IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

A graphical dynamic approach to forecastingroad traffic flow networks

Catriona M. Queen Casper J. Albers

Department of Mathematics and StatisticsThe Open UniversityMilton Keynes U.K.

One-day Conference on Traffic ModellingThe Open University, March 2009

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Setting the sceneData: Geographic locationsData descriptions

Introduction

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Setting the sceneData: Geographic locationsData descriptions

Setting the scene

Want to forecast hourly traffic flows

Model needs:

multivariate,work in real time,accommodate changes, andpreferably interpretable.

Focus on 2 UK networks.

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Setting the sceneData: Geographic locationsData descriptions

LondonM25 / A2 / A296 Junction

Pictures taken from Google Maps

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Setting the sceneData: Geographic locationsData descriptions

ManchesterM60 / M62 / M602 Junctions

Pictures taken from Google Maps

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Setting the sceneData: Geographic locationsData descriptions

LondonM25 / A2 / A296 Junction

Diagram Properties

17 data sites

21 weeks of data

Hourly counts

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Setting the sceneData: Geographic locationsData descriptions

ManchesterM60 / M62 / M602 Junction

Diagram Properties

32 data sites

Over a year’s worth of data

Aggregate hourly counts (byminute)

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Why multivariate?Represent the network by a graphLinear Multiregression Dynamic ModelsExample LMDMLinear relationships

The Model

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Why multivariate?Represent the network by a graphLinear Multiregression Dynamic ModelsExample LMDMLinear relationships

Why multivariate?

Example road

Counts at upstream sites informative about downstream sites.

Suppose:

Minute dataSite 1 → Site 2 takes 1 min

⇒ use Yt−1(1) to forecast Yt(2).

We have hourly counts ⇒ use Yt(1) to forecast Yt(2).

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Why multivariate?Represent the network by a graphLinear Multiregression Dynamic ModelsExample LMDMLinear relationships

Represent the network by a graph

Represent network by a directed acyclic graph (DAG).

Example road with DAG

DAG defines set of conditional distributions: child | parents

→ breaks multivariate problem into univariate conditionals:

Yt(1) Yt(2)|Yt(1) Yt(3)|Yt(2).

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Why multivariate?Represent the network by a graphLinear Multiregression Dynamic ModelsExample LMDMLinear relationships

Linear Multiregression Dynamic Models

Our model is called the Linear Multiregression Dynamic Model (LMDM).

LMDMs:

Represent time series by a DAG.

Model breaks into separate (conditional) univariate models (DLMs).

Each univariate model uses parents as regressors.

Model computations are fast and relatively straightforward nomatter how complex the network.

Interventions are easy.

Graphical structure ⇒ easy to handle changes in network.

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Why multivariate?Represent the network by a graphLinear Multiregression Dynamic ModelsExample LMDMLinear relationships

Example LMDM

DAG

Observation equations:

Yt(1) = θt(1) + vt(1), vt(1) ∼ N(0,Vt(1)),Yt(2) = yt(1)θt(2) + vt(2), vt(2) ∼ N(0,Vt(2)),Yt(3) = yt(2)θt(3) + vt(3), vt(3) ∼ N(0,Vt(3)).

Parameters evolves through system equation:

θt = Gtθt−1 + wt , wt ∼ N(0,Wt).

Model is Bayesian so prior specified for θ0.

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Why multivariate?Represent the network by a graphLinear Multiregression Dynamic ModelsExample LMDMLinear relationships

Linear relationships

LMDM assumes linear relationship between parent and child.

Plot of parent (1431A) vs child (1437A)

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Why multivariate?Represent the network by a graphLinear Multiregression Dynamic ModelsExample LMDMLinear relationships

Do linear relationships depend on day of week?

Plot of parent (1431A) vs child (1437A) — different colour for each day

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Why multivariate?Represent the network by a graphLinear Multiregression Dynamic ModelsExample LMDMLinear relationships

Do linear relationships depend on hour of day?

Plot of parent (1431A) vs child (1437A) — different colour for each hour

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Why multivariate?Represent the network by a graphLinear Multiregression Dynamic ModelsExample LMDMLinear relationships

Linear relationships: conclusion

Linear relationships between parent and child is realisticassumption...

... where the linear relationship depends on hour of day.

⇒ Use different regression parameter for each hour.

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

DAG for a forkDAG for a joinOther types of road layoutDAGs for the data

Creating a DAG for a traffic network

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

DAG for a forkDAG for a joinOther types of road layoutDAGs for the data

DAG for a fork

Fork and DAG Constructing the DAG

Yt(1) ∼ Poi(µt)Yt(1) ' µt + vt(1), vt(1) ∼ N

Yt(2)|Yt(1) ∼ Bin(Yt(1), αt)with αt the proportion 1→ 2

Yt(2) ' αtYt(1)+vt(2), vt(2) ∼ N

Yt(3) = Yt(1)− Yt(2)so Yt(3) is a logical variable

Alternative model possibleEquivalent under assumptionYt(1) = Yt(2) + Yt(3)

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

DAG for a forkDAG for a joinOther types of road layoutDAGs for the data

DAG for a join

Join and DAG Constructing the DAG

Yt(1) = µt(1) + vt(1), vt(1) ∼ N

Yt(2) = µt(2) + vt(3), vt(2) ∼ N

Yt(3) = Yt(1) + Yt(2)so Yt(3) is a logical variable.

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

DAG for a forkDAG for a joinOther types of road layoutDAGs for the data

Other types of road layout

All types of road lay-out can be modelled through a combination of forksand joins.

Example: a junction

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

DAG for a forkDAG for a joinOther types of road layoutDAGs for the data

Other types of road layout

All types of road lay-out can be modelled through a combination of forksand joins.

Example: a junction

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

DAG for a forkDAG for a joinOther types of road layoutDAGs for the data

LondonM25 / A2 / A296 Junction

Because of 6 missing nodes, the network splits into 2 separate DAGs and 2 isolated nodes

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

DAG for a forkDAG for a joinOther types of road layoutDAGs for the data

ManchesterM60 / M62 / M602 Junction

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

ForecastsIntervention requiredThe intervention

Analyses and results

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

ForecastsIntervention requiredThe intervention

Forecasts

Forecast for three days, M25 network

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

ForecastsIntervention requiredThe intervention

Intervention required

Forecast for three days at site 167

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

ForecastsIntervention requiredThe intervention

Intervention required

Same problem seen in Yt(167)’s descendants.

But not seen in Yt(167)’s non-descendants.

Example

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

ForecastsIntervention requiredThe intervention

Intervention required

Same problem seen in Yt(167)’s descendants.

But not seen in Yt(167)’s non-descendants.

Example

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

ForecastsIntervention requiredThe intervention

The intervention

Intervene for Yt(167) only.

Large negative forecast error at time 560.Large positive forecast error at time 561.

Consistent with hold-up at time 560.

Treat Y560(167) as outlier.

Intervene for Yt(167) at time t = 561:Increase forecast mean for Y561(167) — add on forecast error atprevious time.Increase forecast variance for Y561(167) — allows for moreuncertainty.

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

ForecastsIntervention requiredThe intervention

The intervention

The intervention for Y561(167):improves forecast for Yt(167) at time t = 561,also improves forecasts for Yt(167)’s descendants at time t = 561,does not affect the forecasts of non-descendants.

Example

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

ForecastsIntervention requiredThe intervention

The intervention

The intervention for Y561(167):improves forecast for Yt(167) at time t = 561,also improves forecasts for Yt(167)’s descendants at time t = 561,does not affect the forecasts of non-descendants.

Example

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Conclusion

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Conclusion

LMDM’s are

Computationally fast.

Flexible to adapt and easy to intervene.

Promising for forecasting traffic flows.

Easily interpretable by non-statisticians.

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

Conclusion

Future work:

Develop an on-line monitor.

Methods of identifying and estimating missing data.

Automated methods for eliciting the DAG.

Allowing feedback due to queues.

.....

Acknowledgements:

Kent County Council, provided London data

The Highways Agency, provided Manchester data

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks

IntroductionThe Model

Creating a DAG for a traffic networkAnalyses and results

Conclusion

References

Queen, C.M. and Albers, C.J.Intervention and causality in a dynamic Bayesian network.Journal of the American Statistical Society – Applications and CaseStudies, 104(486), 2009.

Queen, C.M. and Albers, C.J.Forecasting Traffic Flows in Road Networks: a Graphical Dynamic ModelApproach.Proceedings of the 28th International Symposium of Forecasting,International Institute of Forecasters, 2008.

Queen, C.M., Wright, B.J. and Albers, C.J.Forecast covariances in the linear multiregression dynamic model.Journal of Forecasting, 27(2), 171–191, 2008.

Queen, C.M., Wright, B.J. and Albers, C.J.Eliciting a Directed Acyclic Graph for a Multivariate Time Series ofVehicle Counts in a Traffic Network.

Australian and New Zealand Journal of Stats, 49(3), 221–239, 2007.

C.M. Queen, C.J. Albers — The Open University Forecasting road traffic flow networks