Geospatial

Misc

Symmetry

  • Most spatial statistics assume reciprocal relationships and will symmetrize weights if they are asymmetric. (e.g. Moran’s I)
  • Symmetrizing weights can indeed introduce bias if the true underlying spatial process is inherently asymmetric (i.e. influence flows in one direction).
    • Examples:
      • In watershed studies where upstream locations affect downstream locations but not vice versa
      • In wind dispersal where particles move predominantly in one direction
      • In economic studies where influence flows from larger to smaller markets but not equally in reverse
  • Statistical inference often assumes undirected (symmetric) relationships
  • Asymmetry creates a logical problem: how can A influence B but B not influence A in a spatial relationship?
  • spdep::is.symmetric.nb can be used to check for symmetry in weight lists (e.g. spdep::graph2nb)

Disjointedness

  • These are groups of points that are connected to each other but completely separated from other groups
  • Think of islands in an archipelago:
    • Points within each island are connected
    • But there are no connections between islands
    • Each island is a “component” or disjoint subgraph
  • Could indicate alternative clustering schemes in mixed effects models
  • Issues
    • Disjoint components can break an assumption of connectivity between observations (aka spatial continuity)
    • Block diagonal spatial weight matrices can affect computation of certain models
    • Might need to analyze each block separately
  • spdep::n.comp.nb gives the number of disjoint connected subgraphs

Spatial Autocorrelation

  • Misc

    • Local metrics can suffer from multiple testing issues when the number of group units is large

  • 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 \(C=\sum_i C_i/2W\)