Spatial Autocorrelation

Overview

• there are scenarios where locations are arbitrary
  • this data cannot be treated as a point process because locations of points are mostly meaningless
    • but they cannot be modeled using off-the-shelf techniques because they break the i.i.d. assumption

Data is often collected by measuring quantities over space

spatial autocorrelation arises when the variation between values

• of the datapoints is affected by their spatial distance
  • the underlying reason for this is that many of the influence act as large spatial scales
• the result is that we can not assume that spatially correlated data are i.i.d.

Source of autocorrelation

anything that causes some data to be more similar to each other than others can result in autocorrelation

• time: data that are close temporally are more related
• space: data that are close spatially are more related

Impact of autocorrelation

• Sample size, \(n\) is the denominator when calculating SEs and CIs

  \(SE = \frac{\sigma}{\sqrt{n}}\) \(95\%CI = \bar{x} \pm 1.96 \frac{\sigma}{\sqrt{n}}\)

  All else equal: \(\uparrow n = \downarrow SE \& \downarrow CI\)

  But with autocorrelated data each new datapoint is related to previously collected datapoint

  • and does not bring a full independent datapoint worth of information
    • e.g. 90% autocorrelation \(\sim\) 10% new info

when data are autocorrelated \(n_{effective} < n\) meaning SEs and CIs shrink faster than they should, resulting in a false sense of confidence

Moran's I

Moran's I is a correlation coefficient that measures the overall spatial autocorrelation of a dataset

• think of it as \(/sim\) weighted covariance

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

  • where \(N\) is the number of spatial units indexed by \(i\) and \(j\)
  • \(x\) is the variable of interest and \(\bar{x}\) is the mean of \(x\)
  • \(w_{ij}\) is a matrix of spatial weights and \(W\) is the sum of all \(w_{ij}\)

Values of \(I\) usually range from -1(dispersed) to 0(random) to +1(clustered)

useful for identifying the presence of autocorrelation

Moran's I is very sensitive to how you define weights

“The idea is to construct a matrix that accurately reflects your assumptions about the particular spatial phenomenon in question. A common approach is to give a weight of 1 if two zones are neighbors, and 0 otherwise, though the definition of ‘neighbors’ can vary. Another common approach might be to give a weight of 1 to k nearest neighbors, 0 otherwise. An alternative is to use a distance decay function for assigning weights. Sometimes the length of a shared edge is used for assigning different weights to neighbors. The selection of spatial weights matrix should be guided by theory about the phenomenon in question.” – Wikipedia

Semi-variograms

functions describing the degree of spatial dependence in a spatial stochastic process \(Z(s)\)

Semi-variance \(\gamma(h)\) is a measure of the degree of similarity

• between pairs of points separated by distance \(h\)
  • given by: \(\gamma(h) = \frac{1}{2V}\int\int_V [f(M+h)-f(M)]^2 dV\)
    • where \(M\) is a point in the spatial field \(V\)
      • \(f(V)\) is the value at point \(M\) (in arbitrary units)
      • \(h\) is the separation distance
        • and the double integral is over 2 dimensions

properties

• \(\gamma(h) \leq 0\) since it is expectation of a square
• since \(Z(s_1)-Z(s_1) = 0\) , \(\gamma(0)\) is always 0
• if the process is stationary, in the limit where \(h \rightarrow \infty\)
• if a stationary process has no spatial dependence
  • the semi-variogram is constant everywhere except at the origin
    • where it is zero
• the semi-variogram might be discontinuous at the origin
  • the height of the jump at the origin is referred to as nugget

To obtain the semi-variogram for a given \(\gamma(h)\) ,

• all pairs of points at that exact distance, \(h\) , would need to be sampled
  • in practice, this is impossible, so the empirical semi-variogram is used instead
• for values of \(Z(s)\) for all pairs separated by distance \(h\)
  • \(\gamma(h)\) is estimated as: \(\hat{\gamma}(h\pm \delta) := \frac{1}{2|N(h\pm\delta)|} \sum_{(i,j)\in N(h\pm\delta)} |z_i-z_j|^2\)
    • where \(\delta\) is some bin width

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