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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.