Moran Eigenvectors
Roger Bivand
Introduction 1
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)
and Griffith and Peres-Neto (2006).
library(spdep)
require("sf", quietly=TRUE)
if (packageVersion("spData") >= "2.3.2") {
NY8 <- sf::st_read(system.file("shapes/NY8_utm18.gpkg", package="spData"))
} else {
NY8 <- sf::st_read(system.file("shapes/NY8_bna_utm18.gpkg", package="spData"))
sf::st_crs(NY8) <- "EPSG:32618"
NY8$Cases <- NY8$TRACTCAS
}## Reading layer `sf_bna2_utm18' from data source
## `/home/rsb/lib/r_libs/spData/shapes/NY8_bna_utm18.gpkg' using driver `GPKG'
## Simple feature collection with 281 features and 12 fields
## Geometry type: MULTIPOLYGON
## Dimension: XY
## Bounding box: xmin: 357628 ymin: 4649538 xmax: 480360.3 ymax: 4808317
## Projected CRS: UTM Zone 18, Northern HemisphereNY_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-16nylm <- 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 ' ' 1Since 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=0.46)nyME## Eigenvector ZI pr(ZI)
## 0 NA NA 0.31
## 1 24 NA 0.46
## 2 164 NA 0.41
## 3 113 NA 0.44
## 4 60 NA 0.50NY8$eigen_1 <- fitted(nyME)[,1]
NY8$eigen_2 <- fitted(nyME)[,2]#gry <- brewer.pal(9, "Greys")[-1]
plot(NY8[,c("eigen_1", "eigen_2")])
nyglmME <- glm(Cases~PEXPOSURE+PCTAGE65P+PCTOWNHOME+offset(log(POP8))+fitted(nyME), data=NY8, family="poisson")
summary(nyglmME)##
## Call:
## glm(formula = Cases ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME + offset(log(POP8)) +
## fitted(nyME), family = "poisson", data = NY8)
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -8.123408 0.184525 -44.023 < 2e-16 ***
## PEXPOSURE 0.144595 0.031469 4.595 4.33e-06 ***
## PCTAGE65P 4.072265 0.607725 6.701 2.07e-11 ***
## PCTOWNHOME -0.392118 0.193313 -2.028 0.0425 *
## fitted(nyME)vec24 1.548573 0.725355 2.135 0.0328 *
## fitted(nyME)vec164 -0.166982 0.767258 -0.218 0.8277
## fitted(nyME)vec113 NA NA NA NA
## fitted(nyME)vec60 -0.001836 0.769811 -0.002 0.9981
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 428.84 on 280 degrees of freedom
## Residual deviance: 349.30 on 274 degrees of freedom
## AIC: Inf
##
## Number of Fisher Scoring iterations: 5nyGLMp <- glm(Cases~PEXPOSURE+PCTAGE65P+PCTOWNHOME+offset(log(POP8)), data=NY8,family="poisson")
anova(nyGLMp, nyglmME, test="Chisq")## Analysis of Deviance Table
##
## Model 1: Cases ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME + offset(log(POP8))
## Model 2: Cases ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME + offset(log(POP8)) +
## fitted(nyME)
## Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1 277 354.0
## 2 274 349.3 3 4.6969 0.1954Figure \[fig:geigen2\] 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 Binghamton 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
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.↩︎