Join count statistic

Given binary data ${\displaystyle x_{i}\in \{0,1\}}$ distributed over ${\displaystyle N}$ spatial sites, where the neighbour relations between regions ${\displaystyle i}$ and ${\displaystyle j}$ are encoded in the spatial weight matrix

${\displaystyle w_{ij}={\begin{cases}1\qquad &i{\text{ neighbor of }}j\\0&{\text{otherwise}}\end{cases}}}$

the join count statistics are defined as ^[8][4]

${\displaystyle J=J_{BB}+J_{BW}+J_{WW}}$

Where

${\displaystyle J_{BB}={\frac {1}{2}}\sum _{ij,i\neq j}w_{ij}x_{i}x_{j}}$

${\displaystyle J_{BW}={\frac {1}{2}}\sum _{ij,i\neq j}w_{ij}(x_{i}-x_{j})^{2}}$

${\displaystyle J_{WW}={\frac {1}{2}}\sum _{ij,i\neq j}w_{ij}(1-x_{i})(1-x_{j})}$

${\displaystyle J={\frac {1}{2}}\sum _{ij,i\neq j}w_{ij}}$

The ${\displaystyle B,W}$ subscripts refer to 'black'=1 and 'white'=0 sites. The relation ${\displaystyle J=J_{BB}+J_{BW}+J_{WW}}$ implies only three of the four numbers are independent. Generally speaking, large values of ${\displaystyle J_{BB}}$ and ${\displaystyle J_{WW}}$ relative to ${\displaystyle J_{BW}}$ imply autocorrelation and relatively large values of ${\displaystyle J_{BW}}$ imply anti-correlation.

To assess the statistical significance of these statistics, the expectation under various null models has been computed.^[9] For example, if the null hypothesis is that each sample is chosen at random according to a Bernoulli process with probability

${\displaystyle p={\frac {\text{number of black cells}}{N}}={\frac {N_{1}}{N}}}$

then Cliff and Ord ^[8] show that

${\displaystyle E(J_{BB})={\frac {1}{2}}S_{0}p^{2}}$

${\displaystyle var(J_{BB})={\frac {p^{2}(1-p)}{4}}([S_{1}(1-p)+S_{2}p])}$

${\displaystyle E(J_{BW})=S_{0}p(1-p)}$

${\displaystyle var(J_{BW})={\frac {p(1-p)}{4}}[4S_{1}+S_{2}(1-4p(1-p))]}$

where

${\displaystyle S_{0}=\sum _{ij}w_{ij}}$

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

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

However in practice^[10] an approach based on random permutations is preferred, since it requires fewer assumptions.

Anselin and Li introduced^[11][12] the idea of the local join count statistic, following Anselin's general idea of a Local Indicator of Spatial Association (LISA).^[13] Local Join Count is defined by e.g.

${\displaystyle J_{BBi}=x_{i}\sum _{j}w_{ij}x_{j}}$

with similar definitions for ${\displaystyle BW}$ and ${\displaystyle WW}$. This is equivalent to the Getis–Ord statistics computed with binary data. Some analytic results for the expectation of the local statistics are available based on the hypergeometric distribution^[11] but due to the multiple comparisons problem a permutation based approach is again preferred in practice.^[12]

When there are ${\displaystyle k\geq 2}$ categories join count statistics have been generalised^[4][8][9]

${\displaystyle J_{rs}={\frac {1}{2}}\sum _{ij}I_{r}(x_{i})I_{s}(x_{j})}$

Where ${\displaystyle I_{r}(x_{i})=\delta _{r,x_{i}}}$ is an indicator function for the variable ${\displaystyle x_{i}}$ belonging to the category ${\displaystyle r}$. Analytic results are available^[14] or a permutation approach can be used to test for significance as in the binary case.