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

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

3. 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!

4. Spatial Dependence
=>
Spatial Autocorrelation

5. 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)

6. Spatial Autocorrelation
Global Spatial Autocorrelation
Local Spatial Autocorrelation
Source:

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

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

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

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

11. Weight Matrix
Neighborhoods (Batty 2005)

12. Weight Matrix
Neighborhoods: irregular
Neighbors

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

14. 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

15. 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

16. 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

17. 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

18. 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

19. Global Spatial Autocorrelation
Variance of I and C:
where
where

20. 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),

21. 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

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