Measure of spatial correlation
Tjøstheim's Coefficient
[1] is a measure of spatial
association that attempts to quantify the degree to which two spatial data sets are related. Developed by Norwegian statistician
Dag Tjøstheim. It is similar to rank correlation coefficients like
Spearman's rank correlation coefficient and the
Kendall rank correlation coefficient but also explicitly considers the spatial relationship between variables.
Consider two variables, and , observed at the same set of spatial locations with co-ordinates and . The Rank of at is
with a similar definition for . Here is a
step function and this formula counts how many values are less than or equal to the value at the target point .
Now define
where is the
Kronecker delta. This is the coordinate of the ranked value. The quantities and can be defined similarly.
Tjøstheim's Coefficient is defined by
[2]
Under the assumptions that and are
independent and identically distributed random variables and are
independent of each other it can be shown that and
The maximum variance of occurs when all points are on a straight line and the minimum variance of occurs for a symmetric cross pattern where and .
[3]
Tjøstheim's Coefficient is implemented as cor.spatial in the
R package SpatialPack.
[4] Numerical simulations suggest that is an effective measure of correlation between variables but is sensitive to the degree of
autocorrelation in and .
[3]
References