Spatial correlation at lag 1

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.