Getis-ord G* on rasters

The Standard Getis-ord is defined on individual points (vector data). In many situations, we want to compute \(G^*_i\) in a raster rather than in a point cloud. This way, the computation can be expressed using map algebra operations and nicely parallelized in frameworks such as Geotrellis.

The rasterized Getis-Ord formula looks as follows:

\[ G^*(R, W, N, M, S) = \frac{ R{\stackrel{\mathtt{sum}}{\circ}}W - M*\sum_{w \in W}{w} }{ S \sqrt{ \frac{ N*\sum_{w \in W}{w^2} - (\sum_{w \in W}{w})^2 }{ N - 1 } } } \]

where:

  • \(R\) is the input raster.
  • \(W\) is a weight matrix of values between 0 and 1. The matrix is square and has odd dimensions, e.g. \(5 \times 5\), \(31 \times 31\) ...
  • \(N\) represents the number of all pixels in \(R\) (because there can be NA values)
  • \(M\) represents the global mean of \(R\).
  • \(S\) represents the global standard deviation of all pixels in \(R\).

It can be seen that the formula can be nicely refactored into:

  • One global operation that computes \(N\), \(M\), \(S\). These values are usually available because they were pre-computed when the raster layer has been ingested into the catalog.
  • One focal operation \(R{\stackrel{\mathtt{sum}}{\circ}}W\) - the convolution of raster \(R\) with the weight matrix \(W\).
  • One local operation that puts all components toghether for each pixel in \(R\).