# How Has the Paris Rail Public-Transportation Network Recovered After the COVID-19 Pandemic? Applying a Mixture of Regressions Model

Hugues Moreau, Étienne Côme, Allou Same, Latifa Oukhellou

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Hugues Moreau, Étienne Côme, Allou Same, Latifa Oukhellou. How Has the Paris Rail Public-Transportation Network Recovered After the COVID-19 Pandemic? Applying a Mixture of Regressions Model. Transportation Research Record, 2025, 2679 (3), pp.330-345. 10.1177/03611981241277817. hal-04777136

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# How Has the Paris Rail Public Transport Network Recovered After the Covid-19 Pandemic? Applying a Mixture of Regressions Model.

Hugues Moreau<fnref n="1" />, Étienne Côme<fnref n="1" />, Allou Samé<fnref n="1" />, and Latifa Oukhellou<fnref n="1" />

## Abstract

Due to a combination of regulations, fear of contagion, and changes in travelers' habits, the Covid-19 pandemic impacted the mobility of public transport ridership worldwide. To understand the longer-term effects of the pandemic on public transport ridership, we focus on the case of Paris, France, thanks to an open five-year record of entries from more than 500 stations. To deal with the large volume of data, we use a statistical model that performs clustering and segmentation simultaneously while incorporating many exogenous variables, such as the day of the week or lockdowns, to account for their effect on the number of entries. We carry out an in-depth analysis of the results regarding the segments and the clusters. Examining and comparing the regression coefficients across clusters and consecutive segments allows us to draw per-cluster and per-segment conclusions. We show that the number of weekday trips decreased in most clusters and that the reduction in weekly variations is proportional to the share of weekday trips in the volume of entries before the pandemic. In addition, we characterize the changes in the weekly profile: Thursday was replaced by Tuesday as the day with the highest ridership. Due to teleworking, Friday became the least crowded weekday in clusters with strong differences between weekdays and weekends, while the lowest ridership weekday remains Monday in the other clusters. **Keywords:** Public transport, Covid-19 pandemic, Ridership, Clustering, Segmentation, Mixture of Regressions, Expectation-Maximization

## Introduction

Public transport is a sustainable set of travel systems widely used in most cities. Recently, the Covid-19 pandemic drastically affected the mobility of the entire world population, particularly in public transport. The restrictions themselves, the fear of contagion, and the change in the habits of travelers (such as teleworking) pushed public transport users to either switch to other transport modes or find ways to travel less. Even after the cities gradually lifted restrictions one after another, there appears to be a noticeable long-term decrease in public transport use (1). This paper aims to understand these long-term trends using a five-year dataset of public rail transport entries in Paris, France.

In France, the first significant measure against the pandemic was closing schools and universities on March 12, 2020. A few days later (on March 17, 2020), the first lockdown was decreed for nearly two months, until May 11. After this date, many measures remained in place for durations ranging from several weeks, such as the closing of non-essential shops, to up to two years for the mask mandate in public transport. A second lockdown was implemented at the end of October 2020 following a second wave of

contaminations. Most restrictions were lifted on December 15, even though some remained in place after this date, such as the early closing of bars. One last lockdown occurred between April 3, 2021, and May 3, 2021. As the vaccine became available for a larger proportion of the population, the government resorted to differentiated measures to prevent non-vaccinated people from further propagating the virus.

Most health measures created to fight the pandemic have been lifted. However, the public transport ridership in Paris has not recovered to its pre-pandemic levels. Teleworking and a modal shift from public transport to private cars or bikes (1) caused a noticeable and lasting impact on public transport ridership. We seek to highlight these changes using an open dataset of entries on the rail public transport of the Paris city and region (2).

Numerous studies have measured the impact of travel restrictions or the longer-term changes during the reopening phase. Due to the difference in data availability, auxiliary

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<footnote n="1">¹University Gustave Eiffel, COSYS, GRETTIA, Paris, France</footnote>

<footnote>Corresponding author:
Hugues Moreau, hugues.moreau.pro@gmail.com</footnote>
data sources, and objectives, many different methodologies have been applied in these analyses. Thanks to its ability to reveal potentially unexpected patterns, the literature has used clustering extensively. However, most publications make little use of exogenous data during the clustering. Many use data like socio-demographic characteristics (age or income distribution of a neighborhood or district, the proportion of foreign-born people, *etc.*) to explain the clusters *after* they have been computed (3, 4), but to the best of our knowledge, none incorporate these variables in the model used to compute the clusters.

In this paper, we used a model named ClustSeg (5), which resembles a mixture of regressions (6) in that it creates clusters with their regression coefficients to model the impact of the covariates on the ridership. The specificity of this model is that it also creates segments (in addition to clusters), i.e. periods between which the regression coefficients may also change. In our case, we will mainly use the difference between successive segments in the post-analysis of the results. This will allow us to compare the stations' behaviors by comparing the coefficients of different segments.

We show that the main impact of the pandemic is a reduction in the number of weekday trips. Even though this trend is the strongest for stations where the difference between weekdays and weekends is the highest, we may see a general decrease in all stations, including the ones for which the differences were small before the pandemic. In addition, we observe changes in the daily patterns of some of the stations: before the pandemic, the weekdays with the lowest number of entries were Mondays across all stations. After this event, in the stations with the most weekday trips, Friday is the day with the most minor travel, which we attribute to this day being favored for teleworking (1).

The rest of this paper is structured as follows: section presents the relevant literature, and section details our methodology (the data and the covariates, along with the mixture of regressions model). Section provides a detailed analysis of the partition obtained, and lastly, section concludes the paper.

## Related work

There is a diversity of publications aiming to understand the impact of the pandemic on public transport usage. This section will focus on publications deploying clustering approaches on ridership data.

### Ridership data

Some use surveys: (7), for example, concluded that one of the determining factors explaining the change in transport habits after the pandemic is the perceived individual responsibility was one of the main reasons explaining the choice to shift from public transport to private ones. Similarly, (8) studied the modal switching of inhabitants of different South

American cities. They found that the estimated gravity of the pandemic mostly pushed individuals to switch to private transport modes (*e.g.*, car). Still, they had little influence over switching to active modes. In general, surveys allow the evaluation of specific aspects of individual behavior, but they require knowing the effects to investigate in advance. In addition, an additional setup is required to avoid the mismatch between *stated* and *revealed* preferences (4, 9).

Many publications analyzing the impact of the pandemic on public transport ridership rely on anonymized recordings of entries. For example, (10) used the origin-destination of public transport users of Santiago, Chile, to examine the changes in trip purposes caused by the pandemic. (11) similarly investigated the origin-destination maps before and after the pandemic to find that work-related trips decreased less than others. Others rely only on the origin without trying to use the destination: (12) compared smart card data with the socio-demographic characteristics of the counties of Stockholm, Sweden. (13) used Automatic passenger counting data to show the variations in ridership that closely follow the number of detected cases in the general population of four cities of Austria and Norway. (14) worked on ticket and card validation data in various cities of Sweden and noticed changes in the type of tickets used to travel: in addition to a net decrease in the total number of travelers, the use of short-term tickets (used mostly by tourists) dropped tremendously. Many users with longer-duration subscription cards (30 or 90 days) switched to single-use tickets, even though the number of trips per traveler remained constant across the pandemic.

(15) worked on several origins of trips, aggregated by census tract. They computed a geographically weighted regression to estimate the impact of demographic variables on transit change for each census tract. They found that the median income had little effect on the ridership change, while race played a significant role. However, their methodology creates one set of regression coefficients for each of the 169 census tracts they used. This complexity illustrates the need to aggregate the data into *clusters*.

Clustering has been used extensively on smart card data, even before the Covid-19 pandemic (see (16) for a review). Surprisingly, there is a great diversity in how clustering is used. For example, even if it is not the main highlight of their respective publications, (10) and (12) used clustering to group the counties or districts into a small number of clusters to better model the impact of the location on ridership.

The most straightforward use of clustering can be found in (17). They started from smart card data of travelers from Santiago, Chile, and computed features such as the number of weekday trips or variations in travelers' trips. Using two clusters, the authors highlighted travelers who returned to their pre-pandemic habits after the reopening and those who adapted by reducing their use of public transport. In a second
step, they sought to explain the cluster assignments using socio-demographic characteristics.

(18) used k-means to cluster public transport data of fifteen European cities extracted from Google mobility reports. They explained the decrease in ridership using three variables: the number of workplace visits, retail or recreational places visits, and the stringency of governmental measures. In the case of Paris, they concluded that the main driving factor of the decrease in public transport use was the number of visits to retail and recreational places.

Another work is the publication of (19), who used a two-step approach: first, they computed clusters of the weeks, where one week is a seven-dimensional vector of normalized records of entries. Then, they grouped the stations into a second set of clusters based on the proportions of weeks each station has in each cluster. They show that center and connection-heavy stops are the most impacted, while stops near hospitals and medical facilities suffer the least from the consequences of the pandemic, ridership-wise. Similarly, (20) are more particular: the authors computed similarities between the graph nodes of the origin and destination of trips in Bogotá, Colombia, for each day of an eleven-month dataset. Then, they clustered the days depending on the similarities of the origin-destination graph. The partition they obtained successfully identified segments corresponding to the lockdown and the start of the recovery. However, given that a day could belong to any cluster, their model had to create one cluster for Sundays and one for Saturdays.

To avoid such change of clusters, (21) applied segmentation. They studied the three-year records of entries in the Montréal public transport network and found, for instance, that business days (weekdays without public holidays) were more impacted than weekends and public holidays.

## Positioning and contributions

Most studies mentioned above use a relatively small period of one year or even a small set of three two-week periods to model the pre-pandemic, lockdown, and recovery periods. Some, such as (18) and (21) worked on more extended periods of two and three years respectively. However, the partition found in the latter work suffers from the fact that it does not take into account any covariate. For example, a period of five months displaying a steady recovery (between April and August 2020) is split into four segments because the number of entries in each segment must remain constant (see Figure 3 in the referenced publication).

The methods we mentioned rely exclusively on ridership data to compute the clusters, using other data (socio-demographic characteristics, etc.) only to explain the clusters in a later step. We argue that there is much to learn by integrating additional data types in the clustering process as covariates. Covariates (or exogenous variables) are values that a model uses to explain an observed variable (in our case, the ridership) without themselves being modeled. For

example, modeling the impact of the day of the week throughout a covariate could prevent a model from creating new clusters for weekends when the public transport ridership is significantly lower than during the week (20). We propose to include covariates such as the day of the week so that the model can create clusters that behave differently during weekends, for instance. This allows us to study the impact of these variables without repeating the analysis once for each variable we wish to study, as in (22).

To summarize, the contributions of this work are as follows:

* We use a statistical model that performs clustering and segmentation simultaneously while incorporating many exogenous variables.

* We work on large amounts of data: five and a half years (ending more than two years after the start of the pandemic), a large public transport network with 3,700,000 trips each day on average, with several transport modes (underground, tramway, long-distance train).

* We group the 542 public transport stations into nine easily-interpretable clusters. At the same time, the segmentation allows a synthetic investigation of the effects of the pandemic by comparing segments.

The dataset (2) is public, and the code is available at https://github.com/HuguesMoreau/GMM Clustering_Segmentation.

## Method

### Dataset

The data used is a public record of entries in Paris city's rail public transport stations (underground, tramway, local trains) and its surrounding region (2). This dataset covers five and a half years (2,004 days), from January 1, 2017, to June 30, 2022. This large window makes it possible to compare the three years before the pandemic with the pandemic period. The dataset only contains the inflow (number of people entering the network), independently of the ticket type (single-use or limited-duration tickets or personal travel card). When changing fare zones, one trip may be accounted for several times (if the journey includes connections between different modes). To estimate the importance of this overrepresentation, we could mention that, between 2009 and 2011, the average number of connections for a trip was 0.6 (see (23), p. 15). However, this includes connections with modes we did not cover (such as connections between underground and bus or between two busses). In the case of entries on the rail public transport network, the connections susceptible to seeing two recordings for a single trip are the ones with a connection between classical underground and farther-reaching trains (RER, Transilien), where a change of fare may occur. As a result, we expect this effect to have a
small impact on the results of our analysis, except for a few train stations (Gare de l'Est, Gare Saint-Lazare, etc.).

The employments in the region are mostly concentrated around a relatively small number of locations ("more than seventy percent of the employments are located on six percent of the surface" (24)). Even though the innermost city has the highest density of employment, this is also the location where the share of employees living in the close vicinity of their workplace is the highest. This means that the individuals working in the center of the city will be more likely to walk or cycle, two modes that are invisible to us. On the other hand, the employment concentrations located in the suburbs are distant from habitation zones, and see a heavy commute using rail public transport (24). Note, however, that commuting is far from representing the entirety of travel: for instance, in the years preceding 2020, 45.4% of the daily travels on public transport were related to professional matters (25). The attribution of a purpose to a trip can be further complicated by the fact that individuals often chain travels (26) (for instance, by going shopping after work). A map of Paris city is available in figure 1, with a focus on the innermost city. The network is constituted by several stations, along with a small number of important hubs (stations such as La Defense, or train stations like Montparnasse, Saint-Lazare, Gare de Lyon). The center (top of fig. 1) is mainly covered with a dense network of underground transport. On the other hand, the suburbs (fig. 1, bottom) are served by lines that go up to the city center, which gives the regional network a 'star' shape.

The raw data contain the daily number of entries for each type of ticket or subscription the travelers used for 699 stations (before any preprocessing). As we did not use the fare type, we considered the sum of all entries for each day and each station. However, only the couples (*station, day*) with at least one entry are present. One might be tempted to say that an absence of data for a given day means no user entered this station (which might have happened during the lockdowns, for instance). However, we cannot know which days are missing due to actual errors in the data collection process and which days are absent because of the lack of users. Thus, we left missing data as-is.

Note that the only daily measure is the total number of entries, i.e. we do not have access to an hourly number of entries each day. Although an hourly profile is available, this profile is computed by averaging the share of entries for one semester for each station. Given that there is no obvious way to merge these two kinds of data without creating artificial trends, we focused on the daily number of entries.

To ensure anonymity, when a station recorded less than five entries for the same type of ticketing in a day, the dataset only mentions "less than five". In such cases, we considered that there were three entries for this day, station, and ticket type. We removed stations with more than 60% missing days (12 stations) or less than 500 entries per day on average (143

stations), leaving 542 stations to cluster. We also computed a two-week moving average of each station's entries to handle the many noisy days in the data. Each value below one-tenth of the result was removed (we removed about 1% of the data this way).

Finally, given the substantial imbalance between stations, we normalized the number of entries of each station. For each station, we divide the number of entries by the average number of entries during the first year, strikes excluded. We consider this period to be representative of usual behavior. Then, similarly to (27), we took the Log in base 10 of the number of entries. Table 1 summarizes the dataset. Out of the 1,086,168 values, some are missing, either because they are missing or because we removed them during the cleaning process.

Figure 2 shows some examples of time series after the preprocessing. The drops in December 2019 due to a critical strike and March 2020, at the start of the pandemic, can be seen. Even though many different measures were implemented (closing of bars and leisure places, contact tracing, verification of vaccination certificates), we will mostly focus on lockdowns. Figure 3 shows the decrease in ridership during the three lockdowns, compared to the average daily number of entries between December 2018 and November 2019. We also display the reduction in the average daily ridership between the start of the pandemic and the end of the dataset (excluding lockdowns). As expected, the first lockdown is much more noticeable than the others.

In the present work, we aim to develop a methodology that helps to draw finer conclusions about the impact of the pandemic than looking at aggregate levels. In particular, we want to understand the effects of the pandemic on public transport ridership. As there are too many stations to examine them all, we used clustering to group them into a small number of easily understandable clusters. One key feature that distinguishes the clusters is their regression coefficients, i.e. parameters that explain the influence of covariates on the stations' ridership. Before presenting the parameters in section, we will present each covariate and its role in the next section.

## Covariates

Covariates, or exogenous variables, allow the model to create clusters whose existence is not due to trivial causes. However, using too many different covariates might cause the model to explain all variations using only covariates without separating clusters. Consequently, one must be careful of the impact a covariate should have on ridership. In this study, we included the following covariates:

* *time* (1 variable): we include time, a scalar varying between 0 and 1. We expect this variable to have two purposes. In the period preceding the pandemic, a linear progression could model small changes, such as
<figure><img src="image_2.png" /><figcaption>**Figure 1.** The physical location of stations within Paris city (top) and the region (bottom).
Prepared using TRR.cls</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Some examples of series in the dataset we process. The zoom-in represents two weeks starting September 11, 2019 (on a Monday). Note the missing data for the *Charles de Gaulle Airport* station (blue curve) during the first and third lockdowns; the strong weekend decrease in the business district of *La Defense*, or the overall strong increase along time of *Le Guichet* station. This illustrates an overview of the impact of the pandemic and the difficulties of modeling the dataset.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3.** The average decrease in ridership during the three lockdowns and during the non-lockdown days between May 11, 2020 and June 30, 2022 (the end of the dataset). The difference in the impact of the lockdowns is the reason why we use one variable per lockdown instead of a single variable for all three lockdowns.</figcaption></figure>

the increase in ridership due to demographic growth. During and after the pandemic, one would expect an initial drop in ridership followed by a slower recovery. We let the model explain this trend by having low mean $\mu_{k,s}$ and a strong coefficient associated with time.

* *year-periodic splines* (20 variables): it can be seen in Figure 2 that for some stations, the recorded number of entries is lower during the summer period (the time when many people leave the region for holidays) or at

the end of the year. To account for these changes and to prevent our model from creating a new segment at the start of summer, we used twenty second-degree splines that are one-year-periodic. The periodicity ensures that slower, longer-term trends are captured by time.

* *weekdays* (7 variables): Figure 2 shows a strong weekly periodicity, which is to be expected given the commute between the workplace and home (23). We created seven dummy variables, one for each day of the week.

* *school holidays and public holidays* (2 variables): the importance of the day of the week shows that commuting plays a significant role in explaining some stations' number of entries. In this context, public holidays falling on a weekday might lead to the model seeing a seemingly unexplainable low value, which impacts the least square linear regression we use to compute the parameters (eq. 5). Even though different public holidays affect rail ridership differently, including one variable for public holidays helps explain potential low ridership during weekdays. As many students use public transport to go to school (23), we also include school holidays.

* *strikes* (2 variables): network-wide strikes sometimes impact the Paris public transport network. Most of these strikes are only a few days long and have a limited impact, but one of them, in December 2019, significantly reduced the rail public transport supply to the point that demand was also reduced. We included one dummy variable specific to the 2019 strike and one for all other strikes. We will see in section that the importance of this strike is too broad for a single covariate to model entirely.

* *lockdowns* (3 variables): Contrary to the Covid-19 pandemic, lockdowns were short-term events that significantly impacted public transport ridership.
Hence, the model will seek to explain the observed drop in public rail ridership, which these variables help accomplish. One could model the effect of lockdowns by creating two extra segments for each lockdown (one segment during and one segment after each lockdown), but doing so would make the interpretation more complex. However, given that the three lockdowns occurred at different stages of the pandemic, we might expect each lockdown to have various consequences. We, therefore, created three variables instead of one. Note that lockdowns are the only exogenous variables in the model related to the Covid-19 pandemic. We expect our model to resort to changing segments to account for changes outside these periods.

With these 35 covariates, we can estimate the model parameters using the EM algorithm (28).

## Definition and notations

The data to model are series observed over discrete time stamps indexed by $t$. For any public transport station $i$, we note $y_{i,t}$ the number of entries of the station $i$ at day $t$. We want to group the stations into $K$ clusters. Then, inside each cluster $k$, we want to group the days inside $S$ segments, a series of timestamps in which the data distribution remains the same. We include exogenous variables, noted $x_t \in \mathbb{R}^L$ as a vector of $L$ variables at day $t$. The model we use is a mixture of regressions: inside every segment, the data are centered around a linear combination of the covariates.

## Model presentation

In this work, we used ClustSeg model (5). Similarly to Gaussian Mixture Models, this model assumes a latent (hidden) cluster membership exists. However, it also assumes that the time series of each station can be partitioned into different segments. Finally, in a similar fashion to mixtures of regressions (6), it provides each segment with its own mean and regression coefficients to model the influence of the covariates. We summarize this model below. Let $z_i$ be the cluster assignment of station $i$, and $w_{i,t}$ the segment assignment of station $i$ at time $t$. We assume that the cluster assignments $z_i$ are i.i.d., following a multinomial distribution whose weights $\pi_k$ are parameters of our model:

$$P(z_i = k) = \pi_k \quad \text{s.t.} \quad \sum_{k=1}^{K} \pi_k = 1. \qquad (1)$$

Given the cluster $k$ of station $i$, we assume that the segment assignment $w_{i,t}$ follows a multinomial law defined by the probabilities:

$$P(w_{i,t} = s | z_i = k) = \kappa_t^{k,s} = \frac{\exp(t.u_{k,s} + v_{k,s})}{\sum_{s'} \exp(t.u_{k,s'} + v_{k,s'})}, \qquad (2)$$

where $u_{k,s}$ and $v_{k,s}$ are model parameters that control how fast the probabilities $\kappa_t^{k,s}$ change.

The linear dependency in time means that the argument $t.u_{k,s} + v_{k,s}$ is monotonous. This ensures, in turn, that if we computed the set of time steps $t$ in which a given segment is the most likely a priori, this set would be an interval. However, our model assumes that the probabilities for a day to belong to a given segment are independent of the other days'. In other words, while the maxima of the priors $\kappa_t^{k,s}$ create segments, the partition of the time axis resulting from the application of the Maximum a Posteriori (MAP) rule may not be a series of contiguous segments. This means, for instance, that the sequence of segments such as 1, 1, 2, 1, 1, 2, 1, 2, 2, 2 is possible as long as both the values $\kappa_t^{k,1}$ and $\kappa_t^{k,2}$ are not too low. To ensure that such fluctuations do not occur for too long, we need the probabilities $\kappa_t^{k,1}$ to progress quickly enough. We do so by applying a linear constraint:

$$\forall k, \forall s \in \{1...S-1\}, \quad u_{k,s+1} - u_{k,s} > \lambda, \qquad (3)$$

Where $\lambda$ is chosen such that the probabilities $\kappa_t^{k,s}$ increase from 0.01 to 0.99 in at most three months.

To explain the observed variable, we use a formulation that is common to all mixtures of regressions (6):

$$y_{i,t} | (z_i = k, w_{i,t} = s) \sim \mathcal{N}(\mu_t^{k,s}, \Sigma_{k,s}), \qquad (4)$$

where $\Sigma_{k,s}$ is the variance in segment $s$ of cluster $k$. Variable $\mu_t^{k,s}$ is the mean of the cluster $k$ and segment $s$ at day $t$. It encompasses a segment average $m_{k,s}$ and the contributions $\alpha_{l,k,s}$ the covariates $x_{l,t}$, as follows:

$$\mu_{i,t}^{k,s} = m_{k,s} + \sum_{l=1}^{L} x_{t,l} \alpha_{l,k,s}. \qquad (5)$$

In this formulation $m_{k,s}$ and the $\alpha_{l,k,s}$ are parameters of the model, and are, respectively the intercept and the coefficients of the linear regression. The model performs one regression per segment and per cluster, which helps to explain the observed rail ridership $y_{i,t}$ from the vector of covariates $x_t$.

To summarize, the model parameters are the mixture weights $\pi_k$, the mean $m_{k,s}$ and regression coefficients $\alpha_{k,s}$, the variance $\Sigma_{k,s}$, and the segment parameters $u_{k,s}$ and $v_{k,s}$. The next section presents how these parameters were estimated.

## Parameter estimation

To estimate the values of the parameters, we maximize the log-likelihood of the observed data:

$$\begin{aligned} \mathcal{L}(\boldsymbol{\theta}) &= \log P(y | \mathbf{X}, \boldsymbol{\theta}) \\ &= \sum_{i} \log \sum_{k} \prod_{t} \pi_{k} \sum_{s} \kappa_{t}^{k,s} \mathcal{N}(y_{i,t}; \mu_{i,t}^{k,s}, \Sigma_{k,s}) \end{aligned} \qquad (6)$$
<table><caption>**Table 1.** An overview of the rail public transport ridership dataset.</caption><tr><td>number of stations</td><td>542</td></tr><tr><td>number of days</td><td>2,004</td></tr><tr><td>average daily number of entries</td><td>3,700,000</td></tr><tr><td>date interval</td><td>January 1, 2017 - June 30, 2022,</td></tr><tr><td>covariates<br>(number of variables)</td><td>year-periodic splines (20), time (1) day of the week (7),<br>holidays (2), strikes (2), lockdowns (3).</td></tr></table>

The formulation of our model allows us to use the Expectation-Maximization algorithm (28), a powerful algorithm aiming to estimate the parameters of latent (hidden) variable models. In our case, the latent variable is the union of the cluster and segment, which we estimate using both the observed variable ($y_{i,t}$) and the exogenous variables ($x_{i,t}$). The EM algorithm consists in alternating two steps until convergence. During the E (Expectation) step, we compute the probability for each station to be in each cluster and for each day to be in any given segment. Then, during the M (Maximization) step, we compute new parameters given the probabilities of cluster and segment assignments.

**E-step** In this step, we compute the posterior distribution of the latent variables using the values of the parameters at the current iteration. As mentioned in (5), this is done in two steps. First, we need to compute the posterior probability of a given station $i$ to belong to cluster $k$, which we note $\rho_i^k$:

$$ \rho_i^k = P(z_i = k | \mathbf{y}_i, \mathbf{x}_i; \boldsymbol{\theta}) \quad (7) $$

$$ \rho_i^k = \frac{\pi_k \prod_t \sum_s \kappa_t^{k,s} \mathcal{N}(y_{i,t}; \mu_{i,t}^{k,s}, \Sigma_{k,s})}{\sum_{k'} \pi_{k'} \prod_t \sum_s \kappa_t^{k',s} \mathcal{N}(y_{i,t}; \mu_{i,t}^{k',s}, \Sigma_{k',s})}, \quad (8) $$

where we note $\mathbf{y}_i = (y_{i,t})_{t \in \{1, \dots, T\}}$.

Then, we can obtain the posterior probability for any day $t$ to belong to segment $s$, named *responsibilities*:

$$ r_{i,t}^{k,s} = P(w_{i,t} = s, z_i = k | \mathbf{y}_i, \mathbf{x}_i; \boldsymbol{\theta}) \quad (9) $$

$$ = \rho_i^k \frac{\kappa_t^{k,s} \mathcal{N}(y_{i,t}; \mu_{i,t}^{k,s}, \Sigma_{k,s})}{\sum_{s'} \kappa_t^{k,s'} \mathcal{N}(y_{i,t}; \mu_{i,t}^{k,s'}, \Sigma_{k,s'})} \quad (10) $$

Once we have computed the responsibilities, we update the parameters during the *Maximization* step.

**M-step** In the maximization step, we use the responsibilities $r_{i,t}^{k,s}$ computed earlier and obtain the parameters that maximize the expectation of the log-likelihood. In this section, we display only the results of the maximization step. We refer the reader to the publication by (5) for more details.

As with many mixture models, the mixture proportions are the proportion of the responsibilities of the samples. For each

cluster $k$:

$$ \pi_k = \frac{\left(\sum_{i,t,s} r_{i,t}^{k,s}\right)}{\left(\sum_{i,t,k',s} r_{i,t}^{k',s}\right)} \quad (11) $$

The segment parameters $u, v$ are found by computing the maximum of the following logistic regression problem:

$$ \sum_{i,t,k,s} r_{i,t}^{k,s} \log \frac{\exp(t.u_{k,s} + v_{k,s})}{\sum_{s'} \exp(t.u_{k,s'} + v_{k,s'})} \quad (12) $$

This problem is convex (even with the constraints mentioned in equation 3), and we solve it using scipy's `optimize` package implementation of the quasi-Newton method from (29). We ensure identifiability by setting $\sum_s u_{k,s} = \sum_s v_{k,s} = 0$ for all $k$.

We can find the optimal values for the regression coefficients separately for each segment. For all $k, s$, the values of $m_{k,s}$, $\alpha_{k,s}$, and $\Sigma_{k,s}$ minimize:

$$ \sum_{i,t} r_{i,t}^{k,s} \left( \frac{(y_{i,t} - \mu_{i,t}^{k,s})^2}{\Sigma_{k,s}} + \log \Sigma_{k,s} \right), \quad (13) $$

where $\mu_{i,t}^{k,s} = m_{k,s} + \sum_t x_{i,t,l} \alpha_{l,k,s}$. This is a common linear regression problem where the $r_{i,t}^{k,s}$ are weights (30).

As mentioned earlier, we use these formulas during each maximization step to estimate new parameter values. We keep alternating the expectation and maximization steps until the log-likelihood stops increasing by more than $10^{-4}$ in ten iterations. Only then is parameter estimation over, and we can start evaluating our model.

**Initialization** We start by setting the mixture proportions equal to $1/K$, where $K$ is the number of clusters. The segment parameters $u, v$ are set in such a way as to respect the constraints we set (see equation 3), and the borders between segments delimit segments of equal duration. The segment means $m_{k,s}$ are estimated by assigning each individual a random cluster and each day its corresponding segment and computing the mean of values inside each segment. Finally, we end the initialization by setting the contribution of exogenous variables to zero.

To choose one model for interpretation, we only need to select the right number of clusters. We present this choice in the next section.
## Model selection

To fix the number of clusters *K*, we relied on the slope heuristic (31), a data-dependent penalization with theoretical guarantees in a non-asymptotic regime. We chose *S* = 2 segments *a priori*, and estimated the parameters with a number of clusters varying between 1 and 20. To reduce the effect of variance, we computed three initializations for each number of clusters and considered the average log-likelihood. For each number of clusters, we obtained one point, defined by the number of parameters and log-likelihood. Then, as recommended by (31), we computed several linear regressions. Figure 4 shows an example of such regressions. The first linear regression included the last three points, and the second regression included the last four points, etc. For each linear regression, we computed a penalized criterion where the penalization is equal to twice the coefficient of the linear regression. Thus, we obtained a series of penalized criteria, each with its maximum. To conclude, (31) recommend selecting the highest number of clusters which is chosen by at least 15% of the criteria. In our case, *K* = 9 was a clear optimum. This number of clusters was, therefore, chosen for the analysis we will present below.

<figure><img src="image_5.png" /><figcaption>**Figure 4.** An example of penalized criterion (in blue), where the penalization equals twice the coefficient of the green line. This line was computed by fitting a linear regression to estimate the log-likelihood from the number of parameters in each model. The slope heuristic described by (31) requires fitting several regression with different numbers of points; we display one example where the regression parameters were estimated with the last nine points. As *K* = 9 clusters is a maximum of the penalized criteria computed with different number of points, this is the number of clusters we choose in our analysis</figcaption></figure>

## Results and Discussion

This section presents the results of the clustering and segmentation performed by the given model, with *S* = 2 segments and *K* = 9 clusters. We begin by explaining the

overall results, which highlight the impact of the pandemic on the rail transport network ridership. In this regard, we will analyze the differences between segments of each cluster in section. We then focus on the specificities of each cluster in section. Note that an extract of the most important model coefficients is available in the supplementary materials (see table A1).

## Effects of the pandemic

As expected, the most important impact of the pandemic is a considerable decrease in the number of entries (fig. 5 and 6), followed by a slower recovery process. However, another difference between the segments can be observed in the influence of the day of the week, as Figure 7 shows. For both segments, the number of entries for all stations is significantly higher for weekdays than for weekends. However, when we compare the coefficients of the two segments (see Figure 7), we notice a decrease in the number of entries during weekdays and an increase during weekends, relative to the same day of the week before the pandemic. It can also be seen that the regression coefficient associated with Friday decreased strongly during the pandemic. On the other hand, the coefficient associated with Tuesdays became the highest, creating a phenomenon similar to rush hours named 'peak day' (1). The peak on Thursdays disappeared, while the coefficient for Tuesdays remained unchanged. Consequently, Tuesdays are the days with the highest ridership in Segment 2 after the pandemic. Finally, we see that the relative reduction of the number of weekday trips is correlated to the share of weekday trips, either before or after the pandemic: Clusters 1 and 2 are the clusters for which the relative decrease in the weekday ridership is the largest, and cluster 9 is the one where the relative increase of the number of entries is the smallest during the weekend.

It should be noted that for certain clusters, the transition between the first and second segments does not occur at the onset of the pandemic but rather a few months before it (fig. 5 and 6). This is due to a major strike that affected the Paris public transport network between December 2019 and January 2020. Even though we encoded this specific strike as a dummy variable, having a single effect for the entire period proved to be insufficient, resulting in an underestimation of the strike's impact. The heaviest-hit days were assigned to Segment 2. For this reason, the transition between the two segments occurred at the beginning of December 2019 or between the strike and the first lockdown.

In summary, all stations were strongly affected during and after the pandemic. The most noticeable effect is, as expected, a substantial decrease at the beginning of the pandemic, followed by a slow three-year recovery. While all clusters register a higher ridership during weekdays and fewer entries during the weekend, this gap became smaller in the second segment of most clusters (which started at the beginning of the pandemic). In addition, the weekly profile changed: there
<figure><img src="image_6.png" /><figcaption>**Figure 5.** The mean and the contribution of time and year-periodic splines, with cluster 2 outlined. For cluster 2, we notice the absence of growth of two to three percent per year, which characterizes most other clusters. For the reason why some clusters begin during the 2019 strike, see section.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 6.** The mean and the contribution of time and year-periodic splines, with cluster 3 outlined. A strong increase during the pre-pandemic segment defines cluster 3. For the reason why some clusters begin during the 2019 strike, see section.</figcaption></figure>

was a peak on Thursdays and Tuesdays to a lower extent in Segment 1. At the same time, Tuesday became a noticeable maximum of ridership in the second segment. Finally, the coefficient associated with Fridays decreased significantly. In some clusters, Friday is the weekday with the fewest recorded number of entries, but the minimum remained on Monday for other clusters. In the next section, we propose an in-depth analysis of the differences between clusters.

## Specificities of each cluster

We sorted the clusters by their share of weekday trips (before the pandemic), but this feature seems to be well correlated with both the relative reduction in the number of entrances during weekdays (fig. 7) and the distance to the center of the city (fig. 8). If we consider each cluster in detail, we see that five of them (clusters 1, 4, 5, 8, and 9) have a low variance (fig. 9): they contain series which vary in ways the model can explain using the provided covariates. The rest each have their peculiarities, meaning that these do not fit the model as

well as the others. Cluster 2 is defined by the absence of the pre-pandemic increase in ridership (most other clusters see an increase of 2% to 3% per year, on par with the demographic growth of the region, see fig. 5). On the other hand, cluster 3 sees an increase during the pre-pandemic which is much larger than most clusters (fig. 6). Cluster 6 groups all stations for which the ridership took longer to recover at the end of the first lockdown (fig. 10), which explains why the model assigns a high variance for segment 2. Cluster 7 is the one cluster for which the change in the weekly patterns does not match the other clusters (fig 7).

However, the main determinant of the changes seems to be the proportion of entries recorded during weekdays. There is a continuity of clusters ranking from largest to smallest over-representation of weekday trips. The relative changes in weekly patterns caused by the pandemic are proportional to the share of entries recorded during weekdays. As suburbs have more commute done via rail than the center (24), this tends to indicate that the clusters for which the proportion of
<figure><img src="image_8.png" /><figcaption>**Figure 7.** The effect of the day of the week on the recorded number of entries. The full dots denote values that differ significantly from zero, using a significance threshold of 0.05 and a Bonferroni correction to account for multiple hypotheses testing. One can see the changes in weekly patterns the pandemic causes, and, in particular, the fact that for clusters 1 and 2, the weekday with the least ridership is now Friday. See the supplementary materials (table A1) for a list of model coefficients.</figcaption></figure>

weekday ridership is the highest are also the ones with the most commute. If this hypothesis were true, we may attribute the changes in the weekly pattern to working-from-home policies. However, definitive conclusions require dedicated studies and additional datasets.

## Conclusion

The ridership in a rail public transport network is a complex signal subject to many fluctuations, even without a pandemic. Using an adequate set of covariates in the proposed model, encompassing calendar variables such as the days of the

week and discrete events such as strikes and lockdowns, we explained the most relevant changes in the data. We presented a mixture of regressions model that simultaneously performed clustering and segmentation. Inside each segment of each cluster, the model estimates regression coefficients. A comparison between the two segments of each cluster provided insight into the impact of the pandemic on ridership at different types of stations. We were able to see that the effect of the pandemic is similar across all clusters: the model creates a sharp drop at the beginning of the first lockdown and allows the ridership to increase steadily
<figure><img src="image_9.png" /><figcaption>**Figure 8.** The histograms of the distance of the stations in each cluster from the center of Paris (log-scaled). We remind the reader that we sorted the clusters according to the effect of the day of the week before the pandemic (first segment), but the model itself had no access to the position of each station.</figcaption></figure>

during the following months. Some effects are similar across clusters but with varying intensities: the weekly variations decreased the most at the stations which had the most over-representation of weekday trips compared to weekend trips. This relative decrease, however, is not impactful enough to even out the differences between weekdays and weekends. The weekly profile also changed: there was a maximum ridership on Thursdays before the pandemic. As Thursdays' influence decreased but Tuesdays remained stable, Tuesdays became the day with the highest recorded number of entries after the pandemic. In clusters 1 and 2, for which the proportion of weekday trips is the highest, Friday became the

weekday with the lowest ridership. The minimum remained on Mondays for other clusters. Finally, some clusters display specific behaviors, such as Cluster 3, with strongly increasing ridership records; Cluster 6, containing stations that took longer to recover after the first lockdown, or Cluster 7, grouping the stations for which the changes in the weekly pattern is different from the other stations.

Our study has some limitations due to the data we processed. The first one is that the data we used do not contain any information concerning the types of travel impacted: did rail public transport users limit their commute or leisure trips? We could only infer effects based on the difference between weekdays and weekends. Another limit stems from using a single data source, which prevents assessing modal shift: we do not know whether the people who stopped taking rail public transport started walking, cycling, taking their car, or canceling their trip.

Similarly, we did not account for the number of transfers in each trip. Excluding the double entries would ensure that large hubs are not overrepresented. This, in turn, would allow us to account for the vast differences between stations by other means than normalizing: one could, for instance, include the population density as a covariate and let the model estimate the share of inhabitants who use public transport.

In addition to using more diverse data sources, one way to extend our work would be to use more than two segments. In our work, as we started with two segments, the period encompassed by the second segment models the ridership both during the apex of the pandemic and after it. Ideally, the distinction between the temporary changes happening during the restrictions and the long-term consequences of the pandemic might have been helpful. However, using three segments instead of two with the current dataset does not lead to satisfying results: the model assigns the whole period between June 2020 and July 2022 to the last segment (the middle segment covers only seven months between December 2019 and June 2020). Including an extra six months or one year of data makes the back-to-normal period longer and, thus, will push the model to detect a segment dedicated to a period after the pandemic.

## Author Contributions

The authors confirm contribution to the paper as follows:

Conceptualization: E. Côme, A. Samé, L. Oukhellou ; Data curation: H. Moreau ; Formal analysis: A. Samé ; Funding acquisition: L. Oukhellou ; Investigation: H. Moreau ; Methodology: A. Samé ; Project administration: L. Oukhellou ; Resources: L. Oukhellou ; Software: H. Moreau, E. Côme ; Supervision: L. Oukhellou ; Validation: E. Côme, A. Samé, H. Moreau ; Visualization: H. Moreau ; Writing – original draft: H. Moreau ; Writing – review & editing: E. Côme, A. Samé, L. Oukhellou ;

All authors reviewed the results and approved the final version of the manuscript.
<figure><img src="image_10.png" /><figcaption>**Figure 9.** The variance inside each cluster. Clusters 5 and 8 are quite stable (they have a low variance on both segments). On the other hand, cluster 6 has a low variance in segment one and a high variance in segment 2, meaning that the post-pandemic variations in this cluster are not explained by the covariates we included.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 10.** A comparison of the mean and standard deviation of two clusters (respectively $\mu_t^{k,s}$ and the square root of $\Sigma_{k,s}$ in eq. 4) to the empirical mean and standard deviation of the stations assigned to the cluster, computed each day. Even though we display only Clusters 5 and 6, the other clusters behave similarly to Cluster 5 for the period after the first lockdown.</figcaption></figure>

## Acknowledgements

This work was supported by Vinci through the funding of the Eco
Conception Chair in École des Ponts ParisTech.

## Declaration of conflicting interests

The authors declare no conflict of interest.

## Data Accessibility Statement

The data is freely downloadable at the following links: https://data.iledefrance-mobilites.fr/explore/dataset/histo-validations-reseau-ferre/ and https://data.iledefrance-mobilites.fr/explore/
dataset/validation-reseau-ferre-nombre-validation-COVID-19-pandemics

The code is also available at the following address:
https://github.com/HuguesMoreau/GMM-
Clustering_Segmentation

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