Geospatial

Misc

• Also see Geospatial, Spatial Weights

Spatial Autocorrelation

• Misc

  • Local metrics can suffer from multiple testing issues when the number of group units is large
  • {spdep::EBImoran.mc} (Vignette) uses empirical bayes to shrink locations counts/rates that have high variance / small populations towards a global average rate. Then tests (via permutation) for spatial autocorrelation.
    • Useful for count data with outliers or overdispersion

• Moran’s I

  \[ I = \frac{N}{W} \frac{\sum_i \sum_j w_{ij}(x_i - \bar{x})(x_j - \bar{x})}{\sum_i (x_i - \bar{x})^2} \]

  • A measure of global spatial autocorrelation or overall clustering of the data
    • If there is no global autocorrelation or no clustering, there can still be clusters at a local level (See Local Moran’s I)
  • Assume homegeneity (i.e. only one statistic is needed to summarize the whole study area)
  • \(N\) is the number of spatial units (e.g. counties)
  • \(w_{ij}\) is an element of the spatial weights matrix
  • \(W\) is the sum of all \(w_{ij}\)
  • Values significantly below the expected value are negatively correlated
  • Values significantly above the exected value are positively correlated
  • Range: \(w_{\text{min}}\frac{N}{W} \lt I \lt w_{\text{max}}\frac{N}{W}\)
    • For a row normalized weight matrix, \(\frac{N}{W} = 1\) (Wiki)
      • In {spdep}, this would be style = “W”
      • I don’t get this. W = 1, but why would N also equal 1? Not sure if this right.

• Local Moran’s I
  \[ \begin{align} &I_i = \frac{x_i - \bar x}{m_2} \sum_{j=1}^N w_{ij} (x_j - \bar x) \\ &\text{where} \;\; m_2 = \frac{\sum_{i=1}^N (x_i - \bar x)^2}{N} \end{align} \]

  • Moran’s I is just the average of all \(I_i s\), \(I = \sum_{i=1}^N I_i /N\)

• Geary’s C
  \[ C = \frac{(N-1) \sum_i \sum_j w_{ij}(x_i-x_j)^2}{2W \sum_i (x_i - \bar x)^2} \]

  • A measure of global spatial autocorrelation or overall clustering of the data
    • More sensitive to local spatial autocorrelation than Moran’s I so it can pick-up on spatial autocorrelation that Moran’s I might have missed.
  • \(N\) is the number of analysis units on the map
  • \(w_{ij}\) is an element of the spatial weights matrix
  • \(W\) is the sum of all \(w_{ij}\)

• Local Geary’s C
  \[ \begin{align} &C_i = \frac{1}{m_2} \sum_j w_{ij}(x_i - xj)^2\\ &\text{where} \;\; m_2 = \frac{\sum_i (x_i - \bar x)^2}{N-1} \end{align} \]

  • Geary’s C is \(C=\sum_i C_i/2W\)