Experiences Modelling a Traffic Network

1
Experiences Modelling a Traffic Network

• Benjamin Wright
• The Open University
• Joint work with Catriona Queen
• (Open University)

2
M25/A2/A282 Junction, Dartford
A282
167
A296
168
165
169
163
170B
164B
170A
164A
171
162
A2
161
172
M25
3
The Traffic Network

• Structure of the network
• A set of data collection sites
• Spatial structure for the network
• Hourly totals for vehicles passing each counting
  point
• The time taken to traverse the network is less
  than an hour.
• We have a multivariate time series for the
  network.

4
Data Collected

• Two weeks data collected at site 167

5
The Traffic Network

• Model requirements
• Forecast future traffic flows
• Discern underlying levels of traffic
• Incorporate expert opinion
• Robustness towards missing data
• Uses of the model
• Analysis of current conditions
• For planning roadworks or new road segments
• Inform expert opinion elsewhere

6
M25/A2/A282 Junction, Dartford
167
170A
168
170B
169
161
171
172
162
163
164B
164A
165
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Dynamic Linear Models in Brief

• The model
• An observation vector Yt and a parameter vector
  ?t
• Current beliefs about ?t Nmt,Ct
• The Observation equation
• Yt FtT ?t vt vt N0,Vt
• The System equation
• ?t Gt ?t-1 wt wt N0,Wt
• At time t we can find priors for ?t1 and Yt1
• We can update beliefs about ?t1 given the
  observation yt1

8
Dynamic Linear Models in Brief

• A DLM is specified by F,G,V,Wt
• In practice, Ft and Gt are often constant
• We can use discounting methods for Wt
• We can estimate Vt in the univariate case
• This Bayesian framework can incorporate
  intervention and missing data with ease.
• It is computationally inexpensive

9
Hierarchical Dynamic Linear Models

• Ft can be a vector of regressors
• Our ?t become coefficients rather than level
  parameters
• Ft is no longer constant
• We can apply this to our network
• If traffic flows from point A to point B, we can
  regress B on A
• We must establish appropriate lags for our
  regression
• However-
• In our network we somehow need lag 0

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Multiregression Dynamic Models in Brief

• MDMs are a multivariate generalisation of DLMs
• The flow of traffic is a causal relationship
• We can define a Directed Acyclic Graph to
  represent conditional independence in the network
• This allows us to decompose the network into
  univariate DLMs where we regress nodes on their
  parents at lag 0.
• Entry points we can model as standard DLMs

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Directed Acyclic Graph
?(167)
Y(167)
?(170)
Y(170)
Y(167)-Y(170)
?(168)
?(170B)
Y(170B)
Y(170A)
Y(L)
Y(168)
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Multiregression Dynamic Models in Brief

• Key features
• Our forecasts for a node are now functions of the
  forecasts for its parents
• Use ?t in place of ?t to show that our parameters
  are now proportions
• We can update our parameters independently
• We can use superposition to create nodes that
  have parents and traffic joining the network.
• Example Observation Equation
• Y(170)t y(167)t ?(170)t v(170)t

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Seasonal Variance

• Consider our forecast variances
• They are complicated functions including seasonal
  variables
• They are seasonal in a way that cannot be
  corrected with a simple transformation
• This seasonality is a product of the problem
  structure
• Implications
• Methodologies that assume constant variance for
  this type of problem are flawed
• We correct this for free in an MDM

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Missing Data

• In a DLM
• Missing data is a trivial problem we can skip
  a time period by letting our posteriors equal our
  priors
• In an MDM
• Occasional missing data is equally trivial
• If there is a large quantity of consecutive
  missing data for a node, we may have to
  restructure the model without that node

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Expert Intervention

• The principle of incorporating outside
  information in the model
• Information from other models or from an expert
• Only as good as the information we use
• Critical when responding to unusual patterns of
  activity
• Goals of intervention
• Improved forecasts
• Parameter integrity
• Rapid adaptation

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Methods of Intervention

• There are several standard methods of
  intervention
• Treat outliers as missing data
• Handle transient change by altering the
  observation equation
• Handle permanent change by altering the system
  equation
• Change discount factor
• Adjust our estimate of the variance Vt
• Change structure of model

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Methods of Intervention

• Overparameterisation
• Changing the observation equation is functionally
  identical to adding a temporary extra parameter
• We can update this parameter and keep it in the
  model to learn about this unusual activity
• We can thus model a systematic change without
  affecting our underlying parameters
• We can use our posteriors for the intervention
  parameter to inform future intervention
• We can only use this technique for a limited time

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Entry Points

• Seasonal variance
• We need some method of modelling the seasonal
  variance at entry points
• One possibility is to use multiple parameters for
  the variance
• Correlation between entry points
• If this exists, as it is likely to, our model may
  break down
• If we can find this correlation, we can build it
  into the model
• We can restructure the model to try and avoid the
  problems caused with little loss of accuracy

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Weekly Pattern

• The data display different patterns of traffic
  depending on the day
• We need to establish what days are different and
  how we can organise them into categories
• We can model this using dummy variables
• For simplicity, we confine ourselves to one of
  the categories

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Comparisons

• The data were modelled using three approaches-
• Seasonal independent univariate ARIMA models
• Independent univariate DLMs
• MDM with simple DLMs for entry points
• For the DLM and MDM approaches, we then model
  incorporating intervention

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ARIMA

• Example Y170B modelled with (1,0,0)x(1,0,0)24

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DLM

• Example DLM for Y170B

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MDM

• Example Y170B modelled with standard DLMs for
  entry points

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Results

• Inspect MSE for each model
• Forecast variances are also lower with MDM method
• This is more pronounced with intervention
• The MDM structure allows intervention to filter
  throughout the network

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Conclusions

• MDM gives
• Competitive forecast performance
• Structure that includes seasonal variances
• Hierarchical intervention
• Easy expert intervention
• robustness to missing data in most circumstances
• parameters that are easily interpreted