Eliciting a Directed Acyclic Graph for a Multivariate Time Series of Vehicle Counts in a Traffic Network

Abstract

SummaryThe problem of modelling multivariate time series of vehicle counts in traffic networks is considered. It is proposed to use a model called the linear multiregression dynamic model (LMDM). The LMDM is a multivariate Bayesian dynamic model which uses any conditional independence and causal structure across the time series to break down the complex multivariate model into simpler univariate dynamic linear models.The conditional independence and causal structure in the time series can be represented by a directed acyclic graph (DAG). The DAG not only gives a useful pictorial representation of the multivariate structure, but it is also used to build the LMDM. Therefore, eliciting a DAG which gives a realistic representation of the series is a crucial part of the modelling process.A DAG is elicited for the multivariate time series of hourly vehicle counts at the junction of three major roads in the UK. A flow diagram is introduced to give a pictorial representation of the possible vehicle routes through the network. It is shown how this flow diagram, together with a map of the network, can suggest a DAG for the time series suitable for use with an LMDM.

Forecast covariances in the linear multiregression dynamic model

“…For example, in a traffc network a temporary diversion may change the DAG temporarily, or a change in the road layout may change the DAG permanently. Because of the structure of the LMDM, much of the posterior information from the original DAG can be used to help form priors in the new DAG (see Queen et al, 2007, for an example illustrating this). In this respect, component covariances may be informative about covariances in the new DAG.…”

“…We shall illustrate why component covariances may be useful by considering two examples. Queen et al (2007) consider the problem of forecasting hourly vehicle counts at various points in a traffc network. The possible routes through the network are used to elicit a DAG for use with an LMDM.…”

Forecast covariances in the linear multiregression dynamic model

“…For example, in a traffc network a temporary diversion may change the DAG temporarily, or a change in the road layout may change the DAG permanently. Because of the structure of the LMDM, much of the posterior information from the original DAG can be used to help form priors in the new DAG (see Queen et al, 2007, for an example illustrating this). In this respect, component covariances may be informative about covariances in the new DAG.…”

“…We shall illustrate why component covariances may be useful by considering two examples. Queen et al (2007) consider the problem of forecasting hourly vehicle counts at various points in a traffc network. The possible routes through the network are used to elicit a DAG for use with an LMDM.…”

Canonical Analysis: Ranks, Ratios and Fits

“…This example relates to four groups each with six samples. The data, based on previous studies by Queen, Wright, and Albers (2007), and Queen and Albers (2009), are vehicle flow counts on different sections of the M25/A296 motorway network in Kent, United Kingdom. The groups refer to the k = 4 slip roads between these motorways and the n 1 = n 2 = n 3 = n 4 = 6 samples refer to different weeks, each about a month apart.…”

Multivariate Forecasting of Road Traffic Flows in the Presence of Heteroscedasticity and Measurement Errors

“…Let Y t ( i ) be the number of vehicles passing site S ( i ) during 15‐min period t . Following Queen et al. (2007), in the LMDMs for and Y t (3), and and Y t (6) can be elicited to have the forms Thus, upstream flows are used in the models for downstream flows.…”

“…In expression (4), the μ t (·) are level parameters, whereas parameter α t represents the proportion of traffic flowing from S (1) to S (2), and v t (·) are normal error terms. In Queen et al. (2007) the normality of the errors v t (·) is justified by appealing to the Poisson approximation to normality for large means.…”

Forecast covariances in the linear multiregression dynamic model

“…For example, in a traffc network a temporary diversion may change the DAG temporarily, or a change in the road layout may change the DAG permanently. Because of the structure of the LMDM, much of the posterior information from the original DAG can be used to help form priors in the new DAG (see Queen et al, 2007, for an example illustrating this). In this respect, component covariances may be informative about covariances in the new DAG.…”

“…We shall illustrate why component covariances may be useful by considering two examples. Queen et al (2007) consider the problem of forecasting hourly vehicle counts at various points in a traffc network. The possible routes through the network are used to elicit a DAG for use with an LMDM.…”

Canonical Analysis: Ranks, Ratios and Fits

“…This example relates to four groups each with six samples. The data, based on previous studies by Queen, Wright, and Albers (2007), and Queen and Albers (2009), are vehicle flow counts on different sections of the M25/A296 motorway network in Kent, United Kingdom. The groups refer to the k = 4 slip roads between these motorways and the n 1 = n 2 = n 3 = n 4 = 6 samples refer to different weeks, each about a month apart.…”

Multivariate Forecasting of Road Traffic Flows in the Presence of Heteroscedasticity and Measurement Errors

“…Let Y t ( i ) be the number of vehicles passing site S ( i ) during 15‐min period t . Following Queen et al. (2007), in the LMDMs for and Y t (3), and and Y t (6) can be elicited to have the forms Thus, upstream flows are used in the models for downstream flows.…”

“…In expression (4), the μ t (·) are level parameters, whereas parameter α t represents the proportion of traffic flowing from S (1) to S (2), and v t (·) are normal error terms. In Queen et al. (2007) the normality of the errors v t (·) is justified by appealing to the Poisson approximation to normality for large means.…”

Forecast covariances in the linear multiregression dynamic model

“…For example, in a traffc network a temporary diversion may change the DAG temporarily, or a change in the road layout may change the DAG permanently. Because of the structure of the LMDM, much of the posterior information from the original DAG can be used to help form priors in the new DAG (see Queen et al, 2007, for an example illustrating this). In this respect, component covariances may be informative about covariances in the new DAG.…”

“…We shall illustrate why component covariances may be useful by considering two examples. Queen et al (2007) consider the problem of forecasting hourly vehicle counts at various points in a traffc network. The possible routes through the network are used to elicit a DAG for use with an LMDM.…”

Canonical Analysis: Ranks, Ratios and Fits

“…This example relates to four groups each with six samples. The data, based on previous studies by Queen, Wright, and Albers (2007), and Queen and Albers (2009), are vehicle flow counts on different sections of the M25/A296 motorway network in Kent, United Kingdom. The groups refer to the k = 4 slip roads between these motorways and the n 1 = n 2 = n 3 = n 4 = 6 samples refer to different weeks, each about a month apart.…”

Multivariate Forecasting of Road Traffic Flows in the Presence of Heteroscedasticity and Measurement Errors

“…Let Y t ( i ) be the number of vehicles passing site S ( i ) during 15‐min period t . Following Queen et al. (2007), in the LMDMs for and Y t (3), and and Y t (6) can be elicited to have the forms Thus, upstream flows are used in the models for downstream flows.…”

“…In expression (4), the μ t (·) are level parameters, whereas parameter α t represents the proportion of traffic flowing from S (1) to S (2), and v t (·) are normal error terms. In Queen et al. (2007) the normality of the errors v t (·) is justified by appealing to the Poisson approximation to normality for large means.…”