Dealing with Spatial AutocorrelationDealing with Spatial Autocorrelation

Spatial Analysis Seminar

Spring 2009

Spatial Autocorrelation Defined

• “…the property of random variables taking values, at pairs of locations a certain distance apart, that are more similar (positive autocorrelation) or less similar (negative autocorrelation) than expected for randomly associated pairs of observations.”– Legendre (1993)

Types of Spatial Autocorrelation

• Inherent autocorrelation: caused by “contagious biotic processes”

vs.

• Induced spatial dependence: biological variables of interest are functionally dependent on one or more autocorrelated exogenous variable(s)

Why Should We Care?

• “natural systems almost always have autocorrelation in the form of patchiness or gradients…over a wide range of spatial and temporal scales.”– Fortin & Dale (2005)

→ Autocorrelation is a “fact of life” for ecologists!

2 Views of Spatial Autocorrelation:

1. It’s a nuisance that complicates statistical hypothesis testing

2. It’s functionally important in many ecosystems, so we must revise our theories and models to incorporate spatial structure

• Either way, the first step involves describing the autocorrelation (i.e., the “spatial structure”)

Describing Spatial Autocorrelation

• Compute Moran’s I or Geary’s c coefficients over multiple distances

• Correlogram: plot distance on X-axis against correlation coefficient on Y-axis

• Mantel correlogram: multivariate response

• Semi-variogram/variogram

Example Data

• Wetland hardwood forest (5 x 5 m cells)

• Response variable: log of non-ground lidar points in 0-1 m vertical height bin

• n1 = 217, n2 = 68

• Welch’s t-test (unequal variance, unequal sample sizes) results: t = 2.33, df = 181, p-value ≈ 0.021

Moran’s I correlograms

Now what do I do???

• Adjusting the effective sample size

• Spatial statistical modeling methods

• Restricted randomization

• Other methods: canonical ordination, partial Mantel tests, etc.

Adjusting the Effective Sample Size

• Estimate of effective sample size (Fortin & Dale 2005, p. 223, Equation 5.15):

n

i

n

jji xxCor

nn

1 1

2

),(

'

• For first-order autocorrelation ρ and large n:

1

1' nn

Adjusting the Effective Sample Size• For the “Recently Burned” example data:

11033.01

33.01217

1

1'

nn

• For the “Long Unburned” example data:

4322.01

22.0168

1

1'

nn

• Welch’s t-test results: t = 1.76, df = 123, p ≈ 0.080• BUT, this is a very simplistic model!

Detour: Autocorrelation Models

• Model 1 (“spatial independence”):

• Model 2 (“first-order autoregressive”):

• Model 3 (“induced autoregressive”):

• Model 4 (“doubly autoregressive”):

ii

iii

z

zx

11,1 iii xx

iii

iii

zz

zx

1

iizi

iixii

zz

xzx

1

1

SOURCE: Fortin & Dale (2005), pp. 213-216

Detour: Autocorrelation Models

• The models on the previous slide were one-dimensional, but most spatial data is two-dimensional (Lat-Long, XY-coordinates, etc.)

• The two-dimensional spatial autocorrelation model incorporates W, a “proximity matrix” of neighbor weights, which in turn affects the variance-covariance matrix (C):

12 )]()[(

)(

WIWIC

ZxWZx

T

Generalized Least Squares (GLS)

• Relatively easy way to introduce spatial autocorrelation structure to linear models

• Fits a parametric correlation function (exponential, Gaussian, spherical, etc.) directly to the variance-covariance matrix

• Assumes normally distributed errors, but errors are allowed to be correlated and/or have unequal variances

• Built-in R package: nlme

GLS Model – No Spatial Structurelibrary(nlme)…## Model A: spatial independenceModelA <- gls(LN_COUNT~BURNED,data=SAC_data)plot(Variogram(ModelA, form=~x+y))

GLS Models with Spatial Structure> ModelB <- gls(LN_COUNT~BURNED,data=SAC_data,corr=corAR1())> ModelC <- gls(LN_COUNT~BURNED,data=SAC_data,corr=corExp(form=~x+y))> ModelD <- gls(LN_COUNT~BURNED,data=SAC_data,corr=corGaus(form=~x+y))> ModelE <- gls(LN_COUNT~BURNED,data=SAC_data,corr=corSpher(form=~x+y))> AIC(ModelA,ModelB,ModelC,ModelD,ModelE)

df AICModelA 3 702.1288ModelB 4 677.3121ModelC 4 591.7996ModelD 4 607.3873ModelE 4 604.7950

> anova(ModelA,ModelC)

Model df AIC BIC logLik Test L.Ratio p-valueModelA 1 3 702.1288 713.0652 -348.0644 ModelC 2 4 591.7996 606.3814 -291.8998 1 vs 2 112.3293 <.0001

→ Exponential GLS model seems to fit best

Other Autocorrelation Models

• Conditional autoregressive (CAR), simultaneous autoregressive (SAR), and moving average (MA) models– See pp. 229-233 of Fortin & Dale (2005)– Implemented in R package spdep, as well as SAM

(Spatial Analysis for Macroecology) software

• Generalized linear mixed models (GLMMs): R built-in packages MASS, nlme

• But wait, there’s more: see Dormann et al. (2007) review paper in Ecography (30) 609-628.

Models and Reality

• “Much of the treatment of spatial autocorrelation in the statistical literature is predicated on the simplest AR model, which produces an exponential decline in autocorrelation as a function of distance (Figure 5.16).”– Fortin & Dale (2005, pp. 247-248)

• BUT, simple corrections based on first-order AR don’t account for effects of potentially negative autocorrelation at greater distances

Restricted Randomization

• PROBLEM: randomization tests based on complete spatial randomness will destroy autocorrelation structure

• POTENTIAL SOLUTIONS:1. “Toroidal shift” randomization (Figure 5.12)

2. Contiguity-constrained permutations (see Legendre et al. 1990 for algorithms)

Conclusion

• Incorporating spatial structure into ecological models was identified by Legendre as a “new paradigm” in 1993, BUT…

• …ecologists are still refining their methods for dealing with spatial autocorrelation

• OUR LAST HOPE?: Dale, M.R.T. and M.-J. Fortin. (in press). Spatial Autocorrelation and Statistical Tests: Some Solutions. Journal of Agricultural, Biological, and Environmental Statistics.

Spatial autocorrelation, don’t make me open this…