1 — Types of spatial structures
A spatial structure may appear in a variable y because the process that has produced the values of y is spatial and has generated autocorrelation in the data; or it may be caused by dependence of y upon one or several causal variables x which are spatially structured; or both. The spatially-structured causal variables x may be explicitly identified in the model, or not; see Table 13.3.
Autocorre- • Model 1: autocorrelation — The value yj observed at site j on the geographic surface lation is assumed to be the overall mean of the process (|y) plus a weighted sum of the centred values (yi - |y) at surrounding sites i, plus an independent error term £j:
The y/s are the values of y at other sites i located within the zone of spatial influence of the process generating the autocorrelation (Fig. 1.4). The influence of neighbouring sites may be given, for instance, by weights Wi which are function of the distance between sites i and j (eq. 13.19); other functions may be used. The total error term is [X f (yi - |y) + ej]; it contains the autocorrelated component of variation. As written here, the model assumes stationarity (Subsection 13.1.1). Its equivalent in time series analysis is the autoregressive (AR) response model (eq. 12.30).
Spatial • Model 2: spatial dependence — If one can assume that there is no autocorrelation in dependence the variable of interest, the spatial structure may result from the influence of some explanatory variable(s) exhibiting a spatial structure. The model is the following:
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