Travel Speed Forecasting by Means of Continuous Conditional Random Fields

interpret and analyze the resulting model. Autoregressive integrated moving average (ARIMA) models, which encompass RW, random-trend models; auto-regressive models; and exponential weighted moving averages are linear time series models that have been quite popular thanks to their ability to exploit temporal dependence in prediction errors (4, 5). Linear models that exploit both spatial and temporal information that have been used in traffic forecasting include Kalman filters (6) and spatial–temporal ARIMA (7 ). Two extensions of the linear models have been quite useful in traffic forecasting. By observing that a single set of parameters might not be suitable over varying traffic conditions, time-varying linear regression models have been employed with considerable success (8–10). As an alternative to a continuous change in model parameters, regime-switching models have been implemented (10) to model different traffic states separately, such as congested and free-flow regimes. In addition to linear models, nonlinear ones have also been extensively studied in traffic-forecasting research. Neural networks (11, 12) and, more recently, support vector machines (13) have been used with some success; this success indicates that the nonlinearities in spatial–temporal traffic behavior can be exploited. The nonparametric approach based on the nearest neighbor algorithm, which is a generalization of the baseline historical method, has also been popular because of its ease of implementation and reasonable forecasting accuracy. For an extensive and quite good overview of traffic forecasting methods see Vlahogianni et al. (14). This paper explores a recently proposed conditional random field (CRF) framework (15) that is extremely successful in modeling dependencies among random variables (such as spatial–temporal correlations) in forecasting problems. This novel method can be easily implemented and is quite flexible, and the goal of this research was to evaluate the method’s applicability in traffic forecasting. Originally, CRFs were proposed for classification of sequential data (15), as an attractive alternative to hidden Markov models (16). Unlike those models, CRFs make no independence assumptions between input variables, and this avoidance of assumptions results in a more flexible modeling framework. Meanwhile, CRFs were applied in a number of fields, including computer vision (17 ) and computational biology (18). Recently, CRFs have been extended to solve regression problems. Continuous CRFs (CCRFs) were first described in Qin et al. (19) in the context of the problem of global document ranking. Following the Qin et al. work, an extension of CCRF that is applicable to spatial–temporal data was proposed by Radosavljevic et al. (20). Because it can easily deal with real-valued, spatial–temporal data, CCRF naturally emerges as an extremely suitable model for travel forecasting. CCRF can combine various, possibly very different, Travel Speed Forecasting by Means of Continuous Conditional Random Fields