A “sums and shares” mixture model to study pedestrian flows
within a multimodal transport hub
P. De Naillya,b,∗ , E. Cômea , L. Oukhelloua , A. Saméa and J. Ferriereb
a Université Gustave Eiffel, Marne La Vallée, France

etienne.come@univ-eiffel.fr, allou.same@univ-eiffel.fr, latifa.oukhellou@univ-eiffel.frb Régie Autonome des Transports Parisiens, Paris, Francepaul.denailly@ratp.fr, jacques.ferriere@ratp.fr∗ Corresponding author
Extended abstract submitted for presentation at the 11th Triennial Symposium onTransportation Analysis conference (TRISTAN XI)
June 19-25, 2022, Mauritius Island

January 13, 2022
Keywords: Mixture model, Sums and shares models, Multimodal transport hub, Mobility data

1INTRODUCTION

This work focuses on a multimodal transportation hub that attracts an average of 140,000 people
on a typical week day (2019). Passenger flows tend to form transportation routes towards areasof interest. However, depending on the time of day, the period of the year, or local events,
these flows do not necessarily go to or transit through the same places. Several sensing systems
positioned at different count locations in the transport hub allow multivariate count data to becollected. Extracting information from these large sets of highly noisy count series is challenging
since these series share common dynamics in response to specific events, but may also have theirown dynamics due to more localized events.
Our goal in this paper is to model these mobility multivariate time series through a set ofmobility patterns, to obtain a synthetic vision which helps understanding how the transport hub
operates, or to use as a basis for prediction work. Because of the non-stationary nature of thedata, the modeling is subdivided into several time segments (Truong et al. (2020)). At the sametime, covariates add distinctive features to these patterns (Zhong et al. (2015)). Overdispersion
and correlations (Winkelmann (2008)) between series are characteristics frequently encounteredwith count data. To take these phenomena into account, we drew on the “Negative binomial sums
and Pólya shares” strategy presented in Jones & Marchand (2019). With this approach, periods
are considered homogeneous if, conditional on covariates and segments, both the totals (i.e.,“sums”) of people observed in the transport hub and their spatial distribution (i.e., “shares”)
among several locations are similar. Combining multiple mobility data sources is a valuableaddition to this kind of study, as it enriches the data available on observed flows in the multimodaltransport hub. To this end, this study relies on counts obtained from both ticketing logs collectedby automated fare collection systems and stereo camera sensors.

2SEGMENTATION MODEL STRUCTURE

The writing of the segmentation model first go through the elicitation of the regression model
before adapting it into a mixture model, able to detect regime changes in the time series.
TRISTAN XI Symposium

Original abstract submittal

P. De Nailly, E. Côme, L. Oukhellou, A. Samé and J. Ferriere

2.1

2

Negative binomial sums and Pólya shares regression model

We consider hereafterPthat Y is a L-vector of counts across L locations : Y = (Yl )l∈(1,..,L) . To
introduce it, let V = l Yl be the sum of these counts; the proposed strategy first models V asa Negative binomial distribution. Then conditionally on V = v, the count vector Y follows aDirichlet-multinomial (i.e. Pólya) distribution. This model can take into account overdispersion
as well as correlations, but these will always be either positive or negative (Jones & Marchand(2019)). The model may be written as follows:
V |x ∼ N B(exp(xT γ), r)
Y |x, v ∼ DM(v, (exp(xT ξl ))l∈1,...L )where N B and DM as the Negative Binomial and Dirichlet Multinomial distributions, respectively. We note x as a D × 1 vector of D exogenous factors. r is the scale parameter of
the N B distribution. γ is the vector (D × 1) of regression parameters of Negative binomial
regression, and ξl the vector (D × 1) of Dirichlet multinomial regression coefficients.

2.2

Negative binomial sums and Pólya shares mixture model

Let us now consider Yj,t as the L-vector of counts for time slot t of day j. We assume that,on day j, the counts at each time slot t can be associated with the dynamics of one segment
among S possible segments, with a certain probability. If we adapt the Negative binomial sumsand Pólya shares regression model to a mixture model framework, we end up with a generativemodel which includes a set of indicator variables, denoted by Zj (Zj ∈ {0, 1}s ) and encoding
the segment membership of the days, with s ∈ {1, ..., S}. The number of segments S is chosen apriori. The following generative model is assumed for the observed data:
Zj |j ∼ M(1, (πs (j; α))s∈1,...,S )Yj,t |Zj = s, xj,t , ζs ∼ D(Yj,t |xj,t , ζs )withP +4exp( Mm=1 αs,m am (j))πs (j; α) = PP
M +4h exp( m=1 αh,m am (j))
with αs,m a weight to be estimated andam (j) = j m−1 , m ∈ {1, ..., 4}
am+4 (j) = (j − κm )3 , m ∈ {1, ..., M }Here D is a Negative binomial sums and Pólya shares regression model with ζs = (rs , γs , ξs )the set of parameters controlling the conditional distributions within segment s. The variable
Zj follows a multinomial distribution of parameter π (i.e the vector of association weights).
The association weights follow a logistic transformation of cubic spline functions with M nodes
and therefore change according to day j. The idea behind this scheme is to help the model
detect regime changes that are otherwise difficult to detect from the data. Each cubic spline
is a piecewise cubic polynomial with knots at κm , m ∈ {1, ..., M } (M = 25 knots here). The
parameters of the model are estimated with the Expectation Maximization (EM) algorithm(Dempster et al. (1977)). The model was built in the R environment using the glm.nb functionin the MASS (Ripley et al. (2013)) package and the MGLM (Kim et al. (2018)) package.

3

EXPERIMENTATION ON REAL-WORLD DATA

The present analysis is based on hourly aggregated count data from sensors and ticketing systems
provided by the Paris public transport operator (i.e. Régie Autonome des Transports Parisiensor RATP) between April 2019 and September 2020. This study focus strictly on working days
TRISTAN XI Symposium

Original abstract submittal

P. De Nailly, E. Côme, L. Oukhellou, A. Samé and J. Ferriere

3

between 7 am and 1 am (the following day), to avoid introducing variables for non-working days,and thus simplify the model. Exogenous factors integrate the xj,t vector and include the 1-hourtime slots. We will focus in the following section on the results associated with the model with
S = 10 segments according to the BIC criterion.
The first result that can be obtained with this model is the period segmentation, as shownin Figure 1. We can observe a richness of segments induced by various context changes suchas maintenance work, strikes, or health measures against the Covid19 pandemic. This diversityof segments, with few returns, underlines the need for urban operators to adapt to a regularlychanging situation. This result highlights that, at time of writing, the regular use of the hub(i.e. “Normal 2019” and “Closure of an exit” in Figure 1) has not returned since the beginningof the Covid19 pandemic.

Figure 1 – Bar plot representation of time segmentation. Each day is associated with the probabilities of belonging to each segment. All 10 segments each have their own colors and labels.
From a spatial point of view, one can study the characteristics of people flow distribution in
the transport hub within each segment. Indeed, each segment s is associated with a set of typical
distributions between the L locations, as displayed in Figure 2. The location codename is labeled“O” for outgoing flows and “I” for incoming flows. Depending on the segments encountered, there
is an overuse (in red) or an underuse (in blue) at certain locations compared to “Normal 2019”.
For example, in segment “Strike + RER A Works” there is an overuse of accesses to and fromthe metro and an underuse of accesses to and from the RER, highlighting an expected transferof users to the metro line when the RER traffic is disrupted.

4CONCLUSION

This paper sets up a model to segment multidimensional mobility time series, whose dynamics
evolve according to characterized periods. This strategy was based on the “sums and shares”
models found in literature. We chose to apply a Negative binomial and Pólya shares mixture
model to analyze mobility data collected at a major transport hub. The regression coefficients ofthese models are dependent on the segments to which they belong. This work reveals how various
restrictions to combat the Covid19 pandemic significantly affected pedestrian flow dynamics inthe transportation hub. These restrictions were not the only events that impacted hub patronage
over the long term. We found that given situations may lead to specific over- and under-use ofparticular locations. This type of study can be applied to any situation where a large set of count
TRISTAN XI Symposium

Original abstract submittal

P. De Nailly, E. Côme, L. Oukhellou, A. Samé and J. Ferriere

1x

0.5x

0.2x
Tram WorksWorks
Tram

022:0

018:0

0
14:0

00.1x

10:0

05x2x

P7 I
P6 O
P6 I
P5 O
P5 I
P2 O
P2 I
P12 O
P12 I
P11 O
P11 I
P1 O
P1 I
T
EO
EIM
RORI
EM
ME
22:0

018:0

010x
lockdown
1st1stlockdown

P7 I
P6 O
P6 I
P5 O
P5 I
P2 O
P2 I
P12 O
P12 I
P11 O
P11 I
P1 O
P1 I
T
EO
EI
M
RO
RI
EM
ME

Normal 2020
Normal2020

10:00

Decrease P7P7
Decrease

P7 I
P6 O
P6 I
P5 O
P5 I
P2 O
P2 I
P12 O
P12 I
P11 O
P11 I
P1 O
P1 I
TEO
EI
M
RO
RI
EMMEStrike
RER AA
Works
Strike+ +RER
Works

P7 I
P6 O
P6 I
P5 O
P5 I
P2 O
P2 I
P12 O
P12 I
P11 O
P11 I
P1 O
P1 I
T
EO
EIM
RO
RIEM
ME
22:00
18:0

00
10:0

Summerholidays
holidays 20192019
Summer0

P7 I
P6 O
P6 I
P5 O
P5 I
P2 O
P2 I
P12 O
P12 I
P11 O
P11 I
P1 O
P1 I
T
EO
EI
M
RO
RI
EM
ME

Relaxed1st
1st lockdown
Relaxedlockdown

14:0

one−hour timestamp

Closure ofof anexitexit
Closure
an

P7 I
P6 O
P6 I
P5 O
P5 I
P2 O
P2 I
P12 O
P12 I
P11 O
P11 I
P1 O
P1 I
TEOEIM
RO
RIEM
ME

P7 I
P6 O
P6 I
P5 O
P5 I
P2 O
P2 I
P12 O
P12 I
P11 O
P11 I
P1 O
P1 I
T
EO
EI
M
RO
RIEM
ME

P7 I
P6 O
P6 I
P5 O
P5 I
P2 O
P2 I
P12 O
P12 I
P11 O
P11 I
P1 O
P1 I
T
EO
EI
M
RO
RI
EM
ME

14:0

X regional trains
X regionaltrains

P7 I
P6 O
P6 I
P5 O
P5 I
P2 O
P2 I
P12 O
P12 I
P11 O
P11 I
P1 O
P1 I
T
EO
EIMRORIEM
ME

4Figure 2 – Typical spatial distributions between the L (=21) locations found in each segment s.
Each cell of the heat maps corresponds to a time slot at a given location. For a given cell, thecolor reflects the log ratio between the proportion of flows in the current segment and that in theNormal 2019 reference segment. Colors thus reflect the differences between the proportions offlows in each segment and those in the reference segment, with regards to spatial distribution.
data is available and there is an interest in synthesizing information from typical spatio-temporalprofiles from distinct periods. In the field of mobility still, these models can be applied to thestudy of a public transport network or a whole city, in which the characterization of humantravel patterns is of great importance. This study leads the way to more advanced clusteringor prediction modeling work. In particular, it allows periods with variable flow dynamics to bedistinguished, which can be helpful when predicting patronage in specific contexts.

References
Dempster, Arthur P, Laird, Nan M, & Rubin, Donald B. 1977. Maximum likelihood from incompletedata via the EM algorithm. Journal of the Royal Statistical Society: Series B (Methodological), 39(1),1–22.
Jones, MC, & Marchand, Éric. 2019. Multivariate discrete distributions via sums and shares. Journal ofMultivariate Analysis, 171, 83–93.
Kim, Juhyun, Zhang, Yiwen, Day, Joshua, & Zhou, Hua. 2018. MGLM: an R package for multivariate
categorical data analysis. The R journal, 10(1), 73.
Ripley, Brian, Venables, Bill, Bates, Douglas M, Hornik, Kurt, Gebhardt, Albrecht, Firth, David, &Ripley, Maintainer Brian. 2013. Package ‘mass’. Cran r, 538, 113–120.
Truong, Charles, Oudre, Laurent, & Vayatis, Nicolas. 2020. Selective review of offline change point
detection methods. Signal Processing, 167, 107299.
Winkelmann, Rainer. 2008. Econometric analysis of count data. Springer Science & Business Media.
Zhong, Chen, Manley, Ed, Arisona, Stefan Mueller, Batty, Michael, & Schmitt, Gerhard. 2015. Measuring
variability of mobility patterns from multiday smart-card data. Journal of Computational Science, 9,125–130.

TRISTAN XI Symposium

Original abstract submittal