# 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 $$R$$.