Stochastic prediction of train delays with dynamic Bayesian networks. Author(s): Kecman, Pavle; Corman, Francesco; Peterson, Anders; Joborn, Martin

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1 Research Collection Other Conference Item Stochastic prediction of train delays with dynamic Bayesian networks Author(s): Kecman, Pavle; Corman, Francesco; Peterson, Anders; Joborn, Martin Publication Date: 2015 Permanent Link: Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library

2 Stochastic prediction of train delays in real-time using Bayesian networks P. Kecman 1 F. Corman 2 A. Peterson 1 M. Joborn 1 1 Department of Science and Technology, Linköping University 2 Section Transport Engineering & Logistics, Maritime and Transport Technology, Delft University of Technology 21 July 2015 Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

3 Outline 1 Introduction 2 Stochastic model based on Bayesian networks 3 Computational experiments 4 Current research 5 Conclusions Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

4 Outline 1 Introduction 2 Stochastic model based on Bayesian networks 3 Computational experiments 4 Current research 5 Conclusions Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

5 Uncertainty in railway traffic Railway traffic typically operates according to a timetable, however... Source: D Ariano, PhD thesis Delay [s] LEDN LAA GV DT SDM RTD RTB RLB DDR Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

6 Importance of tackling uncertainty Quality of railway service depends on accurate predictions of future train movements on many levels: Traffic can be controlled pro-actively by taking actions that prevent delays and delay propagation Timetable, passenger transfer plans, rolling-stock and crew circulation plans can be kept up-to-date Passengers can be provided with accurate information: in-vehicle, on-platform, online Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

7 Existing approaches Deterministic models require frequent updates of train positions and detailed data Stochastic models are based on fixed distributions and do not exploit information available in real time Predictionn error (sec) Dwell time Running time Source: Medeossi et al (2011) JRTPM Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

8 Research objective How can incoming real-time information stream be used to reduce uncertainty? Develop a model that captures dynamics of uncertainty over time and gives predictions that are: Accurate and reliable Dynamic and responsive Stable over longer prediction horizons Source: xkcd.com/612 Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

9 The first step - train delay evolution as a Markov process Realised arrival time Expected arrival time Expected arrival time Expected arrival time Station A Station B Station C Station D Realised arrival time Realised arrival time Expected arrival time Expected arrival time Station A Station B Station C Station D Prediction error [min] Dynamic Static Prediction horizon [min] Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

10 The second step - stochastic delay propagation Station C EET EET EET EET Station B EET EET Realised arrival time EET Station A Realised departure time Time EET Realised departure time Time EET Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

11 Outline 1 Introduction 2 Stochastic model based on Bayesian networks 3 Computational experiments 4 Current research 5 Conclusions Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

12 Bayesian networks Definition Bayesian networks are graphical models for reasoning under uncertainty Variables and the conditional dependencies between them are represented with a directed acyclic graph G = (N, A) Direction of an arc reflects the causality relationship between two variables (nodes) Conditional probability distributions are computed for each node Var 1 Var 2 Var 2 F T T F Var 1 Var 2 T F Var 3 Var 1 F F T T Var 2 F T F T Var 3 T F Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

13 Bayesian networks Properties Compact representation of a joint probability distribution The structure of the network, i.e., the directed arcs between the nodes and conditional probability distributions, can either be learned from the data or determined by expert knowledge Markov property Inference - Probabilistic beliefs updated automatically as new information about a variable becomes available Conditional probability updates - marginal posterior probability Maximum posterior density - configuration q of the variables in Q that has the highest posterior probability Inference P(future traffic state current traffic state) Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

14 Structure of Bayesian network model a) both trains with scheduled stop Arrival Departure b) both trains with scheduled stop and overtaking Arrival Departure Arrival e) overtaking Departure Arrival Departure Arrival Departure Through c) second train without scheduled stop Arrival Departure d) first train without scheduled stop Through b) both trains with scheduled stop and connection Arrival Departure Through Arrival Departure Arrival Departure Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

15 Parameters learning and inference Network structure implication - a local distribution has only a small number parameters to estimate Local distributions are univariate normal random variables Local distributions are conditioned on incoming events Local distributions are in fact linear models in which the parents play the role of explanatory variables Y P(Z X,Y) Z Density X Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

16 Implementation setup Network structure is built based on the route plan and schedule assumed to be known during the prediction horizon The structure of the network is determined assuming that the train routes and orders are known Parameters of the network computed from the historical database (training set) Every train event message represents the new information that is propagated through the graph using statistical inference algorithms The prediction horizon moves, new trains are added to the model Events within prediction horizon Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

17 Outline 1 Introduction 2 Stochastic model based on Bayesian networks 3 Computational experiments 4 Current research 5 Conclusions Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

18 Case study Case study from a busy corridor between Stockholm and Norrköping in Sweden 180 km long double-track line with mixed traffic 27 stations and junctions, 10 of which accommodate scheduled stops of passenger and freight trains Approximately 300 hundred trains per day traverse the corridor (fully or partially). Stockholm Norrköping Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

19 Data set description The database contains the scheduled and realised times for departures, arrivals and through runs for all trains and stations All event times are rounded to full minutes Train position and delay update given only at stations with a frequency of 2 minutes on average Two months (1 January to 28 February 2015) of historical traffic realisation data has been made available by the Swedish infrastructure manager Trafikverket. Randomly selected weekday from the analysis used as a test day to evaluate the performance of the Bayesian network model (test set) Remaining data are used for calibration of network parameters (training set) Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

20 Experimental setup Prediction horizon is 60 minutes A real-time environment for model validation was created by sweeping through the chronologically sorted messages in the test set Every train event message represents the new information received by the monitoring system The graph size does not increase linearly in time as only the nodes representing the last events of all trains are kept in the model due to the inherent Markov property of Bayesian networks Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

21 Model structure, size and parameters The average network size is 137,12 nodes and arcs Arc type Coefficient RSE p value R 2 Headway 0,42 0,33 *** 0,52 Running 0,95 0,06 *** 0,90 Dwell 0,81 0,09 *** 0,79 Table: Average values of predictive power of local models Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

22 Model performance on the test set Model applied on the peak hours (6:30-9:00 and 16:30-19:00) of the test day In total the inference algorithm is executed 563 times The predicted values are compared against the realised event times Prediction error [min] Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

23 Model performance on the test set Mean absolute error Prediction horizon [min] Standad deviation of error Prediction horizon [min] Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

24 Dynamic updates of the marginal distribution An example of how the distribution of arrival time of a train to the final station evolves over time in six discrete steps H =72 min H = 60 min H = 41 min Density Density Density H = 33 min H = 14 min H = 5 min Density Density Density Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

25 Dynamic updates of probability An example of how the probability that the arrival time will be larger than 16 min changes in six discrete steps Probability Time horizon [min] Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

26 Outline 1 Introduction 2 Stochastic model based on Bayesian networks 3 Computational experiments 4 Current research 5 Conclusions Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

27 Stochastic modelling of traffic control decisions Bayesian classification approach How to overcome the assumption that traffic train sequences (orders) are known? Delay train 1 Delay train 2 Secondary delay Classifier Train 1 -> Train 2 Train 2 -> Train 1 Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

28 Stochastic modelling of traffic control decisions Initial results Analysis of traffic control actions when secondary delays occur One point of conflict analysed 1110 delays that may cause secondary delays considered 77 % of cases resulted in reordering Changed order No reordering Frequency Frequency Delay effect of reordering [min] Delay effect of reordering [min] Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

29 Stochastic modelling of traffic control decisions Initial results Analysis of traffic control actions when secondary delays occur One point of conflict analysed 1110 delays that may cause secondary delays considered 77 % of cases resulted in reordering Changed order No reordering Frequency Frequency Delay effect of reordering [min] Delay effect of reordering [min] Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

30 Stochastic modelling of traffic control decisions Bayesian classification approach How to overcome the assumption that traffic train sequences (orders) are known? Delay train 1 Delay train 2 Secondary delay Train priorities Classifier Equity Train 1 -> Train 2 Train 2 -> Train 1 Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

31 Outline 1 Introduction 2 Stochastic model based on Bayesian networks 3 Computational experiments 4 Current research 5 Conclusions Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

32 Summary and conclusions Analysis of train delays and their evolution in real time with respect to the dynamic stochastic phenomena The goal was to determine the impact of real-time information on reducing uncertainty Bayesian networks turned out to be an appropriate method to concisely represent the complex interdependencies between train events The model was evaluated in a simulated real time environment and the computational results indicate that the predictions are reliable for horizons of up to 30 minutes Potential applications include integration in robust rescheduling framework, online traffic control an delay management models and passenger information systems A strong assumption of the model is that there is a perfect knowledge of train orders and routes within the prediction horizon - overcoming this will be in the focus of future research in this direction Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31

33 Thank you for your attention Pavle Kecman (LiU) Stochastic prediction of train delays 21 July / 31