Spatial Autocorrelation and Spatial Filtering

Abstract

This chapter provides an introductory discussion of spatial autocorrelation (SA), which refers to correlation existing and observed in geospatial data, and which characterizes data values that are not independent, but rather are tied together in overlapping subsets within a given geographic landscape. This chapter summarizes the various interpretations of SA, one being map pattern. SA can be quantified in a number of different ways, too, one being with the Moran Coefficient. Spatial filtering is a statistical method whose goal is to obtain enhanced and robust results in a spatial data analysis by decomposing a spatial variable into trend, a spatially structured random component (i.e., spatial stochastic signal), and random noise. Its aim is to separate spatially structured random components from both trend and random noise, and, consequently, leads statistical modeling to sounder statistical inference and useful visualization. This separation procedure can involve eigenfunctions of the matrix version of the numerator of the Moran Coefficient. This chapter summarizes the eigenvector spatial filtering (ESF) conceptual material, and presents the computer code for implementing ESF in R, Matlab, MINITAB, FORTRAN, and SAS. Next, it demonstrates that eigenvector spatial filter estimators are unbiased, efficient, and consistent. Finally, it summarizes an ESF empirical example application, and the extension of ESF to spatial interaction modeling.

Appendix A

The relationship between the MC and a squared product moment correlation coefficient (r²).

MC can be derived as a linear regression solution:

From OLS theory: b = (X ^T X)⁻¹ X ^T Y

• (i) Convert the attribute variable in question to z-scores

• (ii) Let X = z _Y and Y = Cz _Y

• (iii) Regress Cz _Y on z _Y, with a no-intercept option

• (iv) Let X = 1 and Y = C1

• (v) Regress C1 on 1, with a no-intercept option

• (vi) MC = b_numerator/b_denominator

This relationship relates directly to the Moran scatterplot, conveying why it is a useful visualization of spatial autocorrelation.

Next, let MC = (n /1 ^T C1) z ^T Cz/(n−1) and rewrite vector z as the following bivariate regression model specification: z = a1 + bCZ + e, where e is an n-by-1 vector of residuals. Then

$$ \mathrm{ b}=\frac{{{{\mathbf{ z}}^{\mathrm{ T}}}\mathbf{ Cz}}}{{{{\mathbf{ z}}^{\mathrm{ T}}}{{\mathbf{ C}}^2}\mathbf{ z}}}=\frac{{{{\mathrm{ s}}_{\mathrm{ z}}}}}{{{{\mathrm{ s}}_{\mathrm{ Cz}}}}}\mathrm{ r}=\frac{{\sqrt{1}}}{{{{\mathrm{ s}}_{\mathrm{ Cz}}}}}\mathrm{ r} $$

$$ \frac{\mathrm{ M}\mathrm{ C}}{{\mathrm{ M}{{\mathrm{ C}}_1}}}\frac{{\mathrm{ n}{\lambda_1}}}{{{{\mathbf{ 1}}^{\mathrm{ T}}}\mathbf{ C}\mathbf{ 1}}}\frac{{{{\mathbf{ 1}}^{\mathrm{ T}}}\mathbf{ C}\mathbf{ 1}}}{\mathrm{ n}}\frac{{\mathrm{ n}-1}}{{{{\mathbf{ z}}^{\mathrm{ T}}}{{\mathbf{ C}}^2}\mathbf{ z}}}=\frac{1}{{\sqrt{{\frac{{{{\mathbf{ z}}^{\mathrm{ T}}}\mathbf{ C}(\mathbf{ I}-\mathbf{ 1}{{\mathbf{ 1}}^{\mathrm{ T}}}/\mathrm{ n})\mathbf{ C}\mathbf{ z}}}{{\mathrm{ n}-1}}}}}}\mathrm{ r} $$

$$ {{\left( {\frac{\mathrm{ M}\mathrm{ C}}{{\mathrm{ M}{{\mathrm{ C}}_1}}}} \right)}^2}\lambda_1^2\frac{{{{{(\mathrm{ n}-1)}}^2}}}{{{{{\left( {{{\mathbf{ z}}^{\mathrm{ T}}}{{\mathbf{ C}}^2}\mathbf{ z}} \right)}}^2}}}=\frac{{\mathrm{ n}-1}}{{{{\mathbf{ z}}^{\mathrm{ T}}}\mathbf{ C}(\mathbf{ I}-\mathbf{ 1}{{\mathbf{ 1}}^{\mathrm{ T}}}/\mathrm{ n})\mathbf{ C}\mathbf{ z}}}{{\mathrm{ r}}^2} $$

$$ {{\mathrm{ r}}^2}={{\left( {\frac{\mathrm{ M}\mathrm{ C}}{{\mathrm{ M}{{\mathrm{ C}}_1}}}} \right)}^2}\lambda_1^2(\mathrm{ n}-1)\frac{{{{\mathbf{ z}}^{\mathrm{ T}}}\mathbf{ C}(\mathbf{ I}-\mathbf{ 1}{{\mathbf{ 1}}^{\mathrm{ T}}}/\mathrm{ n})\mathbf{ C}\mathbf{ z}}}{{{{{\left( {{{\mathbf{ z}}^{\mathrm{ T}}}{{\mathbf{ C}}^2}\mathbf{ z}} \right)}}^2}}} $$

where MC₁ denotes the maximum value of MC for a given spatial weight matrix C. For a large P-by-Q regular square lattice (i.e., n = PQ) and the rook’s adjacency definition, for which $\mathrm{ M}{{\mathrm{ C}}_1} \approx 1$, if MC = 0.25, then

$$ {{\mathbf{ 1}}^{\mathrm{ T}}}\mathbf{ Cz}\approx 0 $$

$$ {{\mathbf{ z}}^{\mathrm{ T}}}{{\mathbf{ C}}^2}\mathbf{ z}\approx 16\ \left( {\mathrm{ PQ} - 1} \right) $$

$$ {\lambda_1}=2[\cos (\tfrac{\uppi}{{\mathrm{ P}+1}})+\cos (\tfrac{\uppi}{{\mathrm{ Q}+1}})]\approx 4 $$

and, consequently, ${{\mathrm{ r}}^2} \approx 0.05$. Therefore, roughly 5% of the variance in a spatially autocorrelation random variable with MC = 0.25 is attributable to spatial autocorrelation.