Spatial Moving Average Polynomial Model on Irregular Lattice and its Full-Likelihood Based Implementation

Abstract

The simultaneous incorporation of a contiguity-based neighbourhood matrix of first and second order neighbors in the spatial moving average model on an irregular lattice can be specified in two different ways- as a quadratic polynomial or as a product of two linear polynomials. The present article concerns with the second order spatial moving average polynomial model of the latter form. Expression for the joint probability distribution of the response is derived together with expressions for the first and second order moments of the model using characteristic function. Model parameters are estimated using method of maximum likelihood and expression for their standard errors are obtained. In the absence of an analytical solution of the full likelihood function, optimization of the full likelihood function using Differential Evolution technique has been performed. Confidence interval for drawing inference has been constructed based on the derived standard error expression and bootstrapping method. Using likelihood ratio test statistic and bootstrap results, simultaneous confidence regions for the two spatial dependence parameters are also constructed. Implementation of the model has been executed on simulated data, and relevant comparisons made with linear models and first order spatial moving average model. The advantage of using the proposed model lies in its ability to provide statistically valid inference, as it affects producing spatial dependence of the first and the second lag orders in the residuals, insignificant.