## Introduction

The Moran eigenvector approach [@dray+legendre+peres-neto:06; @griffith+peres-neto:06] 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+griffith:07; [ME]{} is based on Matlab code by Pedro Peres-Neto and is discussed in @dray+legendre+peres-neto:06 and @griffith+peres-neto:06.

library(spdep)
require("sf", quietly=TRUE)

## Reading layer NY8_utm18' from data source /home/rsb/lib/r_libs/spData/shapes/NY8_utm18.shp' using driver ESRI Shapefile'
## Simple feature collection with 281 features and 17 fields
## geometry type:  POLYGON
## dimension:      XY
## bbox:           xmin: 358241.9 ymin: 4649755 xmax: 480393.1 ymax: 4808545
## epsg (SRID):    32618
## proj4string:    +proj=utm +zone=18 +ellps=WGS84 +units=m +no_defs

NY_nb <- read.gal(system.file("weights/NY_nb.gal", package="spData"), override.id=TRUE)

library(spatialreg)
nySFE <- SpatialFiltering(Z~PEXPOSURE+PCTAGE65P+PCTOWNHOME, data=NY8, nb=NY_nb, style="W", verbose=FALSE)
nylmSFE <- lm(Z~PEXPOSURE+PCTAGE65P+PCTOWNHOME+fitted(nySFE), data=NY8)
summary(nylmSFE)

##
## Call:
## lm(formula = Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME + fitted(nySFE),
##     data = NY8)
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -1.5184 -0.3523 -0.0105  0.3221  3.1964
##
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)
## (Intercept)        -0.51728    0.14606  -3.542 0.000469 ***
## PEXPOSURE           0.04884    0.03230   1.512 0.131717
## PCTAGE65P           3.95089    0.55776   7.083 1.25e-11 ***
## PCTOWNHOME         -0.56004    0.15688  -3.570 0.000423 ***
## fitted(nySFE)vec13 -2.09397    0.60534  -3.459 0.000630 ***
## fitted(nySFE)vec44 -2.24003    0.60534  -3.700 0.000261 ***
## fitted(nySFE)vec6   1.02979    0.60534   1.701 0.090072 .
## fitted(nySFE)vec38  1.29282    0.60534   2.136 0.033613 *
## fitted(nySFE)vec20  1.10064    0.60534   1.818 0.070150 .
## fitted(nySFE)vec14 -1.05105    0.60534  -1.736 0.083662 .
## fitted(nySFE)vec75  1.90600    0.60534   3.149 0.001826 **
## fitted(nySFE)vec21 -1.06331    0.60534  -1.757 0.080138 .
## fitted(nySFE)vec36 -1.17861    0.60534  -1.947 0.052578 .
## fitted(nySFE)vec61 -1.08582    0.60534  -1.794 0.073986 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6053 on 267 degrees of freedom
## Multiple R-squared:  0.3401, Adjusted R-squared:  0.308
## F-statistic: 10.58 on 13 and 267 DF,  p-value: < 2.2e-16

nylm <- lm(Z~PEXPOSURE+PCTAGE65P+PCTOWNHOME, data=NY8)
anova(nylm, nylmSFE)

## Analysis of Variance Table
##
## Model 1: Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME
## Model 2: Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME + fitted(nySFE)
##   Res.Df     RSS Df Sum of Sq      F    Pr(>F)
## 1    277 119.619
## 2    267  97.837 10    21.782 5.9444 3.988e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Since the [SpatialFiltering]{} approach does not allow weights to be used, we see that the residual autocorrelation of the original linear model is absorbed, or ‘whitened’ by the inclusion of selected eigenvectors in the model, but that the covariate coefficients change little. The addition of these eigenvectors – each representing an independent spatial pattern – relieves the residual autocorrelation, but otherwise makes few changes in the substantive coefficient values.

The [ME]{} function also searches for eigenvectors from the spatial lag variant of the underlying model, but in a GLM framework. The criterion is a permutation bootstrap test on Moran’s $$I$$ for regression residuals, and in this case, because of the very limited remaining spatial autocorrelation, is set at $$\alpha = 0.5$$. Even with this very generous stopping rule, only few eigenvectors are chosen; their combined contribution only just improves the fit of the GLM model.

NYlistwW <- nb2listw(NY_nb, style = "W")
set.seed(111)

nyME <- ME(Cases~PEXPOSURE+PCTAGE65P+PCTOWNHOME, data=NY8, offset=log(POP8), family="poisson", listw=NYlistwW, alpha=0.5)

nyME
NY8$eigen_24 <- fitted(nyME)[,1] NY8$eigen_223 <- fitted(nyME)[,2]

nyglmME <- glm(Cases~PEXPOSURE+PCTAGE65P+PCTOWNHOME+offset(log(POP8))+fitted(nyME), data=NY8, family="poisson")
summary(nyglmME)
nyGLMp <- glm(Cases~PEXPOSURE+PCTAGE65P+PCTOWNHOME+offset(log(POP8)), data=NY8,family="poisson")
anova(nyGLMp, nyglmME, test="Chisq")
`

Figure $fig:geigen2$ shows the spatial patterns chosen to match the very small amount of spatial autocorrelation remaining in the model. As with the other Poisson regressions, the closeness to TCE sites is highly significant. Since, however, many TCE sites are also in or close to more densely populated urban areas with the possible presence of both point-source and non-point-source pollution, it would be premature to take such results simply at their face value. There is, however, a potentially useful contrast between the cities of Binghampton in the south of the study area with several sites in its vicinity, and Syracuse in the north without TCE sites in this data set.

## References

[1]: This vignette formed pp. 302–305 of the first edition of Bivand, R. S., Pebesma, E. and Gómez-Rubio V. (2008) Applied Spatial Data Analysis with R, Springer-Verlag, New York. It was retired from the second edition (2013) to accommodate material on other topics, and is made available in this form with the understanding of the publishers.