Moran’s Autocorrelation Coefficient in Comparative Methods

where x̄ and ȳ are the sample means of both variables. ρ measures whether, on average, xi and yi are associated. For a single variable, say x, I will measure whether xi and xj , with i 6= j, are associated. Note that with ρ, xi and xj are not associated since the pairs (xi, yi) are assumed to be independent of each other. In the study of spatial patterns and processes, we may logically expect that close observations are more likely to be similar than those far apart. It is usual to associate a weight to each pair (xi, xj) which quantifies this [3]. In its simplest form, these weights will take values 1 for close neighbours, and 0 otherwise. We also set wii = 0. These weights are sometimes referred to as a neighbouring function. I’s formula is: