Part B: Spatial Autocorrelation and regression modelling

1
Chapter 5

• Part B Spatial Autocorrelation and regression
  modelling

2
Autocorrelation

• Time series correlation model
• xt,1 t1,2,3n-1 and xt,2 t2,3,4n

3
Spatial Autocorrelation

• Correlation coefficient
• xi i1,2,3n, yi i1,2,3n
• Time series correlation model
• xt,1 t1,2,3n-1 and xt,2 t2,3,4n
• Mean values
  Lag 1 autocorrelation
• large n

4
Spatial Autocorrelation

• Classical statistical model assumptions
• Independence vs dependence in time and space
• Toblers first law
• All things are related, but nearby things are
  more related than distant things
• Spatial dependence and autocorrelation
• Correlation and Correlograms

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

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

7
Spatial Autocorrelation

• Joins counts (50 1s)

A. Completely separated pattern (ve) B. Evenly spaced pattern (-ve)

C. Random pattern

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

9
Spatial Autocorrelation

• Joins counts

A. Completely separated (ve) B. Evenly spaced (-ve)

C. Random

10
Spatial Autocorrelation

• Joins count some issues
• Multiple z-scores
• Binary or k-class data
• Rooks 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

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
1 3 5.54
2 1 2.24
2 2 -5.15
2 3 9.02
3 1 3.1
3 2 -4.39
3 3 -2.09
4 2 0.46
4 3 -3.06
3 7
1 4 8
2 5 9
6 10
Cell numbering
Adjacency matrix, total 1s26
12
Spatial Autocorrelation

• Spatial (auto)correlation coefficient
• Coordinate (x,y,z) data representation for cells
• Spatial weights matrix (binary or other), Wwij
• From last slide S wij26
• 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

13
Spatial Autocorrelation

• Morans I
• TSA model

14
Spatial Autocorrelation
Moran I 1016.19/(26196.68)0.0317 ? 0
A. Computation of variance/covariance-like quantities, matrix C

B. CW Adjustment by multiplication of the weighting matrix, W

15
Spatial Autocorrelation

• Morans 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)

16
Spatial Autocorrelation

• Moran I Correlogram

Source data points Lag distance bands, h Correlogram

17
Spatial Autocorrelation

• Geary C
• Co-variation model uses squared differences
  rather than products
• Similar approach is used in geostatistics

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

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

20
Regression modelling

• Ordinary Least Squares (OLS) model
• Minimise sum of squared errors (or residuals)
• Solved for coefficients by matrix expression

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)

22
Regression modelling

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

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
• yX?u where u is a vector of random variables
  (errors) with mean 0 and variance-covariance
  matrix C
• Solution

24
Regression modelling

• GLS and spatial modelling
• 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

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

26
Regression modelling

• Some regression terminology
• Simple linear
• Multiple
• Multivariate
• SAR
• CAR
• Logistic
• Poisson
• Ecological
• Hedonic
• Analysis of variance
• Analysis of covariance

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

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

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

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

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

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
• Fitting by ILSR methods

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

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

35
Regression modelling

• SAR models
• Pure spatial lag
• Re-arranging
• MRSA model

Spatial weights matrix
Autoregression parameter
Linear regression added
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
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
weighted mean for neighbourhood of i
Expected value at i
38
Regression modelling

• Spatial filtering
• Apply a spatial filter to the data to remove SA
  effects
• Model the filtered data
• Example

Spatial filter