Spatial correlation at lag 1 | Early Warning Signals Toolbox

Near-neighbor spatial correlation (at lag 1) captures the increasingly correlation between neighboring spatial sites. It can be quantified by the spatial correlation function, or Moran’s I, between ecological states z[i, j] and z[m, n] separated by a distance r:

(1) $\begin{equation*} C_2(r) = \frac{MN\sum_{i=1}^M \sum_{m=1}^M \sum_{j=1}^N \sum_{n=1}^N w[i,j; m,n] (z[i,j]-\bar{z})(z[m,n]-\bar{z})} {W\sum_{m=1}^M \sum_{n=1}^N (z[m,n]-\bar{z})^2} \end{equation*}$

where $\bar{z}$ is the spatial mean of the state variable ( $\bar{z}=\sum_{m=1}^M \sum_{n=1}^N z[m,n]/(MN)$ ), $w[i,j; m,n]$ is $1$ if spatial units $[i,j]$ and $[m,n]$ are separated by a distance $r$ , and is 0 otherwise, $W$ is the total number of units separated by the distance $r$ . In the notation $C_2(r)$ , the subscript $2$ stands for 2-dimensional space. The near-neighbor spatial correlation $C_2(\delta)$ , the analog of autocorrelation at lag 1 for time series, is calculated for the distance $\delta$ corresponding nearest neighboring units of the system.