Www.spatialanalysisonline.com Chapter 5 Part B: Spatial Autocorrelation and regression modelling.

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Chapter 5

Part B: Spatial Autocorrelation and regression modelling

3rd edition www.spatialanalysisonline.com 2

Autocorrelation

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

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

x xn.2

x xn 1

x x x x

Classical statistical model assumptionsIndependence vs dependence in time and

spaceTobler’s first law:

“All things are related, but nearby things are more related than distant things”

Spatial dependence and autocorrelationCorrelation and Correlograms

Covariance and autocovarianceLags – fixed or variable intervalCorrelograms and rangeStationary and non-stationary patternsOutliersExtending concept to spatial domain

Transects Neighbourhoods and distance-based models

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

Joins counts (50% 1’s)A. Completely separated pattern (+ve)

B. Evenly spaced pattern (-ve)C. Random pattern

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

Joins countsA. Completely separated (+ve) B. Evenly spaced (-ve) C. Random

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

Irregular lattice – (x,y,z) and adjacency tables

+4.55 +5.54

-5.15 +9.02

-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

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

Cell numbering

Cell data Cell coordinates (row/col) x,y,z view

Adjacency matrix, total 1’s=26

“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

Moran’s I

TSA model

example cell 10our for 1026 hence

where,)(

i jjiij

x x x x

A. Computation of variance/covariance-like quantities, matrix C

B. C*W: Adjustment by multiplication of the weighting matrix, W

Moran I =10*16.19/(26*196.68)=0.0317 0

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)

i jjiij

pI / where,

Moran I Correlogram

Source data points Lag distance bands, h Correlogram

Geary C Co-variation model uses squared differences

rather than products

Similar approach is used in geostatistics

ij i j

w z zC

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

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

y x x x or

y0 1 1 2 2 3 3 ...,

Ordinary Least Squares (OLS) model

Minimise sum of squared errors (or residuals) Solved for coefficients by matrix expression:

0 1 1 2 2 3 3 ... , ori i i i iy x x x y Xβ ε

1T Tβ XX X y ( ) σ2ˆvar

1Tβ XX

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)

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

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

Solution:

1T Tβ XX X y

ˆ 11 T T 1β XC X X C y ˆvar

1T 1 T(β) X C X

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

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

12)ln(2

2)ln(2

kLAICc

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

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

),,( 21 wxxfy

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

Proceed and ignore SA, or Permit the coefficient, , to vary spatially (GWR)

or Modify the regression model to incorporate the SA

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:

y Xβ(t) ε

t t tˆ

1T Tβ( ) XW( )X X W( )y

ˆ 11 T T 1β XC X X C y

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

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

Proceed and ignore SA, or Permit the coefficient, , to vary spatially (GWR)

or Modify the regression model to incorporate the

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

SAR models Pure spatial lag:

Re-arranging:

MRSA model:

y Wy ε

1( ) y I W ε

Autoregression parameter

Spatial weights matrix

εWyXβy ρ

Linear regression added

SAR models Spatial error model:

Substituting and re-arranging:

Spatial weighted error vector

Linear regression + spatial error

y Xβ ε,

ε Wε u

iid error vector

( ) or

y Xβ Wy Xβ u,

y Xβ Wy WXβ u

iid error vectorLinear regression (global)

SAR lag Local trend

CAR models Standard CAR model:

Local weights matrix – distance or contiguity Variance :

Different models for W and M provide a range of CAR models

jjijiiji ywall yyE |

weighted mean for neighbourhood of i

Autoregression parameter

Expected value at i

MW(Iy 1))var(

Spatial filtering Apply a spatial filter to the data to remove SA

effects Model the filtered data Example: y=Xβ+ε

, hence

y Wy=Xβ WXβ+ε

y I W = I W Xβ+ε

y=Xβ+ I W ε

Spatial filter

Www.spatialanalysisonline.com Chapter 5 Part B: Spatial Autocorrelation and regression modelling.

rd editionwww

com18 spatial autocorrelation

com14 spatial autocorrelation

spatial domain

correlograms slide

random slide

hcorrelogram slide

geostatistics slide

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