to formulas (1) (2) and (3) the formula fortime calculation of the road section based on the vehiclemoving track can be divided into two situations

(1) When the speed is fast consider the following

1199040

119894+ (

119881 (119896 119889) (119904119896+1 minus 119904119896)

119881 (119896 + 1 119889) minus 119881 (119896 119889)+ 1199040

119894minus 119904119896)

sdot (119890[(119881(119896+1119889)minus119881(119896119889))(119904119896+1minus119904119896)](119905minus119905

0

119894)minus 1) gt 119904

119896+1

119904lowast

119894= 119904119894(119905lowast

119894) = 119904119896+1

(4)

The approximate result for the travel time of the roadsection is

119905lowast

119894asymp 1199050

119894+

(119904119896+1

minus 119904119896)

119881 (119896 + 1 119889) minus 119881 (119896 119889)

sdot ln(119881 (119896 119889) (119904119896+1 minus 119904

119896) (119881 (119896 + 1 119889) minus 119881 (119896 119889)) + 119904

119896+1minus 119904119896

119881 (119896 119889) (119904119896+1 minus 119904119896) (119881 (119896 + 1 119889) minus 119881 (119896 119889)) + 119904

0

119894minus 119904119896

)

(5)

(2) When the speed is slow consider the followingWhen 119904

119894(119905119889+1

) lt 119904119896+1

the travel time obtained by for-mula (2) is

119905lowast

119894= 1199050

119894+

119904119896+1

minus 1199040

119894

119881 (119896 119889) (6)

This algorithm based on the method mentioned abovecan be summarized as follows

Algorithm 2Step 1 Launch vehicles and select sections (119870 = 1)

Step 2 For vehicle in 1199040

119894 1199050

119894 and 119896 119889 use formulas (2) (3)

(5) and (6) to calculate 119905lowast

119894 and 119904

lowast

119894 when it runs out of the

area If 119904lowast119894

= 119904119896+1

set 119896 = 119896 + 1 and move to next sectionOtherwise 119905lowast

119894= 119905119889+1

set 119889 = 119889+1 and move to the next datacollection period

Step 3 If 119896 ge 119898 it means that the vehicle has arrived atthe destination Record the departure time and arrival timeof the vehicle and then stop Otherwise go back to Step 2 torecalculate the travel time

Step 4 Set 119894 = 119894+1Thenumber of 119894+1 vehicles has continuousheadway at 1199040

119894+1 1199050

119894+1+ ℎ If 1199050

119894+1+ ℎ gt 119899 stop That indicates

that the calculation of the specified time period has beencompleted

This algorithm is firstly assuming themotion trajectory ofthe vehicle on the road and then through using the location-time curve gets the time at which a vehicle runs out of thedetector area to obtain the travel time [15] This method oftravel time estimation through time space motion trajectoryhas high accuracy The error between the results of thecalculation in [13 16] and the result of this algorithm is below6 which means that this algorithmrsquos result is acceptable

4 Forecast Model for Short-Term Travel TimeBased on ARIMA

41 Change-Point Searching Because the traffic data hasdifferent numerical characteristics in different time periodsit can be divided into numbers of similar small states byconditional change-point searching which can effectivelyimprove the fitting degree of the model In [17 18] a newalgorithm for state division based on the demand variationof the observation function is introduced The mean andthe variance of the sequence can be expressed by statisticalformula

Whole sequence is

119909119894

1003816100381610038161003816

119899

1=

sum119899

119894=1119909119894

119899

Dev (119909119894

1003816100381610038161003816

119899

1) =

119899

sum

119894=1

(119909119894minus 119909119894

1003816100381610038161003816

119899

1)2

(7)

Convex (concave) wave is

119909119905

1003816100381610038161003816

119897119895

119897119895minus1=

(119909119897119895minus1

+ sdot sdot sdot + 119909119897119895)

(119897119895minus 119897119895minus1

+ 1)

Dev (119909119894

1003816100381610038161003816

119897119895

119897119895minus1) =

119897119895

sum

119905=119897119895minus1

(119909119905minus 119909119905

1003816100381610038161003816

119897119895

119897119895minus1)

2

119895 = 2 119896

(8)

4 International Journal of Distributed Sensor Networks

119897119895 is valley point of peak carve or peak points of the valley

curveThe observation function decides whether to retain the

possible change-point Before that the control parameter andobservation function values of the minimum state variableare required

Observation function is

119879 (119908) = (

119861 (119908 119909119894

1003816100381610038161003816

119899

1)

119861 (119908 + 1 119909119894

1003816100381610038161003816

119899

1)

) 119908 = 0 1 119869 minus 1

119861 (119908 119909119894

1003816100381610038161003816

119899

1)

=

119908minus1

sum

119892=0

Dev(119909119905

1003816100381610038161003816

119887(119892+1)minus1

119887(119892) ) + Dev (119909

119905

1003816100381610038161003816

119899

119887(119908)) 119908 = 1 119869

Dev (119909119894

1003816100381610038161003816

119899

1) 119908 = 0

(9)

119887(119908)

is the index set of all 119869 points on the curve and 119887(0)

=

1199091198971

= 1199091

The algorithm is as follows

Algorithm 3Step 1 Make the travel time series into carve 119906 represents thetime of cycles119908 represents the number of change-points and119890119890is the auxiliary variable Set 119906 = 1 119908 = 1 119890

119890= 0 and

120573119895= 119897119895 119895 = 1 119896

Step 2 Select two convex waves along the axis of time from120573119906on the carve such as (120573

119906minus 120573119906+1

minus 120573119906+2

) Then calculate 119905lowast

119860 (120573119906 120573119906+1

120573119906+2

)

=

120573119906+1

sum

119905=120573119906

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+1

120573119906)

2

+

120573119906+2

sum

119905=120573119906+1+1

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+2

120573119906+1+1)

2

119872 (120573119906 119905lowast

1 120573119906+2

)

= max(

119905

sum

119905=120573119906

(119909119905minus 119909119905

1003816100381610038161003816

119905

120573119906)2

+

120573119906+2

sum

119905=119905+1

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+2

119905+1)

2

)

119898 (120573119906 119905lowast

2 120573119906+2

)

= min(

119905

sum

119905=120573119906

(119909119905minus 119909119905

1003816100381610038161003816

119905

120573119906)2

+

120573119906+2

sum

119905=119905+1

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+2

119905+1)

2

)

(10)

Step 3 Change-point estimation is

119877119906=

2119860 (120573119906 120573119906+1

120573119906+2

)

(119872 (120573119906 119905lowast

1 120573119906+2

) + 119898 (120573119906 119905lowast

2 120573119906+2

))

119888119903119906=

119872(120573119906 119905lowast

1 120573119906+2

)

119898 (120573119906 119905lowast

2 120573119906+2

)

(11)

Consider the following

(1) If 119872(120573119906 119905lowast

1 120573119906+2

) asymp 119898(120573119906 119905lowast

2 120573119906+2

) 1 le 119888119903119906

lt 13and the sequence change is gently you do not need toset change-point in (119897

119906 119897119906+2

)

(2) If 119877119906

le (14) + 32(119888119903119906+ 1) regardless of whether

119905lowast

1= 120573119906+1

there is 119887(119908) = 120573119906+1

(3) If 1 ge 119877119906

ge (14) + 32(119888119903119906+ 1) seek the nearest

peak or valley point of 120573119906+1

and set the left oneas 120573119871

119906+1and right one as 120573

119877

119906+1 If exist119905

119888isin 120573

119871

119906+1

120573119877

119906+1 have Dev(120573

119906 119905119888 120573119906+2

) = min(Dev(120573119906 120573119871

119906+1

120573119906+2

)Dev(120573119906 120573119877

119906+1 120573119906+2

)) Then 119887(119908)

= 119905119888

(4) If 119877119906gt 1 then 119887

(119908)= 119905lowast

2

(5) Make effective judgments for 119887(119908) offered by (2) (3)

and (4) if 119879(119908 minus 1) minus 1 gt 1205760 keep this change-point

otherwise remove it

Step 4 If you have 119887(119908) then 119906 = 119906 + 1 120573

119906= 119887(119908) 120573119906+1

=

120573119906+1

120573119906+2

= 120573119906+2

119908 = 119908 + 1 and 119890119890= 0 Otherwise 119890

119890=

1 120573119906

= 120573119906 120573119906+1

= 120573119906+1+119890119890

and 120573119906+2+119890119890

= 120573119906+2+119890119890

When120573119906+2

= 120573119896 the searching is complete and the algorithm ends

Otherwise go back to Step 2

The paper [18] has provided a complete method on howto improve this kind of algorithm However there has beena crucial control parameter 119890

0for which the algorithm does

not give the specific processing formula That algorithmsets 119890

0as a constant value such as 05 In this paper we

will carry out several experiments with different 1198900 which

provide reference to the parameter of travel time inferredfrom detectorsrsquo data

42 ARIMA Forecasting Model The preprocessing of thetime series short-term forecasting model includes stationarytest and random test If the time series is nonstationary itneeds to be transformed into stationary series by differentialoperation In this circumstance the ARIMA model is con-verted to ARMA model The sequence of 119889 order differencesis expressed as

nabla119889119909119905=

119889

sum

119894=1

(minus1)119889119862119894

119889119909119905minus119894

(12)

ARIMA (119901 119889 119902) model can be expressed as

Φ (119861) nabla119889119909119905= Φ119901 (119861) 120576119905

119864 (120576119905) = 0

Var (120576119905) = 1205902

120576

119864 (120576119905120576119904) = 0 119904 = 119905

119864119909119904120576119905= 0 forall119904 lt 119905

(13)

International Journal of Distributed Sensor Networks 5

The order number 119901 119902 of ARIMA (119901 119889 119902)model is basedon Autocorrelation Coefficient (ACF) and Partial Auto-correlation Coefficient (PACF) of ARMA after differentialoperation And according to the characteristics of PACF andACF coefficients the model identification is carried out

For random inspection the data collected by the detectoris a large density data point so the calculation result of traveltime is also a large sample of high density which needs to testthe hypothesis by using 119876 statistics

119876 = 119899

119898

sum

119896=1

2

119896sim 1205942(119898) (119898 is delay-stage) (14)

When119876 is less than the quintile of 12059421minus120572

(119898) the sequenceis pure random sequence However when the travel time ismodeled by the ARIMA model if the sample space becomessmall the modified LB statistics can be used

LB = 119899 (119899 + 2)

119898

sum

119896=1

(2

119896

119899 minus 119896) (15)

Because there is only a short-term significant correlationin the sequence the test for the hypothesis is only for 119876 andLBwith short-termdelay-stage which is generally119898 less than10

After differential operation the ARIMA model isdegraded to the standard ARMAmodel its standard form is

119909119905= 120583 +

Φ119901 (119861)

Φ119901 (119861)

120576119905 (16)

In the formula

120576119905sim 119882119873(0 120590

2

119905)

Φ119901 (119861) = 1 minus 120579

1119861 minus sdot sdot sdot minus 120579

119901119861119901

Φ119901 (119861) = 1 minus 120593

1119861 minus sdot sdot sdot minus 120593

119902119861119902

(17)

There are 119901 + 119902 + 2 unknown parameters in ARMAmodel120601

1 1206012 120601

119901 1205791 1205792 120579

119902 120583 1205902

120576Matrix estimation is

usually used to obtain the value of 120583 and 1205902

120576

Calculate the expectation and variance of formula (16) oneach side and get

= 119864 (119909) = 119909 =sum119899

1119909119894

119899

1205902

120576=

sum119899

1(119909119894minus 119909)2

119899=

1 + 1205932

1+ sdot sdot sdot + 120593

2

119901

1 + 1205792

1+ sdot sdot sdot + 1205792

119901

1205902

119909

(18)

The parameters of the equation are reduced to thenumber of 119901 + 119902 Least square estimation is used to estimatethe parameters of the ARMA model which is obtained bydifferential operation

In the case of ARMA (119901 119902) the parameter vector is

= (1206011 120601

119901 1205791 120579

119902)1015840

119865119905() = 120601

1119909119905minus1

+ 1206012119909119905minus2

+ sdot sdot sdot + 120601119901119909119905minus119901

+ 120576119905minus 1205791120576119905minus1

minus 1205792120576119905minus2

minus sdot sdot sdot minus 120579119902120576119905minus119902

(19)

Thus overall observed sum of squared residuals of thesample is

119876() =

119899

sum

1

1205762

119905=

119899

sum

1

(119909119905minus 119865119905())2

(20)

Set the objective function for parameter estimation asmin119876() and use the weighted least squares method to solvethe parameters

The essence of the weighted least square method is totransform the original data to obtain the new explanatoryvariables and explained variable Assume that 119909

119894is time series

data and 120596 is the weight of this point so that weighted leastsquare method is

1199091015840

119894= 119909119894sdot 120596119894

(1199091015840 refers to travel time after transformation)

(21)

Then use (11990910158401 119909

1015840

119899) to do the least square parameter

estimation of ordinaryARIMAmodel we can get the optimalparameter vector

1015840

of themodel inweighted transformationIn this way the formula for the weighted forecast formula is

119865119905(1015840

) = 1205931015840

1119909119905minus1

+ 1205931015840

2119909119905minus2

+ sdot sdot sdot + 1205931015840

119901119909119905minus119901

+ 120576119905

minus 1205791015840

1120576119905minus1

minus 1205791015840

2120576119905minus2

minus sdot sdot sdot minus 1205791015840

119902120576119905minus119902

(22)

After removing theweight the final forecast value is obtained

119865119905() =

119865119905(1015840

)

120596119905

(23)

5 Application Example

51 Preprocessing of the Massive Data from Loop DetectorsThis paper takes the actual data of 2nd ring road in a big cityas an example (detector number is 020lowastlowast line number is Lan1ndashLan 6 date is on Mar 3rd 2013 data collection time is 24hours sampling interval is 2 minutes parameters are trafficvolume speed and occupancy rate and the total number ofdata points is 720) to verify the travel time forecasting modelbased on loop detectors which has been mentioned aboveThe actual data of Lan 1 is shown in Figure 2

In Figure 2 mutation points can be observed in the dataseries of all the three parameters In fact the data of trafficconditions cannot change more than 500 times within twominutes So it can be concluded that there are abnormal ordistorted data in the actual data and it is necessary to filterthose data

According toAlgorithm 1 we can finish the data cleaningFirstly according to the definition the control parametersbased on the basic traffic flow principle and the actualphysical meaning are set up in Table 1

6 International Journal of Distributed Sensor Networks

Table 1 Selection of filtering modelrsquos parameters

Parameter name Abbreviation Experimental valuesSection maximum flow 119902max 119902max = 360030Section maximum speed Vmax Vmax = 150Section maximum occupancy rate 119900max 119900max = 100Constraints of occupancy rate and speed 119867

1 1198811

1198671= 95 119881

1= 5

119902 volume ⟨vehicle volume(lanehour)minus1⟩ V speed ⟨kmh⟩ and 119900 occupancy rate ⟨⟩

00000000 0600 1800120000000000 0600 1800120000000000 0600 18001200

0

20

40

60

80

50

100

150

200

0

10

20

30

40

50

Figure 2 Data distribution of Lan 1rsquos volume-speed-occupancy rate in 24 hours

100 200 300 400 500 600 700

minus04

minus02

02

04

Figure 3 Fault data points of filtering intermediate state search

Under the control of parameters listed in Table 1 wecan finish the data cleaning to find out the data beyond themaximum control range or contrary to the theory of trafficflow The result is shown in Figure 3

Use a one-dimensional matrix to record the effectivenessof each record All the initial value of thematrix is 1When theabnormal data is detected the corresponding matrix valueis changed into 0 From Figure 3 there is a series of errordata points at the time 300ndash600 which is consistent with theoriginal graph shown in Figure 4

Because these error data points do not have actual physi-cal meaning they are replaced by the closest normal recordAfter the cleaning the figure of volume-speed-occupancyrate is shown in Figure 5

Data quality has been improved to a certain extentespecially for the speed data But there still has beenmutationin the filtering results Test the first-order differential ofthe data to determine the mutation data The first-orderdifference graph of intermediate state is shown in Figure 6

Table 2 Result of the filter by different parameters

1198900value 119890

0= 3 119890

0= 4 119890

0= 5

Number of change ranges beyond 83 34 21Rate of the data beyond 115 47 21

Control parameter of differential operational is 1198900

Assuming that 1198900

= 3 is the parameter of the reasonablechange region if the actual value of the first-order is exceededthree times of the standard deviation control range changethe corresponding position of the effective matrix into 0

When 1198900

= 3 there are 83 abnormal data points as isshown in Figure 7 which account for 115 of the total recordThis result is too strict for the data of detectors so we canincrease the value of 119890

0to release the strictness for change

range of the data as is shown in Table 2According to the results shown in the table 119890

0= 4 is more

reasonable Figure 8 is comparison chart between the finalresult of the filter and the original data and it can be seenthat the algorithm has basically achieved the requirements ofthe loop detector datarsquos cleaning and preprocessing

FromFigure 3 it can be seen that the algorithmhas a smallcorrection for the traffic data and the data with the occupancyrate but the algorithm has better effect on the speed data Inthe process of predicting travel time the speed of the detectoris often used only which means that the algorithm can besimplified so that it only needs to produce speed data

Because only the speed data is processed we need to setthe upper and lower limit of speed and the limit of first-orderdifferential change range to restrict data As a result of usingthe detector data to calculate and predict travel time we needthree continuous detectorrsquos data points to simulate the traveltime forecast inwhole roadnetworkThe sketchmap is shownin Figure 9

International Journal of Distributed Sensor Networks 7

00000000 0600 18001200

0

20

40

60

80

00000000 0600 18001200

50

100

150

200

00000000 0600 18001200

0

10

20

30

40

50

Figure 4 Comparison between the actual data and the results of the filterrsquos intermediate state

00000000 0600 18001200 00000000 0600 18001200 00000000 0600 18001200

0

20

40

60

80

20

40

60

80

100

120

0

10

20

30

40

50

Figure 5 Result of filterrsquos intermediate state

Figure 6 First-order difference graph of intermediate state

100 200 300 400 500 600 700

01

02

minus01

minus02

Figure 7 Distortion data distribution when 1198900= 3

Under the condition that 1198900= 3 the filter result of speed

after cleaning of the three detectors is as shown in Figures 1011 and 12

As shown in Figures 10sim12 the red correction curvebasically achieved a reasonable correction of the distortiondata

52 Calculation of Travel Time Parameters Use the traveltime conversion model given by formulas (5) and (6) wecan do the travel time conversion according to the speeddata For example when one calculates the travel timedriving from east to west the vehicles pass through sec-tions Detector 1 Detector 2 and Detector 2 Detector 3The result is shown in Figure 13 and the unit of time is sm

53 Pattern Partition of Travel Time Series Based on Change-Point Analysis Because it is not clear how to choose 119890

0up to

the size of the sample we select 7 values to search the change-point The result is in Table 3

According to the characteristics of the travel time serieswe need to select the result that has 5 to 10 change-points So1198900= 007 119890

0= 005 and 119890

0= 002 are all reasonable control

parameters The results of all the reasonable circumstanceare shown in Figure 14 We use different colors to indicatedifferent state

According to the time sequence diagram the results of thealgorithm are basically completed and the travel time seriesis decomposed into a series of time periods with practicalmeaning

8 International Journal of Distributed Sensor Networks

50

100

150

200

0

10

20

30

40

50

00000000 0600 18001200 00000000 0600 18001200 00000000 0600 18001200

0

20

40

60

80

Figure 8 Comparison of the source data and the filter result

Detector 1Detector 2Detector 3

4494m 157m

Figure 9 Distribution of the three serial detectors on the road

0800 1200 1600 20000

20

40

60

80

Figure 10 Filter result of the speed data of Detector 1

0800 1200 1600 20000

10

20

30

40

50

60

Figure 11 Filter result of the speed data of Detector 2

On the situation of 1198900

= 007 and 1198900

= 005 thereare 90 important change-points that were missed So 119890

0=

002 and 1 25 90 224 375 380 384 390 393 414 are idealcontrolling parameter and result of change-point control

0800 1200 1600 20000

10

20

30

40

50

60

Figure 12 Filter result of the speed data of Detector 3

0800 1200 1600 200000

01

02

03

04

Figure 13 Travel time calculation result from Detector 1Detector 2 to Detector 2 Detector 3

And the travel time is divided into 9 divisions 600sim650650sim900 900sim1328 1328sim1830 1830sim1838 1838sim18501850sim1856 1856sim1948 and 1948sim2300

Then if we assume that this moment is 1700 we canpredict the travel time of 1702 and 1704 The historical dataof the forecast model is shown in Figure 15

54 Travel Time Forecasting Based on ARIMAModel To sumup we choose ARIMA model to build up the forecast modeland regress the parameterUsing the data of 221sim330 collectedby Detector 1 at the time 1328sim1700 and testing the time

International Journal of Distributed Sensor Networks 9

Table 3 Result of change-point searching of each section

1198900

Searching result of change-point Number of change-points001 1 25 59 70 77 90 224 319 326 342 357 376 379 390 393 414 415 17002 1 25 90 224 375 380 384 390 393 414 10005 1 225 375 380 390 393 412 7007 1 225 380 393 412 501 1 411 2012 1 416 2015 1 1

Table 4 ARIMA model and forecast result of different weight functions

Weight functions Fitting equations Forecast value on 1702 Forecast value on 1704Square-root 119909

1015840

119905= 0000321317 + 0726226119909

1015840

119905minus1+ 0191767119909

2

119905minus20208528 0206241

Square 1199091015840

119905= 00000607593 + 0798534119909

1015840

119905minus1+ 0204619119909

2

119905minus20213584 0215573

Growth rate curve 1199091015840

119905= 0000340756 + 0717946119909

1015840

119905minus1+ 0193551119909

2

119905minus20209668 0208116

Liner 1199091015840

119905= 0000948595 + 0617164119909

1015840

119905minus1+ 0110425119909

2

119905minus20206278 0202864

Actual value on 1702 020577Actual value on 1704 0196385

Table 5 Error matrix of different model fittings

Weight functions ME MAE MAPE MSESquare-root (global) minus0000361064 001288 710089 0000279502Square (global) minus0000751537 00133415 734701 0000294168Growth rate curve (global) minus0000697771 00131061 722872 0000289045Liner (global) 136255 lowast 10

minus17 00120611 672481 0000258082Square-root (proximal point) minus000357858 00134934 629296 0000200429Square (proximal point) minus000703095 00144531 674308 0000263613Growth rate curve (proximal point) minus000513248 00133818 624778 0000211068Liner (proximal point) 0000443646 00113057 527639 0000140726

sequence we found it is a nonrandom stationary sequenceThe fix order result of the ARIMA model for the time of1328sim1700 is (2 0 0)

After the weighted least squares are transformed intoordinary least squares we use four kinds of weight functionto do the fitting experiment and also have the error analysisto the output results The forecast results of different weightfunctions are as in Table 4

Error analysis result is shown in Table 5 and we can seethat the crucial indexMAPE has a certain degree of reductionat the proximal point

The forecast results of four different weighting functionscanmeet the basic requirements of the accuracy error of 10At the same time we can know that the linear weight functionhas good fitting and forecasting effect on the experimentaldata The linear weighting function is the optimal weightfunction for this forecast according to the statistics in Table 5

6 ConclusionThis paper uses the change-point detection algorithm todivide travel time series into several patterns and set up

forecasting model through ARIMA for different patternsbased on massive data collected by the loop detectors onthe roads Different from traditional forecasting methodsit is easier to get the more optimized forecasting resultthan to obtain it by using the global search because thedifferent intervals in same pattern have similar statisticalcharacteristics of the mean and variance In the process ofdividing the travel time series the calculation of algorithmis complicated and the derivation of control parameters isonly obtained by experiments which still needs research inthe future

Conflict of Interests

The authors declare no conflict of interests

Authorsrsquo Contribution

Guangyu Zhu designed the forecasting model of travel timebased on the change-point detection algorithm and wrote thepaper Li Wang designed the preprocessing algorithm and

10 International Journal of Distributed Sensor Networks

0800 1200 1600 2000

0800 1200 1600 2000

0800 1200 1600 2000

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

e0 = 002 1 25 90 224 375 380 384 390 393 414

e0 = 005 1 225 375 380 390 393 412

e0 = 007 1 225 380 393 412

Figure 14 Result of the global state change-point searching

0600 0900 1200 1500000

005

010

015

020

025

Figure 15 Historical data of the forecast model

performed the data preprocess and revised the paper PengZhang and Kang Song analyzed the data

Acknowledgments

This work is supported by the National Science Founda-tion of China (nos 61572069 and 61503022) the Funda-mental Research Funds for the Central Universities (no2014JBM211) the Open Project Program of Key Laboratoryof System Control and Information Processing Ministry ofEducation Shanghai Jiaotong University (no Scip201507)Beijing Municipality Key Laboratory of Urban Traffic Oper-ation Simulation and Decision Support Beijing Transporta-tion Research Center the project of the Department of Trafficand Transportation of Hebei Province (no A0201-150505)and The National Key Technology Support Program (no2014BAG01B02)

References

[1] Z-P Li H Yu Y-C Liu and F-Q Liu ldquoAn improved adaptiveexponential smoothing model for short-term travel time fore-casting of urban arterial streetrdquo Acta Automatica Sinica vol 34no 11 pp 1404ndash1409 2008

[2] Y Hollander and R Liu ldquoEstimation of the distribution of traveltimes by repeated simulationrdquo Transportation Research Part CEmerging Technologies vol 16 no 2 pp 212ndash231 2008

[3] G-Y Jiang Technologies and Application of the Identification ofRoad Traffic Condition Communications Press Beijing China2004 (Chinese)

[4] M Gramaglia C J Bernardos and M Calderon ldquoVirtualinduction loops based on cooperative vehicular communica-tionsrdquo Sensors vol 13 no 2 pp 1467ndash1476 2013

[5] X Zhang and J A Rice ldquoShort-term travel time predictionrdquoTransportation Research Part C vol 11 no 3-4 pp 187ndash2102003

[6] U Mori A Mendiburu M Alvarez and J A Lozano ldquoA reviewof travel time estimation and forecasting for Advanced TravellerInformation Systemsrdquo Transportmetrica A Transport Sciencevol 11 no 2 pp 119ndash157 2015

[7] E I Vlahogianni M G Karlaftis and J C Golias ldquoShort-term traffic forecasting where we are and where wersquore goingrdquoTransportation Research Part C Emerging Technologies vol 43part 1 pp 3ndash19 2014

[8] C Shao K Zhang and Y Gu ldquoA study of route traveltime forecast method based on real data of urban expresswaynetworkrdquo China Civil Engineering Journal vol 36 no 1 pp 16ndash20 2003

[9] B R Chilukuri J A Laval and A Guin ldquoMicrosimulation-based framework for freeway travel time forecastingrdquo Trans-portation Research Record vol 2470 pp 34ndash45 2014

[10] E Yao and Y Zhang ldquoA new algorithm of shortmdashterm traveltime prediction for urban expresswayrdquo Journal of WuhanUniversity of Technology (Transportation ScienceampEngineering)vol 6 pp 1133ndash1137 2013

[11] J Zhao H Wang and W Liu ldquoPrediction of expressway traveltime based on adaptive interpolation kalman filteringrdquo Journalof South ChinaUniversity of Technology vol 2 pp 109ndash115 2014

[12] Z Gui and H Yu ldquoTrip travel time forecasting based onselective forgetting extreme learning machinerdquo Mathematical

International Journal of Distributed Sensor Networks 11

Problems in Engineering vol 2014 Article ID 829256 7 pages2014

[13] Y-J Jou Y-H Wen T-T Lee and H-J Cho ldquoMissing datatreatment on travel time estimation for ATISrdquo in Proceedingsof the IEEE International Conference on Systems Man andCybernetics (SMC rsquo03) vol 1 pp 102ndash107 IEEE WashingtonDC USA October 2003

[14] LWang and CWang ldquoA newmethod of real-time informationcollection in intelligent transportation systemrdquo Systems Engi-neering vol 23 no 2 pp 86ndash89 2005

[15] Z Cai Y Duan and A G Bourgeois ldquoDelay efficient oppor-tunistic routing in asynchronous multi-channel cognitive radionetworksrdquo Journal of Combinatorial Optimization vol 29 no 4pp 815ndash835 2015

[16] J W C VanLint and N J Zijpp ldquoAn improved travel timeestimation algorithm using dual-loop detectorsrdquo in Proceedingsof the Annual Meeting of the Transportation Research BoardWashington DC USA 2003

[17] Z Cai R Goebel and G Lin ldquoSize-constrained tree parti-tioning approximating the multicast k-tree routing problemrdquoTheoretical Computer Science vol 412 no 3 pp 240ndash245 2011

[18] G Zhu and H Yan ldquoA kind of demand-forecasting modelbased on analysis of demand booming and principle of naiveforecastingrdquo Systems EngineeringTheory amp Practice vol 24 no5 pp 22ndash33 2004

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AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

4 International Journal of Distributed Sensor Networks

119897119895 is valley point of peak carve or peak points of the valley

curveThe observation function decides whether to retain the

possible change-point Before that the control parameter andobservation function values of the minimum state variableare required

Observation function is

119879 (119908) = (

119861 (119908 119909119894

1003816100381610038161003816

119899

1)

119861 (119908 + 1 119909119894

1003816100381610038161003816

119899

1)

) 119908 = 0 1 119869 minus 1

119861 (119908 119909119894

1003816100381610038161003816

119899

1)

=

119908minus1

sum

119892=0

Dev(119909119905

1003816100381610038161003816

119887(119892+1)minus1

119887(119892) ) + Dev (119909

119905

1003816100381610038161003816

119899

119887(119908)) 119908 = 1 119869

Dev (119909119894

1003816100381610038161003816

119899

1) 119908 = 0

(9)

119887(119908)

is the index set of all 119869 points on the curve and 119887(0)

=

1199091198971

= 1199091

The algorithm is as follows

Algorithm 3Step 1 Make the travel time series into carve 119906 represents thetime of cycles119908 represents the number of change-points and119890119890is the auxiliary variable Set 119906 = 1 119908 = 1 119890

119890= 0 and

120573119895= 119897119895 119895 = 1 119896

Step 2 Select two convex waves along the axis of time from120573119906on the carve such as (120573

119906minus 120573119906+1

minus 120573119906+2

) Then calculate 119905lowast

119860 (120573119906 120573119906+1

120573119906+2

)

=

120573119906+1

sum

119905=120573119906

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+1

120573119906)

2

+

120573119906+2

sum

119905=120573119906+1+1

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+2

120573119906+1+1)

2

119872 (120573119906 119905lowast

1 120573119906+2

)

= max(

119905

sum

119905=120573119906

(119909119905minus 119909119905

1003816100381610038161003816

119905

120573119906)2

+

120573119906+2

sum

119905=119905+1

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+2

119905+1)

2

)

119898 (120573119906 119905lowast

2 120573119906+2

)

= min(

119905

sum

119905=120573119906

(119909119905minus 119909119905

1003816100381610038161003816

119905

120573119906)2

+

120573119906+2

sum

119905=119905+1

(119909119905minus 119909119905

1003816100381610038161003816

120573119906+2

119905+1)

2

)

(10)

Step 3 Change-point estimation is

119877119906=

2119860 (120573119906 120573119906+1

120573119906+2

)

(119872 (120573119906 119905lowast

1 120573119906+2

) + 119898 (120573119906 119905lowast

2 120573119906+2

))

119888119903119906=

119872(120573119906 119905lowast

1 120573119906+2

)

119898 (120573119906 119905lowast

2 120573119906+2

)

(11)

Consider the following

(1) If 119872(120573119906 119905lowast

1 120573119906+2

) asymp 119898(120573119906 119905lowast

2 120573119906+2

) 1 le 119888119903119906

lt 13and the sequence change is gently you do not need toset change-point in (119897

119906 119897119906+2

)

(2) If 119877119906

le (14) + 32(119888119903119906+ 1) regardless of whether

119905lowast

1= 120573119906+1

there is 119887(119908) = 120573119906+1

(3) If 1 ge 119877119906

ge (14) + 32(119888119903119906+ 1) seek the nearest

peak or valley point of 120573119906+1

and set the left oneas 120573119871

119906+1and right one as 120573

119877

119906+1 If exist119905

119888isin 120573

119871

119906+1

120573119877

119906+1 have Dev(120573

119906 119905119888 120573119906+2

) = min(Dev(120573119906 120573119871

119906+1

120573119906+2

)Dev(120573119906 120573119877

119906+1 120573119906+2

)) Then 119887(119908)

= 119905119888

(4) If 119877119906gt 1 then 119887

(119908)= 119905lowast

2

(5) Make effective judgments for 119887(119908) offered by (2) (3)

and (4) if 119879(119908 minus 1) minus 1 gt 1205760 keep this change-point

otherwise remove it

Step 4 If you have 119887(119908) then 119906 = 119906 + 1 120573

119906= 119887(119908) 120573119906+1

=

120573119906+1

120573119906+2

= 120573119906+2

119908 = 119908 + 1 and 119890119890= 0 Otherwise 119890

119890=

1 120573119906

= 120573119906 120573119906+1

= 120573119906+1+119890119890

and 120573119906+2+119890119890

= 120573119906+2+119890119890

When120573119906+2

= 120573119896 the searching is complete and the algorithm ends

Otherwise go back to Step 2

The paper [18] has provided a complete method on howto improve this kind of algorithm However there has beena crucial control parameter 119890

0for which the algorithm does

not give the specific processing formula That algorithmsets 119890

0as a constant value such as 05 In this paper we

will carry out several experiments with different 1198900 which

provide reference to the parameter of travel time inferredfrom detectorsrsquo data

42 ARIMA Forecasting Model The preprocessing of thetime series short-term forecasting model includes stationarytest and random test If the time series is nonstationary itneeds to be transformed into stationary series by differentialoperation In this circumstance the ARIMA model is con-verted to ARMA model The sequence of 119889 order differencesis expressed as

nabla119889119909119905=

119889

sum

119894=1

(minus1)119889119862119894

119889119909119905minus119894

(12)

ARIMA (119901 119889 119902) model can be expressed as

Φ (119861) nabla119889119909119905= Φ119901 (119861) 120576119905

119864 (120576119905) = 0

Var (120576119905) = 1205902

120576

119864 (120576119905120576119904) = 0 119904 = 119905

119864119909119904120576119905= 0 forall119904 lt 119905

(13)

International Journal of Distributed Sensor Networks 5

The order number 119901 119902 of ARIMA (119901 119889 119902)model is basedon Autocorrelation Coefficient (ACF) and Partial Auto-correlation Coefficient (PACF) of ARMA after differentialoperation And according to the characteristics of PACF andACF coefficients the model identification is carried out

For random inspection the data collected by the detectoris a large density data point so the calculation result of traveltime is also a large sample of high density which needs to testthe hypothesis by using 119876 statistics

119876 = 119899

119898

sum

119896=1

2

119896sim 1205942(119898) (119898 is delay-stage) (14)

When119876 is less than the quintile of 12059421minus120572

(119898) the sequenceis pure random sequence However when the travel time ismodeled by the ARIMA model if the sample space becomessmall the modified LB statistics can be used

LB = 119899 (119899 + 2)

119898

sum

119896=1

(2

119896

119899 minus 119896) (15)

Because there is only a short-term significant correlationin the sequence the test for the hypothesis is only for 119876 andLBwith short-termdelay-stage which is generally119898 less than10

After differential operation the ARIMA model isdegraded to the standard ARMAmodel its standard form is

119909119905= 120583 +

Φ119901 (119861)

Φ119901 (119861)

120576119905 (16)

In the formula

120576119905sim 119882119873(0 120590

2

119905)

Φ119901 (119861) = 1 minus 120579

1119861 minus sdot sdot sdot minus 120579

119901119861119901

Φ119901 (119861) = 1 minus 120593

1119861 minus sdot sdot sdot minus 120593

119902119861119902

(17)

There are 119901 + 119902 + 2 unknown parameters in ARMAmodel120601

1 1206012 120601

119901 1205791 1205792 120579

119902 120583 1205902

120576Matrix estimation is

usually used to obtain the value of 120583 and 1205902

120576

Calculate the expectation and variance of formula (16) oneach side and get

= 119864 (119909) = 119909 =sum119899

1119909119894

119899

1205902

120576=

sum119899

1(119909119894minus 119909)2

119899=

1 + 1205932

1+ sdot sdot sdot + 120593

2

119901

1 + 1205792

1+ sdot sdot sdot + 1205792

119901

1205902

119909

(18)

The parameters of the equation are reduced to thenumber of 119901 + 119902 Least square estimation is used to estimatethe parameters of the ARMA model which is obtained bydifferential operation

In the case of ARMA (119901 119902) the parameter vector is

= (1206011 120601

119901 1205791 120579

119902)1015840

119865119905() = 120601

1119909119905minus1

+ 1206012119909119905minus2

+ sdot sdot sdot + 120601119901119909119905minus119901

+ 120576119905minus 1205791120576119905minus1

minus 1205792120576119905minus2

minus sdot sdot sdot minus 120579119902120576119905minus119902

(19)

Thus overall observed sum of squared residuals of thesample is

119876() =

119899

sum

1

1205762

119905=

119899

sum

1

(119909119905minus 119865119905())2

(20)

Set the objective function for parameter estimation asmin119876() and use the weighted least squares method to solvethe parameters

The essence of the weighted least square method is totransform the original data to obtain the new explanatoryvariables and explained variable Assume that 119909

119894is time series

data and 120596 is the weight of this point so that weighted leastsquare method is

1199091015840

119894= 119909119894sdot 120596119894

(1199091015840 refers to travel time after transformation)

(21)

Then use (11990910158401 119909

1015840

119899) to do the least square parameter

estimation of ordinaryARIMAmodel we can get the optimalparameter vector

1015840

of themodel inweighted transformationIn this way the formula for the weighted forecast formula is

119865119905(1015840

) = 1205931015840

1119909119905minus1

+ 1205931015840

2119909119905minus2

+ sdot sdot sdot + 1205931015840

119901119909119905minus119901

+ 120576119905

minus 1205791015840

1120576119905minus1

minus 1205791015840

2120576119905minus2

minus sdot sdot sdot minus 1205791015840

119902120576119905minus119902

(22)

After removing theweight the final forecast value is obtained

119865119905() =

119865119905(1015840

)

120596119905

(23)

5 Application Example

51 Preprocessing of the Massive Data from Loop DetectorsThis paper takes the actual data of 2nd ring road in a big cityas an example (detector number is 020lowastlowast line number is Lan1ndashLan 6 date is on Mar 3rd 2013 data collection time is 24hours sampling interval is 2 minutes parameters are trafficvolume speed and occupancy rate and the total number ofdata points is 720) to verify the travel time forecasting modelbased on loop detectors which has been mentioned aboveThe actual data of Lan 1 is shown in Figure 2

In Figure 2 mutation points can be observed in the dataseries of all the three parameters In fact the data of trafficconditions cannot change more than 500 times within twominutes So it can be concluded that there are abnormal ordistorted data in the actual data and it is necessary to filterthose data

According toAlgorithm 1 we can finish the data cleaningFirstly according to the definition the control parametersbased on the basic traffic flow principle and the actualphysical meaning are set up in Table 1

6 International Journal of Distributed Sensor Networks

Table 1 Selection of filtering modelrsquos parameters

Parameter name Abbreviation Experimental valuesSection maximum flow 119902max 119902max = 360030Section maximum speed Vmax Vmax = 150Section maximum occupancy rate 119900max 119900max = 100Constraints of occupancy rate and speed 119867

1 1198811

1198671= 95 119881

1= 5

119902 volume ⟨vehicle volume(lanehour)minus1⟩ V speed ⟨kmh⟩ and 119900 occupancy rate ⟨⟩

00000000 0600 1800120000000000 0600 1800120000000000 0600 18001200

0

20

40

60

80

50

100

150

200

0

10

20

30

40

50

Figure 2 Data distribution of Lan 1rsquos volume-speed-occupancy rate in 24 hours

100 200 300 400 500 600 700

minus04

minus02

02

04

Figure 3 Fault data points of filtering intermediate state search

Under the control of parameters listed in Table 1 wecan finish the data cleaning to find out the data beyond themaximum control range or contrary to the theory of trafficflow The result is shown in Figure 3

Use a one-dimensional matrix to record the effectivenessof each record All the initial value of thematrix is 1When theabnormal data is detected the corresponding matrix valueis changed into 0 From Figure 3 there is a series of errordata points at the time 300ndash600 which is consistent with theoriginal graph shown in Figure 4

Because these error data points do not have actual physi-cal meaning they are replaced by the closest normal recordAfter the cleaning the figure of volume-speed-occupancyrate is shown in Figure 5

Data quality has been improved to a certain extentespecially for the speed data But there still has beenmutationin the filtering results Test the first-order differential ofthe data to determine the mutation data The first-orderdifference graph of intermediate state is shown in Figure 6

Table 2 Result of the filter by different parameters

1198900value 119890

0= 3 119890

0= 4 119890

0= 5

Number of change ranges beyond 83 34 21Rate of the data beyond 115 47 21

Control parameter of differential operational is 1198900

Assuming that 1198900

= 3 is the parameter of the reasonablechange region if the actual value of the first-order is exceededthree times of the standard deviation control range changethe corresponding position of the effective matrix into 0

When 1198900

= 3 there are 83 abnormal data points as isshown in Figure 7 which account for 115 of the total recordThis result is too strict for the data of detectors so we canincrease the value of 119890

0to release the strictness for change

range of the data as is shown in Table 2According to the results shown in the table 119890

0= 4 is more

reasonable Figure 8 is comparison chart between the finalresult of the filter and the original data and it can be seenthat the algorithm has basically achieved the requirements ofthe loop detector datarsquos cleaning and preprocessing

FromFigure 3 it can be seen that the algorithmhas a smallcorrection for the traffic data and the data with the occupancyrate but the algorithm has better effect on the speed data Inthe process of predicting travel time the speed of the detectoris often used only which means that the algorithm can besimplified so that it only needs to produce speed data

Because only the speed data is processed we need to setthe upper and lower limit of speed and the limit of first-orderdifferential change range to restrict data As a result of usingthe detector data to calculate and predict travel time we needthree continuous detectorrsquos data points to simulate the traveltime forecast inwhole roadnetworkThe sketchmap is shownin Figure 9

International Journal of Distributed Sensor Networks 7

00000000 0600 18001200

0

20

40

60

80

00000000 0600 18001200

50

100

150

200

00000000 0600 18001200

0

10

20

30

40

50

Figure 4 Comparison between the actual data and the results of the filterrsquos intermediate state

00000000 0600 18001200 00000000 0600 18001200 00000000 0600 18001200

0

20

40

60

80

20

40

60

80

100

120

0

10

20

30

40

50

Figure 5 Result of filterrsquos intermediate state

Figure 6 First-order difference graph of intermediate state

100 200 300 400 500 600 700

01

02

minus01

minus02

Figure 7 Distortion data distribution when 1198900= 3

Under the condition that 1198900= 3 the filter result of speed

after cleaning of the three detectors is as shown in Figures 1011 and 12

As shown in Figures 10sim12 the red correction curvebasically achieved a reasonable correction of the distortiondata

52 Calculation of Travel Time Parameters Use the traveltime conversion model given by formulas (5) and (6) wecan do the travel time conversion according to the speeddata For example when one calculates the travel timedriving from east to west the vehicles pass through sec-tions Detector 1 Detector 2 and Detector 2 Detector 3The result is shown in Figure 13 and the unit of time is sm

53 Pattern Partition of Travel Time Series Based on Change-Point Analysis Because it is not clear how to choose 119890

0up to

the size of the sample we select 7 values to search the change-point The result is in Table 3

According to the characteristics of the travel time serieswe need to select the result that has 5 to 10 change-points So1198900= 007 119890

0= 005 and 119890

0= 002 are all reasonable control

parameters The results of all the reasonable circumstanceare shown in Figure 14 We use different colors to indicatedifferent state

According to the time sequence diagram the results of thealgorithm are basically completed and the travel time seriesis decomposed into a series of time periods with practicalmeaning

8 International Journal of Distributed Sensor Networks

50

100

150

200

0

10

20

30

40

50

00000000 0600 18001200 00000000 0600 18001200 00000000 0600 18001200

0

20

40

60

80

Figure 8 Comparison of the source data and the filter result

Detector 1Detector 2Detector 3

4494m 157m

Figure 9 Distribution of the three serial detectors on the road

0800 1200 1600 20000

20

40

60

80

Figure 10 Filter result of the speed data of Detector 1

0800 1200 1600 20000

10

20

30

40

50

60

Figure 11 Filter result of the speed data of Detector 2

On the situation of 1198900

= 007 and 1198900

= 005 thereare 90 important change-points that were missed So 119890

0=

002 and 1 25 90 224 375 380 384 390 393 414 are idealcontrolling parameter and result of change-point control

0800 1200 1600 20000

10

20

30

40

50

60

Figure 12 Filter result of the speed data of Detector 3

0800 1200 1600 200000

01

02

03

04

Figure 13 Travel time calculation result from Detector 1Detector 2 to Detector 2 Detector 3

And the travel time is divided into 9 divisions 600sim650650sim900 900sim1328 1328sim1830 1830sim1838 1838sim18501850sim1856 1856sim1948 and 1948sim2300

Then if we assume that this moment is 1700 we canpredict the travel time of 1702 and 1704 The historical dataof the forecast model is shown in Figure 15

54 Travel Time Forecasting Based on ARIMAModel To sumup we choose ARIMA model to build up the forecast modeland regress the parameterUsing the data of 221sim330 collectedby Detector 1 at the time 1328sim1700 and testing the time

International Journal of Distributed Sensor Networks 9

Table 3 Result of change-point searching of each section

1198900

Searching result of change-point Number of change-points001 1 25 59 70 77 90 224 319 326 342 357 376 379 390 393 414 415 17002 1 25 90 224 375 380 384 390 393 414 10005 1 225 375 380 390 393 412 7007 1 225 380 393 412 501 1 411 2012 1 416 2015 1 1

Table 4 ARIMA model and forecast result of different weight functions

Weight functions Fitting equations Forecast value on 1702 Forecast value on 1704Square-root 119909

1015840

119905= 0000321317 + 0726226119909

1015840

119905minus1+ 0191767119909

2

119905minus20208528 0206241

Square 1199091015840

119905= 00000607593 + 0798534119909

1015840

119905minus1+ 0204619119909

2

119905minus20213584 0215573

Growth rate curve 1199091015840

119905= 0000340756 + 0717946119909

1015840

119905minus1+ 0193551119909

2

119905minus20209668 0208116

Liner 1199091015840

119905= 0000948595 + 0617164119909

1015840

119905minus1+ 0110425119909

2

119905minus20206278 0202864

Actual value on 1702 020577Actual value on 1704 0196385

Table 5 Error matrix of different model fittings

Weight functions ME MAE MAPE MSESquare-root (global) minus0000361064 001288 710089 0000279502Square (global) minus0000751537 00133415 734701 0000294168Growth rate curve (global) minus0000697771 00131061 722872 0000289045Liner (global) 136255 lowast 10

minus17 00120611 672481 0000258082Square-root (proximal point) minus000357858 00134934 629296 0000200429Square (proximal point) minus000703095 00144531 674308 0000263613Growth rate curve (proximal point) minus000513248 00133818 624778 0000211068Liner (proximal point) 0000443646 00113057 527639 0000140726

sequence we found it is a nonrandom stationary sequenceThe fix order result of the ARIMA model for the time of1328sim1700 is (2 0 0)

After the weighted least squares are transformed intoordinary least squares we use four kinds of weight functionto do the fitting experiment and also have the error analysisto the output results The forecast results of different weightfunctions are as in Table 4

Error analysis result is shown in Table 5 and we can seethat the crucial indexMAPE has a certain degree of reductionat the proximal point

The forecast results of four different weighting functionscanmeet the basic requirements of the accuracy error of 10At the same time we can know that the linear weight functionhas good fitting and forecasting effect on the experimentaldata The linear weighting function is the optimal weightfunction for this forecast according to the statistics in Table 5

6 ConclusionThis paper uses the change-point detection algorithm todivide travel time series into several patterns and set up

forecasting model through ARIMA for different patternsbased on massive data collected by the loop detectors onthe roads Different from traditional forecasting methodsit is easier to get the more optimized forecasting resultthan to obtain it by using the global search because thedifferent intervals in same pattern have similar statisticalcharacteristics of the mean and variance In the process ofdividing the travel time series the calculation of algorithmis complicated and the derivation of control parameters isonly obtained by experiments which still needs research inthe future

Conflict of Interests

The authors declare no conflict of interests

Authorsrsquo Contribution

Guangyu Zhu designed the forecasting model of travel timebased on the change-point detection algorithm and wrote thepaper Li Wang designed the preprocessing algorithm and

10 International Journal of Distributed Sensor Networks

0800 1200 1600 2000

0800 1200 1600 2000

0800 1200 1600 2000

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

e0 = 002 1 25 90 224 375 380 384 390 393 414

e0 = 005 1 225 375 380 390 393 412

e0 = 007 1 225 380 393 412

Figure 14 Result of the global state change-point searching

0600 0900 1200 1500000

005

010

015

020

025

Figure 15 Historical data of the forecast model

performed the data preprocess and revised the paper PengZhang and Kang Song analyzed the data

Acknowledgments

This work is supported by the National Science Founda-tion of China (nos 61572069 and 61503022) the Funda-mental Research Funds for the Central Universities (no2014JBM211) the Open Project Program of Key Laboratoryof System Control and Information Processing Ministry ofEducation Shanghai Jiaotong University (no Scip201507)Beijing Municipality Key Laboratory of Urban Traffic Oper-ation Simulation and Decision Support Beijing Transporta-tion Research Center the project of the Department of Trafficand Transportation of Hebei Province (no A0201-150505)and The National Key Technology Support Program (no2014BAG01B02)

References

[1] Z-P Li H Yu Y-C Liu and F-Q Liu ldquoAn improved adaptiveexponential smoothing model for short-term travel time fore-casting of urban arterial streetrdquo Acta Automatica Sinica vol 34no 11 pp 1404ndash1409 2008

[2] Y Hollander and R Liu ldquoEstimation of the distribution of traveltimes by repeated simulationrdquo Transportation Research Part CEmerging Technologies vol 16 no 2 pp 212ndash231 2008

[3] G-Y Jiang Technologies and Application of the Identification ofRoad Traffic Condition Communications Press Beijing China2004 (Chinese)

[4] M Gramaglia C J Bernardos and M Calderon ldquoVirtualinduction loops based on cooperative vehicular communica-tionsrdquo Sensors vol 13 no 2 pp 1467ndash1476 2013

[5] X Zhang and J A Rice ldquoShort-term travel time predictionrdquoTransportation Research Part C vol 11 no 3-4 pp 187ndash2102003

[6] U Mori A Mendiburu M Alvarez and J A Lozano ldquoA reviewof travel time estimation and forecasting for Advanced TravellerInformation Systemsrdquo Transportmetrica A Transport Sciencevol 11 no 2 pp 119ndash157 2015

[7] E I Vlahogianni M G Karlaftis and J C Golias ldquoShort-term traffic forecasting where we are and where wersquore goingrdquoTransportation Research Part C Emerging Technologies vol 43part 1 pp 3ndash19 2014

[8] C Shao K Zhang and Y Gu ldquoA study of route traveltime forecast method based on real data of urban expresswaynetworkrdquo China Civil Engineering Journal vol 36 no 1 pp 16ndash20 2003

[9] B R Chilukuri J A Laval and A Guin ldquoMicrosimulation-based framework for freeway travel time forecastingrdquo Trans-portation Research Record vol 2470 pp 34ndash45 2014

[10] E Yao and Y Zhang ldquoA new algorithm of shortmdashterm traveltime prediction for urban expresswayrdquo Journal of WuhanUniversity of Technology (Transportation ScienceampEngineering)vol 6 pp 1133ndash1137 2013

[11] J Zhao H Wang and W Liu ldquoPrediction of expressway traveltime based on adaptive interpolation kalman filteringrdquo Journalof South ChinaUniversity of Technology vol 2 pp 109ndash115 2014

[12] Z Gui and H Yu ldquoTrip travel time forecasting based onselective forgetting extreme learning machinerdquo Mathematical

International Journal of Distributed Sensor Networks 11

Problems in Engineering vol 2014 Article ID 829256 7 pages2014

[13] Y-J Jou Y-H Wen T-T Lee and H-J Cho ldquoMissing datatreatment on travel time estimation for ATISrdquo in Proceedingsof the IEEE International Conference on Systems Man andCybernetics (SMC rsquo03) vol 1 pp 102ndash107 IEEE WashingtonDC USA October 2003

[14] LWang and CWang ldquoA newmethod of real-time informationcollection in intelligent transportation systemrdquo Systems Engi-neering vol 23 no 2 pp 86ndash89 2005

[15] Z Cai Y Duan and A G Bourgeois ldquoDelay efficient oppor-tunistic routing in asynchronous multi-channel cognitive radionetworksrdquo Journal of Combinatorial Optimization vol 29 no 4pp 815ndash835 2015

[16] J W C VanLint and N J Zijpp ldquoAn improved travel timeestimation algorithm using dual-loop detectorsrdquo in Proceedingsof the Annual Meeting of the Transportation Research BoardWashington DC USA 2003

[17] Z Cai R Goebel and G Lin ldquoSize-constrained tree parti-tioning approximating the multicast k-tree routing problemrdquoTheoretical Computer Science vol 412 no 3 pp 240ndash245 2011

[18] G Zhu and H Yan ldquoA kind of demand-forecasting modelbased on analysis of demand booming and principle of naiveforecastingrdquo Systems EngineeringTheory amp Practice vol 24 no5 pp 22ndash33 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

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RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 5

The order number 119901 119902 of ARIMA (119901 119889 119902)model is basedon Autocorrelation Coefficient (ACF) and Partial Auto-correlation Coefficient (PACF) of ARMA after differentialoperation And according to the characteristics of PACF andACF coefficients the model identification is carried out

For random inspection the data collected by the detectoris a large density data point so the calculation result of traveltime is also a large sample of high density which needs to testthe hypothesis by using 119876 statistics

119876 = 119899

119898

sum

119896=1

2

119896sim 1205942(119898) (119898 is delay-stage) (14)

When119876 is less than the quintile of 12059421minus120572

(119898) the sequenceis pure random sequence However when the travel time ismodeled by the ARIMA model if the sample space becomessmall the modified LB statistics can be used

LB = 119899 (119899 + 2)

119898

sum

119896=1

(2

119896

119899 minus 119896) (15)

Because there is only a short-term significant correlationin the sequence the test for the hypothesis is only for 119876 andLBwith short-termdelay-stage which is generally119898 less than10

After differential operation the ARIMA model isdegraded to the standard ARMAmodel its standard form is

119909119905= 120583 +

Φ119901 (119861)

Φ119901 (119861)

120576119905 (16)

In the formula

120576119905sim 119882119873(0 120590

2

119905)

Φ119901 (119861) = 1 minus 120579

1119861 minus sdot sdot sdot minus 120579

119901119861119901

Φ119901 (119861) = 1 minus 120593

1119861 minus sdot sdot sdot minus 120593

119902119861119902

(17)

There are 119901 + 119902 + 2 unknown parameters in ARMAmodel120601

1 1206012 120601

119901 1205791 1205792 120579

119902 120583 1205902

120576Matrix estimation is

usually used to obtain the value of 120583 and 1205902

120576

Calculate the expectation and variance of formula (16) oneach side and get

= 119864 (119909) = 119909 =sum119899

1119909119894

119899

1205902

120576=

sum119899

1(119909119894minus 119909)2

119899=

1 + 1205932

1+ sdot sdot sdot + 120593

2

119901

1 + 1205792

1+ sdot sdot sdot + 1205792

119901

1205902

119909

(18)

The parameters of the equation are reduced to thenumber of 119901 + 119902 Least square estimation is used to estimatethe parameters of the ARMA model which is obtained bydifferential operation

In the case of ARMA (119901 119902) the parameter vector is

= (1206011 120601

119901 1205791 120579

119902)1015840

119865119905() = 120601

1119909119905minus1

+ 1206012119909119905minus2

+ sdot sdot sdot + 120601119901119909119905minus119901

+ 120576119905minus 1205791120576119905minus1

minus 1205792120576119905minus2

minus sdot sdot sdot minus 120579119902120576119905minus119902

(19)

Thus overall observed sum of squared residuals of thesample is

119876() =

119899

sum

1

1205762

119905=

119899

sum

1

(119909119905minus 119865119905())2

(20)

Set the objective function for parameter estimation asmin119876() and use the weighted least squares method to solvethe parameters

The essence of the weighted least square method is totransform the original data to obtain the new explanatoryvariables and explained variable Assume that 119909

119894is time series

data and 120596 is the weight of this point so that weighted leastsquare method is

1199091015840

119894= 119909119894sdot 120596119894

(1199091015840 refers to travel time after transformation)

(21)

Then use (11990910158401 119909

1015840

119899) to do the least square parameter

estimation of ordinaryARIMAmodel we can get the optimalparameter vector

1015840

of themodel inweighted transformationIn this way the formula for the weighted forecast formula is

119865119905(1015840

) = 1205931015840

1119909119905minus1

+ 1205931015840

2119909119905minus2

+ sdot sdot sdot + 1205931015840

119901119909119905minus119901

+ 120576119905

minus 1205791015840

1120576119905minus1

minus 1205791015840

2120576119905minus2

minus sdot sdot sdot minus 1205791015840

119902120576119905minus119902

(22)

After removing theweight the final forecast value is obtained

119865119905() =

119865119905(1015840

)

120596119905

(23)

5 Application Example

51 Preprocessing of the Massive Data from Loop DetectorsThis paper takes the actual data of 2nd ring road in a big cityas an example (detector number is 020lowastlowast line number is Lan1ndashLan 6 date is on Mar 3rd 2013 data collection time is 24hours sampling interval is 2 minutes parameters are trafficvolume speed and occupancy rate and the total number ofdata points is 720) to verify the travel time forecasting modelbased on loop detectors which has been mentioned aboveThe actual data of Lan 1 is shown in Figure 2

In Figure 2 mutation points can be observed in the dataseries of all the three parameters In fact the data of trafficconditions cannot change more than 500 times within twominutes So it can be concluded that there are abnormal ordistorted data in the actual data and it is necessary to filterthose data

According toAlgorithm 1 we can finish the data cleaningFirstly according to the definition the control parametersbased on the basic traffic flow principle and the actualphysical meaning are set up in Table 1

6 International Journal of Distributed Sensor Networks

Table 1 Selection of filtering modelrsquos parameters

Parameter name Abbreviation Experimental valuesSection maximum flow 119902max 119902max = 360030Section maximum speed Vmax Vmax = 150Section maximum occupancy rate 119900max 119900max = 100Constraints of occupancy rate and speed 119867

1 1198811

1198671= 95 119881

1= 5

119902 volume ⟨vehicle volume(lanehour)minus1⟩ V speed ⟨kmh⟩ and 119900 occupancy rate ⟨⟩

00000000 0600 1800120000000000 0600 1800120000000000 0600 18001200

0

20

40

60

80

50

100

150

200

0

10

20

30

40

50

Figure 2 Data distribution of Lan 1rsquos volume-speed-occupancy rate in 24 hours

100 200 300 400 500 600 700

minus04

minus02

02

04

Figure 3 Fault data points of filtering intermediate state search

Under the control of parameters listed in Table 1 wecan finish the data cleaning to find out the data beyond themaximum control range or contrary to the theory of trafficflow The result is shown in Figure 3

Use a one-dimensional matrix to record the effectivenessof each record All the initial value of thematrix is 1When theabnormal data is detected the corresponding matrix valueis changed into 0 From Figure 3 there is a series of errordata points at the time 300ndash600 which is consistent with theoriginal graph shown in Figure 4

Because these error data points do not have actual physi-cal meaning they are replaced by the closest normal recordAfter the cleaning the figure of volume-speed-occupancyrate is shown in Figure 5

Data quality has been improved to a certain extentespecially for the speed data But there still has beenmutationin the filtering results Test the first-order differential ofthe data to determine the mutation data The first-orderdifference graph of intermediate state is shown in Figure 6

Table 2 Result of the filter by different parameters

1198900value 119890

0= 3 119890

0= 4 119890

0= 5

Number of change ranges beyond 83 34 21Rate of the data beyond 115 47 21

Control parameter of differential operational is 1198900

Assuming that 1198900

= 3 is the parameter of the reasonablechange region if the actual value of the first-order is exceededthree times of the standard deviation control range changethe corresponding position of the effective matrix into 0

When 1198900

= 3 there are 83 abnormal data points as isshown in Figure 7 which account for 115 of the total recordThis result is too strict for the data of detectors so we canincrease the value of 119890

0to release the strictness for change

range of the data as is shown in Table 2According to the results shown in the table 119890

0= 4 is more

reasonable Figure 8 is comparison chart between the finalresult of the filter and the original data and it can be seenthat the algorithm has basically achieved the requirements ofthe loop detector datarsquos cleaning and preprocessing

FromFigure 3 it can be seen that the algorithmhas a smallcorrection for the traffic data and the data with the occupancyrate but the algorithm has better effect on the speed data Inthe process of predicting travel time the speed of the detectoris often used only which means that the algorithm can besimplified so that it only needs to produce speed data

Because only the speed data is processed we need to setthe upper and lower limit of speed and the limit of first-orderdifferential change range to restrict data As a result of usingthe detector data to calculate and predict travel time we needthree continuous detectorrsquos data points to simulate the traveltime forecast inwhole roadnetworkThe sketchmap is shownin Figure 9

International Journal of Distributed Sensor Networks 7

00000000 0600 18001200

0

20

40

60

80

00000000 0600 18001200

50

100

150

200

00000000 0600 18001200

0

10

20

30

40

50

Figure 4 Comparison between the actual data and the results of the filterrsquos intermediate state

00000000 0600 18001200 00000000 0600 18001200 00000000 0600 18001200

0

20

40

60

80

20

40

60

80

100

120

0

10

20

30

40

50

Figure 5 Result of filterrsquos intermediate state

Figure 6 First-order difference graph of intermediate state

100 200 300 400 500 600 700

01

02

minus01

minus02

Figure 7 Distortion data distribution when 1198900= 3

Under the condition that 1198900= 3 the filter result of speed

after cleaning of the three detectors is as shown in Figures 1011 and 12

As shown in Figures 10sim12 the red correction curvebasically achieved a reasonable correction of the distortiondata

52 Calculation of Travel Time Parameters Use the traveltime conversion model given by formulas (5) and (6) wecan do the travel time conversion according to the speeddata For example when one calculates the travel timedriving from east to west the vehicles pass through sec-tions Detector 1 Detector 2 and Detector 2 Detector 3The result is shown in Figure 13 and the unit of time is sm

53 Pattern Partition of Travel Time Series Based on Change-Point Analysis Because it is not clear how to choose 119890

0up to

the size of the sample we select 7 values to search the change-point The result is in Table 3

According to the characteristics of the travel time serieswe need to select the result that has 5 to 10 change-points So1198900= 007 119890

0= 005 and 119890

0= 002 are all reasonable control

parameters The results of all the reasonable circumstanceare shown in Figure 14 We use different colors to indicatedifferent state

According to the time sequence diagram the results of thealgorithm are basically completed and the travel time seriesis decomposed into a series of time periods with practicalmeaning

8 International Journal of Distributed Sensor Networks

50

100

150

200

0

10

20

30

40

50

00000000 0600 18001200 00000000 0600 18001200 00000000 0600 18001200

0

20

40

60

80

Figure 8 Comparison of the source data and the filter result

Detector 1Detector 2Detector 3

4494m 157m

Figure 9 Distribution of the three serial detectors on the road

0800 1200 1600 20000

20

40

60

80

Figure 10 Filter result of the speed data of Detector 1

0800 1200 1600 20000

10

20

30

40

50

60

Figure 11 Filter result of the speed data of Detector 2

On the situation of 1198900

= 007 and 1198900

= 005 thereare 90 important change-points that were missed So 119890

0=

002 and 1 25 90 224 375 380 384 390 393 414 are idealcontrolling parameter and result of change-point control

0800 1200 1600 20000

10

20

30

40

50

60

Figure 12 Filter result of the speed data of Detector 3

0800 1200 1600 200000

01

02

03

04

Figure 13 Travel time calculation result from Detector 1Detector 2 to Detector 2 Detector 3

And the travel time is divided into 9 divisions 600sim650650sim900 900sim1328 1328sim1830 1830sim1838 1838sim18501850sim1856 1856sim1948 and 1948sim2300

Then if we assume that this moment is 1700 we canpredict the travel time of 1702 and 1704 The historical dataof the forecast model is shown in Figure 15

54 Travel Time Forecasting Based on ARIMAModel To sumup we choose ARIMA model to build up the forecast modeland regress the parameterUsing the data of 221sim330 collectedby Detector 1 at the time 1328sim1700 and testing the time

International Journal of Distributed Sensor Networks 9

Table 3 Result of change-point searching of each section

1198900

Searching result of change-point Number of change-points001 1 25 59 70 77 90 224 319 326 342 357 376 379 390 393 414 415 17002 1 25 90 224 375 380 384 390 393 414 10005 1 225 375 380 390 393 412 7007 1 225 380 393 412 501 1 411 2012 1 416 2015 1 1

Table 4 ARIMA model and forecast result of different weight functions

Weight functions Fitting equations Forecast value on 1702 Forecast value on 1704Square-root 119909

1015840

119905= 0000321317 + 0726226119909

1015840

119905minus1+ 0191767119909

2

119905minus20208528 0206241

Square 1199091015840

119905= 00000607593 + 0798534119909

1015840

119905minus1+ 0204619119909

2

119905minus20213584 0215573

Growth rate curve 1199091015840

119905= 0000340756 + 0717946119909

1015840

119905minus1+ 0193551119909

2

119905minus20209668 0208116

Liner 1199091015840

119905= 0000948595 + 0617164119909

1015840

119905minus1+ 0110425119909

2

119905minus20206278 0202864

Actual value on 1702 020577Actual value on 1704 0196385

Table 5 Error matrix of different model fittings

Weight functions ME MAE MAPE MSESquare-root (global) minus0000361064 001288 710089 0000279502Square (global) minus0000751537 00133415 734701 0000294168Growth rate curve (global) minus0000697771 00131061 722872 0000289045Liner (global) 136255 lowast 10

minus17 00120611 672481 0000258082Square-root (proximal point) minus000357858 00134934 629296 0000200429Square (proximal point) minus000703095 00144531 674308 0000263613Growth rate curve (proximal point) minus000513248 00133818 624778 0000211068Liner (proximal point) 0000443646 00113057 527639 0000140726

sequence we found it is a nonrandom stationary sequenceThe fix order result of the ARIMA model for the time of1328sim1700 is (2 0 0)

After the weighted least squares are transformed intoordinary least squares we use four kinds of weight functionto do the fitting experiment and also have the error analysisto the output results The forecast results of different weightfunctions are as in Table 4

Error analysis result is shown in Table 5 and we can seethat the crucial indexMAPE has a certain degree of reductionat the proximal point

The forecast results of four different weighting functionscanmeet the basic requirements of the accuracy error of 10At the same time we can know that the linear weight functionhas good fitting and forecasting effect on the experimentaldata The linear weighting function is the optimal weightfunction for this forecast according to the statistics in Table 5

6 ConclusionThis paper uses the change-point detection algorithm todivide travel time series into several patterns and set up

forecasting model through ARIMA for different patternsbased on massive data collected by the loop detectors onthe roads Different from traditional forecasting methodsit is easier to get the more optimized forecasting resultthan to obtain it by using the global search because thedifferent intervals in same pattern have similar statisticalcharacteristics of the mean and variance In the process ofdividing the travel time series the calculation of algorithmis complicated and the derivation of control parameters isonly obtained by experiments which still needs research inthe future

Conflict of Interests

The authors declare no conflict of interests

Authorsrsquo Contribution

Guangyu Zhu designed the forecasting model of travel timebased on the change-point detection algorithm and wrote thepaper Li Wang designed the preprocessing algorithm and

10 International Journal of Distributed Sensor Networks

0800 1200 1600 2000

0800 1200 1600 2000

0800 1200 1600 2000

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

e0 = 002 1 25 90 224 375 380 384 390 393 414

e0 = 005 1 225 375 380 390 393 412

e0 = 007 1 225 380 393 412

Figure 14 Result of the global state change-point searching

0600 0900 1200 1500000

005

010

015

020

025

Figure 15 Historical data of the forecast model

performed the data preprocess and revised the paper PengZhang and Kang Song analyzed the data

Acknowledgments

This work is supported by the National Science Founda-tion of China (nos 61572069 and 61503022) the Funda-mental Research Funds for the Central Universities (no2014JBM211) the Open Project Program of Key Laboratoryof System Control and Information Processing Ministry ofEducation Shanghai Jiaotong University (no Scip201507)Beijing Municipality Key Laboratory of Urban Traffic Oper-ation Simulation and Decision Support Beijing Transporta-tion Research Center the project of the Department of Trafficand Transportation of Hebei Province (no A0201-150505)and The National Key Technology Support Program (no2014BAG01B02)

References

[1] Z-P Li H Yu Y-C Liu and F-Q Liu ldquoAn improved adaptiveexponential smoothing model for short-term travel time fore-casting of urban arterial streetrdquo Acta Automatica Sinica vol 34no 11 pp 1404ndash1409 2008

[2] Y Hollander and R Liu ldquoEstimation of the distribution of traveltimes by repeated simulationrdquo Transportation Research Part CEmerging Technologies vol 16 no 2 pp 212ndash231 2008

[3] G-Y Jiang Technologies and Application of the Identification ofRoad Traffic Condition Communications Press Beijing China2004 (Chinese)

[4] M Gramaglia C J Bernardos and M Calderon ldquoVirtualinduction loops based on cooperative vehicular communica-tionsrdquo Sensors vol 13 no 2 pp 1467ndash1476 2013

[5] X Zhang and J A Rice ldquoShort-term travel time predictionrdquoTransportation Research Part C vol 11 no 3-4 pp 187ndash2102003

[6] U Mori A Mendiburu M Alvarez and J A Lozano ldquoA reviewof travel time estimation and forecasting for Advanced TravellerInformation Systemsrdquo Transportmetrica A Transport Sciencevol 11 no 2 pp 119ndash157 2015

[7] E I Vlahogianni M G Karlaftis and J C Golias ldquoShort-term traffic forecasting where we are and where wersquore goingrdquoTransportation Research Part C Emerging Technologies vol 43part 1 pp 3ndash19 2014

[8] C Shao K Zhang and Y Gu ldquoA study of route traveltime forecast method based on real data of urban expresswaynetworkrdquo China Civil Engineering Journal vol 36 no 1 pp 16ndash20 2003

[9] B R Chilukuri J A Laval and A Guin ldquoMicrosimulation-based framework for freeway travel time forecastingrdquo Trans-portation Research Record vol 2470 pp 34ndash45 2014

[10] E Yao and Y Zhang ldquoA new algorithm of shortmdashterm traveltime prediction for urban expresswayrdquo Journal of WuhanUniversity of Technology (Transportation ScienceampEngineering)vol 6 pp 1133ndash1137 2013

[11] J Zhao H Wang and W Liu ldquoPrediction of expressway traveltime based on adaptive interpolation kalman filteringrdquo Journalof South ChinaUniversity of Technology vol 2 pp 109ndash115 2014

[12] Z Gui and H Yu ldquoTrip travel time forecasting based onselective forgetting extreme learning machinerdquo Mathematical

International Journal of Distributed Sensor Networks 11

Problems in Engineering vol 2014 Article ID 829256 7 pages2014

[13] Y-J Jou Y-H Wen T-T Lee and H-J Cho ldquoMissing datatreatment on travel time estimation for ATISrdquo in Proceedingsof the IEEE International Conference on Systems Man andCybernetics (SMC rsquo03) vol 1 pp 102ndash107 IEEE WashingtonDC USA October 2003

[14] LWang and CWang ldquoA newmethod of real-time informationcollection in intelligent transportation systemrdquo Systems Engi-neering vol 23 no 2 pp 86ndash89 2005

[15] Z Cai Y Duan and A G Bourgeois ldquoDelay efficient oppor-tunistic routing in asynchronous multi-channel cognitive radionetworksrdquo Journal of Combinatorial Optimization vol 29 no 4pp 815ndash835 2015

[16] J W C VanLint and N J Zijpp ldquoAn improved travel timeestimation algorithm using dual-loop detectorsrdquo in Proceedingsof the Annual Meeting of the Transportation Research BoardWashington DC USA 2003

[17] Z Cai R Goebel and G Lin ldquoSize-constrained tree parti-tioning approximating the multicast k-tree routing problemrdquoTheoretical Computer Science vol 412 no 3 pp 240ndash245 2011

[18] G Zhu and H Yan ldquoA kind of demand-forecasting modelbased on analysis of demand booming and principle of naiveforecastingrdquo Systems EngineeringTheory amp Practice vol 24 no5 pp 22ndash33 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 International Journal of Distributed Sensor Networks

Table 1 Selection of filtering modelrsquos parameters

Parameter name Abbreviation Experimental valuesSection maximum flow 119902max 119902max = 360030Section maximum speed Vmax Vmax = 150Section maximum occupancy rate 119900max 119900max = 100Constraints of occupancy rate and speed 119867

1 1198811

1198671= 95 119881

1= 5

119902 volume ⟨vehicle volume(lanehour)minus1⟩ V speed ⟨kmh⟩ and 119900 occupancy rate ⟨⟩

00000000 0600 1800120000000000 0600 1800120000000000 0600 18001200

0

20

40

60

80

50

100

150

200

0

10

20

30

40

50

Figure 2 Data distribution of Lan 1rsquos volume-speed-occupancy rate in 24 hours

100 200 300 400 500 600 700

minus04

minus02

02

04

Figure 3 Fault data points of filtering intermediate state search

Under the control of parameters listed in Table 1 wecan finish the data cleaning to find out the data beyond themaximum control range or contrary to the theory of trafficflow The result is shown in Figure 3

Use a one-dimensional matrix to record the effectivenessof each record All the initial value of thematrix is 1When theabnormal data is detected the corresponding matrix valueis changed into 0 From Figure 3 there is a series of errordata points at the time 300ndash600 which is consistent with theoriginal graph shown in Figure 4

Because these error data points do not have actual physi-cal meaning they are replaced by the closest normal recordAfter the cleaning the figure of volume-speed-occupancyrate is shown in Figure 5

Data quality has been improved to a certain extentespecially for the speed data But there still has beenmutationin the filtering results Test the first-order differential ofthe data to determine the mutation data The first-orderdifference graph of intermediate state is shown in Figure 6

Table 2 Result of the filter by different parameters

1198900value 119890

0= 3 119890

0= 4 119890

0= 5

Number of change ranges beyond 83 34 21Rate of the data beyond 115 47 21

Control parameter of differential operational is 1198900

Assuming that 1198900

= 3 is the parameter of the reasonablechange region if the actual value of the first-order is exceededthree times of the standard deviation control range changethe corresponding position of the effective matrix into 0

When 1198900

= 3 there are 83 abnormal data points as isshown in Figure 7 which account for 115 of the total recordThis result is too strict for the data of detectors so we canincrease the value of 119890

0to release the strictness for change

range of the data as is shown in Table 2According to the results shown in the table 119890

0= 4 is more

reasonable Figure 8 is comparison chart between the finalresult of the filter and the original data and it can be seenthat the algorithm has basically achieved the requirements ofthe loop detector datarsquos cleaning and preprocessing

FromFigure 3 it can be seen that the algorithmhas a smallcorrection for the traffic data and the data with the occupancyrate but the algorithm has better effect on the speed data Inthe process of predicting travel time the speed of the detectoris often used only which means that the algorithm can besimplified so that it only needs to produce speed data

Because only the speed data is processed we need to setthe upper and lower limit of speed and the limit of first-orderdifferential change range to restrict data As a result of usingthe detector data to calculate and predict travel time we needthree continuous detectorrsquos data points to simulate the traveltime forecast inwhole roadnetworkThe sketchmap is shownin Figure 9

International Journal of Distributed Sensor Networks 7

00000000 0600 18001200

0

20

40

60

80

00000000 0600 18001200

50

100

150

200

00000000 0600 18001200

0

10

20

30

40

50

Figure 4 Comparison between the actual data and the results of the filterrsquos intermediate state

00000000 0600 18001200 00000000 0600 18001200 00000000 0600 18001200

0

20

40

60

80

20

40

60

80

100

120

0

10

20

30

40

50

Figure 5 Result of filterrsquos intermediate state

Figure 6 First-order difference graph of intermediate state

100 200 300 400 500 600 700

01

02

minus01

minus02

Figure 7 Distortion data distribution when 1198900= 3

Under the condition that 1198900= 3 the filter result of speed

after cleaning of the three detectors is as shown in Figures 1011 and 12

As shown in Figures 10sim12 the red correction curvebasically achieved a reasonable correction of the distortiondata

52 Calculation of Travel Time Parameters Use the traveltime conversion model given by formulas (5) and (6) wecan do the travel time conversion according to the speeddata For example when one calculates the travel timedriving from east to west the vehicles pass through sec-tions Detector 1 Detector 2 and Detector 2 Detector 3The result is shown in Figure 13 and the unit of time is sm

53 Pattern Partition of Travel Time Series Based on Change-Point Analysis Because it is not clear how to choose 119890

0up to

the size of the sample we select 7 values to search the change-point The result is in Table 3

According to the characteristics of the travel time serieswe need to select the result that has 5 to 10 change-points So1198900= 007 119890

0= 005 and 119890

0= 002 are all reasonable control

parameters The results of all the reasonable circumstanceare shown in Figure 14 We use different colors to indicatedifferent state

According to the time sequence diagram the results of thealgorithm are basically completed and the travel time seriesis decomposed into a series of time periods with practicalmeaning

8 International Journal of Distributed Sensor Networks

50

100

150

200

0

10

20

30

40

50

00000000 0600 18001200 00000000 0600 18001200 00000000 0600 18001200

0

20

40

60

80

Figure 8 Comparison of the source data and the filter result

Detector 1Detector 2Detector 3

4494m 157m

Figure 9 Distribution of the three serial detectors on the road

0800 1200 1600 20000

20

40

60

80

Figure 10 Filter result of the speed data of Detector 1

0800 1200 1600 20000

10

20

30

40

50

60

Figure 11 Filter result of the speed data of Detector 2

On the situation of 1198900

= 007 and 1198900

= 005 thereare 90 important change-points that were missed So 119890

0=

002 and 1 25 90 224 375 380 384 390 393 414 are idealcontrolling parameter and result of change-point control

0800 1200 1600 20000

10

20

30

40

50

60

Figure 12 Filter result of the speed data of Detector 3

0800 1200 1600 200000

01

02

03

04

Figure 13 Travel time calculation result from Detector 1Detector 2 to Detector 2 Detector 3

And the travel time is divided into 9 divisions 600sim650650sim900 900sim1328 1328sim1830 1830sim1838 1838sim18501850sim1856 1856sim1948 and 1948sim2300

Then if we assume that this moment is 1700 we canpredict the travel time of 1702 and 1704 The historical dataof the forecast model is shown in Figure 15

54 Travel Time Forecasting Based on ARIMAModel To sumup we choose ARIMA model to build up the forecast modeland regress the parameterUsing the data of 221sim330 collectedby Detector 1 at the time 1328sim1700 and testing the time

International Journal of Distributed Sensor Networks 9

Table 3 Result of change-point searching of each section

1198900

Searching result of change-point Number of change-points001 1 25 59 70 77 90 224 319 326 342 357 376 379 390 393 414 415 17002 1 25 90 224 375 380 384 390 393 414 10005 1 225 375 380 390 393 412 7007 1 225 380 393 412 501 1 411 2012 1 416 2015 1 1

Table 4 ARIMA model and forecast result of different weight functions

Weight functions Fitting equations Forecast value on 1702 Forecast value on 1704Square-root 119909

1015840

119905= 0000321317 + 0726226119909

1015840

119905minus1+ 0191767119909

2

119905minus20208528 0206241

Square 1199091015840

119905= 00000607593 + 0798534119909

1015840

119905minus1+ 0204619119909

2

119905minus20213584 0215573

Growth rate curve 1199091015840

119905= 0000340756 + 0717946119909

1015840

119905minus1+ 0193551119909

2

119905minus20209668 0208116

Liner 1199091015840

119905= 0000948595 + 0617164119909

1015840

119905minus1+ 0110425119909

2

119905minus20206278 0202864

Actual value on 1702 020577Actual value on 1704 0196385

Table 5 Error matrix of different model fittings

Weight functions ME MAE MAPE MSESquare-root (global) minus0000361064 001288 710089 0000279502Square (global) minus0000751537 00133415 734701 0000294168Growth rate curve (global) minus0000697771 00131061 722872 0000289045Liner (global) 136255 lowast 10

minus17 00120611 672481 0000258082Square-root (proximal point) minus000357858 00134934 629296 0000200429Square (proximal point) minus000703095 00144531 674308 0000263613Growth rate curve (proximal point) minus000513248 00133818 624778 0000211068Liner (proximal point) 0000443646 00113057 527639 0000140726

sequence we found it is a nonrandom stationary sequenceThe fix order result of the ARIMA model for the time of1328sim1700 is (2 0 0)

After the weighted least squares are transformed intoordinary least squares we use four kinds of weight functionto do the fitting experiment and also have the error analysisto the output results The forecast results of different weightfunctions are as in Table 4

Error analysis result is shown in Table 5 and we can seethat the crucial indexMAPE has a certain degree of reductionat the proximal point

The forecast results of four different weighting functionscanmeet the basic requirements of the accuracy error of 10At the same time we can know that the linear weight functionhas good fitting and forecasting effect on the experimentaldata The linear weighting function is the optimal weightfunction for this forecast according to the statistics in Table 5

6 ConclusionThis paper uses the change-point detection algorithm todivide travel time series into several patterns and set up

forecasting model through ARIMA for different patternsbased on massive data collected by the loop detectors onthe roads Different from traditional forecasting methodsit is easier to get the more optimized forecasting resultthan to obtain it by using the global search because thedifferent intervals in same pattern have similar statisticalcharacteristics of the mean and variance In the process ofdividing the travel time series the calculation of algorithmis complicated and the derivation of control parameters isonly obtained by experiments which still needs research inthe future

Conflict of Interests

The authors declare no conflict of interests

Authorsrsquo Contribution

Guangyu Zhu designed the forecasting model of travel timebased on the change-point detection algorithm and wrote thepaper Li Wang designed the preprocessing algorithm and

10 International Journal of Distributed Sensor Networks

0800 1200 1600 2000

0800 1200 1600 2000

0800 1200 1600 2000

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

e0 = 002 1 25 90 224 375 380 384 390 393 414

e0 = 005 1 225 375 380 390 393 412

e0 = 007 1 225 380 393 412

Figure 14 Result of the global state change-point searching

0600 0900 1200 1500000

005

010

015

020

025

Figure 15 Historical data of the forecast model

performed the data preprocess and revised the paper PengZhang and Kang Song analyzed the data

Acknowledgments

This work is supported by the National Science Founda-tion of China (nos 61572069 and 61503022) the Funda-mental Research Funds for the Central Universities (no2014JBM211) the Open Project Program of Key Laboratoryof System Control and Information Processing Ministry ofEducation Shanghai Jiaotong University (no Scip201507)Beijing Municipality Key Laboratory of Urban Traffic Oper-ation Simulation and Decision Support Beijing Transporta-tion Research Center the project of the Department of Trafficand Transportation of Hebei Province (no A0201-150505)and The National Key Technology Support Program (no2014BAG01B02)

References

[1] Z-P Li H Yu Y-C Liu and F-Q Liu ldquoAn improved adaptiveexponential smoothing model for short-term travel time fore-casting of urban arterial streetrdquo Acta Automatica Sinica vol 34no 11 pp 1404ndash1409 2008

[2] Y Hollander and R Liu ldquoEstimation of the distribution of traveltimes by repeated simulationrdquo Transportation Research Part CEmerging Technologies vol 16 no 2 pp 212ndash231 2008

[3] G-Y Jiang Technologies and Application of the Identification ofRoad Traffic Condition Communications Press Beijing China2004 (Chinese)

[4] M Gramaglia C J Bernardos and M Calderon ldquoVirtualinduction loops based on cooperative vehicular communica-tionsrdquo Sensors vol 13 no 2 pp 1467ndash1476 2013

[5] X Zhang and J A Rice ldquoShort-term travel time predictionrdquoTransportation Research Part C vol 11 no 3-4 pp 187ndash2102003

[6] U Mori A Mendiburu M Alvarez and J A Lozano ldquoA reviewof travel time estimation and forecasting for Advanced TravellerInformation Systemsrdquo Transportmetrica A Transport Sciencevol 11 no 2 pp 119ndash157 2015

[7] E I Vlahogianni M G Karlaftis and J C Golias ldquoShort-term traffic forecasting where we are and where wersquore goingrdquoTransportation Research Part C Emerging Technologies vol 43part 1 pp 3ndash19 2014

[8] C Shao K Zhang and Y Gu ldquoA study of route traveltime forecast method based on real data of urban expresswaynetworkrdquo China Civil Engineering Journal vol 36 no 1 pp 16ndash20 2003

[9] B R Chilukuri J A Laval and A Guin ldquoMicrosimulation-based framework for freeway travel time forecastingrdquo Trans-portation Research Record vol 2470 pp 34ndash45 2014

[10] E Yao and Y Zhang ldquoA new algorithm of shortmdashterm traveltime prediction for urban expresswayrdquo Journal of WuhanUniversity of Technology (Transportation ScienceampEngineering)vol 6 pp 1133ndash1137 2013

[11] J Zhao H Wang and W Liu ldquoPrediction of expressway traveltime based on adaptive interpolation kalman filteringrdquo Journalof South ChinaUniversity of Technology vol 2 pp 109ndash115 2014

[12] Z Gui and H Yu ldquoTrip travel time forecasting based onselective forgetting extreme learning machinerdquo Mathematical

International Journal of Distributed Sensor Networks 11

Problems in Engineering vol 2014 Article ID 829256 7 pages2014

[13] Y-J Jou Y-H Wen T-T Lee and H-J Cho ldquoMissing datatreatment on travel time estimation for ATISrdquo in Proceedingsof the IEEE International Conference on Systems Man andCybernetics (SMC rsquo03) vol 1 pp 102ndash107 IEEE WashingtonDC USA October 2003

[14] LWang and CWang ldquoA newmethod of real-time informationcollection in intelligent transportation systemrdquo Systems Engi-neering vol 23 no 2 pp 86ndash89 2005

[15] Z Cai Y Duan and A G Bourgeois ldquoDelay efficient oppor-tunistic routing in asynchronous multi-channel cognitive radionetworksrdquo Journal of Combinatorial Optimization vol 29 no 4pp 815ndash835 2015

[16] J W C VanLint and N J Zijpp ldquoAn improved travel timeestimation algorithm using dual-loop detectorsrdquo in Proceedingsof the Annual Meeting of the Transportation Research BoardWashington DC USA 2003

[17] Z Cai R Goebel and G Lin ldquoSize-constrained tree parti-tioning approximating the multicast k-tree routing problemrdquoTheoretical Computer Science vol 412 no 3 pp 240ndash245 2011

[18] G Zhu and H Yan ldquoA kind of demand-forecasting modelbased on analysis of demand booming and principle of naiveforecastingrdquo Systems EngineeringTheory amp Practice vol 24 no5 pp 22ndash33 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 7

00000000 0600 18001200

0

20

40

60

80

00000000 0600 18001200

50

100

150

200

00000000 0600 18001200

0

10

20

30

40

50

Figure 4 Comparison between the actual data and the results of the filterrsquos intermediate state

00000000 0600 18001200 00000000 0600 18001200 00000000 0600 18001200

0

20

40

60

80

20

40

60

80

100

120

0

10

20

30

40

50

Figure 5 Result of filterrsquos intermediate state

Figure 6 First-order difference graph of intermediate state

100 200 300 400 500 600 700

01

02

minus01

minus02

Figure 7 Distortion data distribution when 1198900= 3

Under the condition that 1198900= 3 the filter result of speed

after cleaning of the three detectors is as shown in Figures 1011 and 12

As shown in Figures 10sim12 the red correction curvebasically achieved a reasonable correction of the distortiondata

52 Calculation of Travel Time Parameters Use the traveltime conversion model given by formulas (5) and (6) wecan do the travel time conversion according to the speeddata For example when one calculates the travel timedriving from east to west the vehicles pass through sec-tions Detector 1 Detector 2 and Detector 2 Detector 3The result is shown in Figure 13 and the unit of time is sm

53 Pattern Partition of Travel Time Series Based on Change-Point Analysis Because it is not clear how to choose 119890

0up to

the size of the sample we select 7 values to search the change-point The result is in Table 3

According to the characteristics of the travel time serieswe need to select the result that has 5 to 10 change-points So1198900= 007 119890

0= 005 and 119890

0= 002 are all reasonable control

parameters The results of all the reasonable circumstanceare shown in Figure 14 We use different colors to indicatedifferent state

According to the time sequence diagram the results of thealgorithm are basically completed and the travel time seriesis decomposed into a series of time periods with practicalmeaning

8 International Journal of Distributed Sensor Networks

50

100

150

200

0

10

20

30

40

50

00000000 0600 18001200 00000000 0600 18001200 00000000 0600 18001200

0

20

40

60

80

Figure 8 Comparison of the source data and the filter result

Detector 1Detector 2Detector 3

4494m 157m

Figure 9 Distribution of the three serial detectors on the road

0800 1200 1600 20000

20

40

60

80

Figure 10 Filter result of the speed data of Detector 1

0800 1200 1600 20000

10

20

30

40

50

60

Figure 11 Filter result of the speed data of Detector 2

On the situation of 1198900

= 007 and 1198900

= 005 thereare 90 important change-points that were missed So 119890

0=

002 and 1 25 90 224 375 380 384 390 393 414 are idealcontrolling parameter and result of change-point control

0800 1200 1600 20000

10

20

30

40

50

60

Figure 12 Filter result of the speed data of Detector 3

0800 1200 1600 200000

01

02

03

04

Figure 13 Travel time calculation result from Detector 1Detector 2 to Detector 2 Detector 3

And the travel time is divided into 9 divisions 600sim650650sim900 900sim1328 1328sim1830 1830sim1838 1838sim18501850sim1856 1856sim1948 and 1948sim2300

Then if we assume that this moment is 1700 we canpredict the travel time of 1702 and 1704 The historical dataof the forecast model is shown in Figure 15

54 Travel Time Forecasting Based on ARIMAModel To sumup we choose ARIMA model to build up the forecast modeland regress the parameterUsing the data of 221sim330 collectedby Detector 1 at the time 1328sim1700 and testing the time

International Journal of Distributed Sensor Networks 9

Table 3 Result of change-point searching of each section

1198900

Searching result of change-point Number of change-points001 1 25 59 70 77 90 224 319 326 342 357 376 379 390 393 414 415 17002 1 25 90 224 375 380 384 390 393 414 10005 1 225 375 380 390 393 412 7007 1 225 380 393 412 501 1 411 2012 1 416 2015 1 1

Table 4 ARIMA model and forecast result of different weight functions

Weight functions Fitting equations Forecast value on 1702 Forecast value on 1704Square-root 119909

1015840

119905= 0000321317 + 0726226119909

1015840

119905minus1+ 0191767119909

2

119905minus20208528 0206241

Square 1199091015840

119905= 00000607593 + 0798534119909

1015840

119905minus1+ 0204619119909

2

119905minus20213584 0215573

Growth rate curve 1199091015840

119905= 0000340756 + 0717946119909

1015840

119905minus1+ 0193551119909

2

119905minus20209668 0208116

Liner 1199091015840

119905= 0000948595 + 0617164119909

1015840

119905minus1+ 0110425119909

2

119905minus20206278 0202864

Actual value on 1702 020577Actual value on 1704 0196385

Table 5 Error matrix of different model fittings

Weight functions ME MAE MAPE MSESquare-root (global) minus0000361064 001288 710089 0000279502Square (global) minus0000751537 00133415 734701 0000294168Growth rate curve (global) minus0000697771 00131061 722872 0000289045Liner (global) 136255 lowast 10

minus17 00120611 672481 0000258082Square-root (proximal point) minus000357858 00134934 629296 0000200429Square (proximal point) minus000703095 00144531 674308 0000263613Growth rate curve (proximal point) minus000513248 00133818 624778 0000211068Liner (proximal point) 0000443646 00113057 527639 0000140726

sequence we found it is a nonrandom stationary sequenceThe fix order result of the ARIMA model for the time of1328sim1700 is (2 0 0)

After the weighted least squares are transformed intoordinary least squares we use four kinds of weight functionto do the fitting experiment and also have the error analysisto the output results The forecast results of different weightfunctions are as in Table 4

Error analysis result is shown in Table 5 and we can seethat the crucial indexMAPE has a certain degree of reductionat the proximal point

The forecast results of four different weighting functionscanmeet the basic requirements of the accuracy error of 10At the same time we can know that the linear weight functionhas good fitting and forecasting effect on the experimentaldata The linear weighting function is the optimal weightfunction for this forecast according to the statistics in Table 5

6 ConclusionThis paper uses the change-point detection algorithm todivide travel time series into several patterns and set up

forecasting model through ARIMA for different patternsbased on massive data collected by the loop detectors onthe roads Different from traditional forecasting methodsit is easier to get the more optimized forecasting resultthan to obtain it by using the global search because thedifferent intervals in same pattern have similar statisticalcharacteristics of the mean and variance In the process ofdividing the travel time series the calculation of algorithmis complicated and the derivation of control parameters isonly obtained by experiments which still needs research inthe future

Conflict of Interests

The authors declare no conflict of interests

Authorsrsquo Contribution

Guangyu Zhu designed the forecasting model of travel timebased on the change-point detection algorithm and wrote thepaper Li Wang designed the preprocessing algorithm and

10 International Journal of Distributed Sensor Networks

0800 1200 1600 2000

0800 1200 1600 2000

0800 1200 1600 2000

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

e0 = 002 1 25 90 224 375 380 384 390 393 414

e0 = 005 1 225 375 380 390 393 412

e0 = 007 1 225 380 393 412

Figure 14 Result of the global state change-point searching

0600 0900 1200 1500000

005

010

015

020

025

Figure 15 Historical data of the forecast model

performed the data preprocess and revised the paper PengZhang and Kang Song analyzed the data

Acknowledgments

This work is supported by the National Science Founda-tion of China (nos 61572069 and 61503022) the Funda-mental Research Funds for the Central Universities (no2014JBM211) the Open Project Program of Key Laboratoryof System Control and Information Processing Ministry ofEducation Shanghai Jiaotong University (no Scip201507)Beijing Municipality Key Laboratory of Urban Traffic Oper-ation Simulation and Decision Support Beijing Transporta-tion Research Center the project of the Department of Trafficand Transportation of Hebei Province (no A0201-150505)and The National Key Technology Support Program (no2014BAG01B02)

References

[1] Z-P Li H Yu Y-C Liu and F-Q Liu ldquoAn improved adaptiveexponential smoothing model for short-term travel time fore-casting of urban arterial streetrdquo Acta Automatica Sinica vol 34no 11 pp 1404ndash1409 2008

[2] Y Hollander and R Liu ldquoEstimation of the distribution of traveltimes by repeated simulationrdquo Transportation Research Part CEmerging Technologies vol 16 no 2 pp 212ndash231 2008

[3] G-Y Jiang Technologies and Application of the Identification ofRoad Traffic Condition Communications Press Beijing China2004 (Chinese)

[4] M Gramaglia C J Bernardos and M Calderon ldquoVirtualinduction loops based on cooperative vehicular communica-tionsrdquo Sensors vol 13 no 2 pp 1467ndash1476 2013

[5] X Zhang and J A Rice ldquoShort-term travel time predictionrdquoTransportation Research Part C vol 11 no 3-4 pp 187ndash2102003

[6] U Mori A Mendiburu M Alvarez and J A Lozano ldquoA reviewof travel time estimation and forecasting for Advanced TravellerInformation Systemsrdquo Transportmetrica A Transport Sciencevol 11 no 2 pp 119ndash157 2015

[7] E I Vlahogianni M G Karlaftis and J C Golias ldquoShort-term traffic forecasting where we are and where wersquore goingrdquoTransportation Research Part C Emerging Technologies vol 43part 1 pp 3ndash19 2014

[8] C Shao K Zhang and Y Gu ldquoA study of route traveltime forecast method based on real data of urban expresswaynetworkrdquo China Civil Engineering Journal vol 36 no 1 pp 16ndash20 2003

[9] B R Chilukuri J A Laval and A Guin ldquoMicrosimulation-based framework for freeway travel time forecastingrdquo Trans-portation Research Record vol 2470 pp 34ndash45 2014

[10] E Yao and Y Zhang ldquoA new algorithm of shortmdashterm traveltime prediction for urban expresswayrdquo Journal of WuhanUniversity of Technology (Transportation ScienceampEngineering)vol 6 pp 1133ndash1137 2013

[11] J Zhao H Wang and W Liu ldquoPrediction of expressway traveltime based on adaptive interpolation kalman filteringrdquo Journalof South ChinaUniversity of Technology vol 2 pp 109ndash115 2014

[12] Z Gui and H Yu ldquoTrip travel time forecasting based onselective forgetting extreme learning machinerdquo Mathematical

International Journal of Distributed Sensor Networks 11

Problems in Engineering vol 2014 Article ID 829256 7 pages2014

[13] Y-J Jou Y-H Wen T-T Lee and H-J Cho ldquoMissing datatreatment on travel time estimation for ATISrdquo in Proceedingsof the IEEE International Conference on Systems Man andCybernetics (SMC rsquo03) vol 1 pp 102ndash107 IEEE WashingtonDC USA October 2003

[14] LWang and CWang ldquoA newmethod of real-time informationcollection in intelligent transportation systemrdquo Systems Engi-neering vol 23 no 2 pp 86ndash89 2005

[15] Z Cai Y Duan and A G Bourgeois ldquoDelay efficient oppor-tunistic routing in asynchronous multi-channel cognitive radionetworksrdquo Journal of Combinatorial Optimization vol 29 no 4pp 815ndash835 2015

[16] J W C VanLint and N J Zijpp ldquoAn improved travel timeestimation algorithm using dual-loop detectorsrdquo in Proceedingsof the Annual Meeting of the Transportation Research BoardWashington DC USA 2003

[17] Z Cai R Goebel and G Lin ldquoSize-constrained tree parti-tioning approximating the multicast k-tree routing problemrdquoTheoretical Computer Science vol 412 no 3 pp 240ndash245 2011

[18] G Zhu and H Yan ldquoA kind of demand-forecasting modelbased on analysis of demand booming and principle of naiveforecastingrdquo Systems EngineeringTheory amp Practice vol 24 no5 pp 22ndash33 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 International Journal of Distributed Sensor Networks

50

100

150

200

0

10

20

30

40

50

00000000 0600 18001200 00000000 0600 18001200 00000000 0600 18001200

0

20

40

60

80

Figure 8 Comparison of the source data and the filter result

Detector 1Detector 2Detector 3

4494m 157m

Figure 9 Distribution of the three serial detectors on the road

0800 1200 1600 20000

20

40

60

80

Figure 10 Filter result of the speed data of Detector 1

0800 1200 1600 20000

10

20

30

40

50

60

Figure 11 Filter result of the speed data of Detector 2

On the situation of 1198900

= 007 and 1198900

= 005 thereare 90 important change-points that were missed So 119890

0=

002 and 1 25 90 224 375 380 384 390 393 414 are idealcontrolling parameter and result of change-point control

0800 1200 1600 20000

10

20

30

40

50

60

Figure 12 Filter result of the speed data of Detector 3

0800 1200 1600 200000

01

02

03

04

Figure 13 Travel time calculation result from Detector 1Detector 2 to Detector 2 Detector 3

And the travel time is divided into 9 divisions 600sim650650sim900 900sim1328 1328sim1830 1830sim1838 1838sim18501850sim1856 1856sim1948 and 1948sim2300

Then if we assume that this moment is 1700 we canpredict the travel time of 1702 and 1704 The historical dataof the forecast model is shown in Figure 15

54 Travel Time Forecasting Based on ARIMAModel To sumup we choose ARIMA model to build up the forecast modeland regress the parameterUsing the data of 221sim330 collectedby Detector 1 at the time 1328sim1700 and testing the time

International Journal of Distributed Sensor Networks 9

Table 3 Result of change-point searching of each section

1198900

Searching result of change-point Number of change-points001 1 25 59 70 77 90 224 319 326 342 357 376 379 390 393 414 415 17002 1 25 90 224 375 380 384 390 393 414 10005 1 225 375 380 390 393 412 7007 1 225 380 393 412 501 1 411 2012 1 416 2015 1 1

Table 4 ARIMA model and forecast result of different weight functions

Weight functions Fitting equations Forecast value on 1702 Forecast value on 1704Square-root 119909

1015840

119905= 0000321317 + 0726226119909

1015840

119905minus1+ 0191767119909

2

119905minus20208528 0206241

Square 1199091015840

119905= 00000607593 + 0798534119909

1015840

119905minus1+ 0204619119909

2

119905minus20213584 0215573

Growth rate curve 1199091015840

119905= 0000340756 + 0717946119909

1015840

119905minus1+ 0193551119909

2

119905minus20209668 0208116

Liner 1199091015840

119905= 0000948595 + 0617164119909

1015840

119905minus1+ 0110425119909

2

119905minus20206278 0202864

Actual value on 1702 020577Actual value on 1704 0196385

Table 5 Error matrix of different model fittings

Weight functions ME MAE MAPE MSESquare-root (global) minus0000361064 001288 710089 0000279502Square (global) minus0000751537 00133415 734701 0000294168Growth rate curve (global) minus0000697771 00131061 722872 0000289045Liner (global) 136255 lowast 10

minus17 00120611 672481 0000258082Square-root (proximal point) minus000357858 00134934 629296 0000200429Square (proximal point) minus000703095 00144531 674308 0000263613Growth rate curve (proximal point) minus000513248 00133818 624778 0000211068Liner (proximal point) 0000443646 00113057 527639 0000140726

sequence we found it is a nonrandom stationary sequenceThe fix order result of the ARIMA model for the time of1328sim1700 is (2 0 0)

After the weighted least squares are transformed intoordinary least squares we use four kinds of weight functionto do the fitting experiment and also have the error analysisto the output results The forecast results of different weightfunctions are as in Table 4

Error analysis result is shown in Table 5 and we can seethat the crucial indexMAPE has a certain degree of reductionat the proximal point

The forecast results of four different weighting functionscanmeet the basic requirements of the accuracy error of 10At the same time we can know that the linear weight functionhas good fitting and forecasting effect on the experimentaldata The linear weighting function is the optimal weightfunction for this forecast according to the statistics in Table 5

6 ConclusionThis paper uses the change-point detection algorithm todivide travel time series into several patterns and set up

forecasting model through ARIMA for different patternsbased on massive data collected by the loop detectors onthe roads Different from traditional forecasting methodsit is easier to get the more optimized forecasting resultthan to obtain it by using the global search because thedifferent intervals in same pattern have similar statisticalcharacteristics of the mean and variance In the process ofdividing the travel time series the calculation of algorithmis complicated and the derivation of control parameters isonly obtained by experiments which still needs research inthe future

Conflict of Interests

The authors declare no conflict of interests

Authorsrsquo Contribution

Guangyu Zhu designed the forecasting model of travel timebased on the change-point detection algorithm and wrote thepaper Li Wang designed the preprocessing algorithm and

10 International Journal of Distributed Sensor Networks

0800 1200 1600 2000

0800 1200 1600 2000

0800 1200 1600 2000

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

e0 = 002 1 25 90 224 375 380 384 390 393 414

e0 = 005 1 225 375 380 390 393 412

e0 = 007 1 225 380 393 412

Figure 14 Result of the global state change-point searching

0600 0900 1200 1500000

005

010

015

020

025

Figure 15 Historical data of the forecast model

performed the data preprocess and revised the paper PengZhang and Kang Song analyzed the data

Acknowledgments

This work is supported by the National Science Founda-tion of China (nos 61572069 and 61503022) the Funda-mental Research Funds for the Central Universities (no2014JBM211) the Open Project Program of Key Laboratoryof System Control and Information Processing Ministry ofEducation Shanghai Jiaotong University (no Scip201507)Beijing Municipality Key Laboratory of Urban Traffic Oper-ation Simulation and Decision Support Beijing Transporta-tion Research Center the project of the Department of Trafficand Transportation of Hebei Province (no A0201-150505)and The National Key Technology Support Program (no2014BAG01B02)

References

[1] Z-P Li H Yu Y-C Liu and F-Q Liu ldquoAn improved adaptiveexponential smoothing model for short-term travel time fore-casting of urban arterial streetrdquo Acta Automatica Sinica vol 34no 11 pp 1404ndash1409 2008

[2] Y Hollander and R Liu ldquoEstimation of the distribution of traveltimes by repeated simulationrdquo Transportation Research Part CEmerging Technologies vol 16 no 2 pp 212ndash231 2008

[3] G-Y Jiang Technologies and Application of the Identification ofRoad Traffic Condition Communications Press Beijing China2004 (Chinese)

[4] M Gramaglia C J Bernardos and M Calderon ldquoVirtualinduction loops based on cooperative vehicular communica-tionsrdquo Sensors vol 13 no 2 pp 1467ndash1476 2013

[5] X Zhang and J A Rice ldquoShort-term travel time predictionrdquoTransportation Research Part C vol 11 no 3-4 pp 187ndash2102003

[6] U Mori A Mendiburu M Alvarez and J A Lozano ldquoA reviewof travel time estimation and forecasting for Advanced TravellerInformation Systemsrdquo Transportmetrica A Transport Sciencevol 11 no 2 pp 119ndash157 2015

[7] E I Vlahogianni M G Karlaftis and J C Golias ldquoShort-term traffic forecasting where we are and where wersquore goingrdquoTransportation Research Part C Emerging Technologies vol 43part 1 pp 3ndash19 2014

[8] C Shao K Zhang and Y Gu ldquoA study of route traveltime forecast method based on real data of urban expresswaynetworkrdquo China Civil Engineering Journal vol 36 no 1 pp 16ndash20 2003

[9] B R Chilukuri J A Laval and A Guin ldquoMicrosimulation-based framework for freeway travel time forecastingrdquo Trans-portation Research Record vol 2470 pp 34ndash45 2014

[10] E Yao and Y Zhang ldquoA new algorithm of shortmdashterm traveltime prediction for urban expresswayrdquo Journal of WuhanUniversity of Technology (Transportation ScienceampEngineering)vol 6 pp 1133ndash1137 2013

[11] J Zhao H Wang and W Liu ldquoPrediction of expressway traveltime based on adaptive interpolation kalman filteringrdquo Journalof South ChinaUniversity of Technology vol 2 pp 109ndash115 2014

[12] Z Gui and H Yu ldquoTrip travel time forecasting based onselective forgetting extreme learning machinerdquo Mathematical

International Journal of Distributed Sensor Networks 11

Problems in Engineering vol 2014 Article ID 829256 7 pages2014

[13] Y-J Jou Y-H Wen T-T Lee and H-J Cho ldquoMissing datatreatment on travel time estimation for ATISrdquo in Proceedingsof the IEEE International Conference on Systems Man andCybernetics (SMC rsquo03) vol 1 pp 102ndash107 IEEE WashingtonDC USA October 2003

[14] LWang and CWang ldquoA newmethod of real-time informationcollection in intelligent transportation systemrdquo Systems Engi-neering vol 23 no 2 pp 86ndash89 2005

[15] Z Cai Y Duan and A G Bourgeois ldquoDelay efficient oppor-tunistic routing in asynchronous multi-channel cognitive radionetworksrdquo Journal of Combinatorial Optimization vol 29 no 4pp 815ndash835 2015

[16] J W C VanLint and N J Zijpp ldquoAn improved travel timeestimation algorithm using dual-loop detectorsrdquo in Proceedingsof the Annual Meeting of the Transportation Research BoardWashington DC USA 2003

[17] Z Cai R Goebel and G Lin ldquoSize-constrained tree parti-tioning approximating the multicast k-tree routing problemrdquoTheoretical Computer Science vol 412 no 3 pp 240ndash245 2011

[18] G Zhu and H Yan ldquoA kind of demand-forecasting modelbased on analysis of demand booming and principle of naiveforecastingrdquo Systems EngineeringTheory amp Practice vol 24 no5 pp 22ndash33 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 9

Table 3 Result of change-point searching of each section

1198900

Searching result of change-point Number of change-points001 1 25 59 70 77 90 224 319 326 342 357 376 379 390 393 414 415 17002 1 25 90 224 375 380 384 390 393 414 10005 1 225 375 380 390 393 412 7007 1 225 380 393 412 501 1 411 2012 1 416 2015 1 1

Table 4 ARIMA model and forecast result of different weight functions

Weight functions Fitting equations Forecast value on 1702 Forecast value on 1704Square-root 119909

1015840

119905= 0000321317 + 0726226119909

1015840

119905minus1+ 0191767119909

2

119905minus20208528 0206241

Square 1199091015840

119905= 00000607593 + 0798534119909

1015840

119905minus1+ 0204619119909

2

119905minus20213584 0215573

Growth rate curve 1199091015840

119905= 0000340756 + 0717946119909

1015840

119905minus1+ 0193551119909

2

119905minus20209668 0208116

Liner 1199091015840

119905= 0000948595 + 0617164119909

1015840

119905minus1+ 0110425119909

2

119905minus20206278 0202864

Actual value on 1702 020577Actual value on 1704 0196385

Table 5 Error matrix of different model fittings

Weight functions ME MAE MAPE MSESquare-root (global) minus0000361064 001288 710089 0000279502Square (global) minus0000751537 00133415 734701 0000294168Growth rate curve (global) minus0000697771 00131061 722872 0000289045Liner (global) 136255 lowast 10

minus17 00120611 672481 0000258082Square-root (proximal point) minus000357858 00134934 629296 0000200429Square (proximal point) minus000703095 00144531 674308 0000263613Growth rate curve (proximal point) minus000513248 00133818 624778 0000211068Liner (proximal point) 0000443646 00113057 527639 0000140726

sequence we found it is a nonrandom stationary sequenceThe fix order result of the ARIMA model for the time of1328sim1700 is (2 0 0)

After the weighted least squares are transformed intoordinary least squares we use four kinds of weight functionto do the fitting experiment and also have the error analysisto the output results The forecast results of different weightfunctions are as in Table 4

Error analysis result is shown in Table 5 and we can seethat the crucial indexMAPE has a certain degree of reductionat the proximal point

The forecast results of four different weighting functionscanmeet the basic requirements of the accuracy error of 10At the same time we can know that the linear weight functionhas good fitting and forecasting effect on the experimentaldata The linear weighting function is the optimal weightfunction for this forecast according to the statistics in Table 5

6 ConclusionThis paper uses the change-point detection algorithm todivide travel time series into several patterns and set up

forecasting model through ARIMA for different patternsbased on massive data collected by the loop detectors onthe roads Different from traditional forecasting methodsit is easier to get the more optimized forecasting resultthan to obtain it by using the global search because thedifferent intervals in same pattern have similar statisticalcharacteristics of the mean and variance In the process ofdividing the travel time series the calculation of algorithmis complicated and the derivation of control parameters isonly obtained by experiments which still needs research inthe future

Conflict of Interests

The authors declare no conflict of interests

Authorsrsquo Contribution

Guangyu Zhu designed the forecasting model of travel timebased on the change-point detection algorithm and wrote thepaper Li Wang designed the preprocessing algorithm and

10 International Journal of Distributed Sensor Networks

0800 1200 1600 2000

0800 1200 1600 2000

0800 1200 1600 2000

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

e0 = 002 1 25 90 224 375 380 384 390 393 414

e0 = 005 1 225 375 380 390 393 412

e0 = 007 1 225 380 393 412

Figure 14 Result of the global state change-point searching

0600 0900 1200 1500000

005

010

015

020

025

Figure 15 Historical data of the forecast model

performed the data preprocess and revised the paper PengZhang and Kang Song analyzed the data

Acknowledgments

This work is supported by the National Science Founda-tion of China (nos 61572069 and 61503022) the Funda-mental Research Funds for the Central Universities (no2014JBM211) the Open Project Program of Key Laboratoryof System Control and Information Processing Ministry ofEducation Shanghai Jiaotong University (no Scip201507)Beijing Municipality Key Laboratory of Urban Traffic Oper-ation Simulation and Decision Support Beijing Transporta-tion Research Center the project of the Department of Trafficand Transportation of Hebei Province (no A0201-150505)and The National Key Technology Support Program (no2014BAG01B02)

References

[1] Z-P Li H Yu Y-C Liu and F-Q Liu ldquoAn improved adaptiveexponential smoothing model for short-term travel time fore-casting of urban arterial streetrdquo Acta Automatica Sinica vol 34no 11 pp 1404ndash1409 2008

[2] Y Hollander and R Liu ldquoEstimation of the distribution of traveltimes by repeated simulationrdquo Transportation Research Part CEmerging Technologies vol 16 no 2 pp 212ndash231 2008

[3] G-Y Jiang Technologies and Application of the Identification ofRoad Traffic Condition Communications Press Beijing China2004 (Chinese)

[4] M Gramaglia C J Bernardos and M Calderon ldquoVirtualinduction loops based on cooperative vehicular communica-tionsrdquo Sensors vol 13 no 2 pp 1467ndash1476 2013

[5] X Zhang and J A Rice ldquoShort-term travel time predictionrdquoTransportation Research Part C vol 11 no 3-4 pp 187ndash2102003

[6] U Mori A Mendiburu M Alvarez and J A Lozano ldquoA reviewof travel time estimation and forecasting for Advanced TravellerInformation Systemsrdquo Transportmetrica A Transport Sciencevol 11 no 2 pp 119ndash157 2015

[7] E I Vlahogianni M G Karlaftis and J C Golias ldquoShort-term traffic forecasting where we are and where wersquore goingrdquoTransportation Research Part C Emerging Technologies vol 43part 1 pp 3ndash19 2014

[8] C Shao K Zhang and Y Gu ldquoA study of route traveltime forecast method based on real data of urban expresswaynetworkrdquo China Civil Engineering Journal vol 36 no 1 pp 16ndash20 2003

[9] B R Chilukuri J A Laval and A Guin ldquoMicrosimulation-based framework for freeway travel time forecastingrdquo Trans-portation Research Record vol 2470 pp 34ndash45 2014

[10] E Yao and Y Zhang ldquoA new algorithm of shortmdashterm traveltime prediction for urban expresswayrdquo Journal of WuhanUniversity of Technology (Transportation ScienceampEngineering)vol 6 pp 1133ndash1137 2013

[11] J Zhao H Wang and W Liu ldquoPrediction of expressway traveltime based on adaptive interpolation kalman filteringrdquo Journalof South ChinaUniversity of Technology vol 2 pp 109ndash115 2014

[12] Z Gui and H Yu ldquoTrip travel time forecasting based onselective forgetting extreme learning machinerdquo Mathematical

International Journal of Distributed Sensor Networks 11

Problems in Engineering vol 2014 Article ID 829256 7 pages2014

[13] Y-J Jou Y-H Wen T-T Lee and H-J Cho ldquoMissing datatreatment on travel time estimation for ATISrdquo in Proceedingsof the IEEE International Conference on Systems Man andCybernetics (SMC rsquo03) vol 1 pp 102ndash107 IEEE WashingtonDC USA October 2003

[14] LWang and CWang ldquoA newmethod of real-time informationcollection in intelligent transportation systemrdquo Systems Engi-neering vol 23 no 2 pp 86ndash89 2005

[15] Z Cai Y Duan and A G Bourgeois ldquoDelay efficient oppor-tunistic routing in asynchronous multi-channel cognitive radionetworksrdquo Journal of Combinatorial Optimization vol 29 no 4pp 815ndash835 2015

[16] J W C VanLint and N J Zijpp ldquoAn improved travel timeestimation algorithm using dual-loop detectorsrdquo in Proceedingsof the Annual Meeting of the Transportation Research BoardWashington DC USA 2003

[17] Z Cai R Goebel and G Lin ldquoSize-constrained tree parti-tioning approximating the multicast k-tree routing problemrdquoTheoretical Computer Science vol 412 no 3 pp 240ndash245 2011

[18] G Zhu and H Yan ldquoA kind of demand-forecasting modelbased on analysis of demand booming and principle of naiveforecastingrdquo Systems EngineeringTheory amp Practice vol 24 no5 pp 22ndash33 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 International Journal of Distributed Sensor Networks

0800 1200 1600 2000

0800 1200 1600 2000

0800 1200 1600 2000

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

000

005

010

015

020

025

030

035

e0 = 002 1 25 90 224 375 380 384 390 393 414

e0 = 005 1 225 375 380 390 393 412

e0 = 007 1 225 380 393 412

Figure 14 Result of the global state change-point searching

0600 0900 1200 1500000

005

010

015

020

025

Figure 15 Historical data of the forecast model

performed the data preprocess and revised the paper PengZhang and Kang Song analyzed the data

Acknowledgments

This work is supported by the National Science Founda-tion of China (nos 61572069 and 61503022) the Funda-mental Research Funds for the Central Universities (no2014JBM211) the Open Project Program of Key Laboratoryof System Control and Information Processing Ministry ofEducation Shanghai Jiaotong University (no Scip201507)Beijing Municipality Key Laboratory of Urban Traffic Oper-ation Simulation and Decision Support Beijing Transporta-tion Research Center the project of the Department of Trafficand Transportation of Hebei Province (no A0201-150505)and The National Key Technology Support Program (no2014BAG01B02)

References

[1] Z-P Li H Yu Y-C Liu and F-Q Liu ldquoAn improved adaptiveexponential smoothing model for short-term travel time fore-casting of urban arterial streetrdquo Acta Automatica Sinica vol 34no 11 pp 1404ndash1409 2008

[2] Y Hollander and R Liu ldquoEstimation of the distribution of traveltimes by repeated simulationrdquo Transportation Research Part CEmerging Technologies vol 16 no 2 pp 212ndash231 2008

[3] G-Y Jiang Technologies and Application of the Identification ofRoad Traffic Condition Communications Press Beijing China2004 (Chinese)

[4] M Gramaglia C J Bernardos and M Calderon ldquoVirtualinduction loops based on cooperative vehicular communica-tionsrdquo Sensors vol 13 no 2 pp 1467ndash1476 2013

[5] X Zhang and J A Rice ldquoShort-term travel time predictionrdquoTransportation Research Part C vol 11 no 3-4 pp 187ndash2102003

[6] U Mori A Mendiburu M Alvarez and J A Lozano ldquoA reviewof travel time estimation and forecasting for Advanced TravellerInformation Systemsrdquo Transportmetrica A Transport Sciencevol 11 no 2 pp 119ndash157 2015

[7] E I Vlahogianni M G Karlaftis and J C Golias ldquoShort-term traffic forecasting where we are and where wersquore goingrdquoTransportation Research Part C Emerging Technologies vol 43part 1 pp 3ndash19 2014

[8] C Shao K Zhang and Y Gu ldquoA study of route traveltime forecast method based on real data of urban expresswaynetworkrdquo China Civil Engineering Journal vol 36 no 1 pp 16ndash20 2003

[9] B R Chilukuri J A Laval and A Guin ldquoMicrosimulation-based framework for freeway travel time forecastingrdquo Trans-portation Research Record vol 2470 pp 34ndash45 2014

[10] E Yao and Y Zhang ldquoA new algorithm of shortmdashterm traveltime prediction for urban expresswayrdquo Journal of WuhanUniversity of Technology (Transportation ScienceampEngineering)vol 6 pp 1133ndash1137 2013

[11] J Zhao H Wang and W Liu ldquoPrediction of expressway traveltime based on adaptive interpolation kalman filteringrdquo Journalof South ChinaUniversity of Technology vol 2 pp 109ndash115 2014

[12] Z Gui and H Yu ldquoTrip travel time forecasting based onselective forgetting extreme learning machinerdquo Mathematical

International Journal of Distributed Sensor Networks 11

Problems in Engineering vol 2014 Article ID 829256 7 pages2014

[13] Y-J Jou Y-H Wen T-T Lee and H-J Cho ldquoMissing datatreatment on travel time estimation for ATISrdquo in Proceedingsof the IEEE International Conference on Systems Man andCybernetics (SMC rsquo03) vol 1 pp 102ndash107 IEEE WashingtonDC USA October 2003

[14] LWang and CWang ldquoA newmethod of real-time informationcollection in intelligent transportation systemrdquo Systems Engi-neering vol 23 no 2 pp 86ndash89 2005

[15] Z Cai Y Duan and A G Bourgeois ldquoDelay efficient oppor-tunistic routing in asynchronous multi-channel cognitive radionetworksrdquo Journal of Combinatorial Optimization vol 29 no 4pp 815ndash835 2015

[16] J W C VanLint and N J Zijpp ldquoAn improved travel timeestimation algorithm using dual-loop detectorsrdquo in Proceedingsof the Annual Meeting of the Transportation Research BoardWashington DC USA 2003

[17] Z Cai R Goebel and G Lin ldquoSize-constrained tree parti-tioning approximating the multicast k-tree routing problemrdquoTheoretical Computer Science vol 412 no 3 pp 240ndash245 2011

[18] G Zhu and H Yan ldquoA kind of demand-forecasting modelbased on analysis of demand booming and principle of naiveforecastingrdquo Systems EngineeringTheory amp Practice vol 24 no5 pp 22ndash33 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 11

Problems in Engineering vol 2014 Article ID 829256 7 pages2014

[13] Y-J Jou Y-H Wen T-T Lee and H-J Cho ldquoMissing datatreatment on travel time estimation for ATISrdquo in Proceedingsof the IEEE International Conference on Systems Man andCybernetics (SMC rsquo03) vol 1 pp 102ndash107 IEEE WashingtonDC USA October 2003

[14] LWang and CWang ldquoA newmethod of real-time informationcollection in intelligent transportation systemrdquo Systems Engi-neering vol 23 no 2 pp 86ndash89 2005

[15] Z Cai Y Duan and A G Bourgeois ldquoDelay efficient oppor-tunistic routing in asynchronous multi-channel cognitive radionetworksrdquo Journal of Combinatorial Optimization vol 29 no 4pp 815ndash835 2015

[16] J W C VanLint and N J Zijpp ldquoAn improved travel timeestimation algorithm using dual-loop detectorsrdquo in Proceedingsof the Annual Meeting of the Transportation Research BoardWashington DC USA 2003

[17] Z Cai R Goebel and G Lin ldquoSize-constrained tree parti-tioning approximating the multicast k-tree routing problemrdquoTheoretical Computer Science vol 412 no 3 pp 240ndash245 2011

[18] G Zhu and H Yan ldquoA kind of demand-forecasting modelbased on analysis of demand booming and principle of naiveforecastingrdquo Systems EngineeringTheory amp Practice vol 24 no5 pp 22ndash33 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of