MESF\cite{Griffith_2003} is a novel spatial statistical methodology that adds a set of synthetic proxy variables, which are eigenvectors extracted from an n-by-n, usually binary 0-1, spatial weights matrix C that links geographic objects together in space, as control variables to filter SA out of regression residuals and transfer it to the mean response in a model specification (this procedure creates a spatially varying intercept term). These control variables identify and isolate stochastic spatial dependencies among georeferenced observations, thus allowing model parameter estimation to proceed with observations mimicking being independent.
The crucial mathematical quantities from matrix C are eigenfunctions, which are n pairs of quantities computed via the matrix determinant of a modified version of matrix C, \(MCM\), \(M=\left(I-\frac{11^T}{n}\right)\) a scalar (eigenvalue) and a vector (its corresponding eigenvector)—where I denotes the n-by-n identity matrix, and 1 denotes the n-by-1 vector of ones. Eigenvalues are the n scalar solutions to the nth order polynomial matrix equation ; the corresponding eigenvectors E are the non-trivial vector solutions to the matrix equation . These eigenfunctions are the basis of MESF, and are the synthetic variates that account for nonzero SA in spatial regression residuals.
Eigenvector spatial filters (ESFs) are constructed as linear combinations of the eigenvectors from matrix MCM. Appealing properties of these eigenvectors include: (1) they are mutually orthogonal and uncorrelated; (2) one eigenvector is proportional to the vector 1, the intercept covariate in a regression model; and, (3) eigenvalues index, and eigenvectors can be used to visualize, various natures and degrees of SA. Including eigenvectors as covariates, selecting relevant ones with a stepwise procedure, enables SA to be accounted for in a conventional statistical estimation context, in either a linear or a generalized linear model specification.
This moran eigenvector spatial filtering method can also be applied in the context of Poisson regression in order to account for spatial autocorrelation embedded in a geo-referenced count dataset \cite{Griffith_2002}. The basic conceptual framework of eigenvector spatial filtering in Poisson regression can be linked to the random effects model in GLM, which frequently is used to explain overdispersion.