The Moran eigenvector filtering function is intended to remove spatial autocorrelation from the residuals of generalised linear models. It uses brute force eigenvector selection to reach a subset of such vectors to be added to the RHS of the GLM model to reduce residual autocorrelation to below the specified alpha value. Since eigenvector selection only works on symmetric weights, the weights are made symmetric before the eigenvectors are found (from spdep 0.5-50).

ME(formula, data=list(), family = gaussian, weights, offset,
 na.action=na.fail,listw=NULL, alpha=0.05, nsim=99, verbose=NULL,
 stdev=FALSE, zero.policy=NULL)

Arguments

formula

a symbolic description of the model to be fit

data

an optional data frame containing the variables in the model

family

a description of the error distribution and link function to be used in the model

weights

an optional vector of weights to be used in the fitting process

offset

this can be used to specify an a priori known component to be included in the linear predictor during fitting

na.action

a function (default options("na.action")), can also be na.omit or na.exclude with consequences for residuals and fitted values - in these cases the spatial weights list will be subsetted to remove NAs in the data. It may be necessary to set zero.policy to TRUE because this subsetting may create no-neighbour observations. Note that only weights lists created without using the glist argument to nb2listw may be subsetted.

listw

a listw object created for example by nb2listw

alpha

used as a stopping rule to choose all eigenvectors up to and including the one with a p-value exceeding alpha

nsim

number of permutations for permutation bootstrap for finding p-values

verbose

default NULL, use global option value; if TRUE report eigenvectors selected

stdev

if TRUE, p-value calculated from bootstrap permutation standard deviate using pnorm with alternative="greater", if FALSE the Hope-type p-value

zero.policy

default NULL, use global option value; if FALSE stop with error for any empty neighbour sets, if TRUE permit the weights list to be formed with zero-length weights vectors

Details

The eigenvectors for inclusion are chosen by calculating the empirical Moran's I values for the initial model plus each of the doubly centred symmetric spatial weights matrix eigenvectors in turn. Then the first eigenvector is chosen as that with the lowest Moran's I value. The procedure is repeated until the lowest remaining Moran's I value has a permutation-based probability value above alpha. The probability value is either Hope-type or based on using the mean and standard deviation of the permutations to calculate ZI based on the stdev argument.

Value

An object of class Me_res:

selection

a matrix summarising the selection of eigenvectors for inclusion, with columns:

Eigenvector

number of selected eigenvector

ZI

permutation-based standardized deviate of Moran's I if stdev=TRUE

pr(ZI)

probability value: if stdev=TRUE of the permutation-based standardized deviate, if FALSE the Hope-type probability value, in both cases on-sided

The first row is the value at the start of the search

vectors

a matrix of the selected eigenvectors in order of selection

References

Dray S, Legendre P and Peres-Neto PR (2005) Spatial modeling: a comprehensive framework for principle coordinate analysis of neigbbor matrices (PCNM), Ecological Modelling; Griffith DA and Peres-Neto PR (2006) Spatial modeling in ecology: the flexibility of eigenfunction spatial analyses.

Author

Roger Bivand and Pedro Peres-Neto

See also

Examples

#require("spdep", quietly=TRUE)
data(hopkins, package="spData")
hopkins_part <- hopkins[21:36,36:21]
hopkins_part[which(hopkins_part > 0, arr.ind=TRUE)] <- 1
hopkins.rook.nb <- spdep::cell2nb(16, 16, type="rook")
glmbase <- glm(c(hopkins_part) ~ 1, family="binomial")
lw <- spdep::nb2listw(hopkins.rook.nb, style="B")
set.seed(123)
system.time(MEbinom1 <- ME(c(hopkins_part) ~ 1, family="binomial",
 listw=lw, alpha=0.05, verbose=TRUE, nsim=49))
#> eV[,2], I: 0.07978424 ZI: NA, pr(ZI): 0.04
#> eV[,54], I: 0.05704227 ZI: NA, pr(ZI): 0.12
#>    user  system elapsed 
#>   1.590   0.010   1.606 
glmME <- glm(c(hopkins_part) ~ 1 + fitted(MEbinom1), family="binomial")
#anova(glmME, test="Chisq")
coef(summary(glmME))
#>                        Estimate Std. Error   z value     Pr(>|z|)
#> (Intercept)           -1.194892  0.1600429 -7.466073 8.262357e-14
#> fitted(MEbinom1)vec2   9.030818  2.5381594  3.558019 3.736629e-04
#> fitted(MEbinom1)vec54 -8.759610  2.5205844 -3.475230 5.104161e-04
anova(glmbase, glmME, test="Chisq")
#> Analysis of Deviance Table
#> 
#> Model 1: c(hopkins_part) ~ 1
#> Model 2: c(hopkins_part) ~ 1 + fitted(MEbinom1)
#>   Resid. Df Resid. Dev Df Deviance  Pr(>Chi)    
#> 1       255     292.23                          
#> 2       253     266.27  2   25.956 2.311e-06 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
if (FALSE) {
require("sf", quietly=TRUE)
columbus <- st_read(system.file("shapes/columbus.shp", package="spData")[1], quiet=TRUE)
#require("spdep", quietly=TRUE)
col.gal.nb <- spdep::read.gal(system.file("weights/columbus.gal", package="spData")[1])
lw <- spdep::nb2listw(col.gal.nb)
lmbase <- lm(CRIME ~ INC + HOVAL, data=columbus)
lagcol <- SpatialFiltering(CRIME ~ 1, ~ INC + HOVAL, data=columbus,
 nb=col.gal.nb, style="W", alpha=0.1, verbose=TRUE)
lagcol
lmlag <- lm(CRIME ~ INC + HOVAL + fitted(lagcol), data=columbus)
anova(lmbase, lmlag)
set.seed(123)
system.time(lagcol1 <- ME(CRIME ~ INC + HOVAL, data=columbus, family="gaussian",
 listw=lw, alpha=0.1, verbose=TRUE))
lagcol1
lmlag1 <- lm(CRIME ~ INC + HOVAL + fitted(lagcol1), data=columbus)
anova(lmbase, lmlag1)

set.seed(123)
lagcol2 <- ME(CRIME ~ INC + HOVAL, data=columbus, family="gaussian",
 listw=lw, alpha=0.1, stdev=TRUE, verbose=TRUE)
lagcol2
lmlag2 <- lm(CRIME ~ INC + HOVAL + fitted(lagcol2), data=columbus)
anova(lmbase, lmlag2)
NA.columbus <- columbus
NA.columbus$CRIME[20:25] <- NA
COL.ME.NA <- ME(CRIME ~ INC + HOVAL, data=NA.columbus, family="gaussian",
 listw=lw, alpha=0.1, stdev=TRUE, verbose=TRUE,
 na.action=na.exclude)
COL.ME.NA$na.action
summary(lm(CRIME ~ INC + HOVAL + fitted(COL.ME.NA), data=NA.columbus,
 na.action=na.exclude))
nc.sids <- st_read(system.file("shapes/sids.shp", package="spData")[1], quiet=TRUE)
rn <- as.character(nc.sids$FIPS)
ncCC89_nb <- spdep::read.gal(system.file("weights/ncCC89.gal", package="spData")[1],
 region.id=rn)
ncCR85_nb <- spdep::read.gal(system.file("weights/ncCR85.gal", package="spData")[1],
 region.id=rn)
glmbase <- glm(SID74 ~ 1, data=nc.sids, offset=log(BIR74),
 family="poisson")
set.seed(123)
MEpois1 <- ME(SID74 ~ 1, data=nc.sids, offset=log(BIR74),
 family="poisson", listw=spdep::nb2listw(ncCR85_nb, style="B"), alpha=0.2, verbose=TRUE)
MEpois1
glmME <- glm(SID74 ~ 1 + fitted(MEpois1), data=nc.sids, offset=log(BIR74),
 family="poisson")
anova(glmME, test="Chisq")
anova(glmbase, glmME, test="Chisq")
}