One of the most common forms of spatial analysis in geography is the estimation of spatial autocorrelation using correlograms (Cliff and Ord 1973, 1981; Sokal and Oden 1978a). A correlogram consists of a series of estimated autocorrelation coefficients calculated for different spatial relationships. The two common autocorrelation coefficients are Moran’s I and Geary’s c; the Mantel correlation can also be used to construct a correlogram for distance-based data. Correlogram analysis is extremely similar to Join Count analysis, except the data are continuous rather than categorical.
To compute an autocorrelation coefficient, one must choose a system of assigning weights wij to connect the localities. These weights may be based on a binary connections matrix (see sections 2.1.5 and 3.7) or (more commonly) on distance. Usually the weight matrix is binary, such that wij equals 1 if there is a connection between points i and j, and 0 if not. However, the weights do not have to be binary and other weighting schemes can be used; the inverse of the distance (or squared or cubed distance) between points i and j is an obvious potential weight and may allow for more accurate hypothesis testing (Jumars et al. 1977).
Rather than use a single set of weights to calculate an overall measure of spatial autocorrelation, one normally uses an ordered series of weights that depicts different spatial relationships among the localities. A common method is to use a series of classes representing successively larger distances (e.g., 0 to 10 km, 10 to 20 km, etc.), in which a pair of points is given a weight of 1 if the distance between them is within the range of the class and a 0 otherwise. The autocorrelation coefficients are then calculated separately for each distance class. The plot of the autocorrelation coefficients against distance is known as the correlogram.
The significance of individual autocorrelation coefficients can be determined from their moments (Cliff and Ord 1973, 1981; Sokal and Oden 1978a, b), while that of an entire correlogram is usually calculated using a Bonferroni procedure (Oden 1984). A Bonferroni procedure is necessary to deal with the lack of independence of the distance classes; the same points are included within each class, although they are paired differently.