Some explicit conditions for a stationary representation of the unilateral second-order spatial ARMA model

The analysis of the spatial data has been carried out in many disciplines such as demography, meteorology, geology and remote sensing. The spatial data modelling is important because it recognizes the phenomenon of spatial correlation in field experiments. Three main categories of the spatial models, namely, the simultaneous autoregressive (SAR) models (Whittle, 1954), the conditional autoregressive (CAR) models (Bartlett, 1971), and the moving average (MA) models (Haining, 1978) have been studied. Whittle (1954) presented a form of bilateral autoregressive (AR) models, whereas Basu and Reinsel (1993) considered the first-order autoregressive moving average (ARMA) model of the quadrant type. Awang, N. and Mahendran Shitan (2003) presented the second-order ARMA model, and established some explicit stationary conditions for the model. When fitting the spatial models and making prediction, it is assumed that, the properties of the process would not change with sites. Properties like stationarities have to be assumed, and for this reason, it was therefore imperative that the researchers had made certain that the process was stationary. This could be achieved by providing the explicit stationarity conditions for the model. The explicit conditions, for a stationary representation of the second-order spatial unilateral ARMA model denoted as ARMA (2,1;2,1), have been established (Awang, N. and Mahendran Shitan, 2003) and in this paper, some explicit conditions are established for a stationary representation of the second-order spatial unilateral ARMA model, denoted as ARMA(2,2;2,2).

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