research article an elm-based approach for...

Research ArticleAn ELM-Based Approach for Estimating Train Dwell Time inUrban Rail Traffic

Wen-jun Chu Xing-chen Zhang Jun-hua Chen and Bin Xu

State Key Laboratory of Rail Traffic Control and Safety Beijing Jiaotong University Beijing 100044 China

Correspondence should be addressed to Wen-jun Chu 12114228bjtueducn

Received 22 July 2014 Accepted 9 October 2014

Academic Editor Tao Chen

Copyright copy 2015 Wen-jun Chu 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

Dwell time estimation plays an important role in the operation of urban rail system On this specific problem a range of modelsbased on either polynomial regression or microsimulation have been proposed However the generalization performance ofpolynomial regressionmodels is limited and the accuracy of existingmicrosimulationmodels is unstable In this paper a new dwelltime estimation model based on extreme learning machine (ELM) is proposed The underlying factors that may affect urban raildwell time are analyzed first Then the relationships among different factors are extracted and modeled by ELM neural networkson basis of which an overall estimation model is proposed At last a set of observed data from Beijing subway is used to illustratethe proposed method and verify its overall performance

1 Introduction

Dwell time is the time that a public transport vehicle spendsat a station or a stop for passenger alighting and boarding[1] In any mode of public transportation it is an importantparameter which determines the system performance andservice quality to a large extent On one hand dwell timeconstitutes a significant part of the total trip time which isthe key criterion for service quality of public transit On theother hand dwell time determines the capacity utilizationof infrastructure thus affecting the efficiency of the wholetransit systemTherefore reasonable estimation of dwell timeplays an important role in operation of various public transitsystems

A number of studies have been conducted on dwell timeestimation in various types of public transportation and cor-responding research approaches can be roughly classified intotwo categories regression approach and microsimulationapproach

Regression approach is to establish regression model withobserved data to describe the relationship between dwelltime and corresponding factors This approach is first usedin the estimation of bus dwell time Levinson [2] proposeda linear regression model to estimate bus dwell time inwhich the bus dwell time is formulated as a linear function

of two primary contribution factorsmdashnumber of alightingand boarding passengers and the amount of time requiredfor bus doors opening and closing Since then a numberof studies were carried out to take into account some othercontributing factors for the bus dwell time estimation Forexample Guenthner and Hamat [3] investigated the rela-tionship between the bus dwell time and bus fare collectionsystem Levine and Torng [4] analyzed impact of bus floortypes on the bus dwell time Jaiswal et al [5] examinedinfluence of platform walking on bus rapid transit stationson bus dwell time Tirachini [6] studied impact of farepayment technology in urban bus services Most previousstudies on urban rail dwell time estimation also appliedthe regression approach Weston [7] proposed a polynomialregression model using the survey data of London Metroin which various contributing factors including the numberof alighting and boarding passengers passenger distributionand on-board crowdedness are considered Lam et al [8]proposed a linear regression model on basis of observeddata from two LRT stations Lin and Wilson [9] comparedlinear and nonlinear regression models with observed dataof MBTA Green Line and proved that crowdedness has anonlinear effect on urban rail dwell timeOn this basis Puong[10] proposed a nonlinear dwell time model that can fit 90of observed data fromMBTA Red Line

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 473432 9 pageshttpdxdoiorg1011552015473432

2 Mathematical Problems in Engineering

1 n

1 i L

1

w11

w1i

w1Lwn1

wni wn

L

k11

ki1

kL1

Output nodes

Hidden nodes

Input nodes

m

kLm

kimk1m

middot middot middot

middot middot middotmiddot middot middot


Figure 1 Structure of standard SLFN

As can be seen almost all proposed regression models ondwell time estimation are polynomial In these studies themodel structure is first determined through certain hypothe-sis and then corresponding parameters are calibrated Underthis condition though these models fit respective field datawell the generalization performance of them cannot beensured

Microsimulation approach is to calculate the requireddwell time on basis of single passenger behavior descriptionunder computer environment In recent years computer-based pedestrian simulation technology rapidly develops andis gradually introduced into dwell time estimation Li et al[11] appliedMonte Carlo simulation to simulate the bus dwellprocess in which a binary door choice model predictingthe proportion of alighting passengers through front or reardoor is integrated Zhang et al [12] proposed a cellularautomaton based alighting and boarding microsimulationmodel for passengers in Beijing subway stations which isproven effective in estimating urban rail dwell time Baee et al[13] investigated the influence of different boardingalightingstrategies on urban rail dwell time on basis of a microsim-ulation model in which an inclination function governingpassengersrsquo movement in a two-dimensional queue is intro-duced In addition some commercial pedestrian simulationsoftware programs such as VISSIM and Legion are appliedto calculate dwell time in many related studies

Theoretically speaking microsimulation models havebetter generalization performance than regression model Ifthe behavior of passengers is described properly the modelcan be used in any scenario However existing microscopicsimulation theory is still insufficient in describing pedestrianbehavior under crowded condition As a result the accuracyof microsimulation dwell time estimation models cannot beensured at present

In urban rail transit system train operation is typicallybased on timetables which are made in advance and thedwell time at each station is assigned beforehand Under thiscondition the reasonability of preassigned dwell time may

have a significant influence on the performance of the wholesystem If the assigned dwell time is insufficient for passengeralighting and boarding delay will happen and complicatedadjustments need to be made in the predesigned timetable soas to ensure the following train operation On the other sideif the assigned dwell time is too long the headway betweentwo consecutive trains will also be overlong consequentlylimiting the capacity of the whole transit line Thereforein all urban rail transit systems especially in those withheavy traffic such as Beijing subway reasonable estimation ofdwell time is essential to create effective timetables and makea compromise between service quality and transportationcapacity

Artificial neural network is a widely used method ofdata fitting It can approximate complex nonlinear mappingsdirectly from the input samplewithoutmakingmuchhypoth-esis beforehand In this paper a new proposed artificialneural network method ELM is used in urban rail dwell timeestimationThe outline of the paper is as follows In Section 1previous research regarding dwell time estimation of publictransportation is reviewed Section 2 elaborates the principlesand steps of ELM Section 3 makes a detailed analysis on thefactors of train dwell time at urban rail stations and Section 4presents the structure of the proposed model In Section 5several data sets on Beijing subway are used to evaluate theproposed model Conclusions and discussions are given inSection 6

2 Extreme Learning Machine

Single-hidden layer feedforward network (SLFN) is a widelyused type of artificial neural network which has beenproven effective in complex nonlinear approximation [14ndash16] Figure 1 illustrates the structure of a standard SLFNIn this network 119899 input nodes and 119898 output nodes areincluded corresponding to 119899-dimensional input vector and119898-dimensional out vector 119871 nodes are contained in thehidden layer and 120585

119894is the threshold of the 119894th hidden node

Mathematical Problems in Engineering 3

Door opening

Alighting at door 1

Boarding at door 1

ConfirmationDoor closing

Required dwell timeOverlap Overlap

Alighting at door i

Boarding at door i

Alighting at door ilowast

Boarding at door ilowast

t0

Figure 2 Structure of urban rail dwell time

119892(119909) is the activation function w119894= [1199081119894 1199082119894 119908

119899119894]T is

the weight vector connecting the input nodes and the 119894thhidden node and k

119894= [1198961198941 1198961198942 119896

119894119898]T is the weight vector

connecting the 119894th hidden node and the output nodesGiven 119873 arbitrary training samples (x

119895 e119895) where x

119895=

[1199091198951 1199091198952 119909

119895119899]T

isin R119899 and e119895= [1198901198951 1198901198952 119890

119895119898]T

isin R119898the output of the above SLFN is

o119895=

119871

sum

119894

k119894119892 (w119894sdot x119895+ 120585119894) (1)

If this SLFN can approximate these 119873 samples with zeroerror that is sum119873

119895o119895minus e119895 = 0 then there exist k

119894 w119894 and 120585

119894

such that119871

sum

119894

k119894119892 (w119894sdot x119895+ 120585119894) = e119895 (2)

These119873 equations can be written compactly as

HK = E (3)

where

H =[

[

[

119892 (w1x1+ 1205851) 119892 (w

119871x1+ 120585119871)

d

119892 (w1x119873

+ 1205851) sdot sdot sdot 119892 (w

119871x119873

+ 120585119871)

]

]

]119873times119871

(4)

K =[

[

[

k1198791

k119879119871

]

]

]119871times119898

(5)

E =[

[

[

e1198791

e119879119873

]

]

]119873times119898

(6)

As named in Huang and Babri [17]H is called the hiddenlayer output matrix of the SLFN and the 119894th column of itcorresponds to the output of 119894th hidden node with respect to119873 inputs As proven by Huang et al [18] given arbitrary w

119894

and 120585119894 the least square solution of K in formula (3) can be

obtained by formula (7)

K = HdaggerE (7)

whereHdagger is theMoore-Penrose generalized inverse of matrixH On this basis a simple and efficient training algorithm forSLFN called ELM is proposed [18] whose procedure can besummarized as follows

Step 1 Randomly assign input weight w119894and bias 120585

119894 119894 =

1 2 119871

Step 2 Calculate the hidden layer output matrixH accordingto formula (4)

Step 3 Calculate the output weight K according to formula(7)

Due to the fast training speed ELM has been widely usedfor many applications [19] In this paper ELM is applied toapproximate the complex relationship between the factors ofurban rail dwell time

3 Factors of Urban Rail Dwell Time

Urban rail dwell time is typically defined as the time elapsedbetween the door opening and closing of a train sittingat a station [10] In this period several tasks need to beaccomplished as shown in Figure 2

In Figure 2 the horizontal axis represents time and 1199050rep-

resents the timewhen the train stops and doors begin to openOn the vertical axis four types of task are listedThe duration


Platform pattern of this station

Numbers of alighting passengers from each

entrance

Number of boarding passengers

Number of through passengers

Platform pattern of previous stations

Distribution of boardingpassengers on platform

Distribution of alightingpassengers on board

Distribution of throughpassengers on board

Numbers of alightingand boarding

passengers at each door

Crowdedness of each vehicle

Vehicle performance

Operation efficiency

Time of door closing process

Passenger service time at each door

Confirmation time

Required dwell time

Figure 3 Factors of urban rail dwell time

of door opening and closing process is mainly determinedby the mechanism of the vehicles The confirmation processrepresents the interval between the end of passenger alightingat all doors and the beginning of door closing process whichis used for operators confirming the completion of passengeralighting The start time of this process depends on the doorat which passenger boarding completes last that is the door119894lowast The times of alighting and boarding tasks vary acrossdoors According to previous research this is mainly becausethe numbers of alighting through and boarding passengersdiffer from door to door In other words the duration ofalighting and boarding process at a door is mainly decidedby the number of passengers alighting and boarding fromthis door and the crowdedness of corresponding vehicle Andthese parameters will be affected by the passenger flow andplatform pattern of this station and previous stations

Nevertheless in practical terms there exist overlapsbetween some consecutive tasks As shown in Figure 2 theoverlap between door opening and passenger alighting rep-resents that some passengers begin to alight before the dooris fully open and the overlap between passenger alighting andboarding represents that some passengers do not obey theldquoget off and then onrdquo rule Under this condition times of theseprocesses cannot be separately considered no matter fromthe perspective of survey or estimation Therefore an overallconcept passenger service time is proposed here whichrepresents the period from the beginning of door opening tothe end of passenger boarding at single or all doors

On basis of the above analysis the factors of urban raildwell time and their interaction can be concluded which isshown in Figure 3

4 Urban Rail Dwell Time Estimation

41 Notations The key notations used in the dwell timeestimation are shown in Notation Definitions section

42 Problem Statement Generally speaking in practicaloperation of urban rail system the operation-related param-eters that is platform pattern vehicle performance andoperation efficiency are relatively stable Therefore onlythe influence of the traffic-related parameters which is theconcern of most previous research is taken into account hereOn this basis the urban rail dwell time estimation problemcan be described as follows

Consider a 119899-door urban rail train thatwillmake a stop ona station On the train A passengers will alight at the stationand 119862 passengers will not On the platform of the station119861119895passengers who enter the platform through entrance 119895 are

waiting to get on this train In addition the train needs 1205911to

close all its doors and operators need to spend 1205912to confirm

the full close of all doorsThus assign a minimum dwell time119863 for the train which is sufficient for passengers alighting andboarding at the station

According to the analysis in Section 3 the required dwelltime 119863 can be seen as the accumulation of three parts themaximum single-door passenger service time duration ofdoor closing process and confirmation time that is

119863 = max119894

119905119894+ 1205911+ 1205912 (8)

where the passenger service time at 119894th door 119905119894is determined

by the number of boarding alighting and through passen-gers at this door that is

119905119894= 119865 (119886

119894 119887119894 119888119894) (9)

Furthermore for a specific station the distribution ofboarding passengers on the platform is always accordedwith certain rules [18] which means certain mapping existsbetween the vector 120573 = [119887

1 1198872 119887

119899]T and the boarding

passenger vector B = [1198611 1198612 119861

119898]T that is

120573 = 1198921 (B) = 119892

1(1198611 1198612 119861

119898) (10)


Output nodes

Hidden nodes

Input nodes

1

2

3

1

2

p

1

ai bi ci

w11

w12

w1p w21

w22w2p w31

w32 w3p

k11 k21 kp1

ti


(a)

1

1 2 q

i

Bj

wj1wj2

wjq

k11 k21kqn

1

n

k1i k2i

k2nk1n

kqikq1

b1 bi bn

j m

B1 Bm

w11 w12

w1qwm1

wm2 wmq

middot middot middot middot middot middot



(b)

Figure 4 Structure of SDPST model and PPD model

By contrast the distribution of alighting and throughpassengers on board which is determined by platformpattern of previous stations is more complicated In previousresearch the alighting and through passengers on boardare usually assumed to be uniformly distributed [10] ordistributed with constant proportion [7] In this paper theuniform distribution is adopted for 120572 and 120574 that is

120572 = [1198861 1198862 119886

119899]T= 1198922 (

119860) = [

119860

119899

119860

119899

119860

119899

]

T

120574 = [1198881 1198882 119888

119899]T= 1198922 (

119862) = [

119862

119899

119862

119899

119862

119899

]

T

(11)

To summarize the required dwell time 119863 can be de-scribed as follows

119863 = max119894

119865(

119860

119899

[1198921(1198611 1198612 119861

119898)]1198941

119862

119899

) + 1205911+ 1205912

(12)

As can be seen the key to dwell time estimation is toapproximate the mappings 119865 and 119892

1

43 ELM-Based Estimation Model In this section two ELMneural networks are designed to approximate the mappingsshown in formula (12) On this basis an overall estimationmodel is proposed

431 Single-Door Passenger Service Time (SDPST) ModelIn order to approximate the relationship between 119905

119894and

(119886119894 119887119894 119888119894) that is (119886

119894 119887119894 119888119894) an ELM neural network is

designed whose structure is shown in Figure 4(a) As illus-trated in this figure the model has an input vector of threedimensions which represent 119886

119894 119887119894 and 119888

119894 respectively and

a single-dimensional output vector 119905119894 Sigmoid function is

chosen as the activation function of the hidden nodes and thenumber of hidden nodes 119901 needs to be determined through119896-fold cross-validation with training data set

432 Platform Passenger Distribution (PPD)Model AnotherELM neural network is designed to describe the distributionrule of passengers on platform as shown in Figure 4(b)

This model has an input vector of 119898 dimensions whichrepresent the numbers of boarding passengers from eachentrance and an output vector of 119899 dimensions which repre-sent the number of boarding passengers at each door Besidesthe activation function of this model is also sigmoid functionand the number of hidden nodes is 119902 which also needs to bedetermined through cross-validation

433 Overall Dwell Time Estimation Model On basis of theprevious two models an overall model for urban rail dwelltime estimation is proposed which is shown in Figure 5In this model the mappings 119865 and 119892

1in formula (12) are

replaced by SDPST model and PPD model respectively andthis two ELM neural networks need to be trained separatelywith corresponding data sets

5 Model Evaluation

51 Data Collection and Processing A survey is conductedon the outbound platform of Zhichunlu station of Line 13Beijing subway This platform is a typical side platform withthree stairways and one escalator acting as entrances andexits as shown in Figure 6 In the survey 24 recorders areassigned to observe the 24 doors of trains respectively andanother two are assigned to record the number of boardingpassengers entering from the two entrances After 10 daysrsquosurvey a raw data set containing 8304 instances from 346trains is obtained whose structure is illustrated in Table 1 Itshould be noted that the actual number of through passengercannot be observed precisely from platform Therefore theattribute c which is used to describe the crowdedness on thevehicle is replaced by the number of through passengers thatstand on board near the door


A BjB1 Bm C

g2 g2

t1 ti tn

12059111205912Max

An Cnb1 bi bn

PPDmodel

SDPSTmodel

(i)

SDPSTmodel

SDPSTmodel

(1) (n)




Intermediate variableConstantInput variable

OutputFunction and model

D


Figure 5 Overall dwell time estimation model

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Entrance 1Eixt 1 Entrance 2Eixt 2

Figure 6 Layout of the outbound platform of Zhichunlu station(Line 13)

Table 1 Structure of the raw data set

Attribute Description119903 Index of records119894 Index of doors119906 Index of trains119886 Number of alighting passengers at single door119887 Number of boarding passengers at single door

119888

Number of through passengers that stand on boardnear the door

1198611

Number of boarding passengers entering fromEntrance 1

1198612


PST Passenger service time

119879

Actual dwell time of train which spans from thebeginning of door opening to the end of door closing

1198790

Scheduled dwell time of train

From this raw data set the operation-related parametersand three useful data sets are derived

511 Operation-Related Parameters Firstly the confirmationand door closing times are derived Considering the effect ofscheduled dwell time only the records in which actual dwelltime exceeds scheduled dwell time are used and the sum ofconstant parameters 120591

1and 1205912is assigned with the average of

differences between 119879 and PST that is

1205911+ 1205912=

sum119903|119879minus1198790gt60

(119879 minus PST)1003816100381610038161003816119903 | 119879 minus 119879

0gt 60

1003816100381610038161003816

(13)

512 SDPST Data Set This data set has 8304 instances eachof which represents a passenger service process at a singledoor Four attributes a b c and PST are contained andcorresponding data can be extracted directly from the rawdata set This data set can be used to train the SDPST model

513 PPD Data Set 346 instances are contained in this dataset each of which corresponds to an observed train Thereare 26 attributes per instance Two of them are the numbers ofboarding passengers entering from the two entrances (namedas 1198611and 119861

2) and the rest represent the number of boarding

passengers at each door (named as 119887119894 119894 = 1 2 22) In

this way the distribution of boarding passengers for eachobserved train can be described by the instances of this dataset Therefore this data set can be used to train the PPDmodel

514 Dwell Time Data Set This data set concerns therelationship between dwell time of trains and correspondingpassenger flow Therefore 346 records corresponding to 346observed trains are included and each of them has fiveattributes the total number of alighting passengers A thenumbers of boarding passengers entering from Entrance 1and Entrance 2 that is 119861

1and 119861

2 the total number of

through standees 119862 and required dwell time 119863 The formerthree attributes can all be obtained through accumulating thecorresponding single-door data of the raw data set while 119863

is obtained according to

119863 =

119879 119879 minus 1198790gt 60

PST + 1205911+ 1205912

otherwise(14)

52 Training of SDPST Model With the SDPST data setELM is used to train the SDPST model Meanwhile forcomparison another two popular algorithms LMBP andSVM are also applied on this specific regression problem Allthe attributes in this data set are normalized into range [0 1]

and the data set is divided into two parts 4000 observationsare used for training and the rest are used for testing ForELM the number of hidden nodes 119901 is gradually increasedby an interval of 5 and the optimal number 65 is obtainedusing 3-fold cross-validation method which is illustratedin Figure 7 Similarly the number of hidden nodes in theBP network is also determined through repeated cross-validations For SVM RBF is used as kernel function and thecost parameter and kernel parameter are both chosen fromset 2minus10 2minus9 2minus8 29 210 through repeated tests


0 10 20 30 40 50 60 70 80 90 100005

01015

02025

03035

04045

05

Number of hidden nodes

Aver

age R

MSE

Figure 7 Tuning the number of hidden nodes in the ELM-basedSDPST model

Table 2 Comparison of performance of ELM LMBP and SVM onSDPST data set

Algorithms Number ofnodesSVs

Time (s) RMSETraining Testing Training Testing

ELM 65 01358 04128 00865 00945LMBP 20 21722 00986 00904 01218SVM 31024 46375 12519 00853 01046

All the simulations are carried out in MATLAB 82environment running in a Core2 Quad 267GHz CPU andcorresponding results are shown in Table 2 As shown inthis table no matter in training speed or generalizationperformance ELM is remarkably better than the other twoalgorithms In other words the ELM-based SDPST performsbetter in estimating the single-door passenger service time

For further comparison a basic social force model [20]is established to simulate passengers alighting and boardingat single door of urban train The parameters of this modelare calibrated according to the observed data of a basic casein which the numbers of alighting boarding and throughpassengers are all 5 that is 119886 = 119887 = 119888 = 5 On this basisdifferent cases are tested on this microsimulation model andthe results are comparedwith the proposedmodel In the testthe numbers of alighting and through passengers are all set tobe 5 that is 119886 = 119888 = 5 The number of boarding passengers isgradually increased and corresponding PST outputted by themicrosimulationmodel is comparedwith the result estimatedby the ELM-based SDPSTmodel which is shown in Figure 8As can be seen the results of the proposed model are in goodaccordance with the observed data The microsimulationmodel fits the observed data well when 119887 le 16 but it doesnot perform well when 119887 gt 16

Furthermore using the SDPST model trained by ELMthe relationship between passenger service time (PST) andcorresponding factors (a b and c) at single door is alsoinvestigated With the other two factors fixed at 5 thevariation of PST with each factor is tested As shown inFigure 9 PST is in nonlinear relationship with each of the

Table 3 Comparison of performance of ELM BP and SVMon PPDdata set



ELM 25 00057 00075 00987 01015LMBP 10 01167 00029 01077 01102SVM 4232 00764 00828 00972 01023

0 5 10 15 20 250

20

40

60

80

100

120

PST

at si

ngle

doo

r (s)

Observed dataELM-based modelMicrosimulation model

Number of boarding passengers at single door b

Figure 8 Comparison of performance of ELM-based model andmicrosimulation model

three factors which is much different with previous studies[8ndash10]

53 Training of PPD Model With the PPD data set thePPD model is trained to describe the boarding passengerdistribution on the outbound platform of Zhichulu station(Line 13) The data set is also normalized into [0 1] anddivided into two parts 200 observations are used for trainingand the rest are used for testing The other two algorithmsLMBP and SVM are also applied on this data set andtheir performances are compared with ELM in Table 3 Ascan be seen the training speed of ELM is still remark-ably faster than that of the other two algorithms As forgeneralization performance ELM is similar to the SVMand slightly better than LMBP In conclusion the ELM-based model obtains best performance on the PPD dataset

54 Evaluation of Overall Estimation Model With the abovetwo models trained by ELM the overall model can be usedto estimate the train dwell time of Line 13 at Zhichunlustation The proposed overall model is compared with twopolynomial models One is proposed by Lam et al [8] and


0 5 10 15 20 250

102030405060708090

100110

Number of passengers

PST

at si

ngle

doo

r (s)

ab

c

Figure 9 Relationship between PST and corresponding factors

Table 4 Comparison of performance of proposed model and APmodel

Models Coefficient of determination (1198772)Proposed model 08972Lamrsquos model 06711Puongrsquos model 07802

shown as formula (15) The other is proposed by Puong [10]and shown as formula (16)

119863 = 1205831+ 1205832119860 + 120583

3119861 (15)

119863 = ]1+ ]2

119860

119899

+ ]3

119861

119899

+ ]4(

119862

119899

)

3119861

119899

(16)

Using the dwell time data set least squares method isused to calibrate the parameters of the above two modelsConsidering the outputs of these three models are all single-dimensional the coefficient of determinationwhich is usuallydenoted as 1198772 is adopted to evaluate their regression perfor-manceThemodel whose 1198772 is closer to 1 is considered betterThe results are listed in Table 4 As can be seen the ELM-based model proposed in this paper performs much betterthan the other two polynomial models

6 Conclusions

This paper proposed a new model to estimate urban raildwell time In this model two crucial relationships amongthe factors of urban rail dwell time are modeled by twoSLFNs which are trained with ELM Using a set of observeddata from Beijing subway the training of these two networksis illustrated during which ELM is proven more effectivethan other two algorithms and advantage of the proposedapproach is also verified by comparing with an existingestimation model

Notation Definitions

119894 Index of doors119899 Number of doors119895 Index of platform entrances119898 Number of platform entrances119860 Total number of alighting passengers119861119895 Number of boarding passengers enteringthe platform through 119895th entrance

B m-dimensional column vector whose 119895thcomponent is 119861

119895 that is

B = [11986111198612 119861

119898]T

119862 Total number of through passengers119886119894 Number of alighting passengers at 119894th door

119887119894 Number of boarding passengers at 119894thdoor

119888119894 Number of through passengers at 119894th door120572 n-dimensional column vector whose 119894th

component is 119886119894 that is

120572 = [1198861 1198862 119886

119899]T

120573 n-dimensional column vector whose 119894thcomponent is 119887

119894 that is

120573 = [1198871 1198872 119887

119899]T


119894 that is 120574 = [119888

1 1198882 119888

119899]T

119905119894 Passenger service time at 119894th door

1205911 Duration of door closing process

1205912 Confirmation time

119863 Required dwell time

Conflict of Interests

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

Acknowledgments

The authors are grateful to the editor and reviewers for theirvaluable suggestions which improved the paper This work issupported by National Natural Science Foundation of China(U1361114)

References

[1] Q Meng and X Qu ldquoBus dwell time estimation at busbays a probabilistic approachrdquo Transportation Research Part CEmerging Technologies vol 36 pp 61ndash71 2013

[2] H S Levinson ldquoTransit travel time performancerdquo Transporta-tion Research Record vol 915 pp 1ndash6 1983

[3] R P Guenthner and K Hamat ldquoTransit dwell time undercomplex fare structurerdquo Journal of Transportation Engineeringvol 114 no 3 pp 367ndash379 1988

[4] J Levine and G Torng ldquoDwell-time effects of low-floor busdesignrdquo Journal of Transportation Engineering vol 120 no 6pp 914ndash929 1994

[5] S Jaiswal J Bunker and L Ferreira ldquoInfluence of platformwalking on brt station bus dwell time estimation Australiananalysisrdquo Journal of Transportation Engineering vol 136 no 12pp 1173ndash1179 2010


[6] A Tirachini ldquoEstimation of travel time and the benefits ofupgrading the fare payment technology in urban bus servicesrdquoTransportation Research C Emerging Technologies vol 30 pp239ndash256 2013

[7] J G Weston ldquoLondon underground train service model adescription of the model and its usesrdquo in Proceedings of theComputer Applications in Railway Planning and ManagementConference (COMPRAIL rsquo90) pp 133ndash147 Rome Italy 1990

[8] W H K Lam C-Y Cheung and C F Lam ldquoA study ofcrowding effects at the Hong Kong light rail transit stationsrdquoTransportation Research Part A Policy and Practice vol 33 no5 pp 401ndash415 1999

[9] T M Lin and N H M Wilson ldquoDwell time relationships forlight rail systemsrdquoTransportationResearchRecord Journal of theTransportation Research Board vol 1361 pp 287ndash295 1991

[10] A Puong Dwell Time Model and Analysis for the MBTA RedLine MIT OpenCourseWare 2000 httpocwmiteduindexhtm

[11] M T Li F Zhao L F Chow H Zhang and S C Li ldquoSimulationmodel for estimating bus dwell time by simultaneously con-sidering numbers of disembarking and boarding passengersrdquoTransportation Research Record no 1971 pp 59ndash65 2006

[12] Q Zhang B Han and D Li ldquoModeling and simulation ofpassenger alighting and boarding movement in Beijing metrostationsrdquo Transportation Research Part C Emerging Technolo-gies vol 16 no 5 pp 635ndash649 2008

[13] S Baee F Eshghi S M Hashemi and R Moienfar ldquoPassengerboardingalighting management in urban rail transportationrdquoin Proceedings of the Joint Rail Conference (JRC rsquo12) pp 823ndash829Philadelphia Pa USA April 2012

[14] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[15] G B Huang Learning capability of neural networks [PhDthesis] Nanyang Technological University Singapore 1998

[16] G-BHuang Y-Q Chen andHA Babri ldquoClassification abilityof single hidden layer feedforward neural networksrdquo IEEETransactions on Neural Networks vol 11 no 3 pp 799ndash8012000

[17] G-B Huang and H A Babri ldquoUpper bounds on the numberof hidden neurons in feedforward networks with arbitrarybounded nonlinear activation functionsrdquo IEEE Transactions onNeural Networks vol 9 no 1 pp 224ndash229 1998

[18] G-B Huang Q-Y Zhu and C-K Siew ldquoExtreme learningmachine theory and applicationsrdquoNeurocomputing vol 70 no1ndash3 pp 489ndash501 2006

[19] R Rajesh and J S Prakash ldquoExtreme learning machinesmdashareview and state-of-the-artrdquo International Journal of WisdomBased Computing vol 1 no 1 pp 35ndash49 2011

[20] D Helbing and P Molnar ldquoSocial force model for pedestriandynamicsrdquo Physical Review E vol 51 no 5 pp 4282ndash4286 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


1 n

1 i L

1

w11

w1i

w1Lwn1

wni wn

L

k11

ki1

kL1

Output nodes

Hidden nodes

Input nodes

m

kLm

kimk1m




Figure 1 Structure of standard SLFN

As can be seen almost all proposed regression models ondwell time estimation are polynomial In these studies themodel structure is first determined through certain hypothe-sis and then corresponding parameters are calibrated Underthis condition though these models fit respective field datawell the generalization performance of them cannot beensured

Microsimulation approach is to calculate the requireddwell time on basis of single passenger behavior descriptionunder computer environment In recent years computer-based pedestrian simulation technology rapidly develops andis gradually introduced into dwell time estimation Li et al[11] appliedMonte Carlo simulation to simulate the bus dwellprocess in which a binary door choice model predictingthe proportion of alighting passengers through front or reardoor is integrated Zhang et al [12] proposed a cellularautomaton based alighting and boarding microsimulationmodel for passengers in Beijing subway stations which isproven effective in estimating urban rail dwell time Baee et al[13] investigated the influence of different boardingalightingstrategies on urban rail dwell time on basis of a microsim-ulation model in which an inclination function governingpassengersrsquo movement in a two-dimensional queue is intro-duced In addition some commercial pedestrian simulationsoftware programs such as VISSIM and Legion are appliedto calculate dwell time in many related studies

Theoretically speaking microsimulation models havebetter generalization performance than regression model Ifthe behavior of passengers is described properly the modelcan be used in any scenario However existing microscopicsimulation theory is still insufficient in describing pedestrianbehavior under crowded condition As a result the accuracyof microsimulation dwell time estimation models cannot beensured at present

In urban rail transit system train operation is typicallybased on timetables which are made in advance and thedwell time at each station is assigned beforehand Under thiscondition the reasonability of preassigned dwell time may

have a significant influence on the performance of the wholesystem If the assigned dwell time is insufficient for passengeralighting and boarding delay will happen and complicatedadjustments need to be made in the predesigned timetable soas to ensure the following train operation On the other sideif the assigned dwell time is too long the headway betweentwo consecutive trains will also be overlong consequentlylimiting the capacity of the whole transit line Thereforein all urban rail transit systems especially in those withheavy traffic such as Beijing subway reasonable estimation ofdwell time is essential to create effective timetables and makea compromise between service quality and transportationcapacity

Artificial neural network is a widely used method ofdata fitting It can approximate complex nonlinear mappingsdirectly from the input samplewithoutmakingmuchhypoth-esis beforehand In this paper a new proposed artificialneural network method ELM is used in urban rail dwell timeestimationThe outline of the paper is as follows In Section 1previous research regarding dwell time estimation of publictransportation is reviewed Section 2 elaborates the principlesand steps of ELM Section 3 makes a detailed analysis on thefactors of train dwell time at urban rail stations and Section 4presents the structure of the proposed model In Section 5several data sets on Beijing subway are used to evaluate theproposed model Conclusions and discussions are given inSection 6

2 Extreme Learning Machine

Single-hidden layer feedforward network (SLFN) is a widelyused type of artificial neural network which has beenproven effective in complex nonlinear approximation [14ndash16] Figure 1 illustrates the structure of a standard SLFNIn this network 119899 input nodes and 119898 output nodes areincluded corresponding to 119899-dimensional input vector and119898-dimensional out vector 119871 nodes are contained in thehidden layer and 120585

119894is the threshold of the 119894th hidden node


Door opening

Alighting at door 1

Boarding at door 1



Alighting at door i

Boarding at door i



t0



119899119894]T is


119894= [1198961198941 1198961198942 119896



119895 e119895) where x

119895=

[1199091198951 1199091198952 119909

119895119899]T

isin R119899 and e119895= [1198901198951 1198901198952 119890

119895119898]T


o119895=

119871

sum

119894

k119894119892 (w119894sdot x119895+ 120585119894) (1)



119894 w119894 and 120585

119894

such that119871

sum

119894

k119894119892 (w119894sdot x119895+ 120585119894) = e119895 (2)


HK = E (3)

where

H =[

[

[

119892 (w1x1+ 1205851) 119892 (w

119871x1+ 120585119871)

d

119892 (w1x119873

+ 1205851) sdot sdot sdot 119892 (w

119871x119873

+ 120585119871)

]

]

]119873times119871

(4)

K =[

[

[

k1198791

k119879119871

]

]

]119871times119898

(5)

E =[

[

[

e1198791

e119879119873

]

]

]119873times119898

(6)


119894



K = HdaggerE (7)



119894 119894 =

1 2 119871











entrance










Vehicle performance




Confirmation time

Required dwell time













119863 = max119894

119905119894+ 1205911+ 1205912 (8)



119905119894= 119865 (119886

119894 119887119894 119888119894) (9)


1 1198872 119887



119898]T that is

120573 = 1198921 (B) = 119892

1(1198611 1198612 119861

119898) (10)


Output nodes

Hidden nodes

Input nodes

1

2

3

1

2

p

1

ai bi ci

w11

w12

w1p w21

w22w2p w31

w32 w3p

k11 k21 kp1

ti


(a)

1

1 2 q

i

Bj

wj1wj2

wjq

k11 k21kqn

1

n

k1i k2i

k2nk1n

kqikq1

b1 bi bn

j m

B1 Bm

w11 w12

w1qwm1

wm2 wmq




(b)



120572 = [1198861 1198862 119886

119899]T= 1198922 (

119860) = [

119860

119899

119860

119899

119860

119899

]

T

120574 = [1198881 1198882 119888

119899]T= 1198922 (

119862) = [

119862

119899

119862

119899

119862

119899

]

T

(11)


119863 = max119894

119865(

119860

119899

[1198921(1198611 1198612 119861

119898)]1198941

119862

119899

) + 1205911+ 1205912

(12)


1



119894and

(119886119894 119887119894 119888119894) that is (119886



119894 119887119894 and 119888









5 Model Evaluation



A BjB1 Bm C

g2 g2

t1 ti tn

12059111205912Max

An Cnb1 bi bn

PPDmodel

SDPSTmodel

(i)

SDPSTmodel

SDPSTmodel

(1) (n)






D



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24





119888


1198611


1198612



119879


1198790






1205911+ 1205912=

sum119903|119879minus1198790gt60

(119879 minus PST)1003816100381610038161003816119903 | 119879 minus 119879

0gt 60

1003816100381610038161003816

(13)







1and 119861




119863 =

119879 119879 minus 1198790gt 60

PST + 1205911+ 1205912

otherwise(14)




0 10 20 30 40 50 60 70 80 90 100005

01015

02025

03035

04045

05


Aver

age R

MSE





ELM 65 01358 04128 00865 00945LMBP 20 21722 00986 00904 01218SVM 31024 46375 12519 00853 01046







ELM 25 00057 00075 00987 01015LMBP 10 01167 00029 01077 01102SVM 4232 00764 00828 00972 01023

0 5 10 15 20 250

20

40

60

80

100

120

PST

at si

ngle

doo

r (s)








0 5 10 15 20 250

102030405060708090

100110


PST

at si

ngle

doo

r (s)

ab

c





119863 = 1205831+ 1205832119860 + 120583

3119861 (15)

119863 = ]1+ ]2

119860

119899

+ ]3

119861

119899

+ ]4(

119862

119899

)

3119861

119899

(16)


6 Conclusions





119895 that is

B = [11986111198612 119861

119898]T





120572 = [1198861 1198862 119886

119899]T


119894 that is

120573 = [1198871 1198872 119887

119899]T


119894 that is 120574 = [119888

1 1198882 119888

119899]T







Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Door opening

Alighting at door 1

Boarding at door 1



Alighting at door i

Boarding at door i



t0



119899119894]T is


119894= [1198961198941 1198961198942 119896



119895 e119895) where x

119895=

[1199091198951 1199091198952 119909

119895119899]T

isin R119899 and e119895= [1198901198951 1198901198952 119890

119895119898]T


o119895=

119871

sum

119894

k119894119892 (w119894sdot x119895+ 120585119894) (1)



119894 w119894 and 120585

119894

such that119871

sum

119894

k119894119892 (w119894sdot x119895+ 120585119894) = e119895 (2)


HK = E (3)

where

H =[

[

[

119892 (w1x1+ 1205851) 119892 (w

119871x1+ 120585119871)

d

119892 (w1x119873

+ 1205851) sdot sdot sdot 119892 (w

119871x119873

+ 120585119871)

]

]

]119873times119871

(4)

K =[

[

[

k1198791

k119879119871

]

]

]119871times119898

(5)

E =[

[

[

e1198791

e119879119873

]

]

]119873times119898

(6)


119894



K = HdaggerE (7)



119894 119894 =

1 2 119871











entrance










Vehicle performance




Confirmation time

Required dwell time













119863 = max119894

119905119894+ 1205911+ 1205912 (8)



119905119894= 119865 (119886

119894 119887119894 119888119894) (9)


1 1198872 119887



119898]T that is

120573 = 1198921 (B) = 119892

1(1198611 1198612 119861

119898) (10)


Output nodes

Hidden nodes

Input nodes

1

2

3

1

2

p

1

ai bi ci

w11

w12

w1p w21

w22w2p w31

w32 w3p

k11 k21 kp1

ti


(a)

1

1 2 q

i

Bj

wj1wj2

wjq

k11 k21kqn

1

n

k1i k2i

k2nk1n

kqikq1

b1 bi bn

j m

B1 Bm

w11 w12

w1qwm1

wm2 wmq




(b)



120572 = [1198861 1198862 119886

119899]T= 1198922 (

119860) = [

119860

119899

119860

119899

119860

119899

]

T

120574 = [1198881 1198882 119888

119899]T= 1198922 (

119862) = [

119862

119899

119862

119899

119862

119899

]

T

(11)


119863 = max119894

119865(

119860

119899

[1198921(1198611 1198612 119861

119898)]1198941

119862

119899

) + 1205911+ 1205912

(12)


1



119894and

(119886119894 119887119894 119888119894) that is (119886



119894 119887119894 and 119888









5 Model Evaluation



A BjB1 Bm C

g2 g2

t1 ti tn

12059111205912Max

An Cnb1 bi bn

PPDmodel

SDPSTmodel

(i)

SDPSTmodel

SDPSTmodel

(1) (n)






D



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24





119888


1198611


1198612



119879


1198790






1205911+ 1205912=

sum119903|119879minus1198790gt60

(119879 minus PST)1003816100381610038161003816119903 | 119879 minus 119879

0gt 60

1003816100381610038161003816

(13)







1and 119861




119863 =

119879 119879 minus 1198790gt 60

PST + 1205911+ 1205912

otherwise(14)




0 10 20 30 40 50 60 70 80 90 100005

01015

02025

03035

04045

05


Aver

age R

MSE





ELM 65 01358 04128 00865 00945LMBP 20 21722 00986 00904 01218SVM 31024 46375 12519 00853 01046







ELM 25 00057 00075 00987 01015LMBP 10 01167 00029 01077 01102SVM 4232 00764 00828 00972 01023

0 5 10 15 20 250

20

40

60

80

100

120

PST

at si

ngle

doo

r (s)








0 5 10 15 20 250

102030405060708090

100110


PST

at si

ngle

doo

r (s)

ab

c





119863 = 1205831+ 1205832119860 + 120583

3119861 (15)

119863 = ]1+ ]2

119860

119899

+ ]3

119861

119899

+ ]4(

119862

119899

)

3119861

119899

(16)


6 Conclusions





119895 that is

B = [11986111198612 119861

119898]T





120572 = [1198861 1198862 119886

119899]T


119894 that is

120573 = [1198871 1198872 119887

119899]T


119894 that is 120574 = [119888

1 1198882 119888

119899]T







Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












entrance










Vehicle performance




Confirmation time

Required dwell time













119863 = max119894

119905119894+ 1205911+ 1205912 (8)



119905119894= 119865 (119886

119894 119887119894 119888119894) (9)


1 1198872 119887



119898]T that is

120573 = 1198921 (B) = 119892

1(1198611 1198612 119861

119898) (10)


Output nodes

Hidden nodes

Input nodes

1

2

3

1

2

p

1

ai bi ci

w11

w12

w1p w21

w22w2p w31

w32 w3p

k11 k21 kp1

ti


(a)

1

1 2 q

i

Bj

wj1wj2

wjq

k11 k21kqn

1

n

k1i k2i

k2nk1n

kqikq1

b1 bi bn

j m

B1 Bm

w11 w12

w1qwm1

wm2 wmq




(b)



120572 = [1198861 1198862 119886

119899]T= 1198922 (

119860) = [

119860

119899

119860

119899

119860

119899

]

T

120574 = [1198881 1198882 119888

119899]T= 1198922 (

119862) = [

119862

119899

119862

119899

119862

119899

]

T

(11)


119863 = max119894

119865(

119860

119899

[1198921(1198611 1198612 119861

119898)]1198941

119862

119899

) + 1205911+ 1205912

(12)


1



119894and

(119886119894 119887119894 119888119894) that is (119886



119894 119887119894 and 119888









5 Model Evaluation



A BjB1 Bm C

g2 g2

t1 ti tn

12059111205912Max

An Cnb1 bi bn

PPDmodel

SDPSTmodel

(i)

SDPSTmodel

SDPSTmodel

(1) (n)






D



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24





119888


1198611


1198612



119879


1198790






1205911+ 1205912=

sum119903|119879minus1198790gt60

(119879 minus PST)1003816100381610038161003816119903 | 119879 minus 119879

0gt 60

1003816100381610038161003816

(13)







1and 119861




119863 =

119879 119879 minus 1198790gt 60

PST + 1205911+ 1205912

otherwise(14)




0 10 20 30 40 50 60 70 80 90 100005

01015

02025

03035

04045

05


Aver

age R

MSE





ELM 65 01358 04128 00865 00945LMBP 20 21722 00986 00904 01218SVM 31024 46375 12519 00853 01046







ELM 25 00057 00075 00987 01015LMBP 10 01167 00029 01077 01102SVM 4232 00764 00828 00972 01023

0 5 10 15 20 250

20

40

60

80

100

120

PST

at si

ngle

doo

r (s)








0 5 10 15 20 250

102030405060708090

100110


PST

at si

ngle

doo

r (s)

ab

c





119863 = 1205831+ 1205832119860 + 120583

3119861 (15)

119863 = ]1+ ]2

119860

119899

+ ]3

119861

119899

+ ]4(

119862

119899

)

3119861

119899

(16)


6 Conclusions





119895 that is

B = [11986111198612 119861

119898]T





120572 = [1198861 1198862 119886

119899]T


119894 that is

120573 = [1198871 1198872 119887

119899]T


119894 that is 120574 = [119888

1 1198882 119888

119899]T







Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Output nodes

Hidden nodes

Input nodes

1

2

3

1

2

p

1

ai bi ci

w11

w12

w1p w21

w22w2p w31

w32 w3p

k11 k21 kp1

ti


(a)

1

1 2 q

i

Bj

wj1wj2

wjq

k11 k21kqn

1

n

k1i k2i

k2nk1n

kqikq1

b1 bi bn

j m

B1 Bm

w11 w12

w1qwm1

wm2 wmq




(b)



120572 = [1198861 1198862 119886

119899]T= 1198922 (

119860) = [

119860

119899

119860

119899

119860

119899

]

T

120574 = [1198881 1198882 119888

119899]T= 1198922 (

119862) = [

119862

119899

119862

119899

119862

119899

]

T

(11)


119863 = max119894

119865(

119860

119899

[1198921(1198611 1198612 119861

119898)]1198941

119862

119899

) + 1205911+ 1205912

(12)


1



119894and

(119886119894 119887119894 119888119894) that is (119886



119894 119887119894 and 119888









5 Model Evaluation



A BjB1 Bm C

g2 g2

t1 ti tn

12059111205912Max

An Cnb1 bi bn

PPDmodel

SDPSTmodel

(i)

SDPSTmodel

SDPSTmodel

(1) (n)






D



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24





119888


1198611


1198612



119879


1198790






1205911+ 1205912=

sum119903|119879minus1198790gt60

(119879 minus PST)1003816100381610038161003816119903 | 119879 minus 119879

0gt 60

1003816100381610038161003816

(13)







1and 119861




119863 =

119879 119879 minus 1198790gt 60

PST + 1205911+ 1205912

otherwise(14)




0 10 20 30 40 50 60 70 80 90 100005

01015

02025

03035

04045

05


Aver

age R

MSE





ELM 65 01358 04128 00865 00945LMBP 20 21722 00986 00904 01218SVM 31024 46375 12519 00853 01046







ELM 25 00057 00075 00987 01015LMBP 10 01167 00029 01077 01102SVM 4232 00764 00828 00972 01023

0 5 10 15 20 250

20

40

60

80

100

120

PST

at si

ngle

doo

r (s)








0 5 10 15 20 250

102030405060708090

100110


PST

at si

ngle

doo

r (s)

ab

c





119863 = 1205831+ 1205832119860 + 120583

3119861 (15)

119863 = ]1+ ]2

119860

119899

+ ]3

119861

119899

+ ]4(

119862

119899

)

3119861

119899

(16)


6 Conclusions





119895 that is

B = [11986111198612 119861

119898]T





120572 = [1198861 1198862 119886

119899]T


119894 that is

120573 = [1198871 1198872 119887

119899]T


119894 that is 120574 = [119888

1 1198882 119888

119899]T







Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A BjB1 Bm C

g2 g2

t1 ti tn

12059111205912Max

An Cnb1 bi bn

PPDmodel

SDPSTmodel

(i)

SDPSTmodel

SDPSTmodel

(1) (n)






D



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24





119888


1198611


1198612



119879


1198790






1205911+ 1205912=

sum119903|119879minus1198790gt60

(119879 minus PST)1003816100381610038161003816119903 | 119879 minus 119879

0gt 60

1003816100381610038161003816

(13)







1and 119861




119863 =

119879 119879 minus 1198790gt 60

PST + 1205911+ 1205912

otherwise(14)




0 10 20 30 40 50 60 70 80 90 100005

01015

02025

03035

04045

05


Aver

age R

MSE





ELM 65 01358 04128 00865 00945LMBP 20 21722 00986 00904 01218SVM 31024 46375 12519 00853 01046







ELM 25 00057 00075 00987 01015LMBP 10 01167 00029 01077 01102SVM 4232 00764 00828 00972 01023

0 5 10 15 20 250

20

40

60

80

100

120

PST

at si

ngle

doo

r (s)








0 5 10 15 20 250

102030405060708090

100110


PST

at si

ngle

doo

r (s)

ab

c





119863 = 1205831+ 1205832119860 + 120583

3119861 (15)

119863 = ]1+ ]2

119860

119899

+ ]3

119861

119899

+ ]4(

119862

119899

)

3119861

119899

(16)


6 Conclusions





119895 that is

B = [11986111198612 119861

119898]T





120572 = [1198861 1198862 119886

119899]T


119894 that is

120573 = [1198871 1198872 119887

119899]T


119894 that is 120574 = [119888

1 1198882 119888

119899]T







Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 90 100005

01015

02025

03035

04045

05


Aver

age R

MSE





ELM 65 01358 04128 00865 00945LMBP 20 21722 00986 00904 01218SVM 31024 46375 12519 00853 01046







ELM 25 00057 00075 00987 01015LMBP 10 01167 00029 01077 01102SVM 4232 00764 00828 00972 01023

0 5 10 15 20 250

20

40

60

80

100

120

PST

at si

ngle

doo

r (s)








0 5 10 15 20 250

102030405060708090

100110


PST

at si

ngle

doo

r (s)

ab

c





119863 = 1205831+ 1205832119860 + 120583

3119861 (15)

119863 = ]1+ ]2

119860

119899

+ ]3

119861

119899

+ ]4(

119862

119899

)

3119861

119899

(16)


6 Conclusions





119895 that is

B = [11986111198612 119861

119898]T





120572 = [1198861 1198862 119886

119899]T


119894 that is

120573 = [1198871 1198872 119887

119899]T


119894 that is 120574 = [119888

1 1198882 119888

119899]T







Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 250

102030405060708090

100110


PST

at si

ngle

doo

r (s)

ab

c





119863 = 1205831+ 1205832119860 + 120583

3119861 (15)

119863 = ]1+ ]2

119860

119899

+ ]3

119861

119899

+ ]4(

119862

119899

)

3119861

119899

(16)


6 Conclusions





119895 that is

B = [11986111198612 119861

119898]T





120572 = [1198861 1198862 119886

119899]T


119894 that is

120573 = [1198871 1198872 119887

119899]T


119894 that is 120574 = [119888

1 1198882 119888

119899]T







Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article an elm-based approach for...

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