# Spatiotemporal Traffic Prediction Using Hierarchical Bayesian Modeling
## 2 Related Works  
In the literature, various spatiotemporal modeling techniques have been proposed for traffic prediction, which can be mainly classified into two categories: parametric and nonparametric approaches [19]. The parametric approaches make assumptions and define a fixed-parameter for its structural algorithm. Parametric approaches, such as STARIMA, make assumptions on the variables and establish a structural algorithm with fixed parameters [20], while in the nonparametric approaches, such as neural networks (NNs) and Bayesian networks, the number of the model's parameters grows as the size of the training data increases [19; 21]. Having an infinite number of parameters makes nonparametric approaches suitable methods for analyzing and predicting spatiotemporal data. However, due to the complexity of estimating the spatiotemporal traffic data relationships, developing an efficient spatiotemporal traffic model becomes challenging [21]. [PERSON] et al. [22] propose a hybrid modeling framework for short-term traffic prediction that incorporates Bayesian networks (BN) and neural networks (NN) to model the spatiotemporal correlation between traffic variables. However, these two models have several drawbacks, which can be summarized as follows: (1) training data that are reasonable in size are computationally expensive when compared to other spatiotemporal models [23]; and (2) the lack of a clear methodology structure affects the reliability of the results [24]. Another research study for predicting traffic data with missing values using the ST-Kriging approach is proposed in [2]. The study indicates the efficiency of ST-Kriging methods in handling missing values. However, the prediction accuracy might be affected when each road network is considered separately [25]. Furthermore, the ST-Kriging approach faces some challenges that are highlighted in a study by [PERSON] and [PERSON] [26]. First, ST-Kriging prediction lies on the covariance matrix and the inverse covariance matrix; with large-scale data, the matrix inversion is difficult. Therefore, ST-Kriging prediction is implemented on data with relatively small sizes. Another challenge that the ST-Kriging approach faces is optimizing the semivariogram estimation and selecting the optimal lag size and the optimal number of lags [27], as there is no optimal approach for selecting these parameters. [PERSON] et al. [28] and [PERSON] and [PERSON] [29] propose spatiotemporal traffic modeling studies from a microscopic perspective. These studies, however, do not take into account the traffic flow status; instead, these methods focus on simulating the origin-destination (OD) element to develop a singular trajectory model that targets vehicle changing patterns associated with time and location [30]. Although there are numerous studies concerning traffic prediction, there is still a literature gap in modeling large space-time traffic data for short-time prediction considering both the spatial and temporal correlations [20]. On the other hand, a considerable amount of literature has been published on spatiotemporal modeling using a hierarchical Bayesian (HB) approach; however, these models were developed in different application contexts, such as public health, image processing and environmental modeling [31; 32; 33]. The basic idea behind hierarchical Bayesian modeling is to integrate prior knowledge about specific observations, such as air pollution concentrations, traffic flow values, etc., and then analyze the collected observations from each spatial point with the prior knowledge to predict new data that are more accurate and reliable [34]. One of the key features of the Bayesian approach is representing the uncertainty of all the possible predictive distributions, using probability distributions. The uncertainty in Bayesian inferences can be represented in terms of observations level, parameters level, and processes level [16]. Together, this gives a full joint distribution in a hierarchical manner to prove the accuracy of the hypothesis.