Modeling of Passenger Demand using Mixture of Poisson Components

Matej Petrou

1,2

, Ev

zenie Suzdaleva

and Ivan Nagy

1,2

Department of Signal Processing, The Czech Academy of Sciences, Institute of Information Theory and Automation,

Pod vod

arenskou v

ı 4, 18208 Prague, Czech Republic

Faculty of Transportation Sciences, Czech Technical University, Na Florenci 25, 11000 Prague, Czech Republic

Keywords:

Mixture Estimation, Poisson Components, Passenger Demand.

Abstract:

The paper deals with the problem of modeling the passenger demand in the tram transportation network. The

passenger demand on the individual tram stops is naturally inﬂuenced by the number of boarding and disem-

barking passengers, whose measuring is expensive and therefore they should be modeled and predicted. A

mixture of Poisson components with the dynamic pointer estimated by recursive Bayesian estimation algo-

rithms is used to describe the mentioned variables, while their prediction is solved with the help of the Poisson

regression. The main contributions of the presented approach are: (i) the model of the number of boarding

and disembarking passengers; (ii) the real-time data incorporation into the model; (iii) the recursive estimation

algorithm with the normal approximation of the proximity function. The results of experiments with real data

and the comparison with theoretical counterparts are demonstrated.

1 INTRODUCTION

The paper deals with the problem of modeling the

passenger demand in the tram network, which is an

important task in the public transportation. In order

to provide a high-quality and attractive public trans-

port service, it is necessary to minimize the number of

overcrowded vehicles. Moreover, in order to provide

economically effective service and to reduce human

resources needed, it is also necessary to minimize the

number of insufﬁciently occupied vehicles.

In this area, many papers deal with the passenger

demand models in metro systems (Sun et al., 2015;

Roos et al., 2016; Sun et al., 2015(a)). They use data

from turnstiles both at the entrance and the exit at

stations, which could be paired for passengers using

smart cards. In this case, continuous measurements

of the passenger ﬂow are available. However, for ex-

ample, in central Europe, most metro networks are so

called ”open networks” without turnstiles and there-

fore without continuous measuring of the passenger

demand, which means it should be modeled and pre-

dicted. Aside from metro systems, methods of the

demand modeling have been also investigated for bus

networks (Samaras, 2015; Bai et al., 2017; Lijuan and

Chen, 2017; Ma et al., 2014), where continuous mea-

suring of the passenger demand is an expensive task

as well as in tram networks. For tram transportation,

the thesis (Pu

sman, 2013) proposed a method of pro-

portional transit division (PTD) using deterministic

models for the passenger demand calculation.

Generally, the approaches in the discussed ﬁeld

are based on: (i) regression models, (ii) artiﬁcial

neural networks or (iii) hybrid models combining

them. For example, (Milenkovi

c et al., 2016) pro-

posed seasonal autoregressive integrated moving av-

erage model to be used on Serbian railways. (Zhou

et al., 2013) introduced three different models of the

passenger demand in bus networks. The ﬁrst one was

a time varying Poisson model. Secondly, a weighted

time varying Poisson model was proposed to cope

with irregularities in passenger demand. Finally, an

autoregressive integrated moving average model was

also proposed in this paper. All three models were

applied for data from buses in Yantai, China with the

ﬁnal model achieving the most accurate results.

In the area of artiﬁcial neural networks, the fol-

lowing papers were found. (Chen and Wei, 2011)

used back-propagation neural networks for the pas-

senger demand description in the Taipei metro sys-

tem. (Tsai et al., 2009) dealt with two neural net-

work models in the Taiwan railway network. The ﬁrst

one was the multiple temporal units neural network

and the second was parallel ensemble neural network.

The latter provided more accurate results. (Bai et al.,

2017) introduced deep belief networks for the passen-

Petrouš, M., Suzdaleva, E. and Nagy, I.

Modeling of Passenger Demand using Mixture of Poisson Components.

DOI: 10.5220/0007831306170624

In Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2019), pages 617-624

ISBN: 978-989-758-380-3

617

ger ﬂow prediction on a bus line.

A series of studies combine more models to adapt

them to their speciﬁc tasks. (Lijuan and Chen, 2017)

combined stacked auto-encoder (SAE) and deep neu-

ral network (DNN) into SAE-DNN model for the

passenger demand prediction in the Xiamen bus sys-

tem. (Sun et al., 2015) proposed a combination of

wavelet transformation and support vector machine

models in the Beijing subway system. (Jiang et al.,

2014) focused on the empirical mode decomposition

(EEMD) and grey support vector machine (GSVM)

hybrid model of the passenger demand in the high-

speed railway network in China.

Besides methods mentioned above, other ap-

proaches have also been used, e.g., Bayesian net-

works (Roos et al., 2016; Sun et al., 2015(a)), stochas-

tic hybrid automat and Petri nets (Haar and Theissing,

2016) and others.

Despite the signiﬁcant number of studies, current

methods possess a series of disadvantages, such as,

e.g., narrow speciﬁcation, ﬂuctuations in predictions,

etc. Analyzing the above state of the art, it can be

stated that the problem of demand modeling still calls

for novel reliable solutions.

The presented approach is based on the deﬁnition

of the passenger demand at a tram stop as the number

of passengers currently using a tram vehicle at a time

moment. Since the vehicle occupancy can change

only at stops, it is determined by adding the number

of boarding passengers and subtracting the number of

disembarking passengers at each stop, i.e.,

demand = current demand − disembarking +boarding.

However, measuring the number of boarding and

disembarking passengers in the tram network is still

a complicated task, and such measurements are not

available without speciﬁc surveys. Thus, these vari-

ables should be modeled and predicted as well. They

will create the basis for the prediction of the demand.

This paper proposes a novel approach to the mod-

eling of the passenger demand in the tram transporta-

tion network using a mixture of Poisson components

with the dynamic pointer estimated using recursive

Bayesian algorithms primarily based on (K

arn

y et al.,

1998; Peterka, 1981; K

arn

y et al., 2006; Nagy and

Suzdaleva, 2017). The approach is represented in two

parts, where the ﬁrst one deals with the available data

set used for the recursive estimation of the model, and

the second one uses the estimated model for the pre-

diction. The approach is demonstrated for the number

of boarding passengers.

The layout of the paper is organized as follows.

Section 2 introduces the models used and speciﬁes the

problem. Section 3 proposes a recursive algorithm

of the Bayesian estimation, which represents the ﬁrst

part of the proposed approach. Section 4 is devoted to

the prediction part of the presented solution. Results

of experiments with real data and the discussion are

demonstrated in Section 5. Conclusions can be found

in Section 6.

2 MODELS

Let’s observe a system, which represents a tram line,

consisting of n

stops. The individual tram trips de-

part from each stop at non regularly discretized time

periods. In this work, the time of trip departures will

be used as discrete instants of time corresponding to

realizations of random variables and will be denoted

by the index t. At each stop s ∈ {1, 2, . . ., n

}, the sys-

tem generates the number of passengers boarding the

tram trip t, which is denoted by y

s;t

and similarly, the

number of disembarking passengers y

s;t

Having, e.g., three following stops s

, s

and s

the passenger demand denoted by D

23;t

between stops

and s

for a trip t is deﬁned as:

23;t

= D

12;t

− y

2;t

+ y

2;t

. (1)

Let’s assume that for each stop s the variables y

s;t

and

s;t

can be measured for trips t = {1, 2,. . . , T } and are

no longer available for t > T .

In order to be able to use equation (1) recursively

for a line consisting of n

stops, it is necessary to de-

scribe ﬁrstly the number of boarding (similarly dis-

embarking) passengers at a single stop. Here, for the

sake of simplicity the approach will be shown for the

number of boarding passengers y

s;t

. In the case of dis-

embarking passengers, the approach will be the same.

Thus, for the better transparency of the approach, the

superscripts b and d will be omitted.

The available data set of the variables y

s;t

can be

used as the prior knowledge for the preliminary analy-

sis for a choice of the model probability density func-

tion (pdf). Based on the visual analysis of histograms

of the number of boarding passengers at stops in the

considered tram lines, the Poisson distribution as the

discrete distribution with high ﬁnite number of possi-

ble values suitable for non-negative data was chosen

for the description of the data. The example of the

histogram at a selected stop is shown in Figure 1.

In addition, the ﬁgure shows that the variables

are of the multimodal nature. This can be explained

by the behavior of passengers changing probably ac-

cording to a day period, e.g., morning peak time,

lunchtime, afternoon peak time, etc. It means that for

the description of the number of boarding passengers,

a mixture of Poisson pdfs can be a suitable tool.

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

618

Figure 1: Histogram of the number of boarding passengers.

Generally, the mixture model consists of n

com-

ponents and the pointer model (K

arn

y et al., 2006;

arn

y et al., 1998), where the components describe

modes of the observed system behavior and the

pointer variable indicates a component, which is ac-

tive at time t. The active component should be under-

stood as that generating data at the moment.

Within the considered context, each Poisson com-

ponent has the form of the following pdf

f (y

s;t

|λ

, c

= i) = exp

{

−(λ

)

}

(λ

)

s;t

, (2)

where λ

are the parameters for each stop s, and λ

(λ

)

for c

= i, the denotation c

stands for the pointer

variable, described by the categorical distribution, and

i ∈ {1, 2, . . . , n

}. The denotation c

= i means that

at the time instant t, the pointer c

indicates the i-th

component, which is active.

In this paper, switching the active components is

described by the dynamic pointer model (Nagy et al.,

2011) based on (K

arn

y et al., 1998; K

arn

y et al.,

2006) in the form of the following probability func-

tion (also denoted by pdf)

f (c

= i|c

t−1

= j, α), i, j ∈ {1, 2, . . . , n

}, (3)

which is represented by the transition table

= 1 c

= 2 ··· c

= n

t−1

= 1 α

1|1

2|1

··· α

··· ··· ··· ··· ···

t−1

= n

1|n

··· α

where the unknown parameter α is the (n

× n

dimensional matrix, and its entries α

i| j

are non-

negative probabilities of the pointer c

= i (expressing

that the i-th component is active at time t) under con-

dition that the previous pointer c

t−1

= j. The pointer

is supposed to be common for all the stops, which is

explained by connecting the stops into lines. From

this point of view, if a component is active at a stop

(e.g., the morning peak hour happens), it is active as

well at the neighboring stop, etc. For this reason, the

subscript s is omitted for the denotations c

and α.

2.1 Problem Speciﬁcation

Applying the mixture model (2)–(3), the task of mod-

eling the number of boarding passengers y

s;t

(contex-

tually identical in the case of disembarking), which

should be used for the passenger demand prediction

(1) for the time t > T, is speciﬁed as follows:

1) estimate the component parameters (λ

)

which means that the parameter is changing for each

stop within a component,

2) estimate the pointer parameter α

3) and estimate the pointer value c

to be used in

the subsequent prediction of the data.

3 MIXTURE ESTIMATION

To derive the estimation algorithm for (2)–(3), it is

advantageous to recall the estimation of the individual

Poisson pdf (i.e., omitting the subscript i for the sake

of simplicity). The maximum likelihood estimation

of the Poisson distribution, e.g., (Yang and Berdine,

2015) gives the estimate of λ

as the mean value of

s;t

for each stop s, i.e., the likelihood function has

the following form:

(λ) = (exp

{

−λ

}

)

s;T

∏

t=1

s;t

, (4)

where the statistics are

s;t

= S

s;t−1

+ y

s;t

, (5)

s;t

= κ

s;t−1

+ 1 (6)

for each stop s and for t = {1, 2, . . . , T }. Using the

derivation of the likelihood function, the point esti-

mate of λ

of the individual stop s is given by

s;T

. (7)

According to the above relations and recur-

sive Bayesian estimation theory primarily based on

arn

y et al., 1998; Peterka, 1981; K

arn

y et al., 2006;

Nagy and Suzdaleva, 2017), the mixture estimation

algorithm can be derived as follows. Using the joint

pdf for all unknown variables as well as the chain rule

and the Bayes rule (Peterka, 1981), it is obtained

f (λ

, c

= i, c

t−1

= j, α|y

(t))

| {z }

joint posterior pd f

∝ f (y

s;t

, λ

, c

= i, c

t−1

= j, α|y

(t − 1))

| {z }

via chain rule and Bayes rule

Modeling of Passenger Demand using Mixture of Poisson Components

619

= f (y

s;t

|λ

, c

= i)

| {z }

(2)

f (λ

(t − 1))

| {z }

prior pd f o f λ

f (c

= i|α, c

t−1

= j)

| {z }

(3)

× f (α|y

(t − 1))

| {z }

prior pd f o f α

f (c

t−1

= j|y

(t − 1))

| {z }

prior pointer pd f

, (8)

where the denotation y

(t) means the collection

of data {y

s;0

, y

s;1

, . . . , y

s;t

} and y

s;0

corresponds to the

prior knowledge. The parameters λ

and α are as-

sumed to be mutually independent, as well as y

s;t

and

α, and c

and λ

. Generally, the relation (8) should be

marginalized over λ

, α and c

. However, the parame-

ter of the Poisson component (2) cannot be estimated

recursively (Yang and Berdine, 2015), which means

its likelihood function should be placed instead of the

component pdf. This is a complicated task from the

computational and derivation reasons. That’s why the

proximity function (Nagy and Suzdaleva, 2017) giv-

ing the closeness of the measured data element to the

i-th component is proposed to be used here as the nor-

mal approximation of the Poisson pdf, optimal in the

sense of the Kullback-Leibler divergence, see, e.g.,

arn

y et al., 2006). In this case, the expectation

of the approximated Poisson pdf is substituted in the

normal pdf instead of the original one, see for details

(Nagy et al., 2016). The estimation of the parameter

α is solved using the prior Dirichlet pdf according to

arn

y et al., 2006).

Summarizing the derivations, the following steps

of the algorithm should be performed.

3.1 Algorithm

Initialization Part (for t = 1)

• Set the number of stops n

and of components n

• ∀i ∈

{

1, 2, . . . , n

}

and ∀s ∈

{

1, 2 . . . , n

}

1. Set the initial statistics of the components

s;t−1

)

and (κ

s;t−1

)

and of the pointer ν

t−1

2. Compute the initial point estimates of the pa-

rameter (λ

)

according to (7).

• Compute the point estimates of the parameter α

arn

y et al., 2006).

• Set the initial weighting vector w

t−1

Online Part (for t = 2, . . . , T )

• ∀s ∈

{

1, 2 . . . , n

}

1. Load the data item y

s;t

2. ∀i ∈

{

1, 2, . . . , n

}

obtain the proximities de-

noted by m

by the substitution of the previous

point estimate of (λ

)

as the mean and the vari-

ance along with the current data item y

s;t

into

the normal approximating pdf.

3. Construct the weight matrix W

, which contains

the pdfs W

j,i;t

(joint for c

and c

t−1

) using the

previous point estimate of the parameter α and

the obtained proximities, i.e.,

∝



t−1



. ∗

t−1

(9)

and normalize it (Nagy and Suzdaleva, 2017).

4. Obtain the weighting vector w

with the up-

dated entries w

i;t

f (c

= i|d (t)) = w

i;t

∝

∑

j=1

j,i;t

, (10)

which gives probabilities of the component ac-

tivity at time t (Nagy and Suzdaleva, 2017).

5. Update the statistics of all of the components:

s;t

)

= (S

s;t−1

)

+ w

i;t

s;t

, (11)

(κ

s;t

)

= (κ

s;t−1

)

+ w

i;t

. (12)

6. Update the pointer statistic (Nagy et al., 2011):

i| j;t

= ν

i| j;t−1

j,i;t

, i, j ∈

{

1, 2, . . . , n

}

(13)

7. Recompute the point estimates of (λ

)

and α.

8. Declare the active component according to the

maximum entry in the weighting vector w

which gives the point estimate of the pointer.

9. Use the point estimates of (λ

)

and α along

with w

for the ﬁrst step of the online estima-

tion.

More details can be found in (K

arn

y et al., 2006;

Nagy and Suzdaleva, 2017).

This part of the model of the number of board-

ing passengers serves for learning the model for the

case, when measurements of y

s;t

for t ≤ T are avail-

able, which can be e.g., especially undertaken by the

transportation organization after applying some mod-

iﬁcations in the tram network.

4 PREDICTION

The results of the above algorithm are the point esti-

mates of the parameters (λ

)

for each stop and each

component. After the time t > T , the boarding num-

ber y

s;t

is no longer measured and thus should be pre-

dicted. Here, the second part of the stop description

is introduced in the following way, which serves for

the prediction of the boarding number y

s;t

using the

obtained estimates.

Naturally, apart from the boarding number y

s;t

(as

well as disembarking), each stop can be described by

its surroundings, for example, location (e.g., GNSS

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

620

coordinates), demographics around the stop (inhab-

itants, job opportunities etc.), area characteristics

(buildings or important places nearby, etc.), transfer

options, number of available trips from the stop, etc.,

which are measurable online. The surroundings sub-

stantially inﬂuence the behavior of passengers at each

stop.

Let surroundings of each stop s be denoted by x

s;t

and comprise the vector [x

s;t;1

, x

s;t;2

, ..., x

s;t;n

], where

t = {1, 2, . . .} and n

is the number of measured sur-

roundings for each stop. In this paper, the following

entries of the vector x

s;t

are available as the surround-

ings of the stop s: x

s;t;1

– scheduled departure time

of the trip t from the stop s; x

s;t;2

– delay of the trip

t at the stop s; x

s;t;3

– number of trips on other lines

which arrived earlier although they were supposed to

arrive later; x

s;t;4

– number of available trips per hour

on all lines at the stop; x

s;t;5

– scheduled time differ-

ence between the trip and the previous trip; x

s;t;6

–

real time difference between the trip and the previous

trip; x

s;t;7

– transfer for a metro line availability; x

s;t;8

– transfer for a bus line availability; x

s;t;9

– number of

inhabitants living up to 500 m from the stop. Other

variables can be also used.

Let’s assume that the surroundings x

s;t

can inﬂu-

ence the y

s;t

in the following way:

s;t

= b

s;t

+ e

s;t

, (14)

where b are regression coefﬁcients and e

s;t

is a noise.

However, with the Poisson noise distribution, the

Poisson regression should be considered instead, i.e.,

ln(y

s;t

) = b

s;t

+ e

s;t

. (15)

As the variables y

s;t

are no longer measured, the

estimate of the parameter (λ

)

can serve instead of

it (i.e., as the estimate of y

s;t

), which means that the

regression (15) takes the following form:

ln((

)

) = b

s;t

, (16)

where (

)

denotes the point estimate from the active

component for each stop, i.e., for i = c

for the trip t.

Having the point estimates (

)

and surroundings x

s;t

for stops s = {1, 2, . . . , n

} for individual trips, the re-

gression coefﬁcients of (16) can be estimated straight-

forward with the help of the least square method.

For the prediction of the number of boarding pas-

sengers y

s;t

, the regression for each stop s and chosen

trips t is used in the form

ˆy

s;t

= exp{x

s;t

b}. (17)

which then comprises a line of n

stops.

Here, the approach has been presented for the

number of boarding passengers only. For the number

of disembarking passengers, the idea is quite identi-

cal. After predicting both the variables, the prediction

of the passenger demand via relation (1) should be

solved for corresponding stops.

5 EXPERIMENTS

This section provides the results of the experimental

validation of the approach using real data. The vali-

dation was performed according to the following cri-

teria:

1) The predicted values are compared with real

values of the number of boarding passengers.

2) The evolution of component weights, which ex-

press the activity of components, is observed during

the online estimation. The rare activity of components

or its absence indicates the incorrect number of com-

ponents, which is probably too high.

3) The evolution of the point estimates of compo-

nent parameters is monitored during the estimation.

Finding the stabilized values of the point estimates

means the successful estimation.

A series of experiments has been conducted. Here,

typical results are presented.

5.1 Data Collection

For the experiments, the line consisting of four tram

stops was modeled. The real measurements at the

stops were used. The data set was collected manually,

because no automatic passenger data collection sys-

tem exists on trams. A part of data was measured in

all tram trips between 6:00 and 23:00 during 3 week-

days (Tuesday to Thursday) with each trip being mea-

sured exactly once. In addition, the measuring was

taken also at stops during weekdays to cover all possi-

ble modes of the passenger behavior: (i) morning rush

hour between 7:00 and 8:00, (ii) noon between 11:30

and 12:30 and (iii) afternoon rush hour between 16:00

and 17:00. At each stop from the data set, the data

was collected three times for both rush hour times and

once for the time at noon.

Algorithm 3.1 along with the prediction (17) were

applied to the number of boarding passengers y

s;t

and

the stop surroundings x

s;t

from this data set. The num-

ber of components n

has been set equal to 3 based on

the analysis of the evolution of weights. The overall

number of trips used for the estimation was 288 for

each of the four stops.

5.2 Results

For the comparison, the PTD method (Pu

sman, 2013)

was chosen. Figure 2 compares the predicted, PTD

and real values of the passenger boarding for the

tested line consisting of four stops. Each plot rep-

resents one of the selected trips of the line during the

day. Both the predicted and the PTD values are in

Modeling of Passenger Demand using Mixture of Poisson Components

621

the good correspondence with the real ones. The pre-

diction error between the proposed prediction and the

real values is 0.0474633, while between the PTD and

real values it is a bit higher – 0.0499839. It indicates

a slight improvement of the existing method. How-

ever, the model still could be improved by choosing

the more informative data, since in the bottom plot,

the predicted values of both methods lie below the real

ones. The differences can be explained by using the

expectation of the Poisson distribution.

Various surroundings were used for the prediction.

Among them, the variable x

s;t;6

was proven to be the

most signiﬁcantly affecting the prediction quality.

The weight evolution for the three components

and all trips can be found in Figure 3. All of the com-

ponents are regularly active, which conﬁrms that the

model is well established and the number of compo-

nents is set correctly. As it is shown in the ﬁgure, in

most cases the decision about the activity of the com-

ponents is unambiguous.

Figure 4 demonstrates the evolution of the compo-

nent parameter estimates (λ

)

, (λ

)

and (λ

)

over

all of the trips. All the parameters are looking for

their stabilized values in the beginning of the estima-

tion and then remain in the ﬁnal position.

5.3 Discussion

The main aim of the experiments was to validate a

model of the number of boarding passengers (contex-

tually identical to the disembarking case), which is

then assumed to be involved in the model of the pas-

senger demand.

As it was demonstrated in Section 5, the aim was

successfully accomplished. A mixture of three reg-

ularly active components was currently identiﬁed as

the most suitable solution. The components are as-

sumed to correspond to the morning and afternoon

rush hours along with the lunch-time calm trafﬁc.

Currently, the algorithm was tested on the data from

weekdays. However, applying the data from week-

ends as well can bring another components describ-

ing the behavior of passengers at weekends. Another

possibility can be a hierarchical mixture taking into

account a day period in dependence of weekdays or

weekends. Some uncontrollable factors, such as e.g.,

the seasonality, can be also included into the mixture

as the pointer variable , i.e., in this case seasonal com-

ponents should be considered, or seasonable effects

can be covered by the uncertainty.

The main contributions of the presented approach

are as follows: (i) the two-part model of the number

of boarding and disembarking passengers is proposed

to be used for the passenger demand modeling; (ii) the

model can be applied to a small amount of data avail-

able; (iii) the real-time measurements at stops can be

incorporated online into the model; (iv) the recursive

mixture estimation algorithm with the non-trivial ap-

proximation of the proximity function is proposed for

the case of Poisson components.

The potential application of the presented solution

can be expected in the ﬁeld of the transportation net-

work planning and service management. The predic-

tion is solved using the model, which has been esti-

mated with the help of a small available data set. It

means that the data necessary for the prediction af-

ter applying some modiﬁcations in the tram lines and

stops could be shortly measured from time to time as

required by the transportation organization and then

used for further prediction with the stop surroundings

available online.

The limitation of the approach is the necessity to

get the new data sets reﬂecting boarding and disem-

barking of passengers after each line/stop modiﬁca-

tions in the tram network. Otherwise, the changes will

not be covered by the subsequent prediction.

6 CONCLUSIONS

The paper describes a data-based approach to the pas-

senger demand modeling for the tram transportation.

The model has been divided into a model of boarding

passengers and contextually identical model of dis-

embarking passengers, which serve for recursive cal-

culating the passenger demand at stops. The solution

was represented in two phases, including the model

estimation part and the prediction part, where the ﬁrst

of them is solved recursively and the second one is the

regression estimated using the least square method.

The mixture of Poisson components with the dy-

namic pointer model estimated with the help of the

recursive Bayesian algorithm with the non-trivial ap-

proximation of the proximity function was proposed.

A series of validation experiments with real data sets

was conducted for testing the proposed approach. The

prediction of passenger boarding has provided ade-

quate results for most tram trips, however, an im-

provement in prediction still could be achieved.

The open problems, which still remain in the con-

sidered context include the following:

• the optimization of the number of vehicles and

frequency of trips with the consideration to peak

and low seasons on a daily basis;

• an economic analysis of the proposed model com-

pared to current situation;

• the extensive testing on larger data sets.

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Figure 2: The comparison of predictions, PTD and real values of boarding passengers at four stops and selected trips.

Figure 3: The weight evolution for all trips during the day.

Most current models consider ideal trafﬁc condi-

tions (no delays), e.g. (Roos et al., 2016) or do not in-

clude information about lines at all, e.g. (Lijuan and

Chen, 2017), etc. However, in tram networks, traf-

ﬁc conditions signiﬁcantly affect passenger demand

for a speciﬁc trip. For example, if a tram is delayed,

more passengers use it, because apart from its base

passengers it carries passengers who were supposed

to board the next tram. On the other hand, when arriv-

ing shortly after the previous tram, less passengers use

Figure 4: The parameter estimate evolution at a chosen stop.

it. In complex networks with more lines sharing the

same tracks, there are other variables which affect the

passenger demand on a speciﬁc trip (e.g., line rout-

ing). Therefore, incorporating trafﬁc variables mea-

sured in real time can be vital in improving the accu-

racy of the passenger demand model.

ACKNOWLEDGEMENTS

This work has been supported by the project

SILENSE, project number ECSEL 737487 and

MSMT 8A17006.

Modeling of Passenger Demand using Mixture of Poisson Components

623

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