A Bayesian Framework for Post-disruption Travel Time Prediction in Metro Networks

A Ba y esian F ramew ork for P ost-disruption T ra v el Time Prediction in Metro Net w orks Sha yan Nazemi a , Aurélie Labb e a , Stefan Steiner b , Pratheepa Jeganathan c , Martin T répanier d,e , Léo R. Belzile a a Dep artment of De cision Scienc es, HEC Montr é al, 3000, chemin de la Côte-Sainte-Catherine, Montr é al, H3T 2A7, QC, Canada b Dep artment of Statistics and A ctuarial Scienc e, University of W aterlo o, 200, University A venue W est, W aterlo o, N2L 3G1, ON, Canada c Dep artment of Mathematics and Statistics, McMaster University, 1280, Main Str e et W est, Hamilton, L8S 4L8, ON, Canada d Dep artment of Mathematic al and Industrial Engine ering, Polyte chnique Montr é al, Montr é al, H3T 0A3, QC, Canada e Centr e Interuniversitair e de R e cher che sur les rÉse aux d’Entr eprise, la L o gistique et le T r ansp ort (CIRREL T), 2920, chemin de la T our, Montr é al, H3T 1J4, QC, Canada Abstract Disruptions are an inheren t feature of transp ortation systems, o ccurring un- predictably and with v arying durations. Ev en after an incident is rep orted as resolv ed, disruptions can induce irregular train op erations that generate substan tial uncertaint y in passenger waiting and trav el times. A ccurately forecasting p ost-disruption tra v el times therefore remains a critical challenge for transit op erators and passenger information systems. This pap er dev el- ops a Bay esian spatiotemp oral mo deling framew ork for p ost-disruption train tra vel times that explicitly captures train interactions, headwa y im balance, and non-Gaussian distributional c haracteristics observ ed during reco very p e- rio ds. The prop osed mo del decomp oses tra vel times in to delay and journey comp onen ts and incorp orates a mo ving-a verage error structure to represent dep endence b et w een consecutive trains. Sk ew-normal and sk ew- t distribu- tions are emplo y ed to ﬂexibly accommo date heterosk edasticit y , sk ewness, and heavy-tailed behavior in p ost-disruption trav el times. The framew ork is ev aluated using high-resolution trac k-o ccupancy and disruption log data from the Mon tréal metro system, co vering tw o lines in b oth trav el direc- tions. Empirical results indicate that p ost-disruption trav el times exhibit pronounced distributional asymmetries that v ary with trav eled distance, as w ell as signiﬁcan t error dep endence across trains. The prop osed mo dels con- sisten tly outp erform baseline sp eciﬁcations in b oth p oin t prediction accuracy and uncertain t y quan tiﬁcation, with the sk ew- t mo del demonstrating the most robust performance for longer journeys. These ﬁndings underscore the imp ortance of incorp orating b oth distributional ﬂexibility and error dep en- dence when forecasting p ost-disruption tra v el times in urban rail systems. K eywor ds: T ransportation Net w orks, Metro Systems, Spatiotemp oral Statistics, Bay esian Statistics 1. In tro duction T ra vel times exhibit substantial temp oral v ariabilit y driv en b y factors suc h as tra vel distance, passenger demand, service frequency , and train con- gestion [1]. Disruptions, whic h constitute an inheren t and often unav oidable asp ect of metro op erations, t ypically arise without warning and v ary widely in duration. These inciden ts considerably degrade netw ork p erformance by aﬀecting the punctualit y and reliability of the inciden t train, as well as the trains op erating immediately b efore and after it [2, 3]. Accordingly , accu- rate trav el time prediction b ecomes particularly critical during the reco v- ery phase follo wing a disruption, as reliable forecasts supp ort op erational decision-making and impro v e ov erall service quality [4]. Our study devel- ops a probabilistic mo del for predicting the trav el times of p ost-disruption trains, thereb y enabling passengers to mak e more informed trav el decisions and facilitating realistic service expectations. W e emplo y a Ba y esian mo del- ing framework that allows us to obtain probabilistic forecasts of trav el times. In man y high-frequency metro systems, o v ertaking is not possible within a line segmen t. As a result, when a disruption o ccurs, a common op erational resp onse is to susp end train mo vemen ts, after whic h service con trollers man- age train departures to restore regular temp oral spacing once the inciden t is cleared. During the disruption, trains accum ulate along the line, increas- ing congestion and leading to additional dela ys and longer trav el times at do wnstream stations once op erations resume. Because disruptions arise un- predictably and are multi-factorial, providing reliable arriv al-time estimates at the moment the incident b egins is near imp ossible. Our research therefore fo cuses instead on predicting the tra vel time required for trains to mov e from their p ositions at the momen t the disruption resolution is announced until they reach a designated downstream station. 2 In the transp ortation literature, in tercity railw ay systems ha v e received considerably more researc h atten tion than urban metro netw orks, largely b ecause their ﬁxed sc hedules facilitate the mo deling and analysis of delay prediction problems. Central to modeling train operations is the c haracteri- zation of pro cess times, whic h consist of dwel l times , deﬁned as the p erio d a train remains stationary at stations, and running times , deﬁned as the p eriod required to trav el b et w een stations. A complete journey can b e expressed as the sum of dwell and running times across all segmen ts b etw een the origin and destination. Ho wev er, the primary source of v ariabilit y in pro cess-time mo dels arises from the dwell times [5, 6], whereas running times exhibit com- parativ ely lo w v ariabilit y [7]. Existing researc h in transp ortation netw orks has explored pro cess time mo deling and trav el time prediction in rail systems. Li et al. [5] examine b oth parametric and non-parametric approaches to estimating dwell times during p eak and oﬀ-peak p erio ds, incorporating temporal and spatial cov ariates, and conclude that dwell durations are strongly inﬂuenced b y the n um b er of b oarding and aligh ting passengers. Cornet et al. [6] prop ose a data-driv en metho d for estimating dwell-time distributions at a station for a giv en passen- ger demand lev el by decomp osing the pro cess into a deterministic minim um dw ell comp onen t and a sto c hastic comp onen t represen ting v arious disrup- tions during passenger b oarding and aligh ting. In contrast, Lee et al. [1] examine passenger tra vel time rather than dw ell times, decomp osing it into w alking, waiting, and riding comp onents. Their mo del estimates passenger tra vel times using tic ket tap-in and tap-out records. P assenger ﬂo w information is not alw ays a v ailable in real time, limiting its usefulness for real-time prediction and op erational management—an issue that also applies to our case study . More recent researc h has therefore fo cused on incorp orating the complex net work structure of railw ay systems in to mo d- eling frameworks to capture dependencies among op erational even ts such as dela ys, control actions, and pro cess times. Prior w ork on dela y propagation has emplo yed probabilistic net w ork mo dels to represent these in teractions [8, 9, 10], with man y approac hes relying on the Mark ov prop ert y within these netw orks [11, 12]. Using a Bay esian netw ork constructed from the railw ay net work top ology , Corman and Kecman [11] mo del downstream dela ys in a real-time setting. Their approach reduces the uncertaint y asso ciated with predicting future de- la ys as new information b ecomes a v ailable along a train’s tra jectory . F rom the p erspective of w aiting passengers at do wnstream stations, an y up dated 3 arriv al or departure information reﬂecting delays can ero de perceived ser- vice reliabilit y and reduce trust in the system. F or this reason, our study concen trates on generating a single prediction for train trav el time to down- stream stations (during p ost-disruption p erio ds). Mo deling journey times to do wnstream locations is equiv alen t to predicting station arriv al times; the distinction lies only in whether the problem is viewed from the standp oin t of on b oard passengers or those waiting on the platform. This problem is closely tied to delay propagation, which can be deﬁned only in net works op- erating under a ﬁxed timetable. In such systems, delays are measured as the diﬀerence b etw een exp ected and actual arriv al times. Li et al. [12] address dela y propagation by extending the framework of Corman and Kecman [11] to incorp orate prolonged dwell and running times within their conditional Bay esian model, thereb y enabling the estimation of b oth delay distributions and pro cess-time distributions. Ge et al. [13] extends this metho dology b y relaxing the Marko v prop ert y and considering the inﬂuence of more than one preceding trains in their Ba yesian net work. The non-linear and complex dynamics of train op eration v ariables hav e shifted research attention to ward neural-based arc hitectures, esp ecially with the increasing capabilities of deep neural net works. Recurrent Neural Net- w orks (RNNs) are particularly eﬀective in mo deling the sequen tial dynamics of train mov ements and station interactions [14, 15, 16], while Graph Neural Net works (GNNs) exploit the top ological structure of railwa y systems by em- b edding station connectivit y within a graph-based framework [15, 17]. Huang et al. [14] mo dels arriv al delay by considering in ter-train interactions and sta- tion level dep endencies with Long Short-T erm Memory (LSTM) comp onen ts. Li et al. [15] segments time into in terv als and construct an interaction net w ork for the trains in eac h time in terv al and mo dels arriv al and departure dela ys using graph conv olutional netw orks. W ang et al. [17] prop oses a deep rein- forcemen t learning approach to dynamically schedule dwell time at stations with the ob jectiv e of minimizing b oth the total waiting time of passengers on the platform and the in-train tra vel time for on b oard passengers. Although metro systems diﬀer op erationally from road and bus netw orks, the core challenge of predicting tra vel times under congested conditions within an interconnected netw ork is common across mo des. Ma et al. [18] mo del global and lo cal spatial dependencies using a combination of m ulti- atten tion graph neural netw orks and LSTM la yers, enabling accurate predic- tions even on routes with limited data. Chen et al. [19] inv estigate b oth lo cal and long-range correlation structures in bus route net works b y dev eloping a 4 Ba yesian Gaussian framew ork for tra vel-time forecasting. Building on this line of work, Chen et al. [20] prop ose a hierarchical Ba y esian probabilistic forecasting mo del that represen ts link tra vel times and headwa ys with the preceding bus through a m ultiv ariate Gaussian mixture form ulation. Chen et al. [21] further extend this p ersp ectiv e b y mo deling the joint distribution of bus link trav el times and passenger o ccupancy using a Ba yesian Marko v regime-switc hing v ector autoregressiv e mo del capable of capturing sk ewness and multi-modality in bus trav el times. While multiv ariate Gaussian assumptions oﬀer analytical tractabilit y , they ma y b e restrictiv e in real-w orld settings c haracterized b y non-linear, asym- metric, and hea vy-tailed dep endencies. Copula-based mo dels pro vide a more ﬂexible alternativ e by decoupling marginal distributions from the dep endence structure. Multi-mo dalit y is particularly pronounced in road segment trav el times [22, 23]. F or example, Chen et al. [22] address multi-modal marginals using Gaussian mixture mo dels b efore ﬁtting biv ariate copulas, whereas Qin et al. [23] prop ose a K -comp onen t copula mixture mo del that more eﬀectively captures the m ulti-mo dal nature of joint trav el-time distributions. T o the b est of our kno wledge, an imp ortan t research gap in the trans- p ortation literature concerns the eﬀect of disruptions on trav el times during the recov ery phase. T o address this gap, we prop ose a hierarc hical Bay esian framew ork that captures the temp oral dep endence in trav el times among con- secutiv e trains while accommodating the v ariabilit y and sk ewness that arise as a function of trav eled distance. Our method decomp oses p ost-disruption tra vel time into a delay comp onen t and a journey comp onent, mo deling each separately . The delay comp onen t is informed by the spatial separation b e- t ween a train and its predecessors in the netw ork, whereas the journey com- p onen t accoun ts for passenger accumulation resulting from the longer-than- usual headw ays that dev elop during disruptions. The Bay esian structure of our framew ork not only yields p oin t predictions but also provides a prin- cipled quan tiﬁcation of forecast uncertain ty , whic h can improv e passenger information systems and inform service planning decisions. The remainder of this pap er is organized as follows. Section 2 describ es the Mon tréal metro system that serv es as the case study for this research and details the v arious data sources employ ed in the analysis. Section 3 presents the prop osed metho dological framework and outlines the statistical mo deling approac h developed to address the problem of post-disruption trav el time prediction. In Section 4, the empirical results are rep orted and systematically compared with those obtained from the baseline mo dels. Finally , Section 5 5 concludes the pap er by summarizing the main ﬁndings and discussing their implications. 2. Case Study This study fo cuses on the op erations of t w o of the most trav eled metro lines in Mon tréal: the Green and the Orange lines, which comprise resp ec- tiv ely 27 and 31 stations [24]. The analysis includes all stations along the Green line and 26 stations from the Orange line. The excluded stations are either lo cated outside the Island of Mon tréal or exhibit complex net work geometries that prev en t reliable extraction of train trajectories without in- curring a high error rate. Key op erational statistics for these tw o lines are summarized in T able 1. Figure 1 presents the map of the Montréal metro system. Observ ed op erational summaries Line Direction # of trains Median dw ell time (seconds) Median running time (seconds) Median headw ay (min utes) Median line tra vel time (min utes) Green 1 75,134 40.0 43.0 5.1 41.1 2 75,012 40.0 44.0 5.2 40.9 Orange 1 79,098 42.0 46.0 4.9 39.8 2 79,266 42.0 45.0 4.9 39.5 T able 1: Summary of the observ ed op erational data for all train tra jectories on the Green and Orange lines of the Montréal metro system o ver the en tire y ear of 2018. Headw ay is the temp oral gap b et ween t wo consecutive trains. In this researc h, t w o complementary data sources are used to construct a dataset to mo del and predict train tra v el times during post-disruption p eriods. T rain tra jectories are extracted from trac k o ccupancy records (Sec- tion 2.1), and are chronologically aligned with the rep orted disruption on the corresp onding metro line (Section 2.2). F or the remainder of this article, and to a void rep etition, we present results based on data from Direction 1 of the Green metro line. Results for the remaining lines and directions are provided in the App endices. 2.1. T r ack Oc cup ancy Data The track o ccupancy dataset consists of all sensor records from the Mon- tréal metro net w ork. The net work is a ﬁxed blo c k system, with one sensor 6 Down town Area Honoré-Beaugrand Radisson Langelier Cadillac Assomption Viau Pie-IX Joliette Préfontaine F rontenac Papineau Beaudry Saint-Lauren t Place-des-Arts McGill Peel Guy-Concordia At water Charlevoix LaSalle De l’Église V erdun Jolicoeur Monk Angrignon Côte-V ertu Du Collège De La Sav ane Namur Plamondon Côte-Sainte-Catherine Snowdon Villa-Maria V endôme Place-Saint-Henri Lionel-Groulx Georges-V anier Lucien-L’Allier Bonav enture Square-Victoria-OA CI Place-D’Armes Champ-de-Mars Berri-UQAM Sherbrooke Mont-Ro yal Laurier Rosemont Beaubien Jean-T alon Jarry Crémazie Sauve Henri-Bourassa Cartier De la Concorde Montmorency Direction 1 Direction 2 Direction 1 Direction 2 Figure 1: Map of the Montréal metro system. The netw ork comprises four metro lines, of whic h only the Green and Orange lines are analyzed in this study . The op erating direction of each line is indicated. Stations excluded from the Orange line are sho wn in gray . The blue-shaded area represen ts the do wnto wn region, and stations located within this area are referred to as down to wn stations. installed at the b eginning of each section and another at its end. A blo c k o c- cupation signal is recorded when a train enters a blo ck, while a blo ck release signal is recorded once the train leav es it. The dataset includes timestamps for ev ery o ccupation and release even t across all blo cks of the net w ork dur- ing 2018. Eac h station corresp onds to exactly one blo ck, whereas tunnels b et w een stations consist of multiple blo c ks. Blo c k lengths v ary considerably , ranging from sev eral tens of meters to several hundred meters in some sec- tions of the netw ork. By aligning consecutive blo c k o ccupation and release signals and reconstructing the corresp onding sequences, train tra jectories can b e derived. With the inclusion of station blo c k information, exact ar- riv al and departure timestamps for every train mov emen t within the system can b e obtained. 7 2.2. Disruption L o gs Dataset The second dataset contains all of the disruptions rep orted in 2018. Eac h record provides the duration, location, start and end timestamps, as w ell as the rep orted cause of the disruption. Ho w ever, disruptions are not systemat- ically recorded by the staﬀ, so some inciden ts may b e missing. In addition, inaccuracies in the rep orted timestamps and durations are o ccasionally ob- serv ed. Basic descriptiv e statistics for the rep orted disruptions are provided in T able 2. Line Direction # of disruptions A verage disruption length (minutes) A verage # of aﬀected trains Green 1 226 11.05 7.5 2 224 10.96 7.3 Orange 1 171 9.86 7.3 2 179 9.51 7.1 T able 2: Summary statistics of rep orted disruptions on the Green and Orange metro lines. T o accurately mo del p ost-disruption trav el times, it is necessary to deter- mine the precise onset of the p ost-disruption p erio d. Due to the aforemen- tioned inaccuracies in the disruption logging pro cedure, the rep orted ending timestamps cannot b e relied up on exclusively . In practice, a p ost-disruption p eriod b egins only when train op erations resume and the ﬁrst train departs from its station. A ccordingly , the departure of the ﬁrst train following a re- p orted disruption is used to establish the eﬀective ending timestamp of that disruption. This approach reﬂects the op erational pro cedure follow ed by the So ciété de transp ort de Montréal (STM) when managing service disruptions. Our preliminary analysis of train tra jectories around disruption p erio ds indicates that distinct op erational proto cols are enforced dep ending on the time of day , the duration of the inciden t, and the underlying cause of the disruption. In some instances, trains are required to stop immediately at the nearest station, whereas in other cases, trains are p ermitted to contin ue their journey for several stations before b eing forced to stop due to an obstructed path ahead. In yet other situations, some trains contin ue op erating during the disruption itself, visiting intermediate stations within the gap left b y the preceding train. Suc h cases, where trains contin ue to serve gap stations during an ongoing disruption, often lead to the accum ulation of multiple trains along the tracks. This phenomenon subsequen tly increases congestion and results in prolonged tra vel times in the p ost-disruption perio d. 8 3. Metho dology Giv en the unpredictabilit y of disruption durations and the p oten tial for emplo ying diﬀeren t op erational scenarios and management proto cols during a disruption, the most relev an t factor in this context is the instan taneous spa- tial distribution of trains along the trac ks and their relative spacing. When trains are p ositioned closer to one another, they tend to exp erience prolonged pro cess times, as they are required either to dw ell for longer p erio ds at sta- tions or to reduce their sp eed in tunnel sections in order to ensure that the trac k ahead remains clear and that a safe buﬀer distance is preserved relative to the preceding train. In the absence of explicit kno wledge ab out the precise disruption management proto col, the only reliable assumption is that trains resume mov emen t solely when the do wnstream path to subsequent stations is conﬁrmed to b e free of other trains. 3.1. Mo del F ormulation As discussed previously , our ob jective is to mo del the trav el time of trains to downstream stations, as these are display ed on screens to passengers wait- ing on the platform to assist them in making more informed trav el decisions. The p ost-disruption tra vel time can b e conceptually decomp osed in to t wo dis- tinct comp onen ts. The ﬁrst comp onent corresp onds to the dela y pro cess (D) , whic h arises from the accum ulation of trains along the trac ks follo wing a disruption. The second comp onen t represen ts the time required for a train to tra verse all track segmen ts — including b oth stations and tunnel sections — from its origin station to its destination station, which w e refer to as the journey pro cess (J) . W e denote the p ost-disruption trav el time b y Y i,j,k , deﬁned as the time tak en for train i to depart from station j and arrive at station k . Accordingly , the exp ectation of Y i,j,k , denoted by µ i,j,k , can b e expressed as: E [ Y i,j,k ] = µ i,j,k = D i,j + J i,j,k , (1) where D i,j denotes the delay exp erienced by train i at station j due to train bunc hing, and J i,j,k represen ts the time required for train i to complete its journey from departure at station j to arriv al at station k . Illustrations of these parameters are provided in the space–time diagram sho wn in Figure 2. F or eac h of these pro cesses, distinct modeling strategies are emplo y ed, reﬂecting the sp eciﬁc c haracteristics of the underlying dynamics they repre- sen t. 9 Time Station j k T rain i Start of disruption Resolution of disruption D i,j J i,j,k Y i,j,k Figure 2: Space–time diagram illustrating a representativ e p ost-disruption trav el time for train i from station j to station k . The trav el time is decomp osed into a delay comp onent, D i,j , deﬁned as the interv al betw een the time the disruption is rep orted as resolved and the departure of train i from station j , and a journey comp onent, J i,j,k , corresp onding to the time required for train i to trav el from station j to its arriv al at station k . 3.1.1. Mo deling Journey Pr o c ess (J) The journey pro cess corresp onds to the total time a train sp ends trav ers- ing b oth platforms and tunnel segments when trav eling from an origin station to a destination station. In our formulation, the journey is deﬁned as b e- ginning with the departure from the origin station ( j ) and concluding up on arriv al at the destination station ( k ). According to our con ven tion, the tun- nel section immediately follo wing station j is referred to as the j th tunnel (i.e., the segmen t betw een stations j and j + 1 ). With these deﬁnitions, J i,j,k can b e decomposed as the sum of the dwell times at intermediate stations j + 1 , . . . , k − 1 and the running times through the corresp onding tunnel sections. R unning times exhibit relativ ely low v ari- abilit y (right panel of Figure 3) b ecause train sp eeds are tightly regulated b y op erational constrain ts. The primary source of v ariability in journey times rather arises from dw ell times (left panel of Figure 3), whic h are inﬂuenced by the passenger b oarding and aligh ting pro cess [5, 6]. Higher passenger v olumes generally lead to longer dwell times; ho wev er, the relationship b et ween dwell time and passenger volume is not strictly linear, as it can be aﬀected b y n u- merous other factors [6, 12]. Among these, the station la yout — sp eciﬁcally , the num b er and lo cation of platform en trances — is particularly inﬂuential. These structural characteristics shap e the spatial distribution of passengers 10 on the platform, whic h in turn impacts the duration required to complete the passenger exc hange process. 50 75 100 5 10 15 20 25 Hour of da y Dw ell time (seconds) 30 40 50 60 5 10 15 20 25 Hour of da y Running time (seconds) Figure 3: Process times of trains at Guy-Concordia station on the Green metro line based on the hour of the day . Dwell times (left) exhibit substan tial temporal v ariability , with noticeably higher v ariance during p eak p erio ds, while their mean remains relativ ely stable aside from sligh t increases at rush hours. The running times of tunnel segmen t after the station platform (right) remain highly stable throughout the da y , showing no dep endence on time of day and maintaining nearly constan t v ariability . The x-axis indicates the hour of day , with p ost-midnigh t op erations display ed by adding 24 to the corresp onding hour. During a netw ork disruption, trains are typically held at stations un til the issue is resolv ed, whic h leads to the accum ulation of passengers at other platforms. Upon the arriv al of a train at a giv en station, the w aiting cro wd has b een accumulating since the departure of the preceding train. This in- terv al is called the he adway . More formally , w e denote the headw ay h i,j as the time gap b et ween the departure of train i − 1 from station j and the arriv al of train i at the same station. In the Montréal metro netw ork, train arriv als and departures are not go v- erned b y a ﬁxed timetable. Instead, op erations are regulated by adjusting the frequency at whic h trains are dispatched along a metro line. Conse- quen tly , during p eak hours with high passenger demand, headw a ys are rela- tiv ely short, whereas during oﬀ-p eak p erio ds, longer time gaps are observ ed b et w een consecutive trains (left panel of Figure 4). Headw ays are managed to preven t excessive cro wding at an y platform. How ev er, during a disruption, the balance of headwa ys is disrupted: certain stations may remain unvisited for the duration of the incident, leading to passenger accum ulation. These longer-than-usual headw ays result in extended passenger exc hange pro cess, whic h in turn increase the dw ell time of the train arriving at the aﬀected station once the disruption has b een resolv ed. 11 0 5 10 15 20 5 10 15 20 25 Hour of da y Headw a y (minutes) 40 60 80 5 10 15 20 25 Hour of da y Line journe y time (minutes) Figure 4: Headwa ys at Guy-Concordia station (left) and full Green line journey times (righ t) as a function of the hour of da y . Headwa ys sho w a pronounced temp oral pattern corresp onding to op erational adjustments b et w een p eak and oﬀ-p eak p erio ds. Line journey times exhibit higher mean and v ariance during p eak hours, yet remain relatively stable o ver the rest of the day . The x-axis denotes the hour of da y , with p ost-midnigh t operations represen ted by adding 24 to the corresp onding hour. Based on these considerations, the following mo del is prop osed for the journey pro cess during p ost-disruption p erio ds: J i,j,k = t 0 + ˜ T i,j,k + k − 1 X m = j +1 θ m ( h i,m − ˜ h i,m ) , (2) where ˜ T i,j,k denotes the median trav el time from station j to station k un- der normal operating conditions, and ˜ h i,m denotes the median headwa y at station m under normal operations ev aluated at the time train i arriv es at m . Both trav el time and headwa y are inherently time-dep enden t (Figure 4). T o accoun t for this temp oral v ariability , median v alues are computed using a 30-min ute rolling time windo w throughout the day . Importantly , the times- tamp of train i ’s arriv al at each station determines the sp eciﬁc time windo w from which the corresp onding median headw a y and trav el time v alues are extracted. An intercept term, t 0 , is included in this formulation to accoun t for any global oﬀset or systematic discrepancy b et ween p ost-disruption journey times and those observ ed under normal operating conditions. The third term on the righ t-hand side of eqn. 2 represen ts the sum of the added journey time exp erienced b y train i at all intermediate stations m b et ween the origin station j and the destination station k . Prolonged headw ay due to disruption leads to more w aiting passengers at the platform. The higher the headwa y imbalance h i,m − ˜ h i,m is, the more passengers will 12 arriv e at the station platform. The underlying mo deling assumption here is that there is a linear relationship b et w een the excess duration for which passengers hav e b een accum ulating at a platform and the time required for train i to pass the platform of station m during its journey [12]. F or ev ery additional minute of prolonged headw ay at an intermediary station m , the journey pro cess increases by θ m . 3.1.2. Mo deling Delay Pr o c ess (D) When the system resumes its operations after an incident, trains b egin to mov e again; ho wev er, due to the disruption, train bunching may o ccur along the trac ks. Consequen tly , some trains exp erience additional dela ys b efore they can con tinue their journeys. These dela ys, in turn, increase the estimated passenger waiting times for train arriv als at the downstream stations. Our mo deling strategy assumes that train op erations are inﬂuenced by the presence of other trains ahead of them in the direction of trav el. W e deﬁne segmen t j as the com bined track section consisting of station j and its subsequen t tunnel. Since the Montréal metro system aims to maintain a bal- anced headw ay b etw een consecutiv e trains, it in tro duces controlled delays to reestablish uniform spacing during the p ost-disruption p eriod. F or a sp eciﬁc train, w e mo del the dela y pro cess b y assuming that the presence of another train in several segmen ts ahead exerts a constan t eﬀect on the dela y of the curren t train. This eﬀect is b oth distance-dep endent and lo cation-dep enden t, meaning that the inﬂuence of leading trains on delay v aries dep ending on the sp eciﬁc station where the train was stopp ed during the incident. In our mo del, the dela y exp erienced by train i at station j is formulated as a linear com bination of the inﬂuence exerted b y other trains lo cated up to D max segmen ts ahead along the line: D i,j = D max X ℓ =1 γ ℓ,j z i,j,ℓ , (3) where z i,j,ℓ is a binary v ariable that tak es the v alue 1 if, when train i is at station j , another train is lo cated ℓ segmen ts ahead, and 0 otherwise. 3.1.3. V arianc e Sp e ciﬁc ation and T r avel Time Distribution T rav el times consist of the sum of running times and dwell times, so it is reasonable to h yp othesize that its v ariance increases with the tra veled dis- tance. Figure 5 presents the empirical v ariance of tra vel times for all train 13 tra jectories in 2018 as a function of the trav eled distance, measured by the n umber of stations. The data exhibit a clear trend whic h is nearly linear, supp orting the assumption that tra v el-time v ariance accum ulates prop ortion- ally with distance and that a linear functional form in k − j for the v ariance sp eciﬁcation in our mo del is a plausible choice. Additional empirical results supp orting this pattern are reported in App endix A.1. 0. 0 2. 5 5. 0 7. 5 10. 0 1 5 9 13 17 21 25 T ra v el distance from ori gin (in # of stations) V ariance of tra v el time (minutes 2 ) Figure 5: Empirical v ariance of journey times for all train tra jectories in 2018 on the Green line of the Mon tréal metro, shown for direction 1. The horizon tal axis represen ts the trav eled distance from the origin station, measured b y the num b er of stations trav eled, while the vertical axis indicates the v ariance of the corresp onding journey times. While tra v el times can, in theory , b ecome arbitrarily long due to factors suc h as disruptions and congestion, trains cannot complete their journeys faster than a certain lo wer b ound. This limitation arises from op erational constrain ts, including tunnel sp eed limits and the minim um dw ell time re- quired for braking, acceleration, and do or op erations when en tering or leaving stations. Consequently , w e exp ect Y i,j,k to b e asymmetric around its mean µ i,j,k and to exhibit a sk ewed distribution. Baseline mo del T o gain a clearer understanding of the b eha vior of the mo del error terms, w e construct a baseline mo del. In this baseline sp eciﬁcation, the error terms are assumed to follo w a normal distribution with a linearly increasing stan- dard deviation, capturing the heterogeneity present in the data (Figure 5). By combining eqns 1, 2, and 3, w e obtain the following baseline form ulation 14 for p ost-disruption tra v el times: Y i,j,k = µ i,j,k + ε i,j,k (4) µ i,j,k = t 0 + ˜ T i,j,k + k − 1 X m = j +1 θ m ( h i,m − ˜ h i,m ) + D max X ℓ =2 γ ℓ,j z i,j,ℓ (5) ε i,j,k ∼ N { 0 , ω 0 + ω 1 ( k − j ) } , (6) where ε i,j,k represen ts the baseline mo del error terms, assumed to follow a normal distribution N ( µ, σ 2 ) with mean µ and standard deviation σ . Figure 6 displa ys the model residuals against the predicted v alues, along with the empirical sk ewness of the error terms as a function of tra veled dis- tance for Direction 1 of the Green line. The plot of residual vs predicted v alues rev eals a sligh t misﬁt particularly for longer trips, indicating that the baseline mo del do es not adequately capture tra vel times ov er longer dis- tances. This asymmetry implies that the normalit y assumption for the error distribution is insuﬃcient, as the distribution b ecomes increasingly tilted. The residual sk ewness generally decreases with trav el distance, although the pattern is not entirely consisten t. A plausible explanation for the decreasing sk ewness is that dela ys and congestion tend to prolong tra vel times, while op erational constrain ts limit the o ccurrence of unusually short trav el times. A t the same time, minor sp eed adjustments and shorter dw ell pro cesses can b e implemented to partially mitigate the impact of dela ys. Consequen tly , o ver longer journeys, trains ha ve greater opp ortunit y to absorb initial de- la ys, leading to a reduction in sk ewness as tra v eled distance increases. A reasonable adjustment is therefore to allow the skewness parameter to v ary linearly with tra veled distance. In Figure 6, skewness is sho wn only for tra v eled distances up to 20 sta- tions. This restriction is due to op erational pro cedures that t ypically preven t the insertion of new trains along the line during a disruption, resulting in v ery few p ost-disruption observ ations with tra veled distances exceeding 20 stations. Since empirical skewness estimates are unreliable with very small samples, as they b ecome o v erly sensitive to outliers and mo del misﬁt, w e truncate the maxim um distance considered. A dditional empirical results for the remaining line and direction combinations are presen ted in App endix A.2. The presence of numerous extreme residual observ ations suggests that the normalit y assumption is inadequate, indicating the need to adopt a dis- tribution with hea vier tails. In the following sections we prop ose t wo mo dels based on sk ew-normal and skew-studen t- t (skew- t ) distributions. 15 0 10 0 10 20 30 40 Predicted tra v el time (minutes) Residuals (minutes) 2 4 6 5 10 15 20 T ra v eled Distance (k - j) Residual Ske wness Figure 6: Baseline mo del residuals plotted against predicted tra vel times (left) and em- pirical skewness of the residuals (righ t) for Direction 1 of Line 1. The results indicate non-constan t heterogeneity at larger predicted trav el times and pronounced p ositive skew- ness in the residual distributions, with skewness generally decreasing as trav eled distance increases. 3.1.4. Skew-Normal Distribution The skew-normal distribution is a contin uous probability distribution that extends the normal distribution by allo wing for non-zero skewness. Among the v arious existing parametrizations, w e adopt that of Azzalini [25, Sec- tion 2.1]. The sk ew-normal distribution has densit y f Y ( y ) = 2 ω ϕ y − ξ ω ! Φ α y − ξ ω ! , (7) where ξ ∈ R denotes the lo cation parameter, ω ∈ R + the scale parameter, and α ∈ R the skewness parameter, and where ϕ ( x ) and Φ( x ) = R x −∞ ϕ ( t )d t denote the standard normal densit y and distribution functions, resp ectively . The sk ew-normal distribution is a lo cation–scale family , and we may write Y ∼ S N ( ξ , ω , α ) ≡ ξ + ω Z with Z ∼ S N (0 , 1 , α ) ; its exp ectation and v ariance are E ( Y ) = ξ + ω α s 2 π (1 + α 2 ) , (8) V ar( Y ) = ω 2 ( 1 − 2 α 2 π (1 + α 2 ) ) . (9) 3.1.5. Skew- t Distribution The sk ew-normal distribution can b e generalized to accommodate hea vy- tailed data. Indeed, a random scale mixture of standard sk ew-normal with 16 Z = Z 0 / √ V , where Z 0 ∼ S N (0 , 1 , α ) and V ∼ χ 2 ν , yields the skew ed- t distribution [25, Section 4.3], which can b e made into a location-scale family b y taking Y = ξ + ω Z . The skew- t S T ( ξ , ω , α , ν ) density is f ( y ) = 2 ω t ( z ; ν ) T   αz s ν + 1 ν + z 2 ; ν + 1   , z = y − ξ ω ; (10) where ν > 0 , α ∈ R , and t ( · ; ν ) and T ( · ; ν ) are the density and distribution functions of a standard Student- t distribution with ν degrees of freedom, resp ectiv ely . The exp ectation and v ariance of Y ∼ S T ( ξ , ω , α, ν ) are E ( Y ) = ξ + ω α √ ν q π (1 + α 2 ) Γ { ( ν − 1) / 2 } Γ ( ν / 2) , ν > 1; (11) V ar( Y ) = ω 2   ν ν − 2 − 2 α 2 ν π (1 + α 2 ) " Γ { ( ν − 1) / 2 } Γ ( ν / 2) # 2   , ν > 2 . (12) 3.1.6. T r avel Time Err or Dep endenc e As p ost-disruption trains pro ceed tow ard their downstream stations, they ma y encoun ter congestion or dela ys at in termediate stops, resulting in longer tra vel times than those predicted by the mo del. During this recov ery phase, congestion eﬀects can propagate from one train to the next, leading subse- quen t trains to exp erience similar dela ys. Examination of the error terms in our baseline mo del (eqn. 4) for consecutive p ost-disruption trains reveals a p ositiv e correlation b et ween the baseline mo del residuals of successive trains (Figure 7). The additional empirical results for the error dep endence across diﬀeren t direction and line is presen ted in App endix A.3. When a disruption is declared resolved, trains b egin mo ving from diﬀeren t stations along the metro line. F or a giv en downstream destination station k , all p ost-disruption trains that w ere stopped at upstream stations b efore k during the disruption will ev en tually arrive at k . Except for the ﬁrst p ost- disruption train reac hing station k , all subsequen t trains trav erse segmen ts that hav e already b een trav eled by another p ost-disruption train; th us, their journeys from origin to destination partially ov erlap with those of preceding trains. F or instance, consider train i stopp ed at station j during the disruption and train i − 1 stopp ed at station j ′ . If w e wish to display the predicted w aiting times for trains i − 1 and i on the platform screens to passengers 17 0. 0 0. 2 0. 4 0. 6 0. 8 [ 0, 0. 2) [ 0. 2, 0. 4) [ 0. 4, 0. 6) [ 0. 6, 0. 8) [ 0. 8, 1] Ov erlap ratio bin Correlation Corre l a t i on t ype P e a rs on S pe a rm a n Figure 7: P earson and Sp earman correlations b et ween the baseline mo del residuals and those of the preceding trains, binned by the prop ortion of journey ov erlap, for the Green line of the Montréal metro system. The results rev eal p ositive correlations, indicating that delay and congestion propagate from one train to its successor. T rain journeys with a larger ov erlap in trav eled segments with the preceding train exhibit stronger p ositiv e correlations. w aiting at a downstream station k ( j < j ′ < k ), w e must estimate y i − 1 ,j ′ ,k and y i,j,k . The trac k segments b et ween j ′ and k are common to b oth trains, so b oth trains will pass through the same sections on their wa y to k . Since train i − 1 precedes train i , an y congestion encountered b y train i − 1 along these shared segments will app ear in its error term, ε i − 1 ,j ′ ,k , and ma y inﬂuence the tra vel time of train i , in tro ducing correlation b et w een their journeys. Figure 7 sho ws the Pearson and Sp earman correlation b etw een these con- secutiv e errors in our baseline model. Increasing pattern of b oth correlation co eﬃcien ts with journey ov erlap ratio indicates that propagation of delays is b oth monotonic and approximately linear, and b ecomes more pronounced as o verlap ratio increases. The magnitude of this eﬀect dep ends on the prop or- tion of shared segments b et ween the t wo journeys, expressed as ( k − j ′ ) / ( k − j ) whic h w e refer to as o verlap ratio. The closer the p ositions of trains i − 1 and i during the disruption, the greater the ov erlap ratio among their paths to ward station k , and therefore, an y congestion aﬀecting train i − 1 is more lik ely to hav e a pronounced impact on the trav el time of train i . W e extend our baseline model in eqn. 4 to incorporate the error term of the preceding train. There exists a one-to-one corresp ondence b etw een eac h train and its stopping lo cation during the inciden t and that of its preceding train, denoted by ( i, j ) → ( i − 1 , j ′ ) , where j < j ′ . F or an y destination station k lo cated downstream of b oth j and j ′ , the follo wing mean sp eciﬁcation is 18 prop osed: y i,j,k = µ ′ i,j,k + ρ j,j ′ ,k ε i − 1 ,j ′ ,k + ε i,j,k (13) where ρ j,j ′ ,k quan tiﬁes the correlation b etw een the residual of a giv en train and that of its preceding train. Based on the observed increase in correla- tions with the journey-o v erlap ratio, ρ j,j ′ ,k in eqn. 13 should b e an increasing function of the ov erlap ratio. Although v arious functional forms could b e con- sidered, w e adopt an exp onen tial speciﬁcation for the relationship b et w een ρ j,j ′ ,k and the o verlap ratio: ρ j,j ′ ,k = ρ ( 1 − exp − λ k − j ′ k − j !) (14) where ρ ≥ 0 and λ > 0 are mo del parameters. 3.1.7. Mo del Hier ar chy W e integrate our baseline model in eqn. 4 with skew ed error terms, and incorp orate the error correlation term deﬁned in eqn. 14 to mo del the trav el times of p ost-disruption trains tra v eling to ward downstream stations. F rom eqns 8, 9, 11, and 12, it is evident that the mean–v ariance relationship in the skew-normal and skew- t distributions diﬀers from that of the normal and Studen t- t distributions, and to ensure that the exp ectation of Y i,j,k equals µ i,j,k , an adjustmen t term in eqn. 4 m ust b e incorp orated. W e assume a linearly c hanging speciﬁcation for the skewness parameter, denoted by α 0 + α 1 ( k − j ) . F or the scale parameter, our mo deling strategy diﬀers from that of the baseline model. In the baseline mo del of eqn. 4, the motiv ation for a linearly increasing v ariance sp eciﬁcation stems from the ad- ditiv e structure of tra vel times: total tra v el time is the sum of running and dw ell times b et w een the origin and destination stations. Under the assump- tion that these running and dw ell pro cesses are appro ximately indep endent across segments, the ov erall v ariance of trav el time can b e expressed as the sum of segment-lev el v ariances, implying a linear dep endence on the trav eled distance k − j . F or the sk ew-normal and ske w- t distributions, the v ariance in eqns 9 and 12 tak es a more complex form, as it dep ends not only on the scale pa- rameter ω , but also on the skewness parameter and, in the case of the skew- t distribution, the degrees of freedom. Nev ertheless, in b oth distributions the v ariance can be written as ω 2 m ultiplied b y a function of the remaining 19 parameters. T o preserv e the same in tuitive in terpretation of v ariance accu- m ulation with distance while accoun ting for the distribution-sp eciﬁc v ariance structure, w e therefore mo del the squared scale parameter as a linear func- tion of the tra v eled distance, ω 2 kj = ω 0 + ω 1 ( k − j ) . This speciﬁcation allo ws the o verall v ariance to increase approximately linearly with distance, while ﬂexibly accommo dating the eﬀects of skewness and tail heaviness inherent to the skew-normal and sk ew- t distributions. A distinction must also b e made b etw een the ﬁrst p ost-disruption train that reac hes a destination station k and all subsequen t p ost-disruption trains. The ﬁrst p ost-disruption train is the initial train to arriv e at station k fol- lo wing the resolution of an inciden t. Because it has no preceding train with whic h it shares an y p ortion of its journey , no lagged residual term is included; its trav el time is therefore mo deled using the mean sp eciﬁcation in eqn. 4. In contrast, the tra vel times of all subsequent p ost-disruption trains arriving at station k are mo deled according to eqn. 13. W e therefore express y i,j,k as follo ws: Y i,j,k =    µ ′ i,j,k + ε i,j,k , if i is the ﬁrst p ost-disruption train µ ′ i,j,k + ρ j,j ′ ,k ε i − 1 ,j ′ ,k + ε i,j,k , otherwise (15) where, under the prop osed sk ew-normal sp eciﬁcation, µ ′ i,j,k = µ i,j,k − s 2 π { α 0 + α 1 ( k − j ) } q ω 0 + ω 1 ( k − j ) q 1 + ( α 0 + α 1 ( k − j )) 2 (16) ε i,j,k ∼ S N  0 , q ω 0 + ω 1 ( k − j ) , α 0 + α 1 ( k − j )  (17) and under the prop osed sk ew- t sp eciﬁcation, µ ′ i,j,k = µ i,j,k − √ ν Γ { ( ν − 1) / 2 } √ π Γ ( ν / 2) { α 0 + α 1 ( k − j ) } q ω 0 + ω 1 ( k − j ) q 1 + { α 0 + α 1 ( k − j ) } 2 (18) ε i,j,k ∼ S T  0 , q ω 0 + ω 1 ( k − j ) , α 0 + α 1 ( k − j ) , ν  (19) where, in b oth mo dels, µ i,j,k follo ws eqn. 5, and ρ j,j ′ ,k follo ws eqn. 14. The full set of mo del parameters includes θ m , γ ℓ,j , t 0 , ω 0 , ω 1 , α 0 , α 1 , ρ , and λ for b oth mo dels, with the skew- t mo del con taining one additional degree-of-freedom parameter ν . 20 3.2. Pr e dictive distribution with dep endent err ors Recall that the p ost-disruption trav el times are mo deled using a mo ving- a verage structure across consecutiv e trains, conditional on spatial ov erlap, giv en b y Y i,j,k = µ i,j,k + ρ j,k ε i − 1 ,j ′ ,k + ε i,j,k , (20) where ε i,j,k denotes an indep enden t inno v ation term following either a skew- normal or a skew- t distribution with scale ω kj = { ω 0 + ω 1 ( k − j ) } 1 / 2 , skewness α kj = α 0 + α 1 ( k − j ) , and, in the case of skew- t innov ations, degrees of freedom ν . Conditioning on the innov ation of the preceding train, the predictiv e dis- tribution of the realization y i,j,k tak es the form of a lo cation-shifted distribu- tion, y i,j,k | ε i − 1 ,j ′ ,k ∼ S N  µ ′ i,j,k + ρ j,k ε i − 1 ,j ′ ,k , ω kj , α kj  , (21) for skew-normal innov ations, where µ ′ i,j,k is deﬁned in eqn. 16, and y i,j,k | ε i − 1 ,j ′ ,k ∼ S T  µ ′ i,j,k + ρ j,k ε i − 1 ,j ′ ,k , ω kj , α kj , ν  , (22) for skew- t innov ations, where µ ′ i,j,k follo ws from eqn. 18. The unconditional predictiv e distribution is obtained by in tegrating out the latent innov ation of the preceding train, p ( y i,j,k ) = Z p ( y i,j,k | ε i − 1 ,j ′ ,k ) p ( ε i − 1 ,j ′ ,k ) d ε i − 1 ,j ′ ,k , (23) whic h results in a lo cation mixture of skew- t distributions and generally do es not admit a closed-form expression. In practice, p osterior predictiv e inference is carried out via sim ulation by join tly sampling the mo del parameters and laten t error terms, thereb y fully capturing both heavy-tailed b eha vior and dep endence across trains. 3.3. Prior sp e ciﬁc ation W e adopt w eakly informative priors for all mo del parameters to regu- larize estimation while remaining agnostic ab out their precise v alues. Lo ca- tion, intercept, and regression parameters are assigned zero-cen tered Gaus- sian priors, reﬂecting prior symmetry and mo derate uncertaint y . Sp eciﬁ- cally , the baseline trav el-time in tercept t 0 and regression co eﬃcient θ m , for m = 2 , . . . , J − 1 where J is the n umber of stations in eac h case study , 21 are assigned standard normal priors, t 0 , θ m ∼ N (0 , 1) . The cov ariate ef- fect parameters γ ℓ,j are assigned independent zero-centered Gaussian pri- ors with relatively large v ariance, γ ℓ,j ∼ N (0 , 5) , whic h provides suﬃcien t ﬂexibilit y in their magnitudes while main taining mild regularization. Here, ℓ = 1 , . . . , D max and j = 1 , . . . , J index the distance and stations, resp ec- tiv ely . The scale parameters gov erning v ariance accumulation are constrained to b e p ositiv e and are assigned w eakly informativ e Gaussian priors truncated ab o v e zero, namely ω 0 ∼ N + (1 , 1) and ω 1 ∼ N + (1 , 1) , whic h ensures p os- itivit y of the squared scale parameter ov er the observ ed range of tra v eled distances. It is imp ortan t to note that although this constrain t enforces a monotonic increase in the squared scale of b oth the skew-normal and skew- t distributions, it do es not necessarily imply monotonic b eha vior of the v ari- ance itself. As shown in eqns 9 and 12, the v ariance also dep ends on the sk ewness parameter, which, dep ending on the chosen skewness sp eciﬁcation, ma y induce a v ariet y of functional forms with resp ect to the distance trav eled. F or the sk ewness sp eciﬁcation, the intercept and slop e parameters are assigned standard normal priors α 0 , α 1 ∼ N (0 , 1) , which encourages mo der- ate skewness while allo wing the data to determine both its magnitude and direction. The correlation parameter ρ ∈ ( − 1 , 1) is obtained through a logistic trans- formation of an unconstrained parameter, ρ = 2 logit − 1 ( ρ raw ) − 1 , ρ raw ∼ N (0 , 1) , where logit − 1 ( x ) = (1 + e − x ) − 1 , implying a prior that is approximately uni- form ov er the admissible correlation range. The additional regression co- eﬃcien t λ is assigned a standard normal prior λ ∼ N (0 , 1) . Finally , the degrees of freedom ν of the skew- t distribution is assigned a Gamma prior with a shap e–rate parameterization, ν ∼ Gamma (2 , 0 . 1) . This prior fav ors mo derately hea vy-tailed distributions while retaining suﬃcient probability mass o ver larger v alues of ν , thereb y allowing the mo del to appro ximate the Gaussian case when supp orted b y the data. 3.4. Mo del Estimation and Computation W e ﬁtted the prop osed mo del describ ed b y eqns 15, 16, and 18 to b oth the Green and Orange lines of the Montréal metro net work data. Posterior inference for all mo del parameters was carried out using Hamiltonian Monte 22 Carlo [cf. 26, 27] as implemented in Stan through R [28, 29]. The full set of parameters dra wn from the joint p osterior via Marko v c hain Mon te Carlo includes the mean sp eciﬁcation parameters ( t 0 , θ m , γ ℓ,j ) , the scale parameters ( ω 0 , ω 1 ) , the skewness parameters ( α 0 , α 1 ) , the correlation parameter ( ρ , λ ), and, for the skew- t sp eciﬁcation, the degrees of freedom ν . Eac h mo del w as ﬁtted using four indep enden t c hains, with 1,000 w arm- up iterations follo wed by 4,000 sampling iterations p er c hain. Con vergence w as assessed using visual insp ection of trace plots, whic h are rep orted in Ap- p endix A.4. The lo west rep orted eﬀective sample size (ESS) [cf. 30], which accoun ts for auto correlation, exceeded 10,500 for sk ew-normal mo del param- eters and 7,000 for skew- t mo del, indicating satisfactory mixing of the Mark ov c hains. 4. Results T o construct the dataset, the p ositional enco ding was deﬁned using a max- im um lo ok-ahead distance of ﬁv e segments ( D max = 5 ). The analysis fo cused exclusiv ely on the tra jectories of trains that were in op eration at the time of a disruption, as identiﬁed from the disruption logs, and their trav el times to do wnstream stations. Each trav el time to a do wnstream station was treated as an individual observ ations in the dataset. As previously noted, only week- da y op erations w ere considered, with public holida ys excluded according to the oﬃcial So ciété de transport de Montréal (STM) holida y calendar. Eac h metro line and both trav el directions w ere mo deled separately , re- sulting in distinct datasets for eac h case. T o ev aluate mo del p erformance, eac h dataset w as divided in to in-sample and out-of-sample subsets based on the disruption log iden tiﬁers. T o minimize correlations b et ween training and testing samples, the data were split b y disruption ev en t rather than b y in- dividual observ ation. F or eac h dataset, 90% of the disruptions were used for mo del ﬁtting, while the remaining 10% w ere reserved for out-of-sample v alidation. T able 3 shows the num b er of observ ations in our in-sample and out-of-sample subsets in each case study . 4.1. The eﬀe ct of longer-than-usual he adway, θ m T able 4 presen t the estimated θ m parameters in skew-normal and sk ew- t mo dels on the Green line of the Montréal metro system in Direction 1. It is imp ortan t to note that these parameters represent the eﬀect of longer- than-usual headwa y on the trav el time of trains passing through intermediate 23 Green line (line 1) Orange line (line 2) Sample Direction 1 Direction 2 Direction 1 Direction 2 In-sample 18,942 17,362 12,439 12,618 Out-of-sample 2,043 2,092 1,637 1,457 T able 3: Number of in-sample and out-of-sample trav el time observ ations stations. Therefore, the parameters are not deﬁned for the ﬁrst and last sta- tions along eac h metro line. A ccording to eqn. 5, the parameters θ m capture the eﬀects asso ciated with the term h i,m − ˜ h i,m . This quan tity represents the deviation of the realized headwa y during the p ost-disruption perio d at station m from the corresp onding headw a y under normal op erating condi- tions. It reﬂects the excess time gap b et w een successive trains during the p ost-disruption p erio d relative to typical op erations and th us directly relates to the additional time during whic h passengers accum ulate on station plat- forms compared to normal conditions. It is imp ortan t to note, ho wev er, that despite b eing referred to as a “more-than-usual” gap, this quan tit y ma y o c- casionally tak e negativ e v alues, indicating that the realized p ost-disruption headw ay is shorter than the headw ay observ ed during normal op erations. The θ m v alues represen t the additive eﬀect on trav el time (in seconds) for ev- ery additional minute of headw ay caused by a disruption when a train passes a sp eciﬁc station during the p ost-disruption p erio d. F or instance, a p osterior mean of θ Monk = 2 . 4 for the skew-normal mo del implies that trains passing Monk station during p ost-disruption perio ds exp erience an additional 2 . 4 seconds of trav el time for every additional min ute that the headwa y at Monk station was extended due to a disruption. Although these eﬀects ma y app ear small in isolation, they can accum ulate o ver longer journeys follo wing ma jor disruptions, leading to a substan tial increase in total trav el time. An alternative p ersp ectiv e is to compare these eﬀects relativ e to the nor- mal op erational conditions, particularly in terms of dw elling times at sta- tions. As previously stated, total trav el time consists of the sum of running and dw ell times across all intermediate tunnels and stations, with the pri- mary source of v ariabilit y b eing the uncertaint y in dwell times. The "% of median dw ell time" column of in T able 4 expresses θ m as a p ercentage of the median dwell time at the corresp onding station. F or instance in the skew- normal mo del, at Monk station, where θ corresponds to 6 . 39% of the median dw ell time, a 10-minute increase of headwa y (caused b y a disruption) com- 24 pared to normal operation, results in a 63 . 9% increase in the dwell time for the trains passing through Monk once the disruption is resolved. F or the ma jority of stations, the p osterior sample means of the corre- sp onding θ m parameter are observ ed to be close to zero, suggesting that longer-than-usual headwa ys ha ve minimal impact on the trav el time of trains passing through stations during p ost-disruption p erio ds. Although service in- terruptions may initially result in passenger accum ulation on platforms, the w aiting time often allo ws passengers to disp erse to w ard less crowded areas while waiting for the next train, whic h results in a more eﬃcien t passenger exc hange when the next train arriv es. F urthermore, during prolonged dis- ruptions, man y passengers are likely to shift to alternative transp ortation mo des, suc h as buses, leading to a gradual clearing of platforms [31]. There- fore, when normal op erations resume after extended disruptions, platform cro wding is typically limited, whic h explains the near-zero estimates of the θ m parameters. 4.2. The eﬀe ct of tr ain formation p ar ameters, γ ℓ,j The eﬀect of train presence in downstream segments is captured b y the station-sp eciﬁc train formation parameters γ ℓ,j , which quantify the additional dela y exp erienced b y a train departing from station j immediately after an inciden t is resolved, when another train is lo cated ℓ segments ahead. The p osterior means of these parameters are presented in Figure 8 for Direction 1 of the Green metro line for b oth sk ew-normal and skew- t mo dels. γ ℓ,j param- eters capture the magnitude of dela y propagation along the line, reﬂecting the spatial inﬂuence of upstream train formations on p ost-disruption reco very dynamics. T o provide a clearer understanding of the interpretation of these parameters, consider the following example. F or P eel station in the sk ew- t mo del, the p osterior mean of γ ℓ,j at a distance of one segmen t is estimated to b e 70 . 1 seconds, indicating that a train remaining stationary at P eel during an inciden t will exp erience an additional 70 . 1 seconds of trav el time to its do wnstream stations once op erations resume if another train is lo cated one segmen t ahead. W e can observe in Figure 8, that the inﬂuence of other trains on the dela y pro cess of the curren t train v aries substan tially across stations and o ver diﬀeren t distances. Smaller dela y eﬀects are observed near the termi- nal sections of the lines. This pattern can b e attributed to the fact that the endpoints of the lines are t ypically lo cated in less densely populated ar- eas of the city , where in ter-station distances are longer and, consequently , 25 Skew-normal Skew- t m Parameter Posterior mean (seconds) ( 5%, 95%) % of median dwell time Posterior mean (seconds) ( 5%, 95%) % of median dwell time 2 θ Monk 2 . 4 ∗ ( 0 . 8 , 3 . 9 ) 6 . 39% 1 . 3 ( − 0 . 3 , 2 . 9 ) 3 . 49% 3 θ Jolicoeur 1 . 9 ∗ ( 0 . 5 , 3 . 4 ) 5 . 19% 1 . 3 ( 0 . 0 , 2 . 7 ) 3 . 61% 4 θ V erdun − 0 . 7 ( − 1 . 7 , 0 . 2 ) − 1 . 99% − 1 . 7 ∗ ( − 2 . 4 , − 0 . 9 ) − 4 . 47% 5 θ De l’Église 1 . 1 ∗ ( 0 . 2 , 2 . 0 ) 2 . 93% 2 . 5 ∗ ( 1 . 9 , 3 . 1 ) 6 . 55% 6 θ LaSalle − 2 . 6 ∗ ( − 3 . 5 , − 1 . 6 ) − 6 . 76% − 2 . 6 ∗ ( − 3 . 2 , − 2 . 0 ) − 6 . 83% 7 θ Charlevoix − 0 . 4 ( − 1 . 2 , 0 . 4 ) − 1 . 03% 0 . 4 ( − 0 . 2 , 1 . 0 ) 1 . 00% 8 θ Lionel-Groulx 1 . 1 ∗ ( 0 . 6 , 1 . 5 ) 2 . 25% 0 . 9 ∗ ( 0 . 4 , 1 . 4 ) 1 . 94% 9 θ At water 0 . 2 ∗ ( 0 . 0 , 0 . 4 ) 0 . 52% − 0 . 1 ( − 0 . 6 , 0 . 2 ) − 0 . 33% 10 θ Guy-Concordia − 0 . 2 ∗ ( − 0 . 5 , 0 . 0 ) − 0 . 54% 0 . 1 ( − 0 . 1 , 0 . 3 ) 0 . 15% 11 θ Peel 0 . 3 ( 0 . 0 , 0 . 6 ) 0 . 70% − 0 . 3 ( − 0 . 8 , 0 . 3 ) − 0 . 63% 12 θ McGill 0 . 6 ∗ ( 0 . 3 , 0 . 9 ) 1 . 39% 1 . 2 ∗ ( 0 . 7 , 1 . 5 ) 2 . 76% 13 θ Place-des-Arts 0 . 1 ( − 0 . 2 , 0 . 3 ) 0 . 17% − 0 . 3 ∗ ( − 0 . 5 , − 0 . 1 ) − 0 . 81% 14 θ Saint-Lauren t − 0 . 6 ∗ ( − 0 . 9 , − 0 . 3 ) − 1 . 46% − 0 . 1 ( − 0 . 3 , 0 . 1 ) − 0 . 19% 15 θ Berri-UQAM 0 . 1 ( − 0 . 1 , 0 . 4 ) 0 . 27% 0 . 1 ( − 0 . 2 , 0 . 3 ) 0 . 09% 16 θ Beaudry − 0 . 1 ( − 0 . 4 , 0 . 2 ) − 0 . 35% 0 . 3 ∗ ( 0 . 1 , 0 . 6 ) 0 . 85% 17 θ Papineau 0 . 0 ( − 0 . 3 , 0 . 3 ) − 0 . 02% − 0 . 2 ( − 0 . 5 , 0 . 0 ) − 0 . 57% 18 θ F rontenac 0 . 4 ∗ ( 0 . 1 , 0 . 7 ) 1 . 14% 0 . 3 ∗ ( 0 . 0 , 0 . 5 ) 0 . 72% 19 θ Préfontaine − 0 . 4 ∗ ( − 0 . 6 , − 0 . 2 ) − 1 . 04% − 0 . 3 ∗ ( − 0 . 6 , − 0 . 1 ) − 0 . 90% 20 θ Joliette 0 . 2 ( 0 . 0 , 0 . 4 ) 0 . 48% 0 . 0 ( − 0 . 1 , 0 . 2 ) 0 . 10% 21 θ Pie-IX − 0 . 1 ( − 0 . 3 , 0 . 2 ) − 0 . 13% 0 . 1 ( 0 . 0 , 0 . 3 ) 0 . 34% 22 θ Viau 0 . 0 ( − 0 . 2 , 0 . 3 ) 0 . 02% − 0 . 2 ( − 0 . 3 , 0 . 0 ) − 0 . 42% 23 θ Assomption 0 . 1 ( − 0 . 1 , 0 . 4 ) 0 . 36% 0 . 1 ( − 0 . 1 , 0 . 3 ) 0 . 22% 24 θ Cadillac 0 . 4 ∗ ( 0 . 1 , 0 . 7 ) 1 . 07% 0 . 2 ∗ ( 0 . 0 , 0 . 4 ) 0 . 57% 25 θ Langelier 0 . 1 ( − 0 . 3 , 0 . 5 ) 0 . 21% 0 . 0 ( − 0 . 3 , 0 . 3 ) 0 . 02% 26 θ Radisson 0 . 1 ( − 0 . 3 , 0 . 5 ) 0 . 20% 0 . 0 ( − 0 . 3 , 0 . 3 ) 0 . 05% T able 4: P osterior summaries of the θ m parameters for the sk ew-normal and sk ew- t mo dels on the Green line of the Mon treal metro system in Direction 1. The estimates are inter- preted as the change in trav el time (in seconds) asso ciated with a one-minute increase in headw ay . Posterior means marked with an asterisk ( ∗ ) indicate strong evidence of either a p ositiv e or negativ e eﬀect, as determined by the 90% credible interv al. The column “% of median dwell time” rep orts each θ m estimate as a p ercen tage of the median dwell time at the corresp onding station. Rows shown in b old corresp ond to stations lo cated in the do wnto wn area. the impact of other trains’ presence is reduced. A t the v ery end of each line, where no do wnstream segments exist, the corresp onding estimated pa- rameters closely resem ble their prior v alues, eﬀectiv ely zero. The parameter asso ciated with the terminal station Honoré-Beaugrand in b oth mo dels in Figure 8 approac hes zero b ecause no trains can b e lo cated ahead when a train departs from that station, rendering the delay process indep enden t of distance. Similarly , for stations lo cated near the end of the line but with a few do wnstream segmen ts remaining, the induced delay eﬀects from trains ahead remain relativ ely small, consistent with the b eha vior observ ed near 26 3.1 −28.6 −13.1 −28.5 −52.4 2.2 −47.0 −78.5 8.4 1.3 57.8 38.9 28.9 −6.6 −70.9 52.3 17.1 19.5 −16.6 7.7 67.5 9.3 −3.8 −7.6 2.4 45.8 21.7 2.2 18.7 −6.1 62.6 26.0 12.8 −6.0 −24.9 57.5 30.4 18.6 −10.0 −0.2 70.2 35.9 8.8 15.3 1.6 51.3 21.0 38.0 6.5 −1.0 63.1 37.0 4.8 2.5 3.6 78.4 25.0 14.9 −13.4 −9.8 60.7 18.1 19.6 4.6 −7.9 57.4 31.9 25.5 −5.6 −14.0 67.2 25.8 −1.0 5.9 10.4 59.8 −18.8 0.1 38.4 5.9 47.5 11.5 −4.5 4.5 2.1 54.4 21.4 8.5 2.6 2.2 74.3 23.6 9.1 2.8 −9.1 71.5 1.0 2.0 21.9 5.1 −39.3 −6.3 −24.4 −14.2 22.3 34.1 21.7 27.0 −23.2 8.2 43.3 6.4 25.0 18.0 1.2 65.8 19.5 23.4 −1.4 −0.1 66.4 24.1 −1.0 0.7 0.4 86.9 2.9 −1.0 −1.2 1.9 −0.3 −0.8 1.9 0.1 1.6 Downtown Stations 1 2 3 4 5 Angrignon Monk Jolicoeur V erdun De L 'eglise Lasalle Charlevoix Lionel−Groulx Atwater Guy−Concordia Peel Mcgill Place−Des−Arts Saint−Laurent Berri−Uqam Beaudry Papineau Frontenac Prefontaine Joliette Pie−Ix V iau Assomption Cadillac Langelier Radisson Honore−Beaugrand Origin station Distance (segments) Skew−normal model −0.3 −15.6 2.0 −15.9 −22.8 0.1 −13.5 −48.2 14.7 1.3 73.2 39.1 33.1 0.5 −50.8 58.0 40.0 15.4 −16.3 2.5 82.5 3.8 1.9 −2.2 −2.3 54.5 21.1 6.8 8.7 0.7 54.2 18.0 3.2 11.5 −13.4 52.4 22.6 27.3 −5.7 −1.8 71.2 36.6 8.7 13.0 3.7 66.1 34.7 23.2 −5.6 −7.8 70.1 35.3 6.5 7.2 1.5 72.7 32.0 16.9 −12.4 −8.2 72.6 22.6 22.7 −1.6 −10.0 77.8 24.8 21.1 −6.7 −9.3 71.0 29.3 0.2 −0.8 7.3 58.8 −9.3 3.1 24.2 1.3 48.1 6.6 −8.2 0.1 1.2 57.7 18.5 9.4 3.1 1.5 50.3 32.4 9.7 10.7 −8.8 55.4 9.6 −4.1 4.0 −3.5 41.4 22.0 8.7 −4.9 0.9 41.3 13.3 19.6 −11.6 6.5 55.9 −0.4 17.8 12.6 −1.3 63.2 10.4 11.5 1.8 −1.8 52.6 7.2 −1.2 0.7 −3.0 58.5 2.0 1.2 −0.9 −3.1 1.2 −1.8 −3.4 2.4 1.0 Downtown Stations 1 2 3 4 5 Angrignon Monk Jolicoeur V erdun De L 'eglise Lasalle Charlevoix Lionel−Groulx Atwater Guy−Concordia Peel Mcgill Place−Des−Arts Saint−Laurent Berri−Uqam Beaudry Papineau Frontenac Prefontaine Joliette Pie−Ix V iau Assomption Cadillac Langelier Radisson Honore−Beaugrand Origin station Distance (segments) Skew−t model Figure 8: Posterior means of the train formation parameters γ ℓ,j for p ost-disruption op- erations on Direction 1 of the Green line, sho wn for the sk ew-normal mo del (top) and the sk ew- t mo del (bottom). All estimates are rep orted in seconds and represent the av erage additional delay incurred by a train at a given lo cation when another train is p ositioned ℓ segments ahead. Stations in the down to wn area are highlighted in b old, and estimates displa yed in black indicate strong evidence of either a p ositiv e or negative eﬀect based on the 90% credible interv al. Cell colors represent the sign and magnitude of the delay eﬀect, with blue tones corresp onding to p ositiv e v alues and orange tones to negativ e v alues; color in tensity reﬂects magnitude and fades tow ard zero. the opp osite end of the line. A subtle distinction b etw een the proposed models can b e observ ed in Figure 8. In the skew-normal speciﬁcation, the estimated γ ℓ,j parameters exhibit more abrupt v ariations, particularly across distance, but also across lo cation. A clear illustration of this b eha vior is observ ed at the Pie–IX station, where the estimated γ ℓ,j v alues tak e large negativ e magnitudes for distances of ℓ = 1 , 2 , and 4 . This pattern can b e attributed to the presence of outliers in the data, to which the skew-normal mo del is more susceptible and therefore more prone to o v erﬁtting relative to the skew- t mo del. Owing to its heavier tails, the sk ew- t mo del assigns less weigh t to outlier observ ations 27 during parameter estimation, leading to smo other and more uniform v aria- tions across b oth lo cation and distance when compared with the skew-normal mo del. In Figure 9, ridge plots of the posterior samples of the train spacing parameters are presen ted to compare train formation spacing eﬀects across stations for Direction 1 of the Green metro line. The ﬁgure displays the densit y estimates obtained under b oth prop osed mo dels, namely the sk ew- normal and skew- t sp eciﬁcations, for tw o station groups: (i) stations lo cated in the do wnto wn area and (ii) stations lo cated outside the down town area. The comparison is p erformed for spacing distances ranging from 1 to 5. As exp ected, shorter spacing to the preceding train (i.e., smaller distance) is asso ciated with larger dela ys. This eﬀect is more pronounced for down- to wn stations than for non-down town stations under b oth mo del sp eciﬁca- tions. Moreov er, the p osterior samples for down to wn stations exhibit lo wer v ariability relative to those for non-do wnto wn stations. The magnitude of the spacing eﬀect decreases as the distance increases, and the diﬀerence in mean eﬀects b et w een the tw o station groups diminishes accordingly . The t wo prop osed mo dels yield similar mean eﬀects across b oth station groups. Ho wev er, the sk ew- t mo del exhibits relativ ely low er p osterior v ariance in γ ℓ,j compared with the sk ew-normal mo del. Corresponding results for the re- maining case studies, including the opp osite direction of the Green line and b oth directions of the Orange line, are rep orted in App endix B.2. 4.3. Distributional p ar ameters In addition to the mean sp eciﬁcation parameters discussed in the previous subsections ( θ m and γ ℓ,j ), we examine the remaining distributional parame- ters of the mo dels deﬁned in eqn. 17 and eqn. 19. These include the scale sp eciﬁcation parameters ω 0 and ω 1 , as well as the skewness sp eciﬁcation pa- rameters α 0 and α 1 , whic h are common to b oth the sk ew-normal and skew- t mo dels. In addition, the degrees of freedom ν is considered for the skew- t mo del. Posterior summaries of these parameters are rep orted in T able 5 for b oth prop osed mo dels applied to the Green metro line in Direction 1. As describ ed earlier in Section 3.1.7, the squared scale parameter in b oth the sk ew-normal and skew- t models is speciﬁed as a linear function of the tra veled distance k − j , with an in tercept ω 0 and a slop e ω 1 . A comparison of the estimation results rep orted in T able 5 indicates that the sk ew- t mo del yields smaller v alues for b oth ω 0 and ω 1 . This implies that the scale of the error terms is low er across diﬀerent trav el distances and increases more slo wly 28 Skew−normal Skew−t −100−50 0 50 100 150 γ 1 , j (seconds) Model Distance 1 −100 −50 0 50 100 γ 2 , j (seconds) Distance 2 −100 −50 0 50 100 γ 3 , j (seconds) Station group Non−do wntown stations Do wntown stations Distance 3 −50 0 50 γ 4 , j (seconds) Distance 4 −100 −50 0 50 γ 5 , j (seconds) Distance 5 Figure 9: Ridge plots of the train formation parameters γ ℓ,j for Direction 1 of the Green metro line under b oth proposed mo dels and for tw o station groups, namely down to wn and non-down town stations. F or each distance, the corresp onding density represen ts the p osterior samples of γ ℓ,j , where j b elongs to the set of station indices within the resp ective group. The v ertical lines indicate the group-sp eciﬁc mean eﬀect for each mo del, with colors distinguishing the station groups. Sk ew-normal Sk ew-t P arameter P osterior mean (5%, 95%) P osterior mean (5%, 95%) ω 0 2.300 ( 2.168, 2.436) 0.460 ( 0.424, 0.497) ω 1 0.163 ( 0.139, 0.185) 0.081 ( 0.074, 0.089) α 0 2.158 ( 2.064, 2.254) 2.329 ( 2.157, 2.508) α 1 − 0.031 ( − 0.041, − 0.020) − 0.065 ( − 0.076, − 0.054) ν 2.666 ( 2.561, 2.773) T able 5: P osterior summaries of distributional parameters in our prop osed sk ew-normal and skew- t mo dels for the Green metro line in Direction 1. 29 1.0 1.5 2.0 2.5 0 10 20 T raveled Distance (k − j) Scale Parameter (minutes) Skew−normal Ske w−t 0.5 1.0 1.5 2.0 2.5 0 10 20 T raveled Distance (k − j) Ske wness Parameter Skew−normal Ske w−t Figure 10: The left panel illustrates the estimated scale parameter p ω 0 + ω 1 ( k − j ) , while the right panel sho ws the estimated skewness parameter α 0 + α 1 ( k − j ) , together with p oin twise 90% credible in terv als, for the skew-normal and skew- t mo dels applied to the Green metro line in Direction 1 across diﬀerent trav eled distances. with distance in the skew- t model relative to the skew-normal sp eciﬁcation. In contrast, the skew-normal mo del assigns a larger scale parameter, whic h is directly related to v ariance, in order to accommo date the presence of large tra vel time realizations in the data. Ho w ever, due to its heavy-tailed prop erty , the sk ew- t mo del is less inﬂuenced by outlier observ ations, ultimately leading to substantially smaller estimated scale parameters and narrow er prediction in terv als. The estimated scale and sk ewness parameters are display ed in Figure 10. As shown, the scale parameter follows a quadratic functional form in trav eled distance, although its b ehavior is close to linear ov er the relev ant range, with the sk ew- t mo del consistently exhibiting smaller scale v alues. A ccording to the estimates rep orted in T able 5, b oth mo dels estimate a p ositiv e in tercept in the skewness sp eciﬁcation ( α 0 ) along with a small negative slop e ( α 1 ). As illustrated in the righ t panel of Figure 10, the estimated skewness parameter in the sk ew- t model decreases more rapidly as tra v eled distance increases. The negativ e slop e α 1 in the sk ewness sp eciﬁcation indicates that, for short tra vel distances immediately following a disruption, trains are more lik ely to exp erience substan tial dela ys, leading to trav el times that are considerably longer than exp ected while remaining b ounded below. F or longer journeys, ho wev er, trains are able to gradually recov er from schedule deviations, and the resulting trav el time distribution b ecomes closer to symmetric, exhibiting reduced skewness. The estimated degrees of freedom in the skew- t mo del, ν = 2 . 66 , indicates that p ost-disruption tra vel times exhibit pronounced hea vy-tailed b ehavior, 30 reﬂecting the presence of substan tial tail risk in the distribution. 4.4. Err or dep endenc e p ar ameters The ﬁnal comp onent of the prop osed framew ork is the error-dep endence structure introduced in Section 3.1.6. As sp eciﬁed in eqn. 14, the dep endence b et w een the tra vel time errors of t w o consecutiv e p ost-disruption trains is mo deled using an exp onen tially decaying function of journey o verlap, go v- erned by the parameters ρ and λ . Estimation results for Direction 1 of the Green metro line are rep orted in T able 6, while the corresp onding results for the remaining case studies are provided in Appendix A.3. Sk ew-normal Sk ew-t P arameter P osterior mean (5%, 95%) P osterior mean (5%, 95%) ρ 0.966 (0.940, 0.985) 0.947 (0.908, 0.977) λ 1.552 (1.477, 1.643) 1.567 (1.464, 1.701) T able 6: Posterior summaries of error dep endence parameters, ρ and λ in our prop osed sk ew-normal and skew-t mo dels for the Green metro line in Direction 1. Insp ection of T able 6 indicates that the estimated error-dep endence pa- rameters are very similar across the t w o prop osed sp eciﬁcations, namely the sk ew-normal and skew- t models. In particular, the estimated dep endence magnitude ρ is close to one, indicating strong dep endence b et w een trains that share a substantial p ortion of their journeys. A large ov erlap with re- sp ect to a destination station k ma y arise either when t wo trains are lo cated close to eac h other at the time the disruption is declared resolved, or when k is suﬃcien tly distant from both trains, corresp onding to longer journeys. In both situations, it is expected that the tra v el time error of the preceding train largely propagates to the following train. 4.5. Out-of-sample p erformanc e This section ev aluates the out-of-sample p erformance of the ﬁtted skew- normal and sk ew- t mo dels for Direction 1 of the Green metro line. Out-of- sample predictiv e accuracy is assessed using a range of metrics, including the mean squared error (MSE) and mean absolute error (MAE), as well as the length and empirical cov erage of the highest densit y in terv als (HDIs) of the predictive distributions, relative to the baseline mo del deﬁned in eqn. 4. In addition, the unconditional p osterior predictiv e p erformance of the ﬁtted 31 mo dels is examined through the construction of probability-probabilit y (P- P) and quantile-quan tile (Q-Q) plots. Finally , the full p osterior predictive distributions are compared using the con tin uous rank ed probabilit y score (CRPS). 4.5.1. R e alize d vs. pr e dicte d tr avel times Figure 11 presents the realized p ost-disruption tra v el times against the predicted v alues, along with the corresponding mo del residuals, for both prop osed sp eciﬁcations. As illustrated in the left column of Figure 11, the predicted samples cluster closely around the main diagonal for b oth mo d- els, indicating strong predictiv e p erformance. Although the ﬁgure do es not allo w for a clear visual separation b et w een the predictiv e accuracy of the sk ew-normal and skew- t mo dels, the sk ew- t sp eciﬁcation exhibits a slightly narro wer band around the diagonal for smaller trav el times, whic h also cor- resp ond to the most frequen t observ ations in the dataset. Nev ertheless, the impro vemen t app ears marginal when compared with the sk ew-normal mo del. Examining the right panel of Figure 11, b oth mo dels display small residuals across the en tire sample. Moreo v er, the residual patterns provide visual ev- idence of the adopted distributional sp eciﬁcations, including the allow ance for v arying scale and sk ewness, as reﬂected in the structure of the residuals sho wn in Figure 11. The results from the other case studies are presen ted in App endix B.5. Both mo dels exhibit diﬃcult y in predicting the extremely large trav el time realizations in the sample set. W e h yp othesize that trains asso ciated with these observ ations encountered additional disruptions during their p ost- disruption journeys, whic h further prolonged the trav el time to downstream stations. Because such delays cannot b e explained by either the train for- mation pattern along the metro line or the passenger accum ulation at in ter- mediate stations, the prop osed mo del is unable to repro duce these samples accurately , resulting in large prediction errors. An additional and noteworth y observ ation regarding these outliers is that their residuals app ear to follow a dependence pattern with resp ect to the mo del mis-predictions: sp eciﬁcally , there is a sequence of p oints that lies appro ximately parallel to the main diagonal but shifted upw ard by a roughly constan t amoun t. Recall that our ob jectiv e is to forecast, at the momen t a disruption is declared resolved, the tra vel time of a given train to eac h of its do wnstream stations, with the inten t of informing passengers wait- ing at those stations ab out the exp ected arriv al of that train. If, for an y 32 0 10 20 30 40 0 10 20 30 40 Predicted (minutes) Realized (minutes) Skew−normal Model 0 10 0 10 20 30 40 Predicted (minutes) Error (minutes) 0 10 20 30 40 0 10 20 30 40 Predicted (minutes) Realized (minutes) Skew−t 0 10 0 10 20 30 40 Predicted (minutes) Error (minutes) Figure 11: Realized v ersus predicted p ost-disruption trav el times (left panel) and the corresp onding mo del residuals (right panel) for the out-of-sample subset under the skew- normal and skew- t sp eciﬁcations. reason, that particular train exp eriences an unan ticipated delay at some sta- tion after op erations resume, then all subsequent trav el time observ ations to do wnstream stations for that train w ould b e shifted by appro ximately that additional delay . In this situation, the mo del may pro duce tra vel time pre- dictions that are systematically shifted relativ e to the realized v alues. Since the model do es not capture this additional dela y comp onen t, the predicted tra vel times for subsequent stations b ecome inaccurate. Ho wev er, the fact that the slop e of these outlier p oints remains similar to the main diago- nal suggests that, conditional on the realization of the additional dela y , the remaining downstream tra vel times would b e predicted reasonably w ell. Nev- ertheless, b ecause our mo deling framew ork generates predictions at the level of an origin–destination station pair, without conditioning on an in tervening realized dela y , these trav el time predictions remain erroneous for such cases. 33 4.6. Pr e diction ac cur acy and unc ertainty assessment In Figure 12, four ev aluation metrics are rep orted to assess the out-of- sample p erformance of the prop osed mo dels relativ e to the baseline sp eciﬁca- tion introduced in eqn. 4. F or each metric, the ev aluation is conducted sepa- rately for eac h trav eled distance and subsequently compared across models. This approac h is motiv ated by the observ ation that, from a predictive stand- p oin t, an error of a giv en magnitude has diﬀerent implications dep ending on the trav eled distance. F or instance, a tw o-minute error for journeys co ver- ing a single station represen ts substantially p o orer predictiv e performance than the same error for journeys spanning t wen t y stations. Accordingly , all metrics and error measures are ev aluated and compared across journeys that share the same trav eled distance. The results for the remaining case studies are presented in Appendix B.6. 1.5 2.0 2.5 5 10 15 20 T raveled distance (k − j) RMSE (minutes) Model Baseline Sk ew−normal Skew−t Root mean squared error 0.8 1.0 1.2 1.4 5 10 15 20 T raveled distance (k − j) MAE (minutes) Model Baseline Sk ew−normal Skew−t Mean absolute error 2 3 4 5 6 5 10 15 20 T raveled distance (k − j) HDI Length (minutes) Model Baseline Sk ew−normal Skew−t 80% High density interval length 80 85 90 95 5 10 15 20 T raveled distance (k − j) HDI coverage (%) Model Baseline Sk ew−normal Skew−t 80% High density interval co verage Figure 12: Comparison of error-based metrics and uncertain ty assessmen ts for the pro- p osed mo dels relativ e to the baseline sp eciﬁcation. In eac h category , the ev aluation is p erformed separately across trav eled distances. The results indicate that b oth prop osed mo dels outperform the baseline in terms of MAE, HDI length, and empirical cov erage, with the skew- t mo del demonstrating sup erior p erformance compared to the skew-normal sp eciﬁcation. 34 The top-left panel of Figure 12 rep orts the ro ot mean squared error (RMSE) of the predicted p ost-disruption trav el times for all three mo d- els—the baseline, skew-normal, and skew- t sp eciﬁcations. Under this metric, the three mo dels exhibit broadly similar p erformance across tra v eled dis- tances, with no clear out p erformance by any single model. In con trast, the top-right panel presen ts the mean absolute error (MAE), where both prop osed mo dels outperform the baseline across all tra veled distances, with the sk ew- t mo del consistently ac hieving the low est MAE. The discrepancy b et w een the RMSE and MAE results can b e attributed to their diﬀering sen- sitivities to extreme observ ations: RMSE places greater weigh t on outliers, whereas MAE is more robust to their presence. As noted earlier, our ob jective is not only to obtain accurate p oin t predic- tions but also to pro vide a reliable quan tiﬁcation of predictiv e uncertaint y . T o this end, we compare the length of the high-densit y interv als (HDIs) and the empirical co verage of these in terv als across the comp eting mo dels. The HDIs are constructed as the shortest in terv als containing a sp eciﬁed prob- abilit y mass of the p osterior samples, using a data-driven, nonparametric approac h based directly on p osterior draws. F or skew ed distributions, uncer- tain ty is inherently asymmetric, causing equal-tailed in terv als to misrepre- sen t co v erage. In con trast, HDIs concen trate probability mass in regions of highest p osterior densit y , providing a more informativ e and accurate repre- sen tation of uncertain t y . The b ottom-left panel of Figure 12 shows that the 80% HDI produced by the skew- t mo del are consisten tly shorter than those obtained from b oth the skew-normal and the baseline speciﬁcations, while the corresp onding co v erage, shown in the b ottom-righ t panel, remains close to the nominal 80% lev el. When uncertaint y is appropriately quantiﬁed, one exp ects shorter HDIs (at a ﬁxed credibilit y level) paired with empirical co v- erage that closely matc hes the nominal in terv al probability . As illustrated in the b ottom-right panel, the sk ew- t mo del ac hiev es co verage that is closer to 80% than the other tw o mo dels. It is also worth noting that, for b oth HDI length and co v erage, the skew-normal mo del outp erforms the baseline sp eciﬁcation. T o assess the full p osterior predictive distributions within our Bay esian framew ork, w e construct P-P plots and Q-Q plots for the predictiv e out- puts of both the prop osed mo dels and the baseline sp eciﬁcation. Because p osterior predictive samples corresp ond to diﬀerent underlying distributions, a normalization step is required in order to map all samples on to a com- mon scale. This normalization pro cedure, as well as the resulting reference 35 distribution, diﬀers across the considered mo dels and is described b elo w. • Baseline mo del: Using eqns. 4 and 6, the p osterior predictive samples can b e normalized using the following transformation, whic h maps all observ ations to a common scale: z i,j,k = y i,j,k − µ i,j,k q ω 0 + ω 1 ( k − j ) ∼ N (0 , 1) . (24) • Sk ew-normal mo del: F or the sk ew-normal sp eciﬁcation, a normal- ization pro cedure analogous to eqn. 24 is applied. Using eqns. 15, 16 and 17, the normalized predictiv e samples are obtained as z i,j,k =              y i,j,k − µ ′ i,j,k q ω 0 + ω 1 ( k − j ) , if train i is the ﬁrst p ost-disruption train, y i,j,k − µ ′ i,j,k − ρ j,j ′ ,k ε i − 1 ,j ′ ,k q ω 0 + ω 1 ( k − j ) , otherwise , (25) whic h implies that z i,j,k ∼ S N { 0 , 1 , α 0 + α 1 ( k − j ) } . In this setting, the distribution of z i,j,k v aries across samples due to diﬀerences in tra v- eled distance. How ev er, conditional on a ﬁxed tra v eled distance k − j , all samples share the same distribution. This prop erty allows P-P plots and Q-Q plots to b e constructed separately for each trav eled distance and subsequently aggregated in to a single plot. • Sk ew- t mo del: The normalization pro cedure for the sk ew- t mo del follo ws the same structure as in eqn. 25. The distinction lies in the resulting reference distribution, where the normalized samples satisfy z i,j,k ∼ S T { 0 , 1 , α 0 + α 1 ( k − j ) , ν } . Figure 13 displa ys the P-P plots (top ro w) and Q-Q plots (b ottom row) for the baseline, sk ew-normal, and skew- t mo del sp eciﬁcations, constructed using the transformations deﬁned in eqns. 24 and 25. The P-P plots compare the empirical cumulativ e probabilities of the observed data with the cum ula- tiv e probabilities implied b y each mo del. Under correct calibration, the P-P plot is exp ected to align closely with the main diagonal, while systematic deviations indicate under- or ov er-disp ersion in the predictive distribution. 36 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Theoretical uniform(0,1) Empirical PIT values Baseline model, P−P plot 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Theoretical uniform(0,1) Empirical PIT values Skew−normal model, P−P plot 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Theoretical uniform(0,1) Empirical PIT values Skew−t model, P−P plot −5 0 5 10 −2 −1 0 1 2 Theoretical quantiles Empirical quantiles Baseline model, Q−Q plot 0 5 10 −1 0 1 2 3 Theoretical quantiles Empirical quantiles Skew−normal model, Q−Q plot 0 10 0 4 8 Theoretical quantiles Empirical quantiles Skew−t model, Q−Q plot Figure 13: P-P plots (top row) and Q-Q plots (b ottom row) for the baseline, skew-normal, and sk ew- t mo del sp eciﬁcations are sho wn. The empirical and mo del-based cum ulative probabilities and quan tiles are computed using the transformations deﬁned in eqn. 24 and 25 and are plotted against one another. F or the skew-normal and skew- t mo dels, empirical cum ulative probabilities and quantiles are ev aluated separately for each trav eled distance ( k − j ) . As illustrated in the rightmost panels, the skew- t model exhibits comparatively b etter diagnostic performance, with the plotted p oin ts closely aligned with the main di- agonal, indicating improv ed distributional ﬁt relative to the other sp eciﬁcations. As evidenced in Figure 13, the skew- t mo del more accurately assigns prob- abilities that are consistent with the empirical distribution, outp erforming b oth the baseline and sk ew-normal sp eciﬁcations in terms of calibration. In con trast, Q-Q plots are used to assess the agreemen t b et ween the shap es of the empirical and predictive distributions b y comparing their re- sp ectiv e quantiles, thereby pro viding insigh t in to diﬀerences in lo cation, scale, and sk ewness. As with P-P plots, departures from the diagonal indicate dis- tributional mismatch b et ween the mo del and the data. Inspection of the b ot- tom ro w of Figure 13 reveals that, for the baseline and skew-normal mo dels, the predictiv e quan tiles deviate systematically from the empirical quantiles, indicating a lack of ﬁt across substantial p ortions of the distribution. By 37 comparison, the skew- t mo del captures the bulk of the empirical distribution more eﬀectively , with noticeable discrepancies arising primarily in the most extreme observ ations. Lastly , we emplo y the con tinuous rank ed probabilit y score (CRPS) to ev aluate the full p osterior predictive distributions generated by our mo dels [32]. The CRPS is widely used as a quantitativ e measure for assessing prob- abilistic forecasts and a prop er scoring rule. It is deﬁned as a quadratic mea- sure of discrepancy b et w een the predictiv e cumulativ e distribution function (CDF), denoted b y F ( x ) , and the empirical CDF of the realized observ ation y , represented by the indicator function I { x ≥ y } : CRPS ( F , y ) = Z ∞ −∞ [ F ( x ) − I { x ≥ y } ] 2 d x. (26) In our analysis, the av erage CRPS across all observ ations is used as a sum- mary p erformance metric. Lo wer CRPS v alues indicate sup erior predictiv e p erformance, as the score b eha ves analogously to an error metric while ac- coun ting for the en tire predictiv e distribution. Imp ortan tly , the CRPS ex- plicitly accoun ts for uncertain ty quan tiﬁcation, which makes it particularly w ell suited for Ba y esian and probabilistic mo deling framew orks. In the context of p ost-disruption tra vel time mo deling, CRPS is esp ecially relev ant b ecause predictions are inherently distributional, tail b eha vior—such as the heavy tails captured b y the sk ew- t mo del—is of practical imp ortance, and b oth predictiv e accuracy and uncertain ty calibration are cen tral ob jec- tiv es. Moreo ver, CRPS facilitates direct comparisons b et w een competing probabilistic mo dels, ev en when their predictiv e distributions diﬀer in shap e, v ariance, or tail b eha vior. Figure 14 rep orts the mean CRPS for observ ations group ed b y identical tra v eled distances across all three mo dels. The results indicate that the prop osed skew- t mo del consisten tly outp erforms b oth the baseline and the skew-normal sp eciﬁcations. 5. Conclusion This pap er examines the problem of predicting p ost-disruption train tra vel times in urban metro systems. W e prop osed a Bay esian hierarchical mo del that explicitly accounts for train in teractions, headwa y im balance due to disruption, and non-Gaussian features in tra v el time distributions observed during recov ery p erio ds. 38 0.6 0.9 1.2 1.5 1.8 0 10 20 T raveled distance (k − j) Mean CRPS Model Baseline Skew−normal Skew−t Figure 14: Mean CRPS for samples group ed by identical trav eled distance. Empirical results from our case study of the Mon tréal metro system, across tw o lines and in b oth directions, indicate that p ost-disruption tra vel times exhibit pronounced heterosk edasticity , sk ewness, and hea vy-tailed b e- ha vior. In addition, the results provide evidence of meaningful error depen- dence betw een consecutiv e trains, which is w ell captured b y the prop osed mo ving-av erage structure in the error sp eciﬁcation. Both prop osed mo dels outp erform the baseline sp eciﬁcation in terms of p oint prediction accuracy and uncertain t y quantiﬁcation, with the sk ew- t mo del pro viding the most robust p erformance for longer journeys and extreme delays. F rom an op erational p ersp ectiv e, the prop osed framework enables more accurate and b etter-calibrated probabilistic trav el time forecasts following disruptions. Such forecasts can impro ve passenger information systems and supp ort op erational decision-making during reco v ery phases, where uncer- tain ty and dep endence across trains pla y a critical role. Overall, this work pro vides a ﬂexible and interpretable probabilistic framew ork for analyzing p ost-disruption railwa y op erations, con tributing b oth metho dological ad- v ances and practical insigh ts for resilien t transit system managemen t. This study fo cuses on providing a one-time trav el time prediction at the momen t a disruption is rep orted as resolv ed. An imp ortan t direction for future research is to extend the prop osed framework to an online forecast- ing setting, in whic h predictions are con tinuously updated as new real-time op erational information b ecomes av ailable. Such an extension w ould allo w the mo del to adapt dynamically to evolving reco very conditions, incorp orate up dated train p ositions and in teractions, and further impro v e the accuracy and reliability of post-disruption tra vel time forecasts. 39 F unding This pro ject was funded by CANSSI (Canadian Statistical Sciences Insti- tute) with supp ort from NSERC (Natural Sciences and Engineering Research Council of Canada) through Collab orativ e Research T eam (CR T) gran t 25. Declaration of comp eting in terest The authors declare that they hav e no known competing ﬁnancial in ter- ests or p ersonal relationships that could hav e appeared to inﬂuence the w ork rep orted in this pap er. Data a v ailability The data used in this research article are not publicly a v ailable due to conﬁden tiality agreemen ts. CRediT authorship con tribution statement Sha y an Nazemi: Conceptualization, Metho dology , Soft ware, V alida- tion, F ormal Analysis, W riting - Original Draft – A urélie Labb e: Concep- tualization, Metho dology , W riting - Review & Editing, Sup ervision, F unding A cquisition – Stefan Steiner: Conceptualization, Metho dology – Pratheepa Jeganathan: Conceptualization, Metho dology , W riting - Review & Editing – Martin T répanier: Conceptualization, V alidation, W riting - Review & Editing – Léo R. Belzile: Conceptualization, M ethodology , Softw are, W rit- ing - Review & Editing, Sup ervision, F unding Acquisition Declaration of generativ e AI and AI-assisted technologies in the man uscript preparation pro cess During the preparation of this man uscript, the authors used ChatGPT (h ttps://chat.openai.com ) to assist with co de debugging, language reﬁne- men t (without in tro ducing new ideas or conten t), grammatical corrections, and LaT eX syn tax and formatting. All AI-assisted outputs w ere carefully review ed and edited by the authors. The authors tak e full resp onsibility for the accuracy and integrit y of the ﬁnal published work. 40 References [1] H. Lee, D. Zhang, T. He, S. H. Son, Metrotime: T rav el time decomp o- sition under sto c hastic time table for metro netw orks, in: 2017 IEEE In ternational Conference on Smart Computing (SMAR TCOMP), 2017, pp. 1–8. doi: 10.1109/SMARTCOMP.2017.7947021 . [2] T. Dollevoet, D. Huisman, M. Sc hmidt, A. 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The horizontal axis denotes the trav eled distance from the origin station, measured by the num b er of stations trav ersed, whereas the v ertical axis rep orts the v ariance of the asso ciated journey times. 0. 0 2. 5 5. 0 7. 5 10. 0 1 5 9 13 17 21 25 T ra v el distance from ori gin (in # of stations) V ariance of tra v el time (minutes 2 ) Figure A.15: Green metro line – Direction 2 0. 0 2. 5 5. 0 7. 5 1 5 9 13 17 21 25 T ra v el distance from ori gin (in # of stations) V ariance of tra v el time (minutes 2 ) 0. 0 2. 5 5. 0 7. 5 10. 0 1 5 9 13 17 21 25 T ra v el distance from ori gin (in # of stations) V ariance of tra v el time (minutes 2 ) Figure A.16: Orange metro line – Direction 1 (left) and Direction 2 (right) A pp endix A.2. Baseline R esiduals Here, we display the mo del residuals against the predicted v alues, along with the empirical sk ewness of the error terms as a function of trav eled distance for Direction 2 of the Green line in Figure A.17, and the Orange line in Figure A.18. 45 - 10 - 5 0 5 0 10 20 30 40 Predicted tra v el time (minutes) Residuals (minutes) - 1. 0 - 0. 5 0. 0 0. 5 1. 0 1. 5 5 10 15 20 T ra v eled Distance (k - j) Residual Ske wness Figure A.17: Green metro line – Direction 2. The left panel presen ts baseline residuals v ersus predicted trav el times, while the right panel depicts residual skewness as a function of trav eled distance. The residuals-versus-predicted plot reveals noticeable misﬁt for longer tra vel times, accompanied b y a decreasing skewness pattern as tra veled distance increases. - 10 - 5 0 5 0 10 20 30 40 Predicted tra v el time (minutes) Residuals (minutes) - 3 - 2 - 1 0 1 2 5 10 15 20 T ra v eled Distance (k - j) Residual Ske wness - 2. 5 0. 0 2. 5 5. 0 0 10 20 30 40 Predicted tra v el time (minutes) Residuals (minutes) 0. 0 0. 5 1. 0 1. 5 5 10 15 20 T ra v eled Distance (k - j) Residual Ske wness Figure A.18: Orange metro line – Direction 1 (top ro w) and Direction 2 (bottom ro w). The left column shows baseline residuals versus predicted trav el times, while the right column presents residual sk ewness as a function of trav eled distance. Comparing b oth directions reveals diﬀerences in residual v ariabilit y and a shift in the decreasing skewness pattern at longer distances for the Orange line. 46 A pp endix A.3. T r avel Time Err or Dep endenc e P earson and Sp earman correlations betw een the baseline mo del residu- als and those of the preceding trains, binned b y the proportion of journey o verlap. The results rev eal p ositiv e correlations, indicating that dela y and congestion propagate from one train to its successor. T rain journeys with a larger o v erlap in trav eled segments with the preceding train exhibit stronger p ositiv e correlations. 0. 0 0. 2 0. 4 0. 6 0. 8 [ 0, 0. 2) [ 0. 2, 0. 4) [ 0. 4, 0. 6) [ 0. 6, 0. 8) [ 0. 8, 1] Ov erlap ratio bin Correlation Corre l a t i on t ype P e a rs on S pe a rm a n Figure A.19: Green metro line – Direction 2 0. 0 0. 1 0. 2 0. 3 0. 4 [ 0, 0. 2) [ 0. 2, 0. 4) [ 0. 4, 0. 6) [ 0. 6, 0. 8) [ 0. 8, 1] Ov erlap ratio bin Correlation Corre l a t i on t ype P e a rs on S pe a rm a n 0. 1 0. 2 0. 3 0. 4 0. 5 [ 0, 0. 2) [ 0. 2, 0. 4) [ 0. 4, 0. 6) [ 0. 6, 0. 8) [ 0. 8, 1] Ov erlap ratio bin Correlation Corre l a t i on t ype P e a rs on S pe a rm a n Figure A.20: Orange metro line – Direction 1 (left) and Direction 2 (right) A pp endix A.4. Mo del Estimation and Computation Figure A.21 and Figure A.22 displa y trace plots for the Green metro line in Direction 1 under the sk ew-normal and skew- t sp eciﬁcations, co v ering a subset of key mo del parameters. These include the mean parameter t 0 , the scale parameters ( ω 0 , ω 1 ) , the sk ewness parameters ( α 0 , α 1 ) , the dep endence 47 parameters ( ρ, λ ) , and, for the sk ew- t mo del, the degrees-of-freedom param- eter ν . Owing to the large n umber of mean sp eciﬁcation parameters, trace plots for θ m and γ ℓ,j are shown only for selected cases, namely θ Guy–Concordia , θ Peel , and θ McGill , as w ell as γ 1 , Peel , γ 2 , Peel , γ 1 , McGill , and γ 2 , McGill . Ov erall, the trace plots indicate go o d mixing and con vergence across all parameters, with no evidence of div ergence or con vergence issues. T race plots for the remain- ing parameters and for other case studies were also examined and exhibited similar con vergence b eha vior; they are omitted here for brevity and to a void redundancy . Figure A.21: T race plots for a selected subset of mo del parameters under the skew-normal sp eciﬁcation. 48 Figure A.22: T race plots for a selected subset of mo del parameters under the sk ew- t sp eciﬁcation. 49 App endix B. Results A pp endix B.1. The eﬀe ct of longer-than-usual he adway, θ m Analogous to T able 4, T able B.7 rep orts the p osterior summaries of the θ m parameters for the skew-normal and sk ew- t mo dels applied to the Green line of the Montreal metro system in Direction 2. Corresp onding results for the Orange metro line are presen ted in T able B.8 for Direction 1 and in T able B.9 for Direction 2. The estimates are in terpreted as the change in tra vel time (in seconds) asso ciated with a one-minute increase in headwa y . P osterior means mark ed with an asterisk ( ∗ ) indicate strong evidence of either a p ositiv e or negativ e eﬀect, as determined by the 90% credible in terv al. The column “% of median dw ell time” rep orts each θ m estimate as a p ercentage of the median dw ell time at the corresp onding station. Ro ws sho wn in bold corresp ond to stations lo cated in the do wn town area. Skew-normal Skew- t m Parameter Posterior mean (seconds) ( 5%, 95%) % of median dwell time Posterior mean (seconds) ( 5%, 95%) % of median dwell time 2 θ Radisson − 1 . 1 ∗ ( − 2 . 0 , − 0 . 3 ) − 2 . 93% 1 . 6 ∗ ( 1 . 0 , 2 . 2 ) 4 . 20% 3 θ Langelier 0 . 1 ( − 0 . 6 , 0 . 8 ) 0 . 25% − 0 . 7 ∗ ( − 1 . 1 , − 0 . 2 ) − 1 . 71% 4 θ Cadillac − 0 . 4 ( − 1 . 0 , 0 . 2 ) − 1 . 09% 0 . 4 ( 0 . 0 , 0 . 9 ) 1 . 06% 5 θ Assomption 0 . 7 ∗ ( 0 . 1 , 1 . 3 ) 1 . 83% 0 . 2 ( − 0 . 1 , 0 . 6 ) 0 . 67% 6 θ Viau − 2 . 3 ∗ ( − 2 . 8 , − 1 . 8 ) − 6 . 14% − 0 . 4 ∗ ( − 0 . 8 , − 0 . 1 ) − 1 . 13% 7 θ Pie-IX 0 . 5 ( 0 . 0 , 1 . 0 ) 1 . 20% − 0 . 3 ( − 0 . 7 , 0 . 0 ) − 0 . 75% 8 θ Joliette − 0 . 2 ( − 0 . 7 , 0 . 3 ) − 0 . 50% 0 . 9 ∗ ( 0 . 6 , 1 . 3 ) 2 . 46% 9 θ Préfontaine 0 . 8 ∗ ( 0 . 4 , 1 . 3 ) 2 . 19% 0 . 5 ∗ ( 0 . 2 , 0 . 8 ) 1 . 39% 10 θ F rontenac − 0 . 6 ∗ ( − 1 . 2 , − 0 . 1 ) − 1 . 66% − 0 . 3 ( − 0 . 7 , 0 . 0 ) − 0 . 79% 11 θ Papineau 0 . 5 ( 0 . 0 , 1 . 0 ) 1 . 28% 0 . 6 ∗ ( 0 . 2 , 0 . 9 ) 1 . 51% 12 θ Beaudry − 0 . 1 ( − 0 . 6 , 0 . 3 ) − 0 . 36% − 0 . 1 ( − 0 . 4 , 0 . 3 ) − 0 . 16% 13 θ Berri-UQAM − 0 . 1 ( − 0 . 6 , 0 . 4 ) − 0 . 19% − 0 . 2 ( − 0 . 6 , 0 . 2 ) − 0 . 39% 14 θ Saint-Lauren t 0 . 2 ( − 0 . 3 , 0 . 7 ) 0 . 52% 0 . 3 ( − 0 . 1 , 0 . 7 ) 0 . 85% 15 θ Place-des-Arts 1 . 0 ∗ ( 0 . 6 , 1 . 4 ) 2 . 36% 0 . 5 ∗ ( 0 . 1 , 0 . 9 ) 1 . 16% 16 θ McGill 1 . 3 ∗ ( 0 . 8 , 1 . 7 ) 2 . 94% 1 . 1 ∗ ( 0 . 7 , 1 . 4 ) 2 . 52% 17 θ Peel − 0 . 5 ∗ ( − 0 . 9 , 0 . 0 ) − 1 . 13% − 0 . 5 ∗ ( − 0 . 8 , − 0 . 1 ) − 1 . 08% 18 θ Guy-Concordia 0 . 4 ∗ ( 0 . 0 , 0 . 9 ) 1 . 00% 0 . 5 ∗ ( 0 . 2 , 0 . 8 ) 1 . 15% 19 θ At water − 0 . 5 ∗ ( − 1 . 0 , − 0 . 1 ) − 1 . 24% − 0 . 7 ∗ ( − 1 . 1 , − 0 . 3 ) − 1 . 57% 20 θ Lionel-Groulx 0 . 3 ( − 0 . 2 , 0 . 7 ) 0 . 63% 0 . 0 ( − 0 . 4 , 0 . 4 ) − 0 . 06% 21 θ Charlevoix 0 . 8 ∗ ( 0 . 4 , 1 . 2 ) 2 . 18% 1 . 2 ∗ ( 0 . 8 , 1 . 6 ) 3 . 11% 22 θ LaSalle − 0 . 1 ( − 0 . 5 , 0 . 4 ) − 0 . 18% − 0 . 2 ( − 0 . 6 , 0 . 2 ) − 0 . 51% 23 θ De l’Église 0 . 8 ∗ ( 0 . 4 , 1 . 1 ) 1 . 90% 0 . 7 ∗ ( 0 . 3 , 1 . 1 ) 1 . 72% 24 θ V erdun 0 . 1 ( − 0 . 1 , 0 . 3 ) 0 . 25% 0 . 1 ( − 0 . 1 , 0 . 2 ) 0 . 22% 25 θ Jolicoeur − 0 . 2 ( − 0 . 6 , 0 . 3 ) − 0 . 47% − 0 . 1 ( − 0 . 4 , 0 . 2 ) − 0 . 19% 26 θ Monk 0 . 2 ( − 0 . 3 , 0 . 7 ) 0 . 42% − 0 . 1 ( − 0 . 4 , 0 . 2 ) − 0 . 23% T able B.7: Green metro line – Direction 2 50 Skew-normal Skew- t m Parameter Posterior mean (seconds) ( 5%, 95%) % of median dwell time Posterior mean (seconds) ( 5%, 95%) % of median dwell time 2 θ Crémazie − 6 . 9 ∗ ( − 10 . 0 , − 3 . 8 ) − 16 . 47% − 7 . 4 ∗ ( − 10 . 2 , − 4 . 7 ) − 17 . 54% 3 θ Jarry 6 . 0 ∗ ( 5 . 3 , 6 . 6 ) 14 . 90% 4 . 2 ∗ ( 3 . 5 , 6 . 4 ) 10 . 61% 4 θ Jean-T alon − 2 . 1 ∗ ( − 3 . 1 , − 1 . 2 ) − 4 . 74% − 6 . 0 ∗ ( − 7 . 1 , − 5 . 0 ) − 13 . 31% 5 θ Beaubien 3 . 8 ∗ ( 3 . 3 , 4 . 3 ) 9 . 27% 2 . 6 ∗ ( 2 . 2 , 3 . 1 ) 6 . 45% 6 θ Rosemont − 5 . 3 ∗ ( − 6 . 1 , − 4 . 4 ) − 12 . 91% 0 . 7 ( 0 . 0 , 1 . 5 ) 1 . 79% 7 θ Laurier 1 . 0 ∗ ( 0 . 2 , 1 . 8 ) 2 . 45% 1 . 7 ∗ ( 1 . 0 , 2 . 3 ) 4 . 11% 8 θ Mont-Ro yal 0 . 4 ( − 0 . 3 , 1 . 1 ) 1 . 04% 0 . 0 ( − 0 . 5 , 0 . 5 ) 0 . 07% 9 θ Sherbrooke − 1 . 1 ∗ ( − 1 . 8 , − 0 . 4 ) − 2 . 54% 0 . 6 ∗ ( 0 . 1 , 1 . 2 ) 1 . 49% 10 θ Berri-UQAM − 0 . 5 ( − 1 . 3 , 0 . 3 ) − 0 . 97% 0 . 1 ( − 0 . 6 , 0 . 7 ) 0 . 10% 11 θ Champ-de-Mars 1 . 1 ∗ ( 0 . 3 , 1 . 9 ) 2 . 47% 0 . 5 ( − 0 . 1 , 1 . 1 ) 1 . 09% 12 θ Place-d’Armes − 2 . 1 ∗ ( − 3 . 0 , − 1 . 1 ) − 4 . 75% − 1 . 4 ∗ ( − 2 . 2 , − 0 . 7 ) − 3 . 27% 13 θ Square-Victoria-OA CI 1 . 4 ∗ ( 0 . 4 , 2 . 3 ) 3 . 28% 1 . 5 ∗ ( 0 . 7 , 2 . 3 ) 3 . 56% 14 θ Bonav enture − 1 . 1 ∗ ( − 2 . 1 , − 0 . 1 ) − 2 . 79% 1 . 3 ∗ ( 0 . 5 , 2 . 2 ) 3 . 43% 15 θ Lucien-L’Allier 0 . 7 ( − 0 . 3 , 1 . 7 ) 1 . 64% − 1 . 4 ∗ ( − 2 . 3 , − 0 . 6 ) − 3 . 53% 16 θ Georges-V anier 2 . 7 ∗ ( 1 . 8 , 3 . 7 ) 7 . 02% 2 . 1 ∗ ( 1 . 4 , 2 . 9 ) 5 . 47% 17 θ Lionel-Groulx 0 . 8 ( − 0 . 1 , 1 . 6 ) 1 . 63% 0 . 8 ∗ ( 0 . 1 , 1 . 4 ) 1 . 62% 18 θ Place-Saint-Henri 2 . 3 ∗ ( 1 . 7 , 3 . 0 ) 5 . 68% 2 . 6 ∗ ( 2 . 0 , 3 . 1 ) 6 . 27% 19 θ V endôme 0 . 3 ( − 0 . 2 , 0 . 9 ) 0 . 77% 0 . 1 ( − 0 . 3 , 0 . 4 ) 0 . 15% 20 θ Villa-Maria 0 . 2 ( − 0 . 4 , 0 . 7 ) 0 . 43% 0 . 2 ( − 0 . 2 , 0 . 5 ) 0 . 39% 21 θ Snowdon 1 . 1 ∗ ( 0 . 5 , 1 . 7 ) 2 . 47% 0 . 7 ∗ ( 0 . 4 , 1 . 1 ) 1 . 62% 22 θ Côte-Sainte-Catherine − 0 . 6 ( − 1 . 2 , 0 . 0 ) − 1 . 51% − 0 . 8 ∗ ( − 1 . 2 , − 0 . 4 ) − 1 . 89% 23 θ Plamondon 1 . 0 ∗ ( 0 . 2 , 1 . 7 ) 2 . 22% 0 . 9 ∗ ( 0 . 4 , 1 . 4 ) 2 . 13% 24 θ Namur 0 . 8 ∗ ( 0 . 0 , 1 . 5 ) 1 . 96% 0 . 3 ( − 0 . 2 , 0 . 8 ) 0 . 74% 25 θ De La Sav ane 0 . 3 ( − 0 . 5 , 1 . 1 ) 0 . 79% 0 . 1 ( − 0 . 4 , 0 . 7 ) 0 . 34% T able B.8: Orange metro line – Direction 1 51 Skew-normal Skew- t m Parameter Posterior mean (seconds) ( 5%, 95%) % of median dwell time Posterior mean (seconds) ( 5%, 95%) % of median dwell time 2 θ De La Sav ane − 1 . 8 ( − 6 . 5 , 3 . 6 ) − 4 . 50% − 3 . 0 ( − 6 . 8 , 2 . 6 ) − 7 . 56% 3 θ Namur 0 . 3 ( − 1 . 4 , 2 . 0 ) 0 . 81% 1 . 1 ( − 0 . 3 , 2 . 4 ) 2 . 64% 4 θ Plamondon − 2 . 0 ∗ ( − 3 . 6 , − 0 . 3 ) − 4 . 69% − 0 . 1 ( − 1 . 2 , 1 . 2 ) − 0 . 13% 5 θ Côte-Sainte-Catherine 4 . 1 ∗ ( 3 . 4 , 4 . 9 ) 11 . 85% 3 . 3 ∗ ( 2 . 6 , 4 . 0 ) 9 . 41% 6 θ Snowdon 1 . 7 ∗ ( 0 . 8 , 2 . 5 ) 3 . 77% 2 . 0 ∗ ( 1 . 2 , 2 . 8 ) 4 . 48% 7 θ Villa-Maria − 0 . 2 ( − 0 . 9 , 0 . 6 ) − 0 . 36% 0 . 0 ( − 0 . 8 , 0 . 7 ) − 0 . 05% 8 θ V endôme 0 . 3 ( − 0 . 3 , 0 . 9 ) 0 . 60% 0 . 0 ( − 0 . 6 , 0 . 6 ) − 0 . 01% 9 θ Place-Saint-Henri 0 . 3 ( − 0 . 4 , 0 . 9 ) 0 . 62% − 0 . 6 ( − 1 . 2 , 0 . 1 ) − 1 . 40% 10 θ Lionel-Groulx 0 . 7 ∗ ( 0 . 0 , 1 . 3 ) 1 . 42% 2 . 0 ∗ ( 1 . 4 , 2 . 7 ) 4 . 19% 11 θ Georges-V anier − 1 . 3 ∗ ( − 1 . 9 , − 0 . 7 ) − 3 . 20% − 2 . 0 ∗ ( − 2 . 5 , − 1 . 5 ) − 4 . 97% 12 θ Lucien-L’Allier 2 . 2 ∗ ( 1 . 7 , 2 . 7 ) 5 . 35% 3 . 0 ∗ ( 2 . 7 , 3 . 4 ) 7 . 41% 13 θ Bonav enture − 0 . 8 ∗ ( − 1 . 2 , − 0 . 4 ) − 1 . 82% − 0 . 3 ( − 0 . 7 , 0 . 1 ) − 0 . 66% 14 θ Square-Victoria-OA CI − 0 . 3 ( − 0 . 8 , 0 . 3 ) − 0 . 67% − 0 . 4 ( − 0 . 9 , 0 . 1 ) − 0 . 98% 15 θ Place-d’Armes 1 . 3 ∗ ( 0 . 7 , 1 . 9 ) 3 . 04% 0 . 7 ∗ ( 0 . 3 , 1 . 2 ) 1 . 72% 16 θ Champ-de-Mars − 1 . 4 ∗ ( − 2 . 1 , − 0 . 8 ) − 3 . 33% − 0 . 9 ∗ ( − 1 . 4 , − 0 . 4 ) − 2 . 05% 17 θ Berri-UQAM 0 . 1 ( − 0 . 3 , 0 . 5 ) 0 . 17% 0 . 3 ( 0 . 0 , 0 . 6 ) 0 . 53% 18 θ Sherbrooke 1 . 7 ∗ ( 1 . 2 , 2 . 3 ) 3 . 98% 1 . 4 ∗ ( 0 . 9 , 1 . 8 ) 3 . 19% 19 θ Mont-Ro yal − 0 . 6 ∗ ( − 1 . 1 , 0 . 0 ) − 1 . 30% − 0 . 4 ( − 0 . 9 , 0 . 1 ) − 0 . 92% 20 θ Laurier 0 . 8 ∗ ( 0 . 1 , 1 . 4 ) 1 . 75% 0 . 5 ( 0 . 0 , 1 . 1 ) 1 . 20% 21 θ Rosemont 0 . 6 ( − 0 . 1 , 1 . 3 ) 1 . 39% 0 . 6 ∗ ( 0 . 1 , 1 . 2 ) 1 . 56% 22 θ Beaubien 0 . 5 ( 0 . 0 , 1 . 1 ) 1 . 30% 0 . 1 ( − 0 . 4 , 0 . 7 ) 0 . 36% 23 θ Jean-T alon − 0 . 3 ( − 0 . 9 , 0 . 3 ) − 0 . 65% 0 . 0 ( − 0 . 7 , 0 . 8 ) − 0 . 04% 24 θ Jarry 0 . 2 ( − 0 . 4 , 0 . 8 ) 0 . 55% 0 . 8 ∗ ( 0 . 1 , 1 . 4 ) 2 . 06% 25 θ Crémazie 1 . 6 ∗ ( 0 . 8 , 2 . 4 ) 3 . 75% 0 . 7 ( − 0 . 1 , 1 . 7 ) 1 . 71% T able B.9: Orange metro line – Direction 2 52 A pp endix B.2. The eﬀe ct of tr ain formation p ar ameters, γ ℓ,j P osterior means of the train formation parameters γ ℓ,j under p ost-disruption op erations are presented for b oth the skew-normal and skew- t mo dels in Fig- ure B.23 for Direction 2 of the Green line, and in Figure B.24 and Figure B.25 for Directions 1 and 2 of the Orange line, resp ectiv ely . As in Figure 8, the results reveal substan tial v ariation in the induced dela y eﬀects across b oth distance and lo cation. Consistent with earlier ﬁndings, the skew-normal esti- mates display more abrupt changes across distances and stations, whic h may again b e attributed to the presence of outliers in the data—features that are more eﬀectively accommo dated by the hea vier-tailed skew- t distribution. 115.8 −87.3 −86.3 −102.3 94.3 114.0 −4.0 135.8 −48.9 −5.3 −0.3 0.7 −60.5 31.9 112.5 74.2 30.8 12.6 −29.5 1.0 46.6 31.7 −4.3 3.5 −43.8 83.5 22.5 4.6 −5.0 −0.6 69.5 32.8 −2.1 −10.8 0.9 61.6 23.7 −1.1 16.5 −3.5 57.1 16.0 −9.2 −15.8 −6.7 47.7 29.0 4.8 −8.1 12.3 47.2 20.0 −13.4 26.5 9.1 93.8 9.0 18.5 −6.6 −0.6 39.0 57.7 −2.7 18.9 4.6 60.0 10.5 −12.5 −0.1 −3.4 33.0 25.0 12.1 6.1 −1.7 62.3 34.2 15.8 −11.1 −33.2 19.4 7.9 −4.0 14.7 17.4 37.7 2.7 −11.0 16.4 20.1 44.8 −2.5 26.3 8.4 −0.5 53.4 43.5 5.2 20.9 −1.3 61.9 8.1 19.3 17.9 0.6 45.4 24.4 20.7 −7.7 4.3 56.8 21.4 14.7 −5.5 0.9 67.7 6.7 2.3 0.0 0.4 48.9 14.9 2.5 2.7 −0.2 64.0 0.9 −0.6 −1.2 0.9 −1.9 0.7 −0.4 −0.5 −1.1 Downtown Stations 1 2 3 4 5 Honore−Beaugrand Radisson Langelier Cadillac Assomption Viau Pie−Ix Joliette Prefontaine Frontenac Papineau Beaudry Berri−Uqam Saint−Laurent Place−Des−Arts Mcgill Peel Guy−Concordia Atwater Lionel−Groulx Charlevoix Lasalle De L 'eglise V erdun Jolicoeur Monk Angrignon Origin station Distance (segments) Skew−normal model 246.2 −107.8 12.7 −40.1 −0.7 98.6 −0.9 152.2 −17.6 13.6 1.7 −4.8 −24.3 4.7 51.5 72.7 55.3 17.2 −9.5 −17.3 68.2 30.9 10.4 −11.4 −25.8 76.8 27.0 16.8 4.2 −3.9 89.5 35.1 7.3 −10.3 −2.9 66.7 39.0 −0.8 6.5 8.7 64.9 18.7 −0.5 −13.3 −5.3 57.6 23.3 5.9 6.5 10.6 55.1 25.4 8.3 24.0 5.6 70.4 15.9 17.1 −8.9 2.0 51.2 61.7 6.1 14.9 4.4 50.3 23.0 9.5 −2.9 −4.8 54.0 36.9 12.1 4.3 −5.5 89.5 35.1 9.0 −4.5 −22.1 52.3 8.3 −2.4 4.3 19.3 55.1 22.2 −14.5 −1.2 2.7 54.4 −4.6 25.4 6.7 −6.8 57.7 48.9 2.2 12.6 −3.1 67.8 −2.0 −5.3 15.8 13.0 40.1 11.2 20.6 2.1 −5.6 43.6 13.7 10.0 −8.5 1.4 56.9 −2.9 −0.8 2.1 −0.2 39.0 0.6 3.2 −1.3 3.0 42.0 0.6 0.1 −4.3 2.4 −2.7 −0.9 0.9 −3.6 −0.9 Downtown Stations 1 2 3 4 5 Honore−Beaugrand Radisson Langelier Cadillac Assomption Viau Pie−Ix Joliette Prefontaine Frontenac Papineau Beaudry Berri−Uqam Saint−Laurent Place−Des−Arts Mcgill Peel Guy−Concordia Atwater Lionel−Groulx Charlevoix Lasalle De L 'eglise V erdun Jolicoeur Monk Angrignon Origin station Distance (segments) Skew−t model Figure B.23: Green metro line – Direction 2 53 13.0 316.6 14.6 10.3 −95.0 111.0 −50.8 −66.7 15.3 9.6 −5.8 −91.7 −18.7 114.6 66.3 96.5 2.7 0.1 −4.1 −11.7 23.4 28.2 23.7 15.6 −22.0 76.8 8.0 −9.2 −15.4 −21.3 82.7 1.8 23.4 −18.5 −14.4 −15.0 26.2 3.5 5.6 −6.7 78.4 4.1 −37.2 −9.4 22.5 61.4 42.4 23.9 4.5 −22.4 68.9 47.7 19.5 −13.1 8.9 92.2 39.3 14.1 −16.2 −12.2 88.6 40.1 15.3 −9.8 −11.4 68.4 20.6 11.5 11.2 −1.1 99.7 54.0 6.4 −15.1 −29.6 98.1 11.2 −7.1 16.0 −15.6 86.9 44.0 7.1 −8.8 −2.7 64.3 35.1 −15.4 −8.9 −21.5 36.6 −2.3 21.8 24.0 12.2 13.5 −8.5 23.2 37.0 22.7 80.6 16.9 6.4 12.0 −2.8 30.3 −6.7 34.9 13.9 0.5 −49.0 −1.9 36.3 −0.8 −0.8 63.2 6.5 0.7 −2.8 0.2 56.3 −1.1 2.8 0.2 −0.2 −2.3 −0.4 1.3 −1.1 −1.8 Downtown Stations 1 2 3 4 5 Sauve Cremazie Jarry Jean−T alon Beaubien Rosemont Laurier Mont−Royal Sherbrooke Berri−Uqam Champ−De−Mars Place−D'armes Square−Victoria−Oaci Bonaventure Lucien−L 'allier Georges−V anier Lionel−Groulx Place−Saint−Henri V endome Villa−Maria Snowdon Cote−Sainte−Catherine Plamondon Namur De La Savane Du College Origin station Distance (segments) Skew−normal model 21.3 339.4 36.0 20.7 −51.5 76.3 −26.9 −7.8 −6.3 9.8 −19.3 −39.2 −0.1 57.6 56.8 100.5 19.9 2.6 −3.4 −7.6 46.4 24.7 24.1 10.9 −20.6 67.2 10.4 −1.2 −6.8 −8.8 66.0 28.5 20.2 −12.4 −23.9 50.5 40.6 −1.2 −5.4 −17.6 83.3 20.5 −19.7 −1.9 20.5 69.1 43.9 20.5 −1.7 −15.2 75.2 49.8 14.3 −11.8 8.9 90.2 40.4 17.9 −18.4 −18.2 80.2 35.5 19.3 −9.1 −4.7 82.3 12.2 9.4 −1.2 13.5 83.9 41.1 6.7 −12.1 −18.8 88.5 15.7 1.8 7.9 −18.8 82.7 41.9 10.1 −10.0 −3.6 69.6 26.1 −10.0 −9.0 −20.8 44.3 1.0 26.4 20.9 7.5 24.4 −4.8 16.1 34.0 12.9 90.5 16.1 17.2 7.8 −13.7 61.9 −1.2 35.9 6.2 −2.2 38.2 20.9 −0.3 −2.1 0.6 43.2 10.3 −0.9 1.1 1.2 35.9 −2.8 0.1 −0.6 −1.8 0.6 −1.0 0.8 −3.0 −0.1 Downtown Stations 1 2 3 4 5 Sauve Cremazie Jarry Jean−T alon Beaubien Rosemont Laurier Mont−Royal Sherbrooke Berri−Uqam Champ−De−Mars Place−D'armes Square−Victoria−Oaci Bonaventure Lucien−L 'allier Georges−V anier Lionel−Groulx Place−Saint−Henri V endome Villa−Maria Snowdon Cote−Sainte−Catherine Plamondon Namur De La Savane Du College Origin station Distance (segments) Skew−t model Figure B.24: Orange metro line – Direction 1 2.1 −0.4 197.3 0.2 4.1 −37.7 −27.2 53.7 −5.5 0.2 −1.5 59.0 84.8 −70.0 −52.8 54.3 11.0 32.1 −18.6 −25.2 48.2 24.5 −0.2 −19.7 −8.6 45.3 17.6 −1.5 −14.3 −17.4 47.7 −1.7 −22.8 −2.6 24.8 61.6 6.0 7.8 −4.8 14.2 50.1 12.2 32.8 16.7 0.3 29.1 26.8 6.3 15.3 −13.8 71.2 3.3 9.5 −6.7 10.9 48.5 37.8 23.3 5.0 −11.1 49.8 30.5 10.2 −7.4 −8.9 55.5 23.7 −3.7 −12.4 12.9 69.7 7.7 1.8 0.1 −11.9 65.4 12.4 0.4 6.6 −1.5 30.5 10.5 11.4 17.4 29.9 58.7 9.1 −5.8 −9.6 8.3 46.4 24.8 15.9 12.7 −4.4 45.0 22.7 29.3 −5.1 −10.3 48.5 28.6 −1.6 8.3 12.8 55.4 12.3 16.5 −3.8 0.1 41.7 16.8 28.7 −1.7 0.7 46.7 20.0 −2.6 1.0 −1.0 37.8 0.4 0.6 −0.4 1.2 1.4 1.0 0.5 2.1 1.4 Downtown Stations 1 2 3 4 5 Du College De La Savane Namur Plamondon Cote−Sainte−Catherine Snowdon Villa−Maria V endome Place−Saint−Henri Lionel−Groulx Georges−V anier Lucien−L 'allier Bonaventure Square−Victoria−Oaci Place−D'armes Champ−De−Mars Berri−Uqam Sherbrooke Mont−Royal Laurier Rosemont Beaubien Jean−T alon Jarry Cremazie Sauve Origin station Distance (segments) Skew−normal model −0.3 −1.5 205.1 −0.6 1.1 19.7 1.4 20.0 −12.4 1.7 −1.1 58.7 94.0 −76.1 −50.7 47.3 27.9 15.3 −29.7 −13.1 50.9 28.1 1.3 −16.9 2.1 50.2 14.7 0.1 −5.5 4.0 58.1 7.5 −13.3 0.4 13.9 63.6 16.3 15.0 −2.9 17.6 66.6 26.4 35.4 18.0 −2.3 55.5 51.9 9.7 3.6 −3.2 101.8 18.1 3.5 5.1 7.3 78.3 34.9 29.4 3.3 2.0 57.8 43.0 3.0 3.9 −2.1 56.1 25.7 2.5 −5.8 10.5 64.8 18.4 6.1 4.7 −7.2 58.6 18.1 2.7 7.5 3.1 42.2 24.3 19.8 13.0 21.0 55.5 20.6 −2.4 −5.6 9.0 47.3 30.4 14.8 17.5 −4.7 57.0 20.5 28.4 1.3 −5.1 54.1 28.9 4.8 13.2 14.3 50.5 13.5 17.8 −3.2 0.8 40.4 17.5 26.7 −1.0 1.2 47.6 16.5 −0.9 −1.0 −1.3 40.8 −2.2 0.3 −0.5 −2.1 2.0 −0.8 −1.8 1.7 −0.7 Downtown Stations 1 2 3 4 5 Du College De La Savane Namur Plamondon Cote−Sainte−Catherine Snowdon Villa−Maria V endome Place−Saint−Henri Lionel−Groulx Georges−V anier Lucien−L 'allier Bonaventure Square−Victoria−Oaci Place−D'armes Champ−De−Mars Berri−Uqam Sherbrooke Mont−Royal Laurier Rosemont Beaubien Jean−T alon Jarry Cremazie Sauve Origin station Distance (segments) Skew−t model Figure B.25: Orange metro line – Direction 2 54 The ridge plots of the train formation parameters γ ℓ,j for Direction 2 of the Green metro line and b oth directions of the Orange line are sho wn in Figures B.26, B.27, and B.28. The densities of the p osterior samples under b oth prop osed mo dels and for tw o station groups, namely down town and non-down town stations are sho wn. F or each distance, the corresp onding densit y represen ts the p osterior samples of γ ℓ,j , where j b elongs to the set of station indices within the resp ectiv e group. The vertical lines indicate the group-sp eciﬁc mean eﬀect for eac h model, with colors distinguishing the station groups. Skew−normal Skew−t −100−50 0 50 100 150 γ 1 , j (seconds) Model Distance 1 −100 −50 0 50 100 γ 2 , j (seconds) Distance 2 −100 −50 0 50 100 γ 3 , j (seconds) Station group Non−downto wn stations Downto wn stations Distance 3 −50 0 50 γ 4 , j (seconds) Distance 4 −100 −50 0 50 γ 5 , j (seconds) Distance 5 Figure B.26: Green metro line – Direction 2 55 Skew−normal Skew−t −100−50 0 50 100 150 γ 1 , j (seconds) Model Distance 1 −100 −50 0 50 100 γ 2 , j (seconds) Distance 2 −100 −50 0 50 100 γ 3 , j (seconds) Station group Non−downto wn stations Downto wn stations Distance 3 −50 0 50 γ 4 , j (seconds) Distance 4 −100 −50 0 50 γ 5 , j (seconds) Distance 5 Figure B.27: Orange metro line – Direction 1 Skew−normal Skew−t −100−50 0 50 100 150 γ 1 , j (seconds) Model Distance 1 −100 −50 0 50 100 γ 2 , j (seconds) Distance 2 −100 −50 0 50 100 γ 3 , j (seconds) Station group Non−downto wn stations Downto wn stations Distance 3 −50 0 50 γ 4 , j (seconds) Distance 4 −100 −50 0 50 γ 5 , j (seconds) Distance 5 Figure B.28: Orange metro line – Direction 2 56 A pp endix B.3. Distributional Par ameters This section rep orts the estimated distributional parameters for Direc- tion 2 of the Green metro line (T able B.10), Direction 1 of the Orange metro line (T able B.11), and Direction 2 of the Orange metro line (T able B.12). A comparison of the estimated degrees-of-freedom parameters in the skew- t mo dels across all cases indicates pronounced heavy-tailed b eha vior in p ost- disruption tra vel times, as the estimated v alues of ν remain small throughout. Sk ew-normal Sk ew-t P arameter P osterior mean (5%, 95%) P osterior mean (5%, 95%) ω 0 2.875 ( 2.744, 3.008) 0.354 ( 0.326, 0.384) ω 1 0.176 ( 0.157, 0.196) 0.084 ( 0.078, 0.090) α 0 3.990 ( 3.826, 4.162) 2.183 ( 2.027, 2.342) α 1 − 0.229 ( − 0.244, − 0.214) − 0.067 ( − 0.077, − 0.057) ν 2.039 ( 2.003, 2.094) T able B.10: P osterior summaries of distributional parameters in our prop osed skew-normal and skew- t mo dels for the Green metro line in Direction 2. Sk ew-normal Sk ew-t P arameter P osterior mean (5%, 95%) P osterior mean (5%, 95%) ω 0 2.260 ( 2.149, 2.373) 0.332 ( 0.299, 0.366) ω 1 0.029 ( 0.015, 0.043) 0.037 ( 0.031, 0.043) α 0 2.293 ( 2.174, 2.416) 1.442 ( 1.275, 1.614) α 1 − 0.086 ( − 0.097, − 0.075) − 0.037 ( − 0.050, − 0.024) ν 2.485 ( 2.375, 2.601) T able B.11: P osterior summaries of distributional parameters in our prop osed skew-normal and skew- t mo dels for the Orange metro line in Direction 1. In a manner analogous to Figure 10, Figures B.29, B.30, and B.31 present the estimated scale (left panels) and sk ewness (right panels) parameters, with the corresp onding 90% credible interv als, for the skew-normal and skew- t mo dels across the remaining case studies considered in this w ork. A cross all cases, the estimated scale parameter is consisten tly smaller under the sk ew- t sp eciﬁcation than under the skew-normal mo del. In addition, the sk ewness parameter exhibits a decreasing pattern as tra veled distance increases, with the rate of decrease b eing steep er for the sk ew-normal mo dels relative to the sk ew- t models. 57 Sk ew-normal Sk ew-t P arameter P osterior mean (5%, 95%) P osterior mean (5%, 95%) ω 0 1.613 ( 1.524, 1.703) 0.387 ( 0.346, 0.430) ω 1 0.032 ( 0.020, 0.046) 0.032 ( 0.025, 0.039) α 0 2.253 ( 2.116, 2.390) 1.509 ( 1.355, 1.671) α 1 -0.118 ( − 0.134, − 0.100) − 0.054 ( − 0.070, − 0.039) ν 2.720 ( 2.579, 2.867) T able B.12: P osterior summaries of distributional parameters in our prop osed skew-normal and skew- t mo dels for the Orange metro line in Direction 2. 1.0 1.5 2.0 2.5 0 10 20 T raveled Distance (k − j) Scale Parameter (minutes) Skew−normal Skew−t −2 0 2 4 0 10 20 T raveled Distance (k − j) Ske wness Parameter Skew−normal Skew−t Figure B.29: Green metro line – Direction 2 1.0 1.5 0 10 20 T raveled Distance (k − j) Scale Parameter (minutes) Skew−normal Skew−t 0.0 0.5 1.0 1.5 2.0 0 10 20 T raveled Distance (k − j) Ske wness Parameter Skew−normal Skew−t Figure B.30: Orange metro line – Direction 1 58 0.75 1.00 1.25 1.50 0 10 20 T raveled Distance (k − j) Scale Parameter (minutes) Skew−normal Skew−t −1 0 1 2 0 10 20 T raveled Distance (k − j) Ske wness Parameter Skew−normal Skew−t Figure B.31: Orange metro line – Direction 2 59 A pp endix B.4. Err or dep endenc e p ar ameters T ables B.13, B.14, and B.15 rep ort the p osterior summaries of the error- dep endence parameters, ρ and λ , for the prop osed sk ew-normal and skew- t mo dels across the remaining case studies. When these results are compared with those in T able 6, the estimated dep endence parameters are broadly sim- ilar across the tw o prop osed mo del sp eciﬁcations. Nevertheless, the degree of p ost-disruption error dep endence v aries noticeably across metro lines and directions. A comparison b et w een T able 6 and T able B.13 indicates that Direction 2 of the Green line exhibits a faster decay in dep endence with increasing jour- ney o verlap b etw een consecutive trains, while the magnitude of dependence ρ remains comparable. F or the Orange line, Direction 1 is characterized b y a larger deca y rate λ and a smaller dep endence magnitude ρ relativ e to Direction 2. Sk ew-normal Sk ew-t P arameter P osterior mean (5%, 95%) P osterior mean (5%, 95%) ρ 0.943 (0.903, 0.975) 0.952 (0.917, 0.979) λ 1.722 (1.600, 1.875) 1.745 (1.634, 1.887) T able B.13: Green metro line - Direction 2. Sk ew-normal Sk ew-t P arameter P osterior mean (5%, 95%) P osterior mean (5%, 95%) ρ 0.791 (0.735, 0.857) 0.808 (0.742, 0.882) λ 2.112 (1.763, 2.491) 2.039 (1.666, 2.471) T able B.14: Orange metro line - Direction 1 Sk ew-normal Sk ew-t P arameter P osterior mean (5%, 95%) P osterior mean (5%, 95%) ρ 0.874 (0.797, 0.943) 0.860 (0.775, 0.938) λ 1.592 (1.380, 1.900) 1.567 (1.339, 1.933) T able B.15: Orange metro line - Direction 2 60 A pp endix B.5. R e alize d vs. pr e dicte d tr avel times Figures B.32, B.33, and B.34 presen t the realized versus predicted p ost- disruption trav el times (left panels) and the corresp onding mo del residuals (righ t panels) for the out-of-sample subsets under the skew-normal and skew- t sp eciﬁcations for the remaining case studies, namely Direction 2 of the Green line and b oth directions of the Orange line. Ov erall, the plots indicate strong predictive p erformance, as the predicted v alues cluster closely around the main diagonal. Nevertheless, the disp ersion of residuals appears larger for the Green line (Figures 11 and B.32)) than for the Orange line (Figures B.33 and B.34)), which may p oin t to more stable p ost-disruption op erations on the Orange line. 0 10 20 30 40 0 10 20 30 40 Predicted (minutes) Realized (minutes) Skew−normal Model −5 0 5 0 10 20 30 40 Predicted (minutes) Error (minutes) 0 10 20 30 40 0 10 20 30 40 Predicted (minutes) Realized (minutes) Skew−t −4 0 4 8 0 10 20 30 40 Predicted (minutes) Error (minutes) Figure B.32: Green metro line – Direction 2 61 0 10 20 30 40 0 10 20 30 40 Predicted (minutes) Realized (minutes) Skew−normal Model −6 −3 0 3 6 0 10 20 30 40 Predicted (minutes) Error (minutes) 0 10 20 30 40 0 10 20 30 40 Predicted (minutes) Realized (minutes) Skew−t −4 0 4 0 10 20 30 40 Predicted (minutes) Error (minutes) Figure B.33: Orange metro line – Direction 1 0 10 20 30 40 0 10 20 30 40 Predicted (minutes) Realized (minutes) Skew−normal Model −2 0 2 4 6 0 10 20 30 40 Predicted (minutes) Error (minutes) 0 10 20 30 40 0 10 20 30 40 Predicted (minutes) Realized (minutes) Skew−t −2 0 2 4 6 0 10 20 30 40 Predicted (minutes) Error (minutes) Figure B.34: Orange metro line – Direction 2 62 A pp endix B.6. Pr e diction ac cur acy and unc ertainty assessment Comparison of error-based metrics and uncertaint y assessmen ts for the prop osed mo dels relative to the baseline sp eciﬁcation for the Direction 2 of the Green line in Figure B.35, Direction 1 of Orange line in Figure B.36 and Direction 2 of Orange line in Figure B.37. In each category , the ev aluation is p erformed separately across tra veled distances. The results indicate that b oth prop osed mo dels outp erform the baseline in terms of MAE, HDI length, and empirical co verage, with the skew- t mo del demonstrating sup erior p er- formance compared to the sk ew-normal speciﬁcation. Although the sk ew- t mo dels yield smaller HDI lengths across all case studies, for the Orange line case studies the skew-normal mo dels exhibit co verage closer to the 80% lev el for shorter-distance journeys. Nev ertheless, in all cases the skew- t mo dels ac hiev e cov erage closer to the nominal 80% lev el ov erall, indicating that the statistical b eha vior of p ost-disruption trav el times, for longer journeys, exhibits pronounced hea vy-tailed c haracteristics. 1.50 1.75 2.00 2.25 5 10 15 20 T raveled distance (k − j) RMSE (minutes) Model Baseline Skew−normal Skew−t Root mean squared error 1.0 1.2 1.4 5 10 15 20 T raveled distance (k − j) MAE (minutes) Model Baseline Skew−normal Skew−t Mean absolute error 2 4 6 8 5 10 15 20 T raveled distance (k − j) HDI Length (minutes) Model Baseline Skew−normal Skew−t 80% High density interval length 70 80 90 100 5 10 15 20 T raveled distance (k − j) HDI coverage (%) Model Baseline Skew−normal Skew−t 80% High density interval cov erage Figure B.35: Green metro line – Direction 2 63 1.0 1.5 2.0 2.5 5 10 15 20 T raveled distance (k − j) RMSE (minutes) Model Baseline Skew−normal Skew−t Root mean squared error 0.8 1.0 1.2 1.4 5 10 15 20 T raveled distance (k − j) MAE (minutes) Model Baseline Skew−normal Skew−t Mean absolute error 3 4 5 5 10 15 20 T raveled distance (k − j) HDI Length (minutes) Model Baseline Skew−normal Skew−t 80% High density interval length 75 80 85 90 95 5 10 15 20 T raveled distance (k − j) HDI coverage (%) Model Baseline Skew−normal Skew−t 80% High density interval cov erage Figure B.36: Orange metro line – Direction 1 1.00 1.25 1.50 1.75 5 10 15 20 T raveled distance (k − j) RMSE (minutes) Model Baseline Skew−normal Skew−t Root mean squared error 0.8 1.0 1.2 1.4 5 10 15 20 T raveled distance (k − j) MAE (minutes) Model Baseline Skew−normal Skew−t Mean absolute error 2 3 4 5 6 5 10 15 20 T raveled distance (k − j) HDI Length (minutes) Model Baseline Skew−normal Skew−t 80% High density interval length 70 80 90 5 10 15 20 T raveled distance (k − j) HDI coverage (%) Model Baseline Skew−normal Skew−t 80% High density interval cov erage Figure B.37: Orange metro line – Direction 2 64 Figures B.38, B.39, and B.40 present the P–P plots (top rows) and Q–Q plots for the baseline, skew-normal, and sk ew- t mo del sp eciﬁcations for the remaining three case studies, namely Direction 2 of the Green line and b oth directions of the Orange line. The empirical and model-based cumulativ e probabilities and quan tiles are computed using the transformations deﬁned in Equation 24 and Equation 25 and are plotted against one another. F or the skew-normal and skew- t mo dels, empirical cum ulativ e probabilities and quan tiles are ev aluated separately for eac h tra v eled distance ( k − j ) . A cross all case studies, the prop osed mo dels provide a b etter character- ization of b oth the cen tral mass and the tails of the p ost-disruption tra v el time distributions than the baseline speciﬁcation, as evidenced b y the closer alignmen t of p oin ts with the main diagonal in the P–P and Q–Q plots. How- ev er, for Direction 2 of the Green line and Direction 2 of the Orange line, the sk ew-normal mo del oﬀers a more accurate ﬁt to b oth the bulk and the tails than the sk ew- t model, whereas for Direction 1 of the Orange line the sk ew- t speciﬁcation exhibits sup erior p erformance. 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Theoretical uniform(0,1) Empirical PIT values Baseline model, P−P plot 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Theoretical uniform(0,1) Empirical PIT values Skew−normal model, P−P plot 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Theoretical uniform(0,1) Empirical PIT values Skew−t model, P−P plot −5.0 −2.5 0.0 2.5 5.0 −3 −2 −1 0 1 2 3 Theoretical quantiles Empirical quantiles Baseline model, Q−Q plot −2.5 0.0 2.5 −2 −1 0 1 2 3 Theoretical quantiles Empirical quantiles Skew−normal model, Q−Q plot −5 0 5 10 0 5 10 15 Theoretical quantiles Empirical quantiles Skew−t model, Q−Q plot Figure B.38: Green metro line – Direction 2 65 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Theoretical uniform(0,1) Empirical PIT values Baseline model, P−P plot 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Theoretical uniform(0,1) Empirical PIT values Skew−normal model, P−P plot 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Theoretical uniform(0,1) Empirical PIT values Skew−t model, P−P plot −4 0 4 −2 −1 0 1 2 Theoretical quantiles Empirical quantiles Baseline model, Q−Q plot −2.5 0.0 2.5 5.0 −1 0 1 2 3 Theoretical quantiles Empirical quantiles Skew−normal model, Q−Q plot −5 0 5 10 0 4 8 Theoretical quantiles Empirical quantiles Skew−t model, Q−Q plot Figure B.39: Orange metro line – Direction 1 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Theoretical uniform(0,1) Empirical PIT values Baseline model, P−P plot 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Theoretical uniform(0,1) Empirical PIT values Skew−normal model, P−P plot 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Theoretical uniform(0,1) Empirical PIT values Skew−t model, P−P plot −2 0 2 4 −2 −1 0 1 2 Theoretical quantiles Empirical quantiles Baseline model, Q−Q plot −2 0 2 4 −2 −1 0 1 2 3 Theoretical quantiles Empirical quantiles Skew−normal model, Q−Q plot −2.5 0.0 2.5 5.0 7.5 −3 0 3 6 9 Theoretical quantiles Empirical quantiles Skew−t model, Q−Q plot Figure B.40: Orange metro line – Direction 2 66 Figures B.41 and B.42 compare the CRPS v alues across diﬀeren t trav eled distances for the baseline, skew-normal, and skew- t sp eciﬁcations for the remaining three case studies, namely Direction 2 of the Green line and b oth directions of the Orange line. Overall, the prop osed mo dels achiev e lo wer a verage CRPS v alues across distances in all cases. The skew- t mo del generally exhibits sup erior p erformance relativ e to b oth the baseline and the sk ew- normal sp eciﬁcations. While the skew- t mo del consistently outp erforms the alternativ es for Direction 2 of the Green line, its p erformance for shorter journeys on b oth directions of the Orange line (Figure B.42) is comparable to the other mo dels, before clearly outp erforming them for longer journeys. 1.0 1.5 2.0 2.5 0 5 10 15 20 25 T raveled distance (k − j) Mean CRPS Model Baseline Skew−normal Skew−t Figure B.41: Green metro line – Direction 2 0.6 0.9 1.2 1.5 1.8 0 5 10 15 20 25 T raveled distance (k − j) Mean CRPS Model Baseline Skew−normal Skew−t 0.50 0.75 1.00 1.25 0 5 10 15 20 25 T raveled distance (k − j) Mean CRPS Model Baseline Skew−normal Skew−t Figure B.42: Orange metro line – (left) Direction 1 and (right) Direction 2 67

A Bayesian Framework for Post-disruption Travel Time Prediction in Metro Networks

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