Spatial Autocorrelation and Moran Eigenvector Spatial Filtering

Abstract

This chapter provides an introductory discussion of spatial autocorrelation (SA), that 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 with particular emphasis on its explanation as 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., a 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 Moran eigenvector spatial filtering (MESF) conceptual material, and presents the computer code for implementing MESF in R, Matlab, MINITAB, FORTRAN, and SAS. Next, it demonstrates the statistical features of MESF. Finally, it summarizes a MESF empirical example application, and the extensions of MESF to spatial interaction modeling and space-time data analysis.

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^TX)⁻¹X^TY

• (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

$$ \mathrm{MC}={\mathrm{b}}_{\mathrm{numerator}}/{\mathrm{b}}_{\mathrm{denominator}} $$

This relationship links directly to the Moran scatterplot, conveying why it is a useful visualization of SA.

Next, let MC = (n/1^TC1)z^TCz/(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

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

(24)

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

(25)

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

(26)

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

(27)

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 MC₁ ≈ 1, if MC = 0.25, then

$$ \mathbf{1}\mathbf{Cz}\approx 0 $$

(28)

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

(29)

$$ {\uplambda}_1=2\left[\mathrm{COS}\left(\frac{\uppi}{\mathrm{P}+1}\right)+\mathrm{COS}\left(\frac{\uppi}{\mathrm{Q}+1}\right)\right]\approx 4 $$

(30)

and, consequently, r² ≈ 0.05. Therefore, roughly 5% of the variance in a spatially autocorrelation random variable with MC = 0.25 is attributable to spatial autocorrelation.