Spatial correlation

Algorithm

First, distances between all points are calculated. Distance classes are created for point pairs that are more or less at the same distance to each other.

Distance classes are usually based on a user-specified lag spacing.
By using the options 'omni directional' and spherical distance, all distances are calculated over the sphere using the projection of the coordinate system that is used by the input point map.

Then, for all point pairs within a distance group, the spatial autocorrelation, spatial variance and the experimental semi-variogram value is calculated.

In ILWIS, spatial autocorrelation between points is calculated as Moran's I (Odland):

In ILWIS, the spatial variance is calculated as Geary's c (Odland):

Semi-variogram values are defined as:

where:

z	the value of a point
	the average value of all available point values
(z_i - )(z_j - )	the product of: the difference of the value of point i and the average value of all points, and the difference of the value of point j and the average value of all points
(z_i-z_j)²	the squared difference of the values of point i and point j; this is calculated for all point pairs within a distance class and then summed.
(zi- )²	the squared difference of the value of point i and the average value of all points; this is calculated for all points and then summed; this is a constant value (variance).
n	the total number of points
w_ij	weight of a point pair. When using the omnidirectional method, w_ij = 1 when a point pair belongs to a certain distance class, otherwise w_ij = 0. When using the bidirectional method, w_ij = 1 when a point pair belongs to a certain distance class and when within the direction, tolerance and bandwidth as specified by the user (see also Figure 1 in Spatial correlation : functionality); otherwise w_ij = 0.

In the numerators (top of a fraction) of these formulas, the weights assure that only the values of points that have a certain distance towards each other will be taken into account in the calculations for that distance class.

In the denominators of these formulas (bottom of a fraction), i.e. in the standardization parts of the formulas, the weights count the number of valid point pairs within a distance class.

All summations are from i=1 to n and from j=i+1 to n, thus every point pair is counted only once.

References:

Geary, R.C., 1954. The contiguity ratio and statistical mapping.
Moran, P.A.P., 1948. The interpretation of statistical maps.
Odland, J. 1988. Spatial autocorrelation. In: G.I. Thrall (Ed.), Sage University Scientific Geography Series no. 9. Sage Publications, Beverly Hills. 87 pp.