## Introduction

The Moran eigenvector approach (Dray, Legendre, and Peres-Neto 2006; Griffith and Peres-Neto 2006) involved the spatial patterns represented by maps of eigenvectors; by choosing suitable orthogonal patterns and adding them to a linear or generalised linear model, the spatial dependence present in the residuals can be moved into the model.

It uses brute force to search the set of eigenvectors of the matrix \(\mathbf{M W M}\), where

\[\mathbf{M} = \mathbf{I} - \mathbf{X}(\mathbf{X}^{\rm T}
\mathbf{X})^{-1}\mathbf{X}^{\rm T}\] is a symmetric and idempotent projection matrix and \(\mathbf{W}\) are the spatial weights. In the spatial lag form of `SpatialFiltering`

and in the GLM `ME`

form below, \(\mathbf{X}\) is an \(n\)-vector of ones, that is the intercept only.

In its general form, `SpatialFiltering`

chooses the subset of the \(n\) eigenvectors that reduce the residual spatial autocorrelation in the error of the model with covariates. The lag form adds the covariates in assessment of which eigenvectors to choose, but does not use them in constructing the eigenvectors. `SpatialFiltering`

was implemented and contributed by Yongwan Chun and Michael Tiefelsdorf, and is presented in Tiefelsdorf and Griffith (2007); `ME`

is based on Matlab code by Pedro Peres-Neto and is discussed in Dray, Legendre, and Peres-Neto (2006