lec-location-based-services/ex23-24-25.md

Moran's I

A measure of clustering for spatial data, defined as

I = \frac{N}{W} \frac{\sum_{i=1}^N \sum_{j=1}^N w_{ij}(x_i-\overline{x})(x_j - \overline{x})}
{\sum_{i=1}{N}(x_i - x)^2} 

where

• N is the number of spacial units indexed by i and j
• x is the variable of interest
• \overline{x} is the mean of x $w_{ij} are the elements of a matrix of spatial weights that denote adjacency
• W = \sum_{i=1}^N \sum_{j=1}^N w_{ij} is the sum of all w_{ij}

It may be considered time series stationarity-agnostic as the calculation does not make assumptions about temporal behavior of the underlying data. The deviation from the global mean \overline{x} is calculated at a snapshot in time and weighted by w_{ij}, resulting in a value that ranges from [-1;1].

Sources

• https://doi.org/10.2307/2332142
• https://en.wikipedia.org/wiki/Moran%27s_I