Spatial Neighborhood Order Determination for Gaussian Markov Random Fields

In this paper, we propose a new type of the Gaussian Markov random fields (GMRFs) that incorporates a parameter for the order of the spatial neighborhood. The first model is the conditional autoregressive (CAR) model with an additional parameter for neighborhood order and the other is a CAR model parameterized to account for decreasing spatial correlation (association) as the neighborhood order increases. The main characteristics of the new models are that 1) for both models, only one parameter is involved in the neighborhood structure for any order of neighborhood, 2) for the second model, the spatial correlation (association) parameters are represented by the decaying rate of spatial correlation similar to Gaussian geo-statistical models (GGMs), and 3) the most appropriate order of neighborhood is selected instead of being arbitrarily selected by the researcher. A hybrid Gibbs-Metropolis-Hastings algorithm is developed for the parameter estimation. We study the performance of the proposed models via the simulation studies.