Properties and Estimation Fractionally Integrated Spatial Models and Non-Negative Integer-Valued Autoregressive Spatial Models - Ghodsi, Ali Reza (2011)
Spatial modelling has its applications in many ¯elds like geostatistics, geology,geography, agriculture, meteorology, biology, epidemiology, etc. Spatial data can be classi¯ed as geostatistical data, lattice data, or point patterns. This research concentrates on lattice data observed on a regular grid. Examples of spatial data include data collected on a regular grid from satellites ( such as ocean tem-perature) and from agricultural ¯eld trials. Many models have been suggested in modelling spatial dependence like the Simultaneous Autoregressive (SAR),Conditional Autoregressive (CAR), Moving Average (MA) and Autoregressive Moving Average (ARMA). There also exist a class of spatial models that are known as separable models where its correlation structure can be expressed as a product of correlations. In some cases spatial data may exhibit a long memory structure where their autocorrelation function decays rather slowly which can be modelled by fractionally integrated ARMA models. The aim of this research is to introduce and investigate some types of spatial models which have many applications. We ¯rst focus on estimation of the memory parameters of the fractionally inte-grated spatial models. The estimation of the memory parameters by two di®erent methods, namely the regression method and Whittle's method are discussed. Next we consider the Fractionally Integrated Separable Spatial ARMA (FISSARMA) models. The asymptotic properties of the normalised periodogram of the FISSARMA model such as the asymptotic mean and the asymptotic second-order moments of the normalised fourier coe±cients and the asymptotic distribution of the normalised periodogram are established. The third objective of this research is to develop a non-separable counterpart of the FISSAR(1,1) model. We term this model as the ¯rst-order Fractionally Integrated Non-Separable Spatial Autoregressive (FINSSAR(1,1)) model. The theoretical autocovariace function and the spectral function of the model are obtained and some numerical results are presented. Finally, as spatial data may have non-negative integer values, there is a need to introduce non-Gaussian integer-valued spatial models. In this research the ¯rst-order Spatial Integer-valued Autoregressive SINAR(1,1) model with discrete marginal distribution is introduced. Some properties of this model (mean,vari-ance and utocorrelation functions) are established. The Yule-Walker estimator of the parameters of the model is also introduced and the strong consistency of the Yule-Walker estimators of the parameters of the model are also established.