Standards Getis-Ord G*

The standard definition of Getis-Ord \(G^*_i\) statistic assumes a study area with \(n\) points with measurements \(X = [x_1, \ldots, x_n]\). Moreover, it assumes weights \(w_{i,j}\) to be defined between all pairs of points \(i\) and \(j\) (for all \(i,j \in \{ 1, \ldots, n\}\)). The formula to compute \(G^*_i\) at a given point \(i\) is then:

\[ G^*_i = \frac{ \sum_{j=1}^{n}w_{i,j}x_j - \bar{X}\sum_{j=1}^{n}w_{i,j} }{ S \sqrt{ \frac{ n \sum_{j=1}^{n}w_{i,j}^2 - (\sum_{j=1}^{n}w_{i,j})^2 }{n-1} } } \]

where:

  • \(\bar{X}\) is the mean of all measurements,
  • \(S\) is the standard deviation of all measurements.

As it is known, this statistic creates a z-score, which denotes the significance of an area in relation to its surrounding areas.

Note

For \(x \in X\), the \(zscore(x) = \frac{x - mean(X)}{stdev(X)}\)

Todo

Julian Bruns: Add references to papers