Defining weights matrices

The value of ${\displaystyle I}$ can depend quite a bit on the assumptions built into the spatial weights matrix ${\displaystyle w_{ij}}$. The matrix is required because, in order to address spatial autocorrelation and also model spatial interaction, we need to impose a structure to constrain the number of neighbors to be considered. This is related to Tobler's first law of geography, which states that Everything depends on everything else, but closer things more so—in other words, the law implies a spatial distance decay function, such that even though all observations have an influence on all other observations, after some distance threshold that influence can be neglected.

The idea is to construct a matrix that accurately reflects your assumptions about the particular spatial phenomenon in question. A common approach is to give a weight of 1 if two zones are neighbors, and 0 otherwise, though the definition of 'neighbors' can vary. Another common approach might be to give a weight of 1 to ${\displaystyle k}$ nearest neighbors, 0 otherwise. An alternative is to use a distance decay function for assigning weights. Sometimes the length of a shared edge is used for assigning different weights to neighbors. The selection of spatial weights matrix should be guided by theory about the phenomenon in question. The value of ${\displaystyle I}$ is quite sensitive to the weights and can influence the conclusions you make about a phenomenon, especially when using distances.

Expected value

The expected value of Moran's I under the null hypothesis of no spatial autocorrelation is

${\displaystyle E(I)={\frac {-1}{N-1}}}$

The null distribution used for this expectation is that the ${\displaystyle x}$ input is permuted by a permutation ${\displaystyle \pi }$ picked uniformly at random (and the expectation is over picking the permutation).

At large sample sizes (i.e., as N approaches infinity), the expected value approaches zero.

Its variance equals

${\displaystyle \operatorname {Var} (I)={\frac {NS_{4}-S_{3}S_{5}}{(N-1)(N-2)(N-3)W^{2}}}-(E(I))^{2}}$

where

${\displaystyle S_{1}={\frac {1}{2}}\sum _{i}\sum _{j}(w_{ij}+w_{ji})^{2}}$

${\displaystyle S_{2}=\sum _{i}\left(\sum _{j}w_{ij}+\sum _{j}w_{ji}\right)^{2}}$

${\displaystyle S_{3}={\frac {N^{-1}\sum _{i}(x_{i}-{\bar {x}})^{4}}{(N^{-1}\sum _{i}(x_{i}-{\bar {x}})^{2})^{2}}}}$

${\displaystyle S_{4}=(N^{2}-3N+3)S_{1}-NS_{2}+3W^{2}}$

${\displaystyle S_{5}=(N^{2}-N)S_{1}-2NS_{2}+6W^{2}}$^[3]

Values of I usually range from −1 to +1. Values significantly below -1/(N-1) indicate negative spatial autocorrelation and values significantly above -1/(N-1) indicate positive spatial autocorrelation. For statistical hypothesis testing, Moran's I values can be transformed to z-scores.

Moran's I is inversely related to Geary's C, but it is not identical. Moran's I is a measure of global spatial autocorrelation, while Geary's C is more sensitive to local spatial autocorrelation.