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,
, observed at the same set of
spatial locations with co-ordinates
is with a similar definition for
θ
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
δ
is the Kronecker delta.
coordinate of the
value.
The quantities
can be defined similarly.
Tjøstheim's coefficient is defined by[2] Under the assumptions that
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
[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