1. Part B: Spatial Autocorrelation and regression modelling
Chapter 5
Part B: Spatial Autocorrelation and regression modelling

2. Autocorrelation Time series correlation model
{xt,1} t=1,2,3…n‑1 and {xt,2} t=2,3,4…n
3rd edition

3. Spatial Autocorrelation
Correlation coefficient
{xi} i=1,2,3…n, {yi} i=1,2,3…n
Time series correlation model
{xt,1} t=1,2,3…n‑1 and {xt,2} t=2,3,4…n
Mean values: Lag 1 autocorrelation:
large n
3rd edition

4. Spatial Autocorrelation
Classical statistical model assumptions
Independence vs dependence in time and space
Tobler’s first law:
“All things are related, but nearby things are more related than distant things”
Spatial dependence and autocorrelation
Correlation and Correlograms
3rd edition

5. Spatial Autocorrelation
Covariance and autocovariance
Lags – fixed or variable interval
Correlograms and range
Stationary and non-stationary patterns
Outliers
Extending concept to spatial domain
Transects
Neighbourhoods and distance-based models
3rd edition

6. Spatial Autocorrelation
Global spatial autocorrelation
Dataset issues: regular grids; irregular lattice (zonal) datasets; point samples
Simple binary coded regular grids – use of Joins counts
Irregular grids and lattices – extension to x,y,z data representation
Use of x,y,z model for point datasets
Local spatial autocorrelation
Disaggregating global models
3rd edition

7. Spatial Autocorrelation
Joins counts (50% 1’s)
A. Completely separated pattern (+ve)
B. Evenly spaced pattern (-ve)
C. Random pattern
3rd edition

8. Spatial Autocorrelation
Joins count
Binary coding
Edge effects
Double counting
Free vs non-free sampling
Expected values (free sampling)
1-1 = 15/60, 0-0 = 15/60, 0-1 or 1-0 = 30/60
3rd edition

9. Spatial Autocorrelation
Joins counts
A. Completely separated (+ve)
B. Evenly spaced (-ve)
C. Random
3rd edition

10. Spatial Autocorrelation
Joins count – some issues
Multiple z-scores
Binary or k-class data
Rook’s move vs other moves
First order lag vs higher orders
Equal vs unequal weights
Regular grids vs other datasets
Global vs local statistics
Sensitivity to model components
3rd edition

11. Spatial Autocorrelation
Irregular lattice – (x,y,z) and adjacency tables
Cell data
Cell coordinates (row/col)
x,y,z view
+4.55
+5.54
+2.24
-5.15
+9.02
+3.10
-4.39
-2.09
+0.46
-3.06
1,1
1,2
1,3
2,1
2,2
2,3
3,1
3,2
3,3
4,1
4,2
4,3 x y z 1 2
4.55 3
5.54
2.24
‑5.15
9.02
3.1
‑4.39
‑2.09 4
0.46
‑3.06 3 7 1 4 8 2 5 9 6 10
Cell numbering
Adjacency matrix, total 1’s=26
3rd edition

12. Spatial Autocorrelation
“Spatial” (auto)correlation coefficient
Coordinate (x,y,z) data representation for cells
Spatial weights matrix (binary or other), W={wij}
From last slide: Σ wij=26
Coefficient formulation – desirable properties
Reflects co-variation patterns
Reflects adjacency patterns via weights matrix
Normalised for absolute cell values
Normalised for data variation
Adjusts for number of included cells in totals
3rd edition

13. Spatial Autocorrelation
Moran’s I
TSA model
3rd edition

14. Spatial Autocorrelation
Moran I =10*16.19/(26*196.68)=  0
A. Computation of variance/covariance-like quantities, matrix C
B. C*W: Adjustment by multiplication of the weighting matrix, W
3rd edition

15. Spatial Autocorrelation
Moran’s I
Modification for point data
Replace weights matrix with distance bands, width h
Pre-normalise z values by subtracting means
Count number of other points in each band, N(h)
3rd edition

16. Spatial Autocorrelation
Moran I Correlogram
Source data points
Lag distance bands, h
Correlogram
3rd edition

17. Spatial Autocorrelation
Geary C
Co-variation model uses squared differences rather than products
Similar approach is used in geostatistics
3rd edition

18. Spatial Autocorrelation
Extending SA concepts
Distance formula weights vs bands
Lattice models with more complex neighbourhoods and lag models (see GeoDa)
Disaggregation of SA index computations (row-wise) with/without row standardisation (LISA)
Significance testing
Normal model
Randomisation models
Bonferroni/other corrections
3rd edition

19. Regression modelling Simple regression – a statistical perspective
One (or more) dependent (response) variables
One or more independent (predictor) variables
Linear regression is linear in coefficients:
Vector/matrix form often used
Over-determined equations & least squares
3rd edition

20. Regression modelling Ordinary Least Squares (OLS) model
Minimise sum of squared errors (or residuals)
Solved for coefficients by matrix expression:
3rd edition

21. Regression modelling OLS – models and assumptions
Model – simplicity and parsimony
Model – over-determination, multi-collinearity and variance inflation
Typical assumptions
Data are independent random samples from an underlying population
Model is valid and meaningful (in form and statistical)
Errors are iid
Independent; No heteroskedasticity; common distribution
Errors are distributed N(0,2)
3rd edition

22. Regression modelling Spatial modelling and OLS
Positive spatial autocorrelation is the norm, hence dependence between samples exists
Datasets often non-Normal >> transformations may be required (Log, Box-Cox, Logistic)
Samples are often clustered >> spatial declustering may be required
Heteroskedasticity is common
Spatial coordinates (x,y) may form part of the modelling process
3rd edition

23. Regression modelling OLS vs GLS OLS assumes no co-variation
Solution:
GLS models co-variation:
y~ N(,C) where C is a positive definite covariance matrix
y=X+u where u is a vector of random variables (errors) with mean 0 and variance-covariance matrix C
3rd edition

24. Regression modelling GLS and spatial modelling Other models
y~ N(,C) where C is a positive definite covariance matrix (C must be invertible)
C may be modelled by inverse distance weighting, contiguity (zone) based weighting, explicit covariance modelling…
Other models
Binary data – Logistic models
Count data – Poisson models
3rd edition

25. Regression modelling Choosing between models
Information content perspective and AIC
where n is the sample size, k is the number of parameters used in the model, and L is the likelihood function
3rd edition

26. Regression modelling Some ‘regression’ terminology Simple linear
Multiple
Multivariate
SAR
CAR
Logistic
Poisson
Ecological
Hedonic
Analysis of variance
Analysis of covariance
3rd edition

27. Regression modelling
Spatial regression – trend surfaces and residuals (a form of ESDA)
General model:
y - observations, f( , , ) - some function, (x1,x2) - plane coordinates, w - attribute vector
Linear trend surface plot
Residuals plot
2nd and 3rd order polynomial regression
Goodness of fit measures – coefficient of determination
3rd edition

28. Regression modelling Regression & spatial autocorrelation (SA)
Analyse the data for SA
If SA ‘significant’ then
Proceed and ignore SA, or
Permit the coefficient,  , to vary spatially (GWR), or
Modify the regression model to incorporate the SA
3rd edition

29. Regression modelling Regression & spatial autocorrelation (SA)
Analyse the data for SA
If SA ‘significant’ then
Proceed and ignore SA, or
Permit the coefficient,  , to vary spatially (GWR) or
Modify the regression model to incorporate the SA
3rd edition

30. Regression modelling Geographically Weighted Regression (GWR)
Coefficients, , allowed to vary spatially, (t)
Model:
Coefficients determined by examining neighbourhoods of points, t, using distance decay functions (fixed or adaptive bandwidths)
Weighting matrix, W(t), defined for each point
Solution:
GLS:
3rd edition

31. Regression modelling Geographically Weighted Regression
Sensitivity – model, decay function, bandwidth, point/centroid selection
ESDA – mapping of surface, residuals, parameters and SEs
Significance testing
Increased apparent explanation of variance
Effective number of parameters
AICc computations
3rd edition

32. Regression modelling Geographically Weighted Regression
Count data – GWPR
use of offsets
Fitting by ILSR methods
Presence/Absence data – GWLR
True binary data
Computed binary data - use of re-coding, e.g. thresholding
3rd edition

33. Regression modelling Regression & spatial autocorrelation (SA)
Analyse the data for SA
If SA ‘significant’ then
Proceed and ignore SA, or
Permit the coefficient,  , to vary spatially (GWR) or
Modify the regression model to incorporate the SA
3rd edition

34. Regression modelling Regression & spatial autocorrelation (SA)
Modify the regression model to incorporate the SA, i.e. produce a Spatial Autoregressive model (SAR)
Many approaches – including:
SAR – e.g. pure spatial lag model, mixed model, spatial error model etc.
CAR – a range of models that assume the expected value of the dependent variable is conditional on the (distance weighted) values of neighbouring points
Spatial filtering – e.g. OLS on spatially filtered data
3rd edition

35. Regression modelling SAR models Pure spatial lag: Re-arranging:
MRSA model:
Spatial weights matrix
Autoregression parameter
Linear regression added
3rd edition

36. Regression modelling SAR models Spatial error model:
Substituting and re-arranging:
Linear regression + spatial error
iid error vector
Spatial weighted error vector
Linear regression (global)
iid error vector
SAR lag
Local trend
3rd edition

37. Regression modelling CAR models Standard CAR model:
Local weights matrix – distance or contiguity
Variance :
Different models for W and M provide a range of CAR models
Autoregression parameter
Expected value at i
weighted mean for neighbourhood of i
3rd edition

38. Regression modelling Spatial filtering
Apply a spatial filter to the data to remove SA effects
Model the filtered data
Example:
Spatial filter
3rd edition