In this section, we define a new space–time model, the localized STARIMA (LSTARIMA) model. Like the traditional STARIMA model, LSTARIMA makes use of a spatial weight matrix W to model the influence of the spatiotemporal neighborhood. However, it relaxes the globally fixed parametric structure of STARIMA by allowing the AR and MA parameters to vary by location, which allows it to account for spatial heterogeneity. Furthermore, it accounts for temporal nonstationarity by allowing the size of the spatial neighborhood to vary with time. In this sense, it is a locally dynamic space–time model.
Let z(t) be an N-dimensional column vector containing the observations zi(t) on each link i, where i = 1, 2, … , N, during each time interval t, where t = 1, 2, … , T. The conventional STARIMA model can be defined as
The first term in equation (3) is the AR component, whereas the second term is the MA. The parameters p and q are the AR and MA orders, respectively. The term ε(·) is an N-dimensional column vector of residuals on each link, and h is the spatial order, which represents the order of spatial separation between two locations. The parameters mk and nl are the spatial orders associated with the kth and lth temporally lagged terms in the AR and MA components, respectively. They specify the size of the spatial neighborhood that could influence the link of interest i within temporal lags k and l. The notation ϕkh and θlh are the AR and MA parameters, respectively, to be calibrated for the entire network. The matrix W(h) is an N × N spatial weight matrix for spatial lag h, containing the set of weights wij specifying the assumed relationship between i and j (see Kamarianakis and Prastacos 2005; Getis 2009). The number of parameters to be calibrated in equation (3) is p × mk + q × nl.
We extend the standard STARIMA model to account for spatial heterogeneity and temporal nonstationarity using the following formulation, which we call LSTARIMA:
where W(h,t–k,i) and W(h,t–l,i) are the elements of dynamic spatial weight matrix W(h,t) pertaining to link i at temporal lags k and l. LSTARIMA has a separate set of AR and MA parameters for each link i, which are stored in N × N diagonal matrices Φkh and Θlh such that
where [ϕi,kh] and [θi,lh] are the parameters for each link i (i = 1, 2, … , N). The number of parameters that needs to be calibrated in equation (4) is (p × mk(t−k,i) + q × nl(t−l,i), although the whole model needs to calibrate N links of the entire network.
The STARIMA and ARIMA models can be viewed as special cases of the LSTARIMA model. For example, if p1 = p2 = pN and q1 = q2 = qN (i.e., p and q are spatially fixed), mk(t − 1, i) = mk(t − 2, i) = … = mk(t − k, i) and nl(t − 1, i) = nl(t − 2, i) = … = nl(t − l, i) (i.e., the spatial influence of adjacent links does not change over time), and [ϕ1,kh] = [ϕ2,kh] = [ϕN,kh] and [θ1,lh] = [θ2,lh] = … = [θN,lh] (i.e., all parameters are the same for all of the links), then LSTARIMA (equation (4)) becomes a STARIMA model (equation (3)). This is unlikely for road network data, but for other data sets such as annual temperature, spatial autocorrelations may not change rapidly over time. Moreover, if mk(t − k, i) = 0 and nl(t − l, i) = 0 (i.e., the adjacent links produce no spatial influence), then the LSTARIMA becomes an ARIMA. This reduction might happen when traffic is flowing freely (or possibly is highly congested), resulting in the speeds on all the links being more or less the same.
In the next sections, we use LSTARIMA (pi, qi), ARIMA (pi, di, qi), and STARIMA (p, d, q) specifications to represent the models because differencing (which is described by parameter d) is not needed for the LSTARIMA model.
Equation (4) is a time series model that considers the spatial influence of adjacent links in networks. Thus, its parameters can be estimated by means of standard time series calibration algorithms. The procedure of parameter estimation in equation (4) can be regarded as the minimization of the following sum of squared errors function:
where T is the number of observations in time, zi(t) is the observation vector at time t and link i, εi(t) is the random error vector at time t, and .
Equation (6) presents a nonlinear least squares minimization problem because εi(t) is required for the calibration of the MA parameter θlh, but is unknown a priori. Thus, εi(t) must be estimated first in order to determine ϕkh and θlh. Furnishing appropriate starting values is important to ensure convergence of the optimization procedure. Although trial and error can be used to implement the parameter optimization process, it cannot guarantee convergence. Moreover, an exhaustive search is needed, which is time consuming.
Hannan and Rissanen (1982) demonstrated that the Hannan–Rissanen (HR) algorithm is an effective approach for parameter optimization of time series models. It makes use of the residuals of a high-order AR model to feed the εi(t) as initialized values. Then, parameters ϕkh and θlh are calibrated using the linear least squares method. The methodology has been proven valid in their work and has been broadly accepted in practice. Here, we use the same procedure for LSTARIMA model calibration by considering εi(t) to be an independent random variable after the spatiotemporal autocorrelation has been fully modeled by the dynamic weight matrix. This result is verified by testing for spatial, temporal, and spatiotemporal autocorrelation in the predictive residuals in the subsequent case study.
Although Box and Jenkins's algorithm (Box, Jenkins, and Reinsel 1970, pp. 498–505) was proposed earlier than the HR algorithm, it does not provide the details of how to furnish appropriate initial values (epsilon) to ensure convergence of the optimization procedure of model calibration. Given the implementation method and that mathematical proof has been provided (Hannan and Rissanen 1982), HR algorithm is commonly used in practice and is chosen here to calibrate our model.