Skip to contents

Global Moran's I with its inference

Source: R/sarml.R
gmoran.Rd

Moran's (1950) I of values on units with spatial weights w: with z = x - mean(x), I = n / S0 sum_ij w_ij z_i z_j / sum z_i^2, its expectation -1 / (n - 1) and its variance under normality and under randomisation (Cliff and Ord 1981), the z statistics and their upper normal tails (positive autocorrelation: neighbours alike), and, with nsim > 0, the permutation p-value (1 + #{I* >= I}) / (nsim + 1), the values shuffled over the units from seed. These are the statistics of spdep::moran.test() and spdep::moran.mc(). Given a matrix or data frame, each column is tested on the same weights: one row per category or year, a sweep of a variable across a polygon surface. lmorn() gives the local statistics, which add up to S0 I with row-standardised weights.

Usage

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

Arguments

x

The values: finite numbers, one per unit; or a matrix or data frame with one column per variable.

w

The weights: spwt()'s list, an spdep listw object, or an n x n matrix of weights from 0 (zero diagonal).

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 data frame with one row per variable: var (its name), i, e, vn and vr, zn and zr, pn and pr, pperm (NaN without permutations) and s0 (the sum of the weights); NA but s0 for a constant variable.

References

Moran, P. A. P. (1950). Notes on continuous stochastic phenomena. Biometrika, 37(1-2), 17-23. doi:10.2307/2332142

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))
w <- spwt(p, k = 4)
gmoran(p[, 1] + rnorm(50, sd = 0.2), w, nsim = 99, seed = 1)
#>   var         i           e          vn          vr      zn       zr
#> 1   x 0.6251582 -0.02040816 0.007802795 0.007875543 7.30829 7.274458
#>             pn           pr pperm s0
#> 1 1.352813e-13 1.739075e-13  0.01 50
# a sweep over three yearly counts
gmoran(data.frame(y1 = rpois(50, 3), y2 = rpois(50, 3 + 4 * p[, 1]),
                  y3 = rpois(50, 9 * p[, 1])), w)
#>   var             i           e          vn          vr        zn        zr
#> 1  y1 -0.0008085382 -0.02040816 0.007802795 0.007858451 0.2218823 0.2210952
#> 2  y2  0.1023809524 -0.02040816 0.007802795 0.007581203 1.3900640 1.4102328
#> 3  y3  0.3831195986 -0.02040816 0.007802795 0.007584193 4.5682339 4.6336020
#>             pn           pr pperm s0
#> 1 4.122028e-01 4.125092e-01   NaN 50
#> 2 8.225473e-02 7.923548e-02   NaN 50
#> 3 2.459255e-06 1.796788e-06   NaN 50