imation capability and can better model the complicated relationship between historical and future traffic flow data than other methods (19) . The application of neural networks does not require an explicit model formulation to be specified, as is usually required. Despite the many attractive features of neural networks, their application is not an easy task. Model training and selection involve tricky decisions with regard to network architectures, type of transfer (activation) functions, learning rate, and number of hidden neurons (20). Cautions must be taken during the training of neural networks to prevent overfitting the training data and to avoid local minima. To address these problems, SVMs have been introduced (16) (17) . Similar to neural networks, SVMs have superior function approximation capability and do not require the specification of model formulations. In addition, they are developed on the structural risk minimization (SRM) principle (21), as opposed to the empirical risk minimization (ERM) principle used in conventional neural networks. Theoretically then, SVMs can better solve the overfitting problem, and they have better generalization capabilities than do conventional neural networks. Another important feature is the SVM capacity to guarantee a globally optimal solution for a given training data set (16) (17) . A v-support vector machine (v-SVM) model was compared with multilayer, feed-forward neural networks by using traffic volume data collected from interstates (I-5, I-90, and I-405) in the Seattle area. The comparison resulted in favor of the v-SVM model (16). Gaussian processes (GPs) are another important class of kernelbased learning algorithms that have attracted attention in the machine learning community (22-23). Similar to other popular kernel machines, such as SVMs, GP models are powerful tools to explore implicit relationships between a set of variables based on a training data set, which make GPs especially useful to address difficult nonlinear regression and classification problems (20) . A particularly attractive feature of GPs is their formulation in a full Bayesian framework, which allows for explicit probabilistic interpretation of model outputs (22) . Moreover, GP model parameters (e.g., kernel parameters) can be computed naturally by means of Bayesian learning, as opposed to the grid-searching, trial-and-error method commonly used to optimize classical SVMs (22). Hence, some researchers refer to GPs as Bayesian SVMs (24). The superior performance of GPs for difficult supervised learning problems has been demonstrated in many domain-specific applications when compared against conventional methods such as neural networks and other advanced learning algorithms such as SVMs (20, 25) . In the study reported on here, a GP regression model was adopted to model and predict traffic volume data. This GP model can be used to forecast travel speed and travel time as well. Like SVMs, GP models are kernel-based machine learning methods and possess many of the same desirable features. More important, they can produce better informative outputs than SVMs and neural networks, which makes it easier to interpret GP prediction results. The accurate modeling and forecasting of traffic flow data, such as volume, speed, and travel time, are critical to intelligent transportation systems (ITS), especially advanced traveler information systems (ATIS) and advanced traffic management systems (ATMS). Given reliable real-time traffic flow predictions, travelers can choose the best routes dynamically. Also, such information can be used by traffic management personnel to develop proactive traffic control strategies that make better use of the available road network resources. The success of many ATIS and ATMS applications depends largely on the accuracy of the selected traffic flow modeling and forecasting algorithms. Numerous methods have been developed and compared since the 1970s to improve the accuracy of traffic flow forecasting. These methods can generally be categorized into the following groups: autoregressive integrated moving average (ARIMA) models (1-3), nonparametric regression (4-5), Kalman filtering theory (6-8), neural networks (9-15), support vector machines (SVMs) (16) (17) , and hybrid models (18). Of the existing traffic flow forecasting methods, neural networks are the most widely used ones. One major reason is that neural networks have a strong function approx-
Xie et al. REF used Gauss regression process to model and forecast traffic flow.
Gaussian Processes for Short-Term Traffic Volume Forecasting
Transportation Research Record
Engineering