Multivariate Short-term Traffic Flow Forecasting using Bayesian Vector Autoregressive Mov-

1 Short-term Traffic Flow Forecasting (STFF), the process of predicting future traffic conditions 2 based on historical and real-time observations is an essential aspect of Intelligent Transportation 3 Systems (ITS). The existing well-known algorithms used for STFF include time-series analysis 4 based techniques, among which the seasonal Autoregressive Moving Average (ARMA) model 5 is one of the most precise method used in this field. In the existing literature, ARMA model 6 is mostly used in its univariate multiplicative form and the parameters of the model are mostly 7 estimated using a frequentist approach. The effectiveness of STFF in an urban transport network 8 can be fully be realized only in its multivariate form where traffic flow is predicted at multiple 9 sites simultaneously. In this paper, this concept in explored utilizing an Additive Seasonal Vector 10 ARMA (A-SVARMA) model to predict traffic flow in short-term future considering the spatial 11 dependency among multiple sites. The Dynamic Linear Model (DLM) representation of the A12 SVARMA model has been used here to reduce the number of latent variables. The parameters 13 of the model have been estimated in a Bayesian inference framework employing a Markov Chain 14 Monte Carlo (MCMC) sampling method. The serial correlation problem of MCMC sampling is 15 relaxed by using marginalization and adaptive MCMC. Multiple variations of A-SVARMA, such 16 as differenced process and mean process, have been studied to identify the most suitable prediction 17 methodology. The efficiency of the proposed prediction algorithm has been evaluated by modelling 18 real-time traffic flow observations available from a certain junction in the city-centre of Dublin. 19 Tiep Mai, Bidisha Ghosh and Simon Wilson 2 INTRODUCTION 1 Intelligent Transportation Systems (ITS) is an emerging concept which has been utilized to im2 prove efficiency and sustainability of existing transportation systems. Short-term traffic forecast3 ing, the process of predicting future traffic conditions in short-term or near-term future, based on 4 current and the past observations is an essential aspect of ITS. In the last decade, considerable 5 research attention has been focused on developing precise, flexible, adaptable and universal short6 term prediction algorithms for traffic variable observations. Several parametric and non-parametric 7 techniques have been utilized to develop successful Short-Term Traffic Forecasting (STTF) algo8 rithms (1, 2). 9 The predominant parametric approach in STTF is time-series analysis techniques. Time10 series analysis techniques which are popular in STFF are smoothing techniques (1), Autoregressive 11 linear processes (3) and Kalman filtering (4). Among these, the Autoregressive linear processes are 12 the most developed and well-documented in this field. Ahmed and Cook (5) introduced the Auto13 Regressive Moving Average (ARMA) class of models to the traffic flow forecasting literature. The 14 next seminal step was extending the simple ARMA model to a seasonal format and accounting for 15 the daily and/or weekly variability (6, 7). The aforementioned studies on applying ARMA model 16 in developing STTF algorithms mainly focused on univariate structure; traffic data from any single 17 station were modeled. In the last decade, the research attention has been shifted in developing more 18 efficient prediction algorithms through the utilization of multivariate ARMA techniques which can 19 model the spatial dependency and temporal evolution of traffic variables (such as, volume, speed 20 and travel time) simultaneously. 21 One of the initial multivariate models was developed by Stathopoulos and Karlaftis (8) 22 using state-space methodology. The model provided superior forecasts to equivalent univariate 23 ARMA models. A multivariate ARMA technique called space-time autoregressive integrated mov24 ing average (STARIMA) methodology was applied to develop a model to account for the spatial 25 dependency of traffic data in an urban network (9). The spatial dependency of the network were 26 incorporated in the STARIMA model through the use of weighting matrices estimated based on the 27 distances among the data collection points. A new class of time-series model called the Strutural 28 Time-Series Model was used to develop multivariate STTF algorithm; these models outperformed 29 univariate seasonal ARMA models (10). A direct multivariate extension of autoregressive linear 30 processes has been first attempted by Chandra and Al-Deek (11). This model utilizes a Vector 31 Auto-Regressive (VAR) structure for STTF. Freeway traffic speed and volume had been predicted 32 in this study. The model did not consider the correlation of the noise among multiple stations or 33 data collection points as there does not exist a Moving Average (MA) part. Also, the seasonal na34 ture of the traffic data has been modeled by eliminating seasonality through a seasonal difference 35 and not by direct modeling of seasonality in a seasonal ARMA form. In this study the authors com36 pared VAR with other univariate models and concluded that adding correlations among different 37 locations improves the prediction result. Still, the results are restricted to VAR structure which is 38 only a subclass of seasonal Vector ARMA (VARMA) model. In the same year, Multi-Regression 39 Dynamic Model (MDM) was adopted to develop a multivariate algorithm for STTF by Queen and 40 Albers (12). The MDM consists of multiple independent regression equations often represented in 41 Dynamic Linear Model (DLM) form. Similar to the previous study (11), MDM did not include the 42 noise cross-correlation or the MA coefficients and moreover, MDM assumed spatially indepen43 dent noise, which allows separate statistical inference for each station. Also, the spatial correlation 44 Tiep Mai, Bidisha Ghosh and Simon Wilson 3 among neighbouring stations evolved contemporaneously in the model and a temporal evolution 1 of spatial cross-correlation was not modelled. 2 In this paper, a full multivariate extension of the most efficient univariate time-series model 3 i.e. the seasonal ARMA has been proposed. Unlike the past studies this model involves a noise 4 cross-correlation along with a seasonal form. In particular, an Additive Seasonal VARMA (A5 SVARMA) model has been developed to predict traffic flow in short-term future in urban signal6 ized arterial networks. A Bayesian framework has been proposed to estimate the parameters of 7 the A-SVARMA model. The inference framework utilizes a Markov chain Monte Carlo (MCMC) 8 sampling method. One serious problem of MCMC is the slow convergence caused by serial corre9 lation. Hence, in order to have a better sampling of MA parameters, marginalization and adaptive 10 MCMC are used. The proposed method has been applied to model traffic volume observations 11 from multiple junctions situated at the city-centre of Dublin, Ireland. The results indicate that 12 the proposed forecasting algorithm is an effective approach in predicting real-time traffic flow at 13 multiple junctions within an urban transport network. 14 VARMA MODEL 15 Time series theory, including VARMA and DLM, is discussed in detail in (13, 14, 15). A brief 16 summary is as follows. The vector of time-series observations is denoted by Yt (k × 1) where k 17 is the number of variables observed at time instants t = 1, 2, ..., n. A k-variate VARMA(p, q) is 18 considered with k × 1 common mean β and identical independent (iid) Multivariate Normal noise 19 Et ∼ N(0,Σe): 20 Φ(B)(Yt − β) = Θ(B)Et (1) with 21