Chapter 11 SPATIAL AUTOREGRESSION

In the previous sections, Moran’s-\(I\) has been discussed, and can be thought of as a measure of spatial autocorrelation. However, up to this point no consideration has been given to a model of a spatially autocorrelated process. In this section, two spatial models will be considered - these are term spatial autoregressive models - essentially they regress the \(z_i\) value for any given polygon on values of \(z_j\) for neighbouring polygons. The two models that will be considered are the simultaneous autoregressive (SAR) and conditional autoregressive (CAR) models. In each case, the models can also be thought of as multivariate distributions for \(\mathbf{z}\), with the variance/covariance matrix being dependent on the \(\mathbf{W}\) matrix considered earlier.

The SAR model may be specified as \[ z_i = \mu + \sum_{j=1}^{j=n} b_{ij} \left(z_j - \mu \right) + \epsilon_i \label{sareq} \]

where \(\epsilon_i\) has a Gaussian distribution with mean 0 and variance \(\sigma_i^2\) (often \(\sigma_i^2 = \sigma^2 \ \forall \ i\), so that the variance of \(\epsilon_i\) is constant across zones), \(\textrm{E}(z_i) = \mu\) and \(b_{ij}\) are constants, with \(b_{ii} = 0\) and usually \(b_{ij} = 0\) if polygon \(i\) is not adjacent to polygon \(j\) - thus, one possibility is that \(b_{ij}\) is \(\lambda w_{ij}\). Here, \(\lambda\) is a parameter specifying the degree of spatial dependence. When \(\lambda=0\) there is no dependence, when it is positive then positive autocorrelation exists, and when it is negative, negative correlation exists. \(\mu\) is an overall level constant (as it is in a standard normal distribution model). If the rows of \(\mathbf{W}\) are normalised to sum to one, then the deviation from \(\'mu\) for \(z_i\) is dependent on the deviation from \(\mu\) for the \(z_j\) values for its neighbours.

The CAR model is specified by

\[ z_i| \left\{ z_j : j \ne i \right\} \sim N \left( \mu + \sum_{j=1}^{j=n} c_{ij} \left(z_j - \mu \right), \tau_i^2 \right) \label{careq} \]

where, in addition to the above definitions, \(N(.,.)\) denotes a normal distribution with the usual mean and variance parameters, \(\tau_i^2\) is the conditional variance of \(z_i\) given \(\left\{ z_j : j \ne i \right\}\) and \(c_{ij}\) are constants such that \(c_{ii} = 0\) and, as with \(b_{ij}\) in the SAR model, typically \(c_{ij} = 0\) if polygon \(i\) is not adjacent to polygon \(j\). Again, a common model is to set \(c_{ij} = \lambda w_{ij}\). \(\mu\) and \(\lambda\) have similar interpretations to the SAR model. A detailed discussion in (???) refers to the matrices \(\mathbf{B} = [b_{ij}]\) and \(\mathbf{C}=[c_{ij}]\) as `spatial dependence’ matrices. Note that this model can be expressed as a multivariate normal distribution in \(\textbf{z}\) as

\[ \textbf{z} \sim N(\mu \mathbf{1}, (\mathbf{I} - \mathbf{C})^{-1}\mathbf{T}) \label{carmat} \]

where \(\mathbf{1}\) is a column vector of 1’s (of size \(n\)) and \(\mathbf{T}\) is a diagonal matrix composed of the \(\tau_i\)’s - see (???) for example. Note that this suggests that the matrix \((\mathbf{I} - \mathbf{C})^{-1}\mathbf{T}\) must be symmetrical. If the \(\mathbf{W}\) matrix is row-normalised, and the \(c_{ij} = \lambda w_{ij}\) model is used, then this implies that
\(\tau_i\) must be proportional to \(\left[ \sum_j c_{ij} \right]^{-1}\).