Tjøstheim's coefficient

(Redirected from Tjøstheim’s Coefficient)

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]

See also

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References

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  1. ^ D. Tjøstheim (1978). "A measure of association for spatial variables". Biometrika. 65 (1): 109–114. doi:10.1093/biomet/65.1.109. JSTOR 2335284.
  2. ^ Vallejos, Ronny; Osorio, Feilpe; Bevilacqua, Moreno (2020). Spatial relationships between two georeferenced variables: With applications in R. Springer Cham. doi:10.1007/978-3-030-56681-4. ISBN 978-3-030-56681-4.
  3. ^ a b B.J. Glick (1982). "A Spatial Rank-Order Correlation Measure". Geographical Analysis. 14 (2): 177–181. doi:10.1111/j.1538-4632.1982.tb00066.x.
  4. ^ http://spatialpack.mat.utfsm.cl