Spatial Autocorrelation GRAD6104/8104 INES 8090 Spatial Statistic- Spring 2017 Spatial Autocorrelation

So, why spatial statistics? Spatial dependence Spatial non-stationarity Spatial isotropy

The First Law of Geography Waldo Tober “Everything is related to everything else, but near things are more related than distant things.” (Tobler 1970) Spatial dependence!

Spatial Dependence => Spatial Autocorrelation

Spatial Autocorrelation The correlation of a spatial variable with itself Correlation(Z(si), Z(sj)) Global structure Characteristics of a phenomenon across the entire study area (e.g., clustering) Local structure Individual-level characteristics of a feature and their relationship to nearby features (e.g., hotspots or coldspots)

Spatial Autocorrelation Global Spatial Autocorrelation Local Spatial Autocorrelation Source: https://www.census.gov/popest/data/maps/2011/PopDensity_11.jpg

Global Spatial Autocorrelation Indices of (Global) Spatial Autocorrelation Black-White Join Counts Moran’s I statistics Geary’s C statistics

Global Spatial Autocorrelation Weight matrix Based on adjacency (contiguity) or distance Regular or irregular

Global Spatial Autocorrelation Weight matrix Based on adjacency or distance Regular or irregular

Von Neumann Neighborhood Weight Matrix Neighborhoods: first-order regular Neighbors Moore Neighborhood Von Neumann Neighborhood

Weight Matrix Neighborhoods (Batty 2005)

Weight Matrix Neighborhoods: irregular Neighbors

Global Spatial Autocorrelation Black-White Join Counts For binary spatial variable

Global Spatial Autocorrelation Black-White Join Counts For binary spatial variable Z(s) Special type of generalized cross-product statistic The Black-Black joint count statistic The Black-White joint count statistic Two sites are joined if they share an edge

Global Spatial Autocorrelation Moran’s I statistics (Moran 1950): very popular For continuous spatial variables Resemblance of the ordinary correlation coefficient I statistic varies between -1 (negative) and 1 (positive) Similarity measure Where Where

Global Spatial Autocorrelation Geary’s C statistics (Geary, 1954) For continuous spatial variables Expected value of C: 1 Range: >=0 Less than 1: positive spatial autocorrelation Large than 1 (less than 2): Negative Dissimilarity measure Where

Global Spatial Autocorrelation Significance testing Permutation test (randomization distribution) Monte Carlo approach Normal approximation I ~ N(E(I),Var(I)) Z=(I-E(I))/(Var(I))1/2 Rule of thumb: n >= 25

Global Spatial Autocorrelation Mean values of I and C: I = E[I] => no spatial autocorrelation I > E[I] => positive autocorrelation I < E[I] => negative autocorrelation C is inversely related to I

Global Spatial Autocorrelation Variance of I and C: where where

Spatial Autocorrelation Function Divide the range of distance into a set of classes Compute spatial autocorrelation measure once for each distance class Plot of autocorrelation indices against distance (spatial correlogram) Fortin et al. 1989, Figure 2b Fortin, M. J., Drapeau, P., & Legendre, P. (1989). Spatial autocorrelation and sampling design in plant ecology. Vegetatio, 83(1-2), 209-222.

Spatial Autocorrelation Testing spatial autocorrelation in regression residuals Linear Model: Z(s)=X(s)β+e Residual: Conduct Moran’s I or Geary’s C statistics on residuals

Reading Assignment Chapter 2 by Chun and Griffith 2014 Anselin, L. (1995). Local indicators of spatial association—LISA. Geographical analysis, 27(2), 93-115.