# Short-term traffic flow forecasting

Short-term traffic flow forecasting (STFF) is an essential aspect of intelligent transportation systems (ITS). Even though auto-regressive moving average (ARMA) has been widely used as one of the most precise methods in such a problem, it has several weaknesses.

In this work, a spatial temporal model is proposed and the joint forecast is performed on the traffic flow and the vehicle number of the link. Using an additional variable, vehicle number, allows better forecasts, and adaption to the dynamic traffic scenarios. Applying this model on an expanded traffic network enables the prediction of spatial effects, from one source to another. Real-time prediction might be obtained by parallel and independent inference on separate links and junctions.

## ARMA

A 4-day simulated traffic flow of 1-minute data collection interval with a traffic accident at time index 3100 is pictured. Several weaknesses of ARMA limit its usage in traffic flow, according to the images. The model adapts slowly to the dynamic situation with he multi-step prediction suffering from exponential decay.

The intractability of moving average, and the non-parallelisable inference make multivariate ARMA impractical, and not scalable to a large traffic network.

## Traffic Networks and the General Model

4-day data with a traffic blocking incident is obtained from transportation micro-simulator VISSIM with the below network map. The dataset consists of link in-flows z_{1,t} and out-flows z_{2,t} and the vehicle numbers, v_{t} of all directed links

The general model consists of 4 sub-models

- The root link: z
_{1,t}=f_{rlk}(z_{1,t-1}) - The link: z
_{2,t}=f_{lk}(v_{t-1},z_{1,t-1}) - The junction: ζ
_{2,t}=f_{jc}(ζ_{1,t}) where ζ_{1,t},ζ_{2,t}are junction inflows and outflows - The occupancy: v
_{t}=f_{o}(v_{t-1},z_{2,t-1},z_{1,t-1})

## The Link Sub-Model

Piecewise polynomial smoothing is used for the link module. In the analysis of 1-minute dataset, the effect of z_{1,t-1} to z_{2,t} seems negligible. One possible candidate model is:

y=xΒ_{0}+(x-k_{1})+Β_{1}+(x-k_{2})+Β_{2}

with 2 knots k_{1}, k_{12}.y=z_{2,t}; x=v_{t-1}. The model is fitted with t=1..2880 but the prediction seems satisfactory even during the interval of the blocking incident (L-H0203).

## The Junction Sub-Model

A junction example is shown on the right. Denote the out-flows and in-flows of the junction by ζ_{2,t}=ζ_{2,t,1:2} and ζ_{1,t}=ζ_{1,t,1:2} correspondingly. The dynamic model for the junction is:

ζ_{2,t}=A_{t}(x_{t})ζ_{1,t}+v_{t}

x_{t}=x_{t-1}+u_{t}

with u_{t}~N(0,(λ_{u}I)^{-1}) and v_{t}~N(0,(λ_{v}I)^{-1})

For the example in the figure, x_{t}=x_{t,1:2} is a vector of length 2. A_{t} is a 2x2 turning matrix: A_{t,i,1}=logistic(x_{t,i}), A_{t,i,2}=1-A_{t,i,1}. Sequential inference is applied on both state vector and unknown parameters, based on smooth functional approximation iterLap.

The results for junction J-0103 are as follows:

The top 2 graphs are a summary of the inferred distribution of x_{t}. The blue and purple solid lines represents the mean and the mode of the distribution. The blue dashed lines mark the interval 2σ limits (roughly a 95% probability interval for x_{t}). The red dashed lines represent the optimised series x_{t}, using all data. The brown dashed vertical lines mark a duration when a traffic incident occurs. We see that the method has inferred values of x_{t} that follow the optimal values quite closely, even during the traffic incident.

The bottom 2 graphs show the fit of ζ_{ζ2;t} and ζ_{2;t} to the data. Purple solid lines and red dashed lines are ζ_{ζ2;t} and ζ_{2;t} respectively. The brown dashed vertical lines mark the starting point of the traffic incident.

## People

Tiep Mai, Simon P. Wilson, Bidisha Ghosh