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+# Moran Eigenvectors
+
+## Introduction [^1]
+
+The Moran eigenvector approach ([Dray, Legendre, and Peres-Neto
+2006](#ref-dray+legendre+peres-neto:06); [Griffith and Peres-Neto
+2006](#ref-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 and Griffith ([2007](#ref-tiefelsdorf+griffith:07)); `ME` is
+based on Matlab code by Pedro Peres-Neto and is discussed in Dray,
+Legendre, and Peres-Neto ([2006](#ref-dray+legendre+peres-neto:06)) and
+Griffith and Peres-Neto ([2006](#ref-griffith+peres-neto:06)).
+
+``` r
+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 `NY8_utm18' from data source 
+    ##   `/home/rsb/lib/r_libs/spData/shapes/NY8_utm18.gpkg' using driver `GPKG'
+    ## Simple feature collection with 281 features and 17 fields
+    ## Geometry type: POLYGON
+    ## Dimension:     XY
+    ## Bounding box:  xmin: 358241.9 ymin: 4649755 xmax: 480393.1 ymax: 4808545
+    ## Projected CRS: WGS 84 / UTM zone 18N
+
+``` r
+NY_nb <- read.gal(system.file("weights/NY_nb.gal", package="spData"), override.id=TRUE)
+```
+
+``` r
+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
+
+``` r
+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.
+
+``` r
+NYlistwW <- nb2listw(NY_nb, style = "W")
+set.seed(111)
+```
+
+``` r
+nyME <- ME(Cases~PEXPOSURE+PCTAGE65P+PCTOWNHOME, data=NY8, offset=log(POP8), family="poisson", listw=NYlistwW, alpha=0.46)
+```
+
+``` r
+nyME
+```
+
+    ##   Eigenvector ZI pr(ZI)
+    ## 0          NA NA   0.31
+    ## 1          24 NA   0.44
+    ## 2         223 NA   0.42
+    ## 3         206 NA   0.43
+    ## 4         169 NA   0.48
+
+``` r
+NY8$eigen_1 <- fitted(nyME)[,1]
+NY8$eigen_2 <- fitted(nyME)[,2]
+```
+
+``` r
+#gry <- brewer.pal(9, "Greys")[-1]
+plot(NY8[,c("eigen_1", "eigen_2")])
+```
+
+![](SpatialFiltering_files/figure-html/unnamed-chunk-6-1.png)
+
+``` r
+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:
+    ##                    Estimate Std. Error z value Pr(>|z|)    
+    ## (Intercept)        -8.13431    0.18388 -44.237  < 2e-16 ***
+    ## PEXPOSURE           0.14136    0.03134   4.511 6.45e-06 ***
+    ## PCTAGE65P           4.16875    0.60149   6.931 4.19e-12 ***
+    ## PCTOWNHOME         -0.39290    0.19222  -2.044  0.04096 *  
+    ## fitted(nyME)vec24   1.62658    0.72243   2.252  0.02435 *  
+    ## fitted(nyME)vec223  0.92941    0.70391   1.320  0.18671    
+    ## fitted(nyME)vec206 -0.11559    0.68987  -0.168  0.86693    
+    ## fitted(nyME)vec169  1.82674    0.68142   2.681  0.00735 ** 
+    ## ---
+    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+    ## 
+    ## (Dispersion parameter for poisson family taken to be 1)
+    ## 
+    ##     Null deviance: 428.25  on 280  degrees of freedom
+    ## Residual deviance: 340.08  on 273  degrees of freedom
+    ## AIC: Inf
+    ## 
+    ## Number of Fisher Scoring iterations: 5
+
+``` r
+nyGLMp <- 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     353.35                       
+    ## 2       273     340.08  4   13.269  0.01003 *
+    ## ---
+    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+
+Figure 
+``` math
+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
+
+Dray, S., P. Legendre, and P. R. Peres-Neto. 2006. “Spatial Modeling: A
+Comprehensive Framework for Principle Coordinate Analysis of Neighbor
+Matrices (PCNM).” *Ecological Modelling* 196: 483–93.
+
+Griffith, D. A., and P. R. Peres-Neto. 2006. “Spatial Modeling in
+Ecology: The Flexibility of Eigenfunction Spatial Analyses.” *Ecology*
+87: 2603–13.
+
+Tiefelsdorf, M., and D. A. Griffith. 2007. “Semiparametric Filtering of
+Spatial Autocorrelation: The Eigenvector Approach.” *Environment and
+Planning A* 39: 1193–1221.
+
+[^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.

docs/articles/SpatialFiltering.html

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+# Articles
+
+### All vignettes
+
+- [Spatial weights objects as sparse matrices and
+  graphs](https://r-spatial.github.io/spatialreg/articles/nb_igraph.md):
+- [North Carolina SIDS data set
+  (models)](https://r-spatial.github.io/spatialreg/articles/sids_models.md):
+- [Moran
+  Eigenvectors](https://r-spatial.github.io/spatialreg/articles/SpatialFiltering.md):