Geary's C

Geary's (1954) contiguity ratio of values on units with spatial weights w: C = (n - 1) sum w_ij (x_i - x_j)^2 / (2 S0 sum (x_i - xbar)^2), 1 under no autocorrelation, below 1 when neighbours are alike. Its variance is given under normality and under randomisation (Cliff and Ord 1981), with z statistics (1 - C) / sd and their upper normal tails, and, with nsim > 0, a permutation p-value against positive autocorrelation. That C has expectation 1 under randomisation is proved in Lean: see proof("gearc").

Usage

gearc(x, w, nsim = 0, seed = NULL)

Arguments

x

The values: finite numbers, one per unit.

w

The weights: spwt()'s list, or an spdep listw object, for as many units as x.

nsim

The number of permutations: a whole number from 0.

seed

NULL to draw the simulations' seed from R's generator (so set.seed() fixes them), or one whole number from 0.

Value

A named numeric vector: c, vn and vr (its variance under normality and randomisation), zn, zr, pn, pr, pperm ((1 + #{C* <= C}) / (nsim + 1), NaN without permutations) and s0 (the sum of the weights).

References

Geary, R. C. (1954). The contiguity ratio and statistical mapping. The Incorporated Statistician, 5(3), 115-145. doi:10.2307/2986645

Cliff, A. D. and Ord, J. K. (1981). Spatial Processes: Models and Applications. Pion, London.

Examples

set.seed(1)
p <- cbind(x = runif(50), y = runif(50))
gearc(p[, 1] + p[, 2] + rnorm(50, sd = 0.1), spwt(p, k = 4))
#>            c           vn           vr           zn           zr           pn 
#> 9.221870e-02 9.592157e-03 9.036261e-03 9.268791e+00 9.549637e+00 9.413378e-21 
#>           pr        pperm           s0 
#> 6.507588e-22          NaN 5.000000e+01