Guerry/0000755000176200001440000000000014516023372011532 5ustar liggesusersGuerry/NAMESPACE0000644000176200001440000000013014053052354012741 0ustar liggesusers# all the rest is data # v 1.6: removed thinnedSpatialPoly #export(thinnedSpatialPoly)Guerry/demo/0000755000176200001440000000000014513000475012452 5ustar liggesusersGuerry/demo/guerry_density.R0000644000176200001440000000264414425214446015666 0ustar liggesusers#' --- #' title: Density plots of Guerry data #' --- library(dplyr) # A Grammar of Data Manipulation library(tidyr) # tidy methods library(ggplot2) # Create Elegant Data Visualisations Using the Grammar of Graphics library(scales) # Scale Functions for Visualization library(Guerry) # Maps, Data and Methods Related to Guerry (1833) "Moral Statistics of France" data("Guerry", package="Guerry") #' ## Convert to long format guerry_long <- Guerry |> filter(!is.na(Region)) |> select(dept:Suicides) |> pivot_longer(cols = Crime_pers:Suicides, names_to = "variable", values_to = "value") guerry_long #' ## density plot of each variable ggplot(data = guerry_long, aes(x=value, fill=TRUE)) + geom_density(alpha=0.2) + geom_rug() + facet_wrap(~variable, scales="free") + theme_bw(base_size = 14) + theme(legend.position = "none", axis.ticks.y=element_blank(), axis.text.y=element_blank()) #' ## by region col.region <- colors()[c(149, 254, 468, 552, 26)] # colors for region ggplot(data = guerry_long, aes(x=value, fill=Region)) + geom_density(alpha=0.2) + geom_rug() + facet_wrap(~variable, scales="free") + scale_fill_manual(values=col.region) + theme_bw(base_size = 14) + theme(legend.position = "bottom", axis.ticks.y=element_blank(), axis.text.y=element_blank()) Guerry/demo/guerry_mra.R0000644000176200001440000000132114425217565014762 0ustar liggesusers#' --- #' title: Predicting crime, multiple regression #' --- library(car) library(effects) #' ## Fit models predicting crime variables crime.mod1 <- lm(Crime_pers ~ Region + Literacy + Donations + Infants + Suicides, data=Guerry) crime.mod2 <- lm(Crime_prop ~ Region + Literacy + Donations + Infants + Suicides, data=Guerry) Anova(crime.mod1) Anova(crime.mod2) #' ## diagnostic plots op <- par(mfrow=c(2,2)) plot(crime.mod1) par(op) op <- par(mfrow=c(2,2)) plot(crime.mod2) par(op) #' ## Effect plots plot(predictorEffects(crime.mod1, ~ Region + Literacy + Infants + Suicides), lwd=2, main="") plot(predictorEffects(crime.mod2, ~ Region + Literacy + Infants + Suicides), lwd=2) Guerry/demo/guerry_manova.R0000644000176200001440000000522014426762677015500 0ustar liggesusers#' --- #' title: MANOVA, candisc models for Guerry data #' --- #' Load packages library(heplots) library(candisc, warn.conflicts = FALSE) library(car) library(Guerry) library(dplyr) #library(sp) data(Guerry, package="Guerry") #' ## extract pieces df <- data.frame(Guerry)[, 4:9] # the 6 variables dep.names <- data.frame(Guerry)[, 3] # departement names region.names <- data.frame(Guerry)[, 2] # region names col.region <- colors()[c(149, 254, 468, 552, 26)] # colors for region #' ## MANOVA: What variables predict crime? crime.mod <- lm(cbind(Crime_pers, Crime_prop) ~ Region + Literacy + Donations + Infants + Suicides, data=Guerry) Anova(crime.mod) #' ## bivariate confidence intervals for coefficients coefplot(crime.mod, fill = TRUE, fill.alpha = 0.1, level=0.68) coefplot(crime.mod, fill = TRUE, fill.alpha = 0.1, level=0.68, parm = 6:9) #' ## diagnostic plot for multivariate normality and outliers labels <- paste0(Guerry$dept,":", Guerry$Department) cqplot(crime.mod, id.n=4, labels=labels) heplot(crime.mod, fill=TRUE, fill.alpha=0.05, cex=1.4, cex.lab=1.3 ) #' ## Canonical discriminant analysis crime.can <- candisc(crime.mod) crime.can #plot(crime.can) heplot(crime.can, fill=TRUE, fill.alpha=0.1, var.col = "black", var.cex = 1.3, cex=1.4, cex.lab=1.3) #' ## Use ranks #' Guerry recognized the extreme skewness of some variables and so used ranks on these measures Guerry_ranks <- Guerry |> select(1:9) |> mutate( Crime_pers = dense_rank(Crime_pers), Crime_prop = dense_rank(Crime_prop), Literacy = dense_rank(Literacy), Donations = dense_rank(Donations), Infants = dense_rank(Infants), Suicides = dense_rank(Suicides) ) # Guerry_ranks <- Guerry |> # select(1:9) |> # mutate(across(Crime_pers:Suicides, dense_rank)) crime.modr <- lm(cbind(Crime_pers, Crime_prop) ~ Region + Literacy + Donations + Infants + Suicides, data=Guerry_ranks) Anova(crime.modr) heplot(crime.modr, fill=TRUE, fill.alpha=0.05, cex=1.4, cex.lab=1.2 ) crime.canr <- candisc(crime.modr) crime.canr #plot(crime.can) heplot(crime.canr) #' ## What variables discriminate among Regions ? region.mod <- lm(cbind(Crime_pers, Crime_prop, Literacy, Donations, Infants, Suicides) ~ Region, data=Guerry) Anova(region.mod) heplot(region.mod, fill=TRUE, fill.alpha=0.1) pairs(region.mod, variables=1:3, fill=TRUE, fill.alpha=0.1) region.can <- candisc(region.mod) region.can plot(region.can) Guerry/demo/guerry_bivar.R0000644000176200001440000000507414425220641015304 0ustar liggesusers#' --- #' title: Bivariate data ellipses for crime rates vs. literacy #' --- #' library(Guerry) library(car) data(Guerry) set.seed(12315) with(Guerry,{ dataEllipse(Literacy, Crime_pers/1000, levels = 0.68, ylim = c(0,40000), xlim = c(0, 80), ylab="Pop. per crime against persons", xlab="Percent who can read & write", pch = 16, grid = FALSE, id = list(method="mahal", n = 8, labels=Department, location="avoid", cex=1.2), center.pch = 3, center.cex=5, cex.lab=1.5) dataEllipse(Literacy, Crime_pers/1000, levels = 0.95, add=TRUE, # ylim = c(0,40000), xlim = c(0, 80), lwd=2, lty="longdash", col="gray", center.pch = FALSE ) abline( lm(Crime_pers ~ Literacy), lwd=2) lines(loess.smooth(Literacy, Crime_pers), col="red", lwd=3) } ) with(Guerry,{ dataEllipse(Literacy, Crime_prop, levels = 0.68, ylim = c(0,20000), xlim = c(0, 80), ylab="Pop. per crime against poperty", xlab="Percent who can read & write", pch = 16, grid = FALSE, id = list(method="mahal", n = 8, labels=Department, location="avoid", cex=1.2), center.pch = 3, center.cex=5, cex.lab=1.5) dataEllipse(Literacy, Crime_prop, levels = 0.95, add=TRUE, ylim = c(0,40000), xlim = c(0, 80), lwd=2, lty="longdash", col="gray", center.pch = FALSE ) abline( lm(Crime_prop ~ Literacy), lwd=2) lines(loess.smooth(Literacy, Crime_prop), col="red", lwd=3) } ) # redo this as crimes per 100,000 persons Guerry$Crime_pers_rate <- 100000 / Guerry$Crime_pers set.seed(12315) with(Guerry,{ dataEllipse(Literacy, Crime_pers_rate, levels = 0.68, ylim = c(-5,45), xlim = c(-5, 80), ylab = "Crimes against persons / 100k pop", xlab="Percent who can read & write", pch = 16, grid = FALSE, id = list(method="mahal", n = 8, labels=Department, location="avoid", cex=1.2), center.pch = 3, center.cex=5, cex.lab=1.5) dataEllipse(Literacy, Crime_pers_rate, levels = 0.95, add=TRUE, ylim = c(-5,45), xlim = c(-5, 80), lwd=2, lty="longdash", col="gray", center.pch = FALSE ) abline( lm(Crime_pers_rate ~ Literacy), lwd=2) lines(loess.smooth(Literacy, Crime_pers_rate), col="red", lwd=3) } ) Guerry/demo/00Index0000644000176200001440000000037314513000475013607 0ustar liggesusersguerry_bivar Bivariate data ellipses for crime rates vs. literacy guerry_density Density plots of Guerry data guerry_manova MANOVA, candisc models for Guerry data guerry_mra Predicting crime, multiple regression Guerry/.Rinstignore0000644000176200001440000000006214053052354014032 0ustar liggesusersdoc/figs/.*[.]pdf$ doc/figs doc/figs/.*[.]png$ Guerry/data/0000755000176200001440000000000014347134633012450 5ustar liggesusersGuerry/data/gfrance85.RData0000644000176200001440000055624414117675226015172 0ustar liggesusers7zXZi"6!X])TW"nRʟF\ Xv +.DJwFkZ/oah.,d<ю2D%Мq+t)~ߖxl)8!|,a|clv :[Z;hvω7}\U5Gċ&6l@2mR,D`/!(Z1 ijbNNyM=!pR=GL`цt.XBLtz׈5A-@`gĥ,s/޿)ѹ"28huqu4+#հ>.Il:vǪK(ϝH]I5Vw7]%w]<3]$6'iWO[{ Bӱ+HқQ{&JFWzg}ʠW_6D;6s/#@xMn Idڹ6Q#~ pR2q?S(2R8ÉX'6jk'5ciWRv! ӑ#sMuy #VR$T 6g+?|藒s$\!Q)b^Y@~nȑhtKx2_/kI!HN齱b<^R Z }[YM*u t&?h9$b(6=V;nl:z(nԡy6GuFȔ?&Hh)}N6bM;7OhNgHh!iz^*IPl~VSMj t@75z?/h,wئMETI*Xߠ[F3h'-HFtEq|c.C4Tw IhqPI i3W;y& rط~'}N1 VSq4+?18ozfOihy+C (9648 vhP)лwOlHT]wR쑆#eTbGgO,>X&[u5T%sr&֕LOxW^[^lɍ݁0>{P1@fd;Glط+,8u_1}-rʎ (NԚJ*ڦ5rVjaJȵfc@0:ȉ#9E4c 1뙨!Nehf 9 Y@E:޵a7pwҍmyCr);Gl(?qd%mP/ 5Do_K1іmslN^ }B ;Y8S#S_`nO Fi/3@bs_s/L7Sn}Iyx.W(:yeTbe tX`e{*K߀Ghxݣr<ٷr@R̗RjmǬёN.62m{$Q~!F{S;Tǐ wn2 2RadU?$k,DǏ j>LF ޟ}A.kWן>ry%{/j'h.o,cIK[{PI&1J5DkJZ(#\h-A#Gt3xZӌw/4ܓʂۇjQ^2ezX2yN#BKJip[q3sct*YR}o8UMtW?8g?Vkˮu@Le\x69c;(O<1o!PXywfkmNi}6 WD;An7ݐH7+\x:7^lx+%5o* cQ3`NuHa1UY UNr E#M-XgINۂVD 'YЫ`Q#Tɪej4:G-gW|s E:wÊyo1_QjxXa>-3MUQQyli2?tQthAs%"_ Uh$-#K  Z1TP XŻ7'#Š~A,YvMp8a_ _OJg"i޼9∙-֪Zǔz~#{.N`3eyv --u'r f.+VfASchdPpVQGp 񒂔;v-ËEBgztaO|ck:^Ll DEՏʁGG pV9H=raA/ I)eI>o?ʏ ǤV\f3-9Z'.4Yy1 uE)³=.miH#~F5xh3ٳ$ K}2ڀݼe+(Mfk[ $-7{%S\*59b9W +\\dōXʓ $ؾ]^Tyz%oJ%"vhlCVyjU? 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It gives the rank ordering of crimes against persons in seven age groups, in long form. } \usage{data("propensity")} \format{ A data frame with 124 observations on the following 4 variables. \describe{ \item{\code{age}}{a character vector, with 7 age groups, \code{<21}, \code{21-30}, \code{30-40} ... \code{60-70}, \code{70-}} \item{\code{rank}}{a numeric vector, rank of the crime within each age group} \item{\code{crime}}{a character vector, label of the crime} \item{\code{share}}{a numeric vector, share (frequency) of the crime in a population of 1000} } } \details{ For each age group (both males and females), the 17 most frequent crimes are listed in rank order, followed by an 'Other crime' category. } \source{ H. P. Whitt and V. W. Reinking (2002). A Translation of Andr\'e-Michel Guerry's \emph{Essay on the Moral Statistics of France}, Lewiston, N.Y.: Edwin Mellen Press, 2002. } \references{ Guerry, A.-M. (1833). \emph{Essai sur la statistique morale de la France} Paris: Crochard. } \examples{ data(propensity) ## maybe str(propensity) ; plot(propensity) ... } \keyword{datasets} Guerry/man/gfrance.Rd0000644000176200001440000000502514425246450014206 0ustar liggesusers%\encoding{latin1} \name{gfrance} \Rdversion{1.1} \alias{gfrance} \docType{data} \title{ Map of France in 1830 with the Guerry data } \description{ \code{gfrance} is a \code{SpatialPolygonsDataFrame} object created with the \code{sp} package, containing the polygon boundaries of the map of France as it was in 1830, together with the \code{\link{Guerry}} data frame. } \usage{data(gfrance)} \format{ The format is: Formal class 'SpatialPolygonsDataFrame' [package "sp"] with 5 slots: \itemize{ \item \code{gfrance@data}, \item \code{gfrance@polygons}, \item \code{gfrance@plotOrder}, \item \code{gfrance@bbox}, \item \code{gfrance@proj4string}. } See: \code{\link[sp]{SpatialPolygonsDataFrame}} for descriptions of some components. The analysis variables, represented in \code{gfrance@data} are described in \code{\link{Guerry}}. } \details{ In the present version, the PROJ4 projection is not specified. } \source{ Friendly, M. (2007). Supplementary materials for Andre-Michel Guerry's Moral Statistics of France: Challenges for Multivariate Spatial Analysis, \url{http://www.datavis.ca/gallery/guerry/}. } \references{ Friendly, M. (2007). A.-M. Guerry's Moral Statistics of France: Challenges for Multivariable Spatial Analysis. \emph{Statistical Science}, 22, 368-399. } \seealso{ \code{\link{Guerry}} for description of the analysis variables \code{\link{Angeville}} for other analysis variables } \examples{ library(sp) data(gfrance) names(gfrance) ## list @data variables plot(gfrance) ## just show the map outline # Show basic choropleth plots of some of the variables spplot(gfrance, "Crime_pers") # use something like Guerry's pallete, where dark = Worse my.palette <- rev(RColorBrewer::brewer.pal(n = 9, name = "PuBu")) spplot(gfrance, "Crime_pers", col.regions = my.palette, cuts = 8) spplot(gfrance, "Crime_prop") # Note that spplot assumes all variables are on the same scale for comparative plots # transform variables to ranks (as Guerry did) \dontrun{ local({ gfrance$Crime_pers <- rank(gfrance$Crime_pers) gfrance$Crime_prop <- rank(gfrance$Crime_prop) gfrance$Literacy <- rank(gfrance$Literacy) gfrance$Donations <- rank(gfrance$Donations) gfrance$Infants <- rank(gfrance$Infants) gfrance$Suicides <- rank(gfrance$Suicides) spplot(gfrance, c("Crime_pers", "Crime_prop", "Literacy", "Donations", "Infants", "Suicides"), layout=c(3,2), as.table=TRUE, main="Guerry's main moral variables") }) } } \keyword{datasets} \keyword{spatial} Guerry/man/figures/0000755000176200001440000000000014513000475013745 5ustar liggesusersGuerry/man/figures/README-gfrance4-1.png0000644000176200001440000003475514502051152017247 0ustar liggesusersPNG  IHDRMR/APLTE6:af6:::f:aff8XZp6666666666::::f:::::::f:::ff:f:f::aa6aaaff:fff:f::f:fffftψ6aaϬ::::fff::fې۬aψff::fې۶۶ψ6ې:ېf۶f۶۶۶aψfې۶K" pHYsod IDATx #uߟp:!6aX"D 8'k'Dc-EIvH]|tQjta ]ޯ_p0;b:o?<ǿ '=|xĿN^~bpx|y|?~r#>u8ɑҟ?k {'/ Α,h~N"r.ſYkrO}4{H17apH a=|O돋?Nuߑ44=Z}+}_+,K#1Hoi$[,D'{g )8$~ ?w>sߑ44=Z=n@C1-uP2F!Z9FJtm')8|?OܻhE7W4$FX;?}@1|#tL}VHӣx$ӋdV:;ŠF,w{(yl:{4=Z=r@IG=ٿ4?I@FX˩L-EPъRMǦoϞQ -L=$ @Hҿ'Zx>u+>|GRE?:AM+4c7>E=!ku=t"L^ihejZihejZiheV P0=<p|e gd:߂@|Ŵ-h?͟|v{i^@IM4, :?h7R 5Ն/ˉ\Ȁ{dHv5Ӷ,CN4h)_f h ZIH6Uc#G/h hZh"gWXN~p̉kΟݔ%_O?>_`}Wq?~omyvx'=&?7&}_M/~ˀE;hԪs.$}8鿯?>pȔ οӁS}1Nږ7m'|1|O?7mJǤ Qu AWDUgz$ynXpѽlO'/| ޿ptY>?U8W{HJ>uP~ W޿?MLqL4'.|Aj7  Pn$ L^|5T`~Gcu;=ExOږ7MGS%mlÀnrEW`LV\|*&EE5Di;< APfPC0&p" +Rsձ/!Sw^y:S WW:_ۃݽ:Z)~1|5w?A޿xҶi:ZdM:!hlL;ޔ-Gg `xҶi;\>'D>:Ikwy44|.!фokZȄmReaN2@˺_,VPNXЪ~LKrX;^㩯ނKp5luu&j}=LrMRNr݀MZ@]8Y ӳM&tI˕<zLJx6I*T=bN(Jo?}vxe.B] . 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R|fmJ|tQPtYWX/BhQɥ/|Ӊŧ-4l]\tH|s<''y a:RΪ*,63;J}ϵb>bRbc nr:D2IӚpj S8耮+&ُ @U"JА0uYԍbN)CrQa3ʌvz Y$ 'ϲ'!r)π+/栫 >4m$i>P!CSW@-[;k[R<-l:|"L :d1;A>wYCt7YPūYU<[)YHBfA ?$PvlCjtj6ַ5 ʀ;AJ<"5#$F'SLSyЙkH Z+B%ԑא  x9iyLuhBmb4:cvyʉ:#.xD'Ȯ>3v-Q[aʱǹ1#ҝu2s]G}jJhv㜑 k[;녣?=kR$2 2C=8g$-vtN N-ISϊ:YꐸIŔO_/ ]K~i1man%•JA<됮йeuqC ]W9dP׷('P~ZK'0vL~`n8(_U1S7/&spPr[hy`kB h)ww E0vRtw|iʒ&AeWZ}?k>ͦEBʀ+dAww_b˴oH3H5#H7>#6-R^@N/ifܓBf<?ooǯ}*Pt O)Ze6]4/$1A3P, aNNs&Ԩ|U!N05M8I嫲@Yz 4MY/UYXr/2h Et}jZX%|rӒ//|u UPYI+2YI+2YI+2YI+2YI+2YI+2YI+2YI+2YI+2YIqcR}IENDB`Guerry/man/gfrance85.Rd0000644000176200001440000000417314430775011014362 0ustar liggesusers\encoding{latin1} \name{gfrance85} \Rdversion{1.1} \alias{gfrance85} \docType{data} \title{ Map of France in 1830 with the Guerry data, excluding Corsica } \description{ \code{gfrance85} is a SpatialPolygonsDataFrame object created with the \code{sp} package, containing the polygon boundaries of the map of France as it was in 1830, together with the \code{\link{Guerry}} data frame. This version excludes Corsica, which is an outlier both in the map and in many analyses. } \usage{data(gfrance85)} \format{ The format is: Formal class 'SpatialPolygonsDataFrame' [package "sp"] with 5 slots: \code{gfrance85@data}, \code{gfrance85@polygons}, \code{gfrance85@plotOrder}, \code{gfrance85@bbox}, \code{gfrance85@proj4string}. See: \code{\link[sp]{SpatialPolygonsDataFrame}} for descriptions of some components. The analysis variables are described in \code{\link{Guerry}}. } \details{ In the present version, the PROJ4 projection is not specified. } \source{ Friendly, M. (2007). Supplementary materials for Andr?-Michel Guerry's Moral Statistics of France: Challenges for Multivariate Spatial Analysis, \url{http://datavis.ca/gallery/guerry/}. } \references{ Dray, S. and Jombart, T. (2009). A Revisit Of Guerry's Data: Introducing Spatial Constraints In Multivariate Analysis. Unpublished manuscript. Friendly, M. (2007). A.-M. Guerry's Moral Statistics of France: Challenges for Multivariable Spatial Analysis. \emph{Statistical Science}, 22, 368-399. } \examples{ data(gfrance85) require(sp) require(scales) plot(gfrance85) # plot the empty outline map # extract some useful components df <- data.frame(gfrance85)[,7:12] # main moral variables xy <- coordinates(gfrance85) # department centroids dep.names <- data.frame(gfrance85)[,6] region.names <- data.frame(gfrance85)[,5] col.region <- colors()[c(149,254,468,552,26)] |> scales::alpha(alpha = 0.2) # plot the map showing regions by color with department labels op <-par(mar=rep(0.1,4)) plot(gfrance85,col=col.region[region.names]) text(xy, labels=dep.names, cex=0.6) par(op) } \keyword{datasets} \keyword{spatial} Guerry/man/Guerry-package.Rd0000644000176200001440000000761014425246471015454 0ustar liggesusers%\encoding{latin1} \name{Guerry-package} \alias{Guerry-package} \docType{package} \title{ \packageTitle{Guerry} } \description{ Andre-Michel Guerry (1833) was the first to systematically collect and analyze social data on such things as crime, literacy and suicide with the view to determining social laws and the relations among these variables. He provided the first essentially multivariate and georeferenced spatial data on socially important questions, e.g., Is the rate of crime related to education or literacy? How does this vary over the departments of France? Are the rates of crime or suicide within departments stable over time? In an age well before the idea of correlation had been invented, Guerry used graphics and statistical maps to try to shed light on such questions. In a later work (Guerry, 1864), he explicitly tried to entertain larger questions, but with still-limited statistical tools: Can rates of various crimes be related to multiple causes or predictors? Are the rates and ascribable causes in France similar or different to those found in England? The \pkg{Guerry} package comprises maps of France in 1830, multivariate data from A.-M. Guerry and others (Angeville, 1836), and statistical and graphic methods related to Guerry's \emph{Moral Statistics of France}. The goal of providing these as an R package is to facilitate the exploration and development of statistical and graphic methods for multivariate data in a geo-spatial context. } \details{ The DESCRIPTION file: \packageDESCRIPTION{Guerry} \packageIndices{Guerry} Data from Guerry and others is contained in the data frame \code{\link{Guerry}}. Because Corsica is often considered an outlier both spatially and statistically, the map of France circa 1830, together with the Guerry data is provided as \code{SpatialPolygonsDataFrame}s in two forms: \code{\link{gfrance}} for all 86 departments, and and \code{\link{gfrance85}}, for the 85 departments excluding Corsica. } \author{ \packageAuthor{Guerry} Maintainer: \packageMaintainer{Guerry} } \references{ d'Angeville, A. (1836). \emph{Essai sur la Statistique de la Population francaise}, Paris: F. Darfour. Dray, S. and Jombart, T. (2011). A Revisit Of Guerry's Data: Introducing Spatial Constraints In Multivariate Analysis. \emph{The Annals of Applied Statistics}, 5(4). Brunsdon, C. and Dykes, J. (2007). Geographically weighted visualization: interactive graphics for scale-varying exploratory analysis. Geographical Information Science Research Conference (GISRUK 2007). NUI Maynooth, Ireland, April, 2007. \url{https://www.maynoothuniversity.ie/national-centre-geocomputation-ncg}. Friendly, M. (2007). A.-M. Guerry's Moral Statistics of France: Challenges for Multivariable Spatial Analysis. \emph{Statistical Science}, 22, 368-399. \url{http://www.datavis.ca/papers/guerry-STS241.pdf} Friendly, M. (2007). Supplementary materials for Andre-Michel Guerry's Moral Statistics of France: Challenges for Multivariate Spatial Analysis, \url{http://www.datavis.ca/gallery/guerry/}. Friendly, M. (2022). The life and works of Andre-Michel Guerry, revisited. \emph{Sociological Spectrum}, \bold{42}, 233–259. \doi{10.1080/02732173.2022.2078450} Guerry, A.-M. (1833). \emph{Essai sur la statistique morale de la France} Paris: Crochard. English translation: Hugh P. Whitt and Victor W. Reinking, Lewiston, N.Y.: Edwin Mellen Press, 2002. Guerry, A.-M. (1864). \emph{Statistique morale de l'Angleterre compar?e avec la statistique morale de la France, d'apres les comptes de l'administration de la justice criminelle en Angleterre et en France, etc.} Paris: J.-B. Bailliere et fils. } %~~ Optionally other standard keywords, one per line, from file KEYWORDS in ~~ %~~ the R documentation directory ~~ \keyword{spatial} \keyword{data} %\examples{ %#~~ simple examples of the most important functions ~~ %} Guerry/man/partials/0000755000176200001440000000000014427014144014122 5ustar liggesusersGuerry/man/partials/_ggradar.Rmd0000644000176200001440000000547414427016662016355 0ustar liggesusers ```{r echo=FALSE} data(Guerry, package = "Guerry") library(dplyr) library(ggplot2) ``` ## Plots of means In his discussion, Guerry often referred to differences among the Regions of France. One useful display to address this is a **profile** plot of the means of the moral variables for each `Region`, typically shown as a line plot of the mean value over the variables. Another term used for this is a **parallel coordinates** plot. First, find the means of each variable. ```{r sumry} guerry_sumry <- Guerry |> select(Region, Crime_pers:Suicides) |> filter(!is.na(Region)) |> mutate(Region = factor(Region, labels =c("Central", "East", "North", "South", "West"))) |> group_by(Region) |> summarise(Crime_pers= mean(Crime_pers), Crime_prop = mean(Crime_prop), Literacy = mean(Literacy), Donations = mean(Donations), Infants = mean(Infants), Suicides = mean(Suicides), ) |> ungroup() |> mutate_at(vars(-Region), scales::rescale) guerry_sumry |> knitr::kable(digits = 2) ``` Even in this simple table, the regions that stand out as best (1) and worst (0) can be seen. In particular, the Central, North and West are highest on personal crime, while personal crime is highest in the Central region and lowest in the South. A plot of these means by region can be produced with the `ggpcp` package. ```{r ggpcp} #| fig.width = 8 library(ggpcp) guerry_sumry |> pcp_select(Region, Crime_pers:Suicides) |> pcp_scale(method="uniminmax") |> pcp_arrange() |> ggplot(aes_pcp()) + geom_pcp_axes() + geom_pcp(aes(colour = Region), linewidth = 2) + geom_pcp_labels() + xlab("Variable") + ylab("Percent of maximum") + theme_bw(base_size = 14) + theme(legend.position = 'bottom') ``` #### Radar chart A more attractive form for this display shows the axes for each variable in polar coordinates, known as a **radar chart**. Because the variables are on different scales, a first step is to normalize them, by scaling each to a range of (0,1). The `scales::rescale()` function handles this. Here, I use the `ggradar` package by Ricardo Bion. It is not on CRAN, so you may need to install it to reproduce the figure here. ```{r ggradar} #| fig.width = 8 if(!require(ggradar)) remotes::install_github("ricardo-bion/ggradar") library(ggradar) # Create radar charts using ggplot2 guerry_sumry |> ggradar( grid.label.size = 4, axis.label.size = 3, axis.label.offset = 1.2, group.point.size = 4, plot.extent.x.sf = 1, plot.extent.y.sf = 1.2, fill = TRUE, fill.alpha = 0.2, legend.position = "bottom", legend.title = "Region", legend.text.size = 12, plot.title = "Guerry data: Means by Region") ``` Guerry/man/Guerry.Rd0000644000176200001440000001633714502054406014060 0ustar liggesusers%\encoding{latin1} %\encoding{UTF-8} \name{Guerry} \Rdversion{1.1} \alias{Guerry} \docType{data} \encoding{latin1} \title{ Data from A.-M. Guerry, "Essay on the Moral Statistics of France" } \description{ Andre-Michel Guerry (1833) was the first to systematically collect and analyze social data on such things as crime, literacy and suicide with the view to determining social laws and the relations among these variables. The Guerry data frame comprises a collection of 'moral variables' on the 86 departments of France around 1830. A few additional variables have been added from other sources. } \usage{data(Guerry)} \format{ A data frame with 86 observations (the departments of France) on the following 23 variables. \describe{ \item{\code{dept}}{Department ID: Standard numbers for the departments, except for Corsica (200)} \item{\code{Region}}{Region of France ('N'='North', 'S'='South', 'E'='East', 'W'='West', 'C'='Central'). Corsica is coded as NA } \item{\code{Department}}{Department name: Departments are named according to usage in 1830, but without accents. A factor with levels \code{Ain} \code{Aisne} \code{Allier} ... \code{Vosges} \code{Yonne}} \item{\code{Crime_pers}}{Population per Crime against persons. Source: A2 (Comptes general, 1825-1830)} \item{\code{Crime_prop}}{Population per Crime against property. Source: A2 (Compte general, 1825-1830)} \item{\code{Literacy}}{Percent Read & Write: Percent of military conscripts who can read and write. Source: A2 } \item{\code{Donations}}{Donations to the poor. Source: A2 (Bulletin des lois)} \item{\code{Infants}}{Population per illegitimate birth. Source: A2 (Bureau des Longitudes, 1817-1821)} \item{\code{Suicides}}{Population per suicide. Source: A2 (Compte general, 1827-1830)} \item{\code{MainCity}}{Size of principal city ('1:Sm', '2:Med', '3:Lg'), used as a surrogate for population density. Large refers to the top 10, small to the bottom 10; all the rest are classed Medium. Source: A1. An ordered factor with levels \code{1:Sm} < \code{2:Med} < \code{3:Lg}} \item{\code{Wealth}}{Per capita tax on personal property. A ranked index based on taxes on personal and movable property per inhabitant. Source: A1} \item{\code{Commerce}}{Commerce and Industry, measured by the rank of the number of patents / population. Source: A1} \item{\code{Clergy}}{Distribution of clergy, measured by the rank of the number of Catholic priests in active service / population. Source: A1 (Almanach officiel du clergy, 1829)} \item{\code{Crime_parents}}{Crimes against parents, measured by the rank of the ratio of crimes against parents to all crimes-- Average for the years 1825-1830. Source: A1 (Compte general) } \item{\code{Infanticide}}{Infanticides per capita. A ranked ratio of number of infanticides to population-- Average for the years 1825-1830. Source: A1 (Compte general) } \item{\code{Donation_clergy}}{Donations to the clergy. A ranked ratio of the number of bequests and donations inter vivios to population-- Average for the years 1815-1824. Source: A1 (Bull. des lois, ordunn. d'autorisation) } \item{\code{Lottery}}{Per capita wager on Royal Lottery. Ranked ratio of the proceeds bet on the royal lottery to population--- Average for the years 1822-1826. Source: A1 (Compte rendus par le ministre des finances)} \item{\code{Desertion}}{Military desertion, ratio of the number of young soldiers accused of desertion to the force of the military contingent, minus the deficit produced by the insufficiency of available billets-- Average of the years 1825-1827. Source: A1 (Compte du ministere du guerre, 1829 etat V) } \item{\code{Instruction}}{Instruction. Ranks recorded from Guerry's map of Instruction. Note: this is inversely related to \code{Literacy} (as defined here)} \item{\code{Prostitutes}}{Prostitutes in Paris. Number of prostitutes registered in Paris from 1816 to 1834, classified by the department of their birth Source: Parent-Duchatelet (1836), \emph{De la prostitution en Paris}} \item{\code{Distance}}{Distance to Paris (km). Distance of each department centroid to the centroid of the Seine (Paris) Source: calculated from department centroids } \item{\code{Area}}{Area (1000 km^2). Source: Angeville (1836) } \item{\code{Pop1831}}{1831 population. Population in 1831, taken from Angeville (1836), \emph{Essai sur la Statistique de la Population francaise}, in 1000s } } } \details{ Note that most of the variables (e.g., \code{Crime_pers}) are scaled so that 'more is better' morally. Values for the quantitative variables displayed on Guerry's maps were taken from Table A2 in the English translation of Guerry (1833) by Whitt and Reinking. Values for the ranked variables were taken from Table A1, with some corrections applied. The maximum is indicated by rank 1, and the minimum by rank 86. } \source{ Angeville, A. (1836). \emph{Essai sur la Statistique de la Population fran?aise} Paris: F. Doufour. Guerry, A.-M. (1833). \emph{Essai sur la statistique morale de la France} Paris: Crochard. English translation: Hugh P. Whitt and Victor W. Reinking, Lewiston, N.Y. : Edwin Mellen Press, 2002. Parent-Duchatelet, A. (1836). \emph{De la prostitution dans la ville de Paris}, 3rd ed, 1857, p. 32, 36 } \references{ Dray, S., & Jombart, T. (2011). Revisiting Guerry's data: Introducing spatial constraints in multivariate analysis. \emph{Annals of Applied Statistics}, \bold{5}, 2278-2299 Brunsdon, C. and Dykes, J. (2007). Geographically weighted visualization: Interactive graphics for scale-varying exploratory analysis. \emph{Geographical Information Science Research Conference (GISRUK 07)}, NUI Maynooth, Ireland, April, 2007. Friendly, M. (2007). A.-M. Guerry's Moral Statistics of France: Challenges for Multivariable Spatial Analysis. \emph{Statistical Science}, 22, 368-399. Friendly, M. (2007). Data from A.-M. Guerry, Essay on the Moral Statistics of France (1833), \url{https://www.datavis.ca/gallery/guerry/guerrydat.html}. } \seealso{ \code{\link{Angeville}} for other analysis variables } \examples{ library(car) data(Guerry) # Is there a relation between crime and literacy? # Plot personal crime rate vs. literacy, using data ellipses. # Identify the departments that stand out set.seed(12315) with(Guerry,{ dataEllipse(Literacy, Crime_pers, levels = 0.68, ylim = c(0,40000), xlim = c(0, 80), ylab="Pop. per crime against persons", xlab="Percent who can read & write", pch = 16, grid = FALSE, id = list(method="mahal", n = 8, labels=Department, location="avoid", cex=1.2), center.pch = 3, center.cex=5, cex.lab=1.5) # add a 95% ellipse dataEllipse(Literacy, Crime_pers, levels = 0.95, add=TRUE, ylim = c(0,40000), xlim = c(0, 80), lwd=2, lty="longdash", col="gray", center.pch = FALSE ) # add the LS line and a loess smooth. abline( lm(Crime_pers ~ Literacy), lwd=2) lines(loess.smooth(Literacy, Crime_pers), col="red", lwd=3) } ) # A corrgram to show the relations among the main moral variables # Re-arrange variables by PCA ordering. library(corrgram) corrgram(Guerry[,4:9], upper=panel.ellipse, order=TRUE) } \keyword{datasets} Guerry/man/Angeville.Rd0000644000176200001440000001054714425210321014500 0ustar liggesusers\name{Angeville} \Rdversion{1.1} \alias{Angeville} \docType{data} \title{ Data from d'Angeville (1836) on the population of France } \description{ Adolph d'Angeville (1836) presented a comprehensive statistical summary of nearly every known measurable characteristic of the French population (by department) in his \emph{Essai sur la Statistique de la Population francaise}. Using the graphic method of shaded (choropleth) maps invented by Baron Charles Dupin and applied to significant social questions by Guerry, Angeville's \emph{Essai} became the first broad and general application of principles of graphic representation to national industrial and population data. The collection of variables in the data frame \code{Angeville} is a small subset of over 120 columns presented in 8 tables and many graphic maps. } \usage{data(Angeville)} \format{ A data frame with 86 observations on the following 16 variables. \describe{ \item{\code{dept}}{a numeric vector} \item{\code{Department}}{Department name: a factor with levels \code{Ain} \code{Aisne} ... \code{Vosges} \code{Yonne}} \item{\code{Mortality}}{Mortality: Number of births to give 100 people at age 21 (T1:13)} \item{\code{Marriages}}{Number of marriages per 1000 men aged 21 (T1:15)} \item{\code{Legit_births}}{Annual no. of legitimate births (T2:17)} \item{\code{Illeg_births}}{Annual no. of illegitimate births (T2:18)} \item{\code{Recruits}}{Number of people registered for military recruitment from 1825-1833 (T3:32)} \item{\code{Conscripts}}{Number of inhabitants per military conscript (T3:33)} \item{\code{Exemptions}}{Number of military exemptions per 1000 all of physical causes (T3:47)} \item{\code{Farmers}}{Number of farmers during the census in 1831 (T4:65)} \item{\code{Recruits_ignorant}}{Average number of ignorant recruits per 1000 (T5:69)} \item{\code{Schoolchildren}}{Number of schoolchildren per 1000 inhabitants (T5:71)} \item{\code{Windows_doors}}{Number of windows & doors in houses per 100 inhabitants (T5:72). This is sometimes taken as an indicator of household wealth.} \item{\code{Primary_schools}}{"Number of primary schools (T5:74)} \item{\code{Life_exp}}{Life expectancy in years (T1:9a,9b)} \item{\code{Pop1831}}{Population in 1831} } } \details{ ID codes for \code{dept} were modified from those in Angeville's tables to match those used in \code{\link{Guerry}}. Angeville's variables are recorded in a variety of different ways and some of these were calculated from other columns in his tables not included here. As well, the variable names and labels used here were often shortened from the more complete descriptions given by d'Angeville. The notation "(Tn:k)" indicates that the variable used here came from Table n, Column k. } \source{ Angeville, A. d' (1836). \emph{Essai sur la Statistique de la Population francaise}, Paris: F. Darfour. The data was digitally scanned from Angeville's tables using OCR software, then extensively edited to correct obvious errors and finally subjected to some consistency checks using the column totals and ranked values he provided. } \references{ Whitt, H. P. (2007). Modernism, internal colonialism, and the direction of violence: suicide and crimes against persons in France, 1825-1830. Unpublished ms. } \examples{ library(Guerry) library(sp) library(RColorBrewer) data(Guerry) data(gfrance) data(Angeville) gf <- gfrance # the SpatialPolygonsDataFrame # Add some Angeville variables, transform them to ranks gf$Mortality <- rank(Angeville$Mortality) gf$Marriages <- rank(Angeville$Marriages) gf$Legit_births <- rank(Angeville$Legit_births) gf$Illeg_births <- rank(Angeville$Illeg_births) gf$Farmers <- rank(Angeville$Farmers) gf$Schoolchildren <- rank(Angeville$Schoolchildren) # plot them on map of France my.palette <- rev(brewer.pal(n = 9, name = "PuBu")) spplot(gf, c("Mortality", "Marriages", "Legit_births", "Illeg_births", "Farmers", "Schoolchildren"), names.attr = c("Mortality", "Marriages", "Legit_births", "Illeg_births", "Farmers", "Schoolchildren"), layout=c(3,2), as.table=TRUE, col.regions = my.palette, cuts = 8, # col = "transparent", main="Angeville variables") } \keyword{datasets} Guerry/DESCRIPTION0000644000176200001440000000362414516023372013245 0ustar liggesusersPackage: Guerry Type: Package Title: Maps, Data and Methods Related to Guerry (1833) "Moral Statistics of France" Version: 1.8.3 Date: 2023-10-24 Authors@R: c( person(given = "Michael", family = "Friendly", role=c("aut", "cre"), email="friendly@yorku.ca", comment = c(ORCID = "0000-0002-3237-0941")), person(given = "Stephane", family = "Dray", role="aut", email="stephane.dray@univ-lyon1.fr", comment = c(ORCID = "0000-0003-0153-1105")), person(given = "Roger", family = "Bivand", role="ctb", email = "Roger.Bivand@nhh.no") ) Maintainer: Michael Friendly Encoding: UTF-8 Language: en-US Depends: R (>= 3.5.0) Imports: sp Suggests: knitr, sf, spdep, ade4, adegraphics, adespatial, RColorBrewer, corrgram, car, effects, rmarkdown, here, ggplot2, ggpcp, ggrepel, heplots, patchwork, candisc, colorspace, scales, remotes, dplyr, tidyr Description: Maps of France in 1830, multivariate datasets from A.-M. Guerry and others, and statistical and graphic methods related to Guerry's "Moral Statistics of France". The goal is to facilitate the exploration and development of statistical and graphic methods for multivariate data in a geospatial context of historical interest. License: GPL URL: https://github.com/friendly/Guerry BugReports: https://github.com/friendly/Guerry/issues LazyLoad: yes LazyData: yes VignetteBuilder: knitr NeedsCompilation: no Packaged: 2023-10-24 16:55:48 UTC; friendly Author: Michael Friendly [aut, cre] (), Stephane Dray [aut] (), Roger Bivand [ctb] Repository: CRAN Date/Publication: 2023-10-24 20:20:10 UTC Guerry/build/0000755000176200001440000000000014515773424012642 5ustar liggesusersGuerry/build/vignette.rds0000644000176200001440000000046014515773424015201 0ustar liggesusersR]K0֏}DC_`o¨2݃4j F:/_MmG}IϹ9{ӷ)!dHa@CH0KlߋO&vJiL2$/?Juu\ ÛpW{.IW-}AV(PqQziD&UCjۛ AOZ{0IĴPX 5 (ǩyt]͟n'}:9(=c$TH%Z e?ђoy?jٱDov7y~ۥqELB+-ɻz?@ZGuerry/build/partial.rdb0000644000176200001440000001621114515773401014763 0ustar liggesusers=vF,Y-qIlDz"%J'^Zn3" hq?`Ιyߘy/0=u " {oݭPT*՗;bzw": p}?5[E$w`+q*/h e .monlz h4Tܸ.4a]쮒P} B4_ Ön(:?,֫k'`Os.-WVO5C CڏK/Ljْ!SUR]_,!ݥU=KSl~&+8s٧5jlUPq_(L^w yXk{ի4T}PK{qn;Ὥw5jm=zzxA\M>3D˕m(/!~) mUˈ_%H-_ ^hC/#JD* ,= di+@t BsM&u?ǖxo *Q@p!^!GY7Ԭ)А,0C;4|^P~dK7wp/G/ +ͅ[n;& UA44OE$QMA70TlRI&SQXq$33wf )X,3-Y4V"t<`u02<"D/l9lhTKAF: ƲiN"K \Wkyra(5ӵ9(U2,VⱿ$VlY7-Ze2hNDMQh,Rej pL(3!Tb uLx 9JE!u~&DɁlUl (75"s$/QFkUV S&5ʚʾj d<,)1*⫑yZD7L[4jɶVUYm-jW{ 5XdmP\/p <}f ^a E~84IL9&5 d: ?*DuY=GfkA3&5Hd4N԰Y.V5!e,pl<]0hCfS;e 1AY̜C>EicGak*Lr=r@ђRfl!kHEXY}ˤK4\@_LfeC+cM9b1 +rb5"K3Av@J>rxbzD7FElk- btSFgZK  YaIeK#0bMlLCu5`|qDREӡk}ȌX2㾘g y g\ڤ@XXiзֱD!?A|jN5b9C ^'I9 T~ x 8R~}gw}{AiGzFOuǨQq6Ru`A1oGSܞlz%`9aʱB#䣑IuX^τ1HBz#E$`ñj28DvBML9#scdp| ~dKjs+gf)OQ9O1gs}QLNOMf)OQ=Gj,BmDHFLe-UCbEx:8P{JBUV@*.I w}54>'ƅgwI ziUb'uدeDa<̉͢VG23//vL$r3SN߹.ZkAQ[]pc>:6 *;[XTRLk3Ss,GK ~Zї] pQ~+Rb ^Rt8qDT26~'t Sm 3NqIAƀ} f件Q")ñ¯ۺ"a$sCP˵azW;:HI|9LucLAz'\f8=M۫9uVbç#ZLcմ"l yGoBɠKTO1_ M'3%MьEФf2&QHhiDE+#7/ijB:UM-岮hV.! 2rĤF&u+*1fBX Bа(l !gL[2EAIџaV5+1 JROJdn|4RuAe];U`agwt>a68>>fV@dHcQ,M?kñM}<4Ͱ'FԴ2O o"U*6tg#rUsE~X-ibY -VjTωڱZ9O_ByitJS &H n&K7anftKRˢijWgj@"%)*鐳X7\kFYiL|禮!RF3Yՙ0lw&6/%K똈u G&(%/|DMUouxfU07m$*"5c]Ёv۪mdzNQ{ǓQ0TiF)N,ME3;:;y4zئܳuѺtYf뀁0]kJvz̸ϚZK8Y>MWL(v{2a|VAZM n:@dT.93a+t={t{ʉB,*5{%0e1!_ M)[otސj$ 2eAVM w`[o <퇹*[nyeUbov3YuV]66 ^0 RNy8h"mT[JBn"YP"-ķbV$󋡺n#Y; ,.R77Hk%̦,1O f)25Ȟ'9Sݔz|Qocp`D>V{ -(=g)X7pȼ\rjgIAj^BR$hLO/d\d~0xmp69JL>Wsyߦ^Dчn%CYS05-DK{"3sdp:7Ү !ʀ[! 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E?D]F6@fRRXʯoo#)GCgy[\@GvC.l~9 dq6x|9e9'w8ev)gFHK{GOppeof[Ojt0ُnֽtRIx硩?{Guerry/vignettes/0000755000176200001440000000000014515773424013553 5ustar liggesusersGuerry/vignettes/guerry-multivariate.Rmd0000644000176200001440000005455514513000475020241 0ustar liggesusers--- title: "Guerry data: Multivariate Analysis" author: "Michael Friendly" date: "`r Sys.Date()`" package: Guerry output: rmarkdown::html_vignette: toc: true toc_depth: 2 bibliography: ["refs.bib", "packages.bib"] link-citations: yes vignette: > %\VignetteIndexEntry{Guerry data: Multivariate Analysis} %\VignetteKeywords{crime, literacy, suicide, France, multivariate analysis} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r setup, include = FALSE} knitr::opts_chunk$set( collapse = TRUE, warning = FALSE, message = FALSE, # suppress package loading messages comment = "#>", fig.height = 5, fig.width = 5, dpi = 96 ) # packages to be cited here. Code at the end automatically updates packages.bib to.cite <- c("car", "effects", "ggplot2", "GGally", "ggbiplot") ``` André-Michel Guerry's _Essai sur la Statistique Morale de la France_ [@Guerry:1833] collected data on crimes, suicide, literacy and other "moral statistics" for various départements in France. He provided the first real social data analysis, using graphics and maps to summarize this multivariate dataset. One of his main goals in this ground-breaking study was to determine if the prevalence of crime in France could be explained by other social variables. In 1833, the scatterplot had not yet been invented; the idea of a correlation or a regression was still 50 years in the future [@Galton:1886]. Guerry displayed his data in shaded choropleth maps and semi-graphic tables and argued how these could be seen as implying systematic, lawful relations among moral variables. In this analysis, we ignore the spatial context of the départements and focus on multivariate analyses of the the data set. # Load data and packages We will primarily use the following packages, so load them now. ```{r load} library(Guerry) # Guerry data library(car) # better scatterplots library(effects) # Effect Displays for Linear Models library(ggplot2) # Elegant Data Visualisations Using the Grammar of Graphics library(ggrepel) # better handling of text labels library(patchwork) # combine plots library(heplots) # Hypothesis-Error plots library(candisc) # Visualizing Generalized Canonical Discriminant Analysis library(dplyr) # A Grammar of Data Manipulation library(tidyr) # Tidy Messy Data data(Guerry) ``` ## `Guerry` data set Guerry's (1833) data consisted of six main moral variables shown in the table below. He wanted all of these to be recorded on aligned scales so that **larger** numbers consistently reflected "**morally better**". Thus, four of the variables are recorded in the inverse form, as "Population per ...". | Name | Description | |:------------|:-----------------------------------------------------| |`Crime_pers` | Population per crime against persons | |`Crime_prop` | Population per crime against property | |`Literacy` | Percent of military conscripts who can read and write| |`Donations` | Donations to the poor | |`Infants` | Population per illegitimate birth | |`Suicides` | Population per suicide | The `Guerry` data set also contains: * `dept` and `Department`, the French ID numbers and names for the 86 départements of metropolitan France in 1830, including Corsica. * `Region`: a factor with main levels "N", "S", "E", "W", "C". Corsica is coded as `NA`. * A collection of 14 other related variables from other sources at the same time. See `?Guerry` for their precise definitions. ```{r guerry-more-vars} names(Guerry)[-(1:9)] ``` Among these, as other aspects of criminal behavior, we see crime against parents, `Infanticide` and `Prostitutes`. `Clergy` and `Donations_clergy` are considered to be measures of moral rectitude, potentially counteracting crime. ## Guerry's questions The main questions that concerned Guerry were whether indicators of crime could be shown to be related to factors which might be considered to ameliorate crime. Among these, Guerry focused most on `Literacy` defined as the number of military conscripts who could do more than mark an "X" on their enrollment form. A related variable is `Instruction`, the rank recorded from Guerry's map; as defined, it is inversely related to `Literacy`. Other potential explanatory variables are: : `Donations` (a measure of donations to the poor), : `Donation_clergy` (a measure of donations to clergy) : `Clergy` (the rank of number of Catholic priests in active service, per population) # Multivariate visualization methods Visualization methods for multivariate data take an enormous variety of forms simply because more than two dimensions of data offer exponentially increasingly possibilities. It is useful to distinguish several broad categories: * __data plots__ : primarily plot the raw data, often with annotations to aid interpretation (regression lines and smooths, data ellipses, marginal distributions) * __model plots__ : primarily plot the results of a fitted model, considering that the fitted model may involve more variables than can be shown in a static 2D plot. Some examples are: Added variable plots, effect plots, coefficient plots, ... * __diagnostic plots__ : indicating potential problems with the fitted model. These include residual plots, influence plots, plots for testing homogeneity of variance and so forth. * __dimension reduction plots__ : plot representations of the data into a space of fewer dimensions than the number of variables in the data set. Simple examples include principal components analysis (PCA) and the related biplots, and multidimensional scaling (MDS) methods. # Data plots Data plots portray the data in a space where the coordinate axes are the observed variables. * 1D plots include line plots, histograms and density estimates * 2D plots are most often scatterplots, but contour plots or hex-binned plots are also useful when the sample size is large. * For higher dimensions, biplots, showing the data in principal components space, together with vectors representing the correlations among variables, are often the most useful. ## Density plots It is useful to examine the distributions of the variables and **density** plots are quite informative. I want to do this for each of the 6 main variables, so I'll use this trick of tidy data analysis with `ggplot2`: 1. Reshape the data from wide to long. This gives `guerry_long`, where the different variables are in a column labeled `variable` and the values are in `value`. ```{r guerry-long} data("Guerry", package="Guerry") guerry_long <- Guerry |> filter(!is.na(Region)) |> select(dept:Suicides) |> pivot_longer(cols = Crime_pers:Suicides, names_to = "variable", values_to = "value") guerry_long ``` 2. Plot the density, but make a different subplot by `facet_wrap(~ variable)`. These plots all have different scales for the X and Y (density) values, so it is important to use `scales="FREE"`. Moreover, I'm primarily interested in the **shape** of these distributions, so I suppress the Y axis tick marks and labels. ```{r guerry-density1} #| out.width = "100%" ggplot(data = guerry_long, aes(x=value, fill=TRUE)) + geom_density(alpha=0.2) + geom_rug() + facet_wrap(~variable, scales="free") + theme_bw(base_size = 14) + theme(legend.position = "none", axis.ticks.y=element_blank(), axis.text.y=element_blank()) ``` You can see that all variables are positively skewed, `Donations`, `Infants` and `Suicides` particularly so, but not so much as to cause alarm. It is also of interest to see whether and how these distributions differ according to `Region`. This is easy to do, using `aes(... fill=Region)` ```{r guerry-density2} #| out.width = "100%", #| fig.height = 6 col.region <- colors()[c(149, 254, 468, 552, 26)] # colors for region ggplot(data = guerry_long, aes(x=value, fill=Region)) + geom_density(alpha=0.2) + geom_rug() + facet_wrap(~variable, scales="free") + scale_fill_manual(values=col.region) + theme_bw(base_size = 14) + theme(legend.position = "bottom", axis.ticks.y=element_blank(), axis.text.y=element_blank()) ``` For some variables, like `Infants` and `Suicides` the differences do not seem particularly large. However, both crime variables and `Literacy` show marked differences across region. ## Bivariate relations Let's start with plots of crime (`Crime_pers` and `Crime_prop`) in relation to `Literacy`. A simple scatterplot is not very informative. All that can be seen is that there is not much of a relation between personal crime and literacy. ```{r lit-pers-scat0} ggplot(aes(x=Literacy, y=Crime_pers/1000), data=Guerry) + geom_point(size=2) ``` More useful scatterplots are annotated with additional statistical summaries to aid interpretation: * linear regression line, * smoothed non-parametric (loess) curve, to diagnose potential non-linear relations, * data ellipses, to highlight the overall trend and variability, * point labels for potentially outlying or influential points. I use `ggplot2` here. It provides most of these features, except that to label unusual points, I calculate the Mahalanobis squared distance of all points from the grand means. ```{r lit-pers-scat} gdf <- Guerry[, c("Literacy", "Crime_pers", "Department")] gdf$dsq <- mahalanobis(gdf[,1:2], colMeans(gdf[,1:2]), cov(gdf[,1:2])) ggplot(aes(x=Literacy, y=Crime_pers/1000, label=Department), data=gdf) + geom_point(size=2) + stat_ellipse(level=0.68, color="blue", size=1.2) + stat_ellipse(level=0.95, color="gray", size=1, linetype=2) + geom_smooth(method="lm", formula=y~x, fill="lightblue") + geom_smooth(method="loess", formula=y~x, color="red", se=FALSE) + geom_label_repel(data = gdf[gdf$dsq > 4.6,]) + theme_bw() ``` The flat (blue) regression line and the nearly circular data ellipses show that the correlation is nearly zero; the smoothed (red) curve indicates that there is no tendency for a nonlinear relation. Doing the same for crimes against property: ```{r lit-prop-scat} gdf <- Guerry[, c("Literacy", "Crime_prop", "Department")] gdf$dsq <- mahalanobis(gdf[,1:2], colMeans(gdf[,1:2]), cov(gdf[,1:2])) ggplot(aes(x=Literacy, y=Crime_prop/1000, label=Department), data=gdf) + geom_point(size=2) + stat_ellipse(level=0.68, color="blue", size=1.2) + stat_ellipse(level=0.95, color="gray", size=1, linetype=2) + geom_smooth(method="lm", formula=y~x, fill="lightblue") + geom_smooth(method="loess", formula=y~x, color="red", se=FALSE) + geom_label_repel(data = gdf[gdf$dsq > 4.6,]) + theme_bw() ``` So, somewhat surprisingly, increased literacy is associated with an increase in property crime (greater population per crime) as opposed to the situation with personal crime, which seems unrelated to literacy. Creuse again stands out as an unusual point, one that is likely to be influential in regression models. ## Reconnaisance plots Reconnaisance plots attempt to give a bird's-eye overview of a multivariate data set. For example, to see the relations among more than two variables we could turn to a scatterplot matrix or some other display to show all pairwise bivariate relations. For these, my preferred package is `car` [@R-car] with the `scatterplotMatrix` function. `GGally` [@R-GGally] works within the the `ggplot2` framework, but doesn't have the flexibility I'd like. ```{r spm1} #| out.width="100%", #| echo=-1 par(mar=rep(2,4)) library(car) # Companion to Applied Regression scatterplotMatrix(Guerry[,4:9], ellipse=list(levels=0.68), smooth=FALSE) ``` ### Corrgrams Sometimes, particularly with more variables than this, we want to see a more schematic overview. A _correlation diagram_ or "corrgram" [@Friendly:02:corrgram] is a graphic display of a correlation matrix, allowing different renderings of the correlation between each pair of variables: as a shaded box, a pie symbol, a schematic data ellipse, and other options. This is implemented in the `corrgram` package [@R-corrgram]. The panels in the upper and lower triangles can be rendered differently. ```{r corrgram1} #| echo=-1 par(mar=rep(1,4)+.1) library(corrgram) # Plot a Correlogram corrgram(Guerry[,4:9], upper=panel.pie) ``` Or, the data in each pairwise tile can be rendered with data ellipses and smoothed curves to show possible nonlinear relations. Another feature is that the rows/column variables can be permuted to put similar variables together, using the `order` option, which arranges the variables according to similarity of their correlations. ```{r corrgram2} #| echo=-1 par(mar=rep(1,4)+.1) corrgram(Guerry[,4:9], upper=panel.ellipse, order=TRUE, lwd=2) ``` Here, there are a number of pairwise plots that appear markedly nonlinear. For the main crime variables, the most nonlinear are that of personal crime vs. donations to the poor, and property crime vs. infants born out of wedlock and suicides. `Literacy` stands out here as having negative relations with all other variables. An alternative analysis might include: * converting the data to ranks. * considering transformations of some of the variables ## Biplots Rather than viewing the data in **data space**, a biplot shows the data in the **reduced-rank PCA space** that explains most of the variation of the observations. This is essentially a plot of the observation scores on the first principal component overlaid with vectors representing the variables projected into PCA space. First, we use `prcomp()` to carry out the PCA. We'd like to visualize the result in relation to `Region`, so delete Corsica where `Region` is missing. ```{r guerry.pca} gdata <- Guerry |> select(Region, Crime_pers:Suicides) |> # keep only main variables filter(!is.na(Region)) # delete Corsica (Region==NA) guerry.pca <- gdata |> select(-Region) |> prcomp(scale = TRUE) print(guerry.pca, digits=3) ``` In the `ggplot2` framework, biplots can be produced by the `ggbiplot` package [@R-ggbiplot], but this package is not on CRAN, so cannot be directly used in this vignette. Instead, the code below was run locally and the result included. ```{r biplot1} #| eval=FALSE if(!require(ggbiplot)) remotes::install_github("vqv/ggbiplot") library(ggbiplot) # A ggplot2 based biplot ggbiplot(guerry.pca, groups=gdata$Region, ellipse=TRUE, var.scale = 3, varname.size = 5) + theme_bw() + labs(color="Region") + theme(legend.position = c(0.1, 0.8)) ``` ```{r ggbiplot} knitr::include_graphics("figures/ggbiplot.png") ``` This is OK, but there are many features of such plots that cannot be customized (line widths, colors, ... ). I prefer those created using the `heplots` package. ```{r biplot2, out.width="95%"} op <- par(mar=c(5,4,1,1)+.1) cols = colorspace::rainbow_hcl(5) covEllipses(guerry.pca$x, group=gdata$Region, pooled=FALSE, fill=TRUE, fill.alpha=0.1, col=cols, label.pos=c(3,0,1,1,3), cex=2, xlim=c(-4,4), ylim=c(-4,4), xlab = "Dimension 1 (35.7 %)", ylab = "Dimension 2 (20.0 %)", cex.lab=1.4 ) points(guerry.pca$x, pch=(15:19)[Guerry$Region], col=cols[Guerry$Region]) candisc::vectors(guerry.pca$rotation, scale=5, col="black", lwd=3, cex=1.4, pos = c(4,2,4,2,2,2), xpd=TRUE) abline(h=0, v=0, col=gray(.70)) ``` An interpretation can be read from both the directions of the variable arrows and the relative positions of the ellipses representing the scatter of the component scores for the different regions. * The first component is largely aligned positively with `Literacy` and negatively with property crime, suicides and children born out of wedlock (`Infants`) * The second dimension reflects mainly the correlation of personal crime and donations to the poor. * The South region is generally lower on PC2, the West, generally higher. * The North stands out as being higher than the others on PC1; the West somewhat higher on PC2. # Models Here we illustrate: * Model based plots for linear regression models predicting personal crime and property crime * Multivariate analysis of variance (MANOVA) and HE plots for the joint relation of the crime variables to other predictors. ## Predicting crime: Univariate regression The simplest approach to predicting the crime variables would be to fit a separate multiple regression to each. ```{r mra-models} crime.mod1 <- lm(Crime_pers ~ Region + Literacy + Donations + Infants + Suicides, data=Guerry) crime.mod2 <- lm(Crime_prop ~ Region + Literacy + Donations + Infants + Suicides, data=Guerry) ``` Tests for the predictors are best obtained using `car::Anova()` which gives **partial** (Type II) tests, adjusting for other predictors, rather than the **sequential** (Type I) tests provided by `stats::anova()` ```{r mra-anova} Anova(crime.mod1) Anova(crime.mod2) ``` These are somewhat disappointing if you look only at the significance stars: Only `Region` is significant for personal crime and only `Suicides` for property crime. There is no evidence for the argument, supported by the liberal hygenicists of Guerry's time, that increased `Literacy` would reduce crime. For such models, we can understand the nature of the predicted effects using the `effects` package. The (marginal) effect for a given term gives the predicted values, averaging over all other terms in the model. ```{r mra-effect1-code} #| eval=FALSE plot(predictorEffects(crime.mod1, ~ Region + Literacy + Infants + Suicides), lwd=2, main="") ``` ```{r mra-effect1} #| echo = FALSE knitr::include_graphics("figures/mra-effect1.png") ``` Doing the same for property crime, we get: ```{r mra-effect2-code} #| eval=FALSE plot(predictorEffects(crime.mod2, ~ Region + Literacy + Infants + Suicides), lwd=2, main="") ``` ```{r mra-effect2} #| echo = FALSE knitr::include_graphics("figures/mra-effect2.png") ``` ## Predicting crime: Multivariate regression The two regression models can be fit together in a multivariate regression for both crime variables jointly. ```{r manova1} crime.mod <- lm(cbind(Crime_pers, Crime_prop) ~ Region + Literacy + Donations + Infants + Suicides, data=Guerry) Anova(crime.mod) ``` As a quick check on the assumption that the residuals are bivariate normally distributed and a check for outliers, a $\chi^2$ Q-Q plot graphs the squared Mahalanobis distances of the residuals against the corresponding $\chi^2_2$ quantiles these would have in a bivariate normal distribution. The data for Creuse stands out as a potential outlier. ```{r cqplot} #| echo=-1 par(mar=c(3,3,1,1)+.1) labels <- paste0(Guerry$dept,":", Guerry$Department) cqplot(crime.mod, id.n=4, labels=labels) ``` ### HE plots Hypothesis-Error (HE) plots [@Friendly:2007:heplots; @FoxFriendlyMonette:09:compstat; @heplots] provide a convenient graphical summary of hypothesis tests in multivariate linear model. They plot a data ellipse for the residuals in the model, representing the $\mathbf{E}$ matrix in the test statistics (Roy's maximum root test, Pillai and Hotelling trace criteria and Wilks' Lambda). Overlaid on this are $\mathbf{H}$ ellipses for each term in the model, representing the data ellipses for the fitted values. Using Roy's test, these have a convenient interpretation: a term is significant _iff_ the H ellipse projects anywhere outside the E ellipse. For a 1 df (regression) variable, the H ellipse collapses to a line. ```{r heplot} #| echo=-1 par(mar = c(3,3,1,1)+.1) heplot(crime.mod, fill=TRUE, fill.alpha=0.05, cex=1.4, cex.lab=1.3 ) ``` In this plot, the effect of `Suicides` is completely aligned with crimes against property. The effect of `Region` is positively correlated with both types of crime. The means for the regions show that the South of France is lower (worse) on personal crime; the other regions vary most in property crime, with the North being lower and the Center being higher. ### Canonical plots The HE plot displays these relations in **data space**. An alternative is provided by canonical discriminant analysis, which finds the weighted sums of the response variables leading to the largest test statistics for the terms, which can be visualized in **canonical space**. The analysis below reflects the effect of `Region` in relation to both crime variables. ```{r candisc} crime.can <- candisc(crime.mod) crime.can ``` The HE plot for this analysis is shown below. Variable vector represent the correlations of the crime variables with the canonical dimension. ```{r hecan} #| echo=-1 par(mar = c(3,3,1,1)+.1) heplot(crime.can, fill=TRUE, fill.alpha=0.1, var.col = "black", var.cex = 1.3, cex=1.4, cex.lab=1.3) ``` This gives a simple interpretation of the differences in `Region` on the crime variables. The first canonical dimension accounts for 83% of differences among the regions, and this is nearly perfectly aligned with personal crime, with the largest difference between the South and the other regions. The second canonical dimension, accounting for the remaining 17%, is perfectly aligned with property crime. On this dimension, the North stands out compared to the other regions. ```{r write-bib, echo = FALSE} # write a packages.bib file of the packages (.packages()) that have been used here pkgs <- unique(c(to.cite, .packages())) knitr::write_bib(pkgs, file = here::here("vignettes", "packages.bib")) ``` # References Guerry/vignettes/figures/0000755000176200001440000000000014425210321015175 5ustar liggesusersGuerry/vignettes/figures/mra-effect1.png0000644000176200001440000002567314425210321020012 0ustar liggesusersPNG  IHDRRS/PLTE:f&:::f:ff::::f::::::::ff:f:fffff:::ff۶ffېې:fېT pHYsod IDATx c:zl;xMl;:q^MO'>f| ^#@H ()P R@)HR J@ ()P R@)HR J@ ()P R@)HR J@ ()P R@)HR J@ ()P R@)HRO-nF|e,ko϶Z͈;tDx{-nFߡ ۳mw3"ޞm˵w%l[݌ȿC/AgrfD: "aC`گYxjGk/vuNT*5ڻ xFoE(U j~yx_y0h5>~GQT)(j'yRzf/rg ooU,TP*;jUj^G}X.G jP>( (5c*O,~fq Pj+ժRʘ7HuJ͋rvop(ѕxz֯گR0Gٶ9uunN)UTΥ!: JݚWaP*R/Թ#Xkn0Wif+kip IJM)q$pn9FD~+YHT׌k>pn@)MQ!ΥE_*7?絺 I'pj<7({o-ʧ4v-[ ~RP!P܉NNVh|U+۷mvNMOډT;.`͘x+ C}\ݾa8 VJEnM%|NP~hϋY3h0|_~;:etGQj9(O Y3hɋ˗zno)WxN&=ʗze֌eEƋRpԷXJ~#YWnA9{KXJݹYW;byb3";q?Q jG[PNJYRQ~PJt#ZWnA9)ew{ l*)x,N]XnA5(A)߂ɔrk{XA@)j RR*jGWnAA)⦟(z#.2!uneQ ڑ{7[P9ZkK)ʦ1 ڑy/`[PRaDS)^RW;23r{TaT)#λNv[PzNEq87RPw[+2ye֌eğ5J9x:R R7nmȤ.'{{>Yx_=q=Ô#Vx7{bz_^)k`=K!AK!dPoI8R}CIXoTu'9QZJo.=~S~;HG&Vŭfg8R݉Dv$c ϪWjKqRR:V}Y)h"aV bJRUrz9:#Ց}n6YHϻfJ\DJoS^0 [bS@b]=K7+8+36 jTsJ b0y_!3$5DZ`Rkhާ=~AIBzۤqa(Ţj^Ւ#ɲ(F*>z$&wetEEY-~]<{V`eWZ9,ps$B^~E_{ K˄m߄Y_Fb:Xv٫XC3{k%BgqRzuߪQ11YRehnTDxgl=ڕVvl B%*{4:gu gM[{:zAӱYHR J@ ()P R@)HR J@ ()P RZ)9{dj50L[&ӡ|p/*9apIf-s<᧞;Gr,̷/dә&5m5(%3s}Ĵ]mBI<虋ez.%S35鹓S2AVz#g򧚫R3u?o_+5f%_Ie4übS`\sz/jVN)$/^11Nd8~9rpE4 ʮvT JedԐ ?ɑILi)WJ ;N;R*#cq>TMA^ZUE \>8wNԨQR J@ ()P R@)HR J@ ()P R@)HR J@ ()P R@)HR J@ ()P R@)HR J@ ()P R@)HR J@ ()P R@)HR J@ ()P R@)HR J@ ; oa6HOմNvrٕ#=\wR._.2rZ:eWRpUKY'|\ʑj)d/]K9uW-e+s)Gz3Y X#Bm}>Q e]ݧS*Ն#B|B$^[x8JBxz_|}8?^/ @RV3!7}%HURy{/˟ݗ?NJ&^ܽ A朔3]R|>ꓦŰIxAUJ]Rq6ӉމA,[x{ J;)GFwL\~NR*A5o33Brm^俗C'-OrYjRR.!A)%B83n?03Ր;ȒJ`'( ʤR8 J:T0+NTl|RLO*̊ K $ fʼnNP*;ђJIԤR87T0+>ALOʉ=篿sOD(NntR9H 'T7$R,)w_Np_}͆[=ԨɬjnOFsq^ wS4!'#9LJKY A<@yAE(b=A^U̹'dBSdbD. 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It illustrates some classical methods for analysis of multivariate spatial data that focus *either* on the multivariate aspect or on the spatial one, as well as some more modern methods that attempt to integrate geographical and multivariate aspects *simultaneously*. # Preliminary steps Several packages are required to run the different analyses and should be loaded. The `ade4` package for multivariate analysis is supplemented by `adegraphics` (for associated graphical methods) and `adespatial` for multivariate spatial analysis. For further information on this spatial analysis approach, see `vignette(package="adespatial", "tutorial")`. ```{r} library(Guerry) # Guerry's data library(sp) # management of spatial data library(ade4) # multivariate analysis library(adegraphics) # graphical representation library(spdep) # spatial dependency library(adespatial) # multivariate spatial analysis ``` Guerry gathered data on crimes, suicide, literacy and other moral statistics for various départements (i.e., counties) in France. He provided the first real social data analysis, using graphics and maps to summarize this georeferenced multivariate dataset. We use the dataset `gfrance85` and consider six key quantitative variables (shown in the table below) for each of the 85 départements of France in 1830 (Corsica, an island and often an outlier, was excluded). Data are recorded on aligned scales so that **larger numbers** consistently reflect "morally better". Thus, four of the "bad" variables are recorded in the inverse form, as "Population per ...". With this scaling, it would be expected that all correlations be $\ge 0$. | Name | Description | |:------------|:------------| |`Crime_pers` | Population per crime against persons| |`Crime_prop` | Population per crime against property| |`Literacy` | Percent of military conscripts who can read and write| |`Donations` | Donations to the poor| |`Infants` | Population per illegitimate birth| |`Suicides` | Population per suicide| The dataset `gfrance85` is actually a `SpatialPolygonsDataFrame` object created with the `sp` package. It contains the polygon boundaries of the map of France in 1830, as well the variables in the `Guerry` data frame. As an ordinary data.frame, it has these components ```{r names} names(gfrance85) ``` To simplify analyses, we extract the components to be used below. ```{r components} data(gfrance85) df <- data.frame(gfrance85)[, 7:12] # the 6 variables france.map <- as(gfrance85, "SpatialPolygons") # the map xy <- coordinates(gfrance85) # spatial coordinates dep.names <- data.frame(gfrance85)[, 6] # departement names region.names <- data.frame(gfrance85)[, 5] # region names col.region <- colors()[c(149, 254, 468, 552, 26)] # colors for region ``` # Standard approaches In this section, we focus on classical approaches that consider either the multivariate or the spatial aspect of the data. ## Multivariate analysis Here we consider $p=6$ variables measured for $n=85$ individuals (départements of France). As only quantitative variables have been recorded, principal component analysis [PCA, @SD308] is well adapted. PCA summarizes the data by maximizing simultaneously the variance of the projection of the individuals onto the principal axes and the sum of the squared correlations between the principal component and the variables. ```{r dudi-pca} pca <- dudi.pca(df, scannf = FALSE, nf = 3) ``` The biplot is simply obtained by ```{r} biplot(pca, plabel.cex = 0.8) ``` The first two PCA dimensions account for 35.7% and 20% ,respectively, of the total variance. ```{r} pca$eig/sum(pca$eig) * 100 ``` Correlations between variables and principal components can be represented on a correlation circle. The first axis is negatively correlated to literacy and positively correlated to property crime, suicides and illegitimate births. The second axis is aligned mainly with personal crime and donations to the poor. ```{r} s.corcircle(pca$co) ``` Spatial information can be added on the factorial map representing the projections of départements on principal axes by coloring according to colors representing the different regions of France. ```{r, fig.width = 4, fig.height = 4} s.label(pca$li, ppoint.col = col.region[region.names], plabel.optim = TRUE, plabel.cex = 0.6) s.Spatial(france.map, col = col.region[region.names], plabel.cex = 0) s.class(xy, region.names, col = col.region, add = TRUE, ellipseSize = 0, starSize = 0) ``` For the first axis, the North and East are characterized by negative scores, corresponding to high levels of literacy and high rates of suicides, crimes against property and illegitimate births. The second axis mainly contrasts the West (high donations to the the poor and low levels of crime against persons) to the South. ## Spatial autocorrelation Spatial autocorrelation statistics, such as @SD455 Coefficient (MC) and @SD223 Ratio, aim to measure and analyze the degree of dependency among observations in a geographical context [@SD577]. ### The spatial weighting matrix The first step of spatial autocorrelation analysis is to define a spatial weighting matrix $\mathbf{W}=[w_{ij}]$ . In the case of Guerry's data, we simply defined a binary neighborhood where two départements are considered as neighbors if they share a common border. The spatial weighting matrix is then obtained after row-standardization (`style = "W"`): ```{r} nb <- poly2nb(gfrance85) lw <- nb2listw(nb, style = "W") ``` We can represent this neighborhood on the geographical map: ```{r} s.Spatial(france.map, nb = nb, plabel.cex = 0, pSp.border = "white") ``` ### Moran's Coefficient Once the spatial weights have been defined, the spatial autocorrelation statistics can then be computed. Let us consider the $n$-by-1 vector $\mathbf{x}=\left[ {x_1 \cdots x_n } \right]^\textrm{T}$ containing measurements of a quantitative variable for $n$ spatial units. The usual formulation for Moran's coefficient of spatial autocorrelation [@SD577;@SD455] is \begin{equation} \label{eq1} MC(\mathbf{x})=\frac{n\sum\nolimits_{\left( 2 \right)} {w_{ij} (x_i -\bar {x})(x_j -\bar {x})} }{\sum\nolimits_{\left( 2 \right)} {w_{ij} } \sum\nolimits_{i=1}^n {(x_i -\bar {x})^2} }\mbox{ where }\sum\nolimits_{\left( 2 \right)} =\sum\limits_{i=1}^n {\sum\limits_{j=1}^n } \mbox{ with }i\ne j. \end{equation} MC can be rewritten using matrix notation: \begin{equation} \label{eq2} MC(\mathbf{x})=\frac{n}{\mathbf{1}^\textrm{T}\mathbf{W1}}\frac{\mathbf{z}^\textrm{T}{\mathbf{Wz}}}{\mathbf{z}^\textrm{T}\mathbf{z}}, \end{equation} where $\mathbf{z}=\left ( \mathbf{I}_n-\mathbf{1}_n\mathbf{1}_n^\textrm{T} /n \right )\mathbf{x}$ is the vector of centered values ($z_i=x_i-\bar{x}$) and $\mathbf{1}_n$ is a vector of ones (of length $n$). The significance of the observed value of MC can be tested by a Monte-Carlo procedure, in which locations are permuted to obtain a distribution of MC under the null hypothesis of random distribution. An observed value of MC that is greater than that expected at random indicates the clustering of similar values across space (positive spatial autocorrelation), while a significant negative value of MC indicates that neighboring values are more dissimilar than expected by chance (negative spatial autocorrelation). We computed MC for the Guerry's dataset. A positive and significant autocorrelation is identified for each of the six variables. Thus, the values of literacy are the most covariant in adjacent departments, while illegitimate births (Infants) covary least. ```{r} moran.randtest(df, lw) ``` ### Moran scatterplot If the spatial weighting matrix is row-standardized, we can define the lag vector $\mathbf{\tilde{z}} = \mathbf{Wz}$ (i.e., $\tilde{z}_i = \sum\limits_{j=1}^n{w_{ij}x_j}$) composed of the weighted (by the spatial weighting matrix) averages of the neighboring values. Thus, we have: \begin{equation} \label{eq3} MC(\mathbf{x})=\frac{\mathbf{z}^\textrm{T}{\mathbf{\tilde{z}}}}{\mathbf{z}^\textrm{T}\mathbf{z}}, \end{equation} since in this case $\mathbf{1}^\textrm{T}\mathbf{W1}=n$. This shows clearly that MC measures the autocorrelation by giving an indication of the intensity of the linear association between the vector of observed values $\mathbf{z}$ and the vector of weighted averages of neighboring values $\mathbf{\tilde{z}}$. @SD566 proposed to visualize MC in the form of a bivariate scatterplot of $\mathbf{\tilde{z}}$ against $\mathbf{z}$. A linear regression can be added to this *Moran scatterplot*, with slope equal to MC. Considering the Literacy variable of Guerry's data, the Moran scatterplot clearly shows strong autocorrelation. It also shows that the Hautes-Alpes département has a slightly outlying position characterized by a high value of Literacy compared to its neighbors. ```{r, fig.width = 4, fig.height=4} x <- df[, 3] x.lag <- lag.listw(lw, df[, 3]) moran.plot(x, lw) text(x[5], x.lag[5], dep.names[5], pos = 1, cex = 0.8) ``` ## Indirect integration of multivariate and geographical aspects The simplest approach considered a two-step procedure where the data are first summarized with multivariate analysis such as PCA. In a second step, univariate spatial statistics or mapping techniques are applied to PCA scores for each axis separately. One can also test for the presence of spatial autocorrelation for the first few scores of the analysis, with univariate autocorrelation statistics such as MC. We mapped scores of the départements for the first two axes of the PCA of Guerry's data. Even if PCA maximizes only the variance of these scores, there is also a clear spatial structure, as the scores are highly autocorrelated. The map for the first axis corresponds closely to the split between *la France éclairée* (North-East characterized by an higher level of Literacy) and *la France obscure*. ```{r, fig.dim = c(6,3)} moran.randtest(pca$li, lw) s.value(xy, pca$li[, 1:2], Sp = france.map, pSp.border = "white", symbol = "circle", pgrid.draw = FALSE) ``` # Spatial multivariate analysis Over the last decades, several approaches have been developed to consider both geographical and multivariate information simultaneously. The multivariate aspect is usually treated by techniques of dimensionality reduction similar to PCA. On the other hand, several alternatives have been proposed to integrate the spatial information. ## Spatial partition One alternative is to consider a spatial partition of the study area. In this case, the spatial information is coded as a categorical variable, and each category corresponds to a region of the whole study area. For instance, Guerry's data contained a partition of France into 5 regions. We used the between-class analysis [BCA, @SD148], to investigate differences between regions. BCA maximizes the variance between groups. ```{r} bet <- bca(pca, region.names, scannf = FALSE, nf = 2) ``` Here, 28.8 % of the total variance (sum of eigenvalues of PCA) corresponds to the between-regions variance (sum of the eigenvalues of BCA). ```{r} bet$ratio ``` The main graphical outputs are obtained by the generic `plot` function: ```{r, fig.dim=c(5,5)} plot(bet) ``` The barplot of eigenvalues indicates that two axes should be interpreted. The first two BCA dimensions account for 59 % and 30.2 %, respectively, of the between-regions variance. ```{r} barplot(bet$eig) bet$eig/sum(bet$eig) * 100 ``` The coefficients used to construct the linear combinations of variables are represented: ```{r} s.arrow(bet$c1, plabel.cex = 0.8) ``` The first axis opposed literacy to property crime, suicides and illegitimate births. The second axis is mainly aligned with personal crime and donations to the poor. Projections of départements on the BCA axes can be represented on the factorial map: ```{r, fig.dim = c(4,4)} s.label(bet$ls, as.character(dep.names), ppoint.cex = 0, plabel.optim = TRUE, plabel.col = col.region[region.names], plabel.cex = 0.5) s.class(bet$ls, fac = region.names, col = col.region, ellipse = 0, add = TRUE) ``` The scores can be mapped to show the spatial aspects: ```{r, fig.dim = c(6,3)} s.value(xy, bet$ls, symbol = "circle", Sp = france.map, pSp.col = col.region[region.names], pSp.border = "transparent") ``` The results are very close to those obtained by PCA: the first axis contrasted the North and the East (*la France éclairée*) to the other regions while the South is separated from the other regions by the second axis. The high variability of the region Centre is also noticeable. In contrast, the South is very homogeneous. ## Spatial explanatory variables Principal component analysis with respect to the instrumental variables [PCAIV, @SD540], and related methods, have been often used in community ecology to identify spatial relationships. The spatial information is introduced in the form of spatial predictors and the analysis maximized the "spatial variance" (i.e., the variance explained by spatial predictors). Note that BCA can also be considered as a particular case of PCAIV, where the explanatory variables are dummy variables indicating group membership. ### Trend surface of geographic coordinates @SD626 proposed to express observed values in time series as a polynomial function of time, and mentioned that this could be done for spatial data as well. @SD59 extended this approach to the spatial and multivariate case by introducing polynomial functions of geographic coordinates as predictors in PCAIV. We call this approach PCAIV-POLY. The centroids of départements of France were used to construct a second-degree orthogonal polynomial. ```{r, fig.dim = c(6,4)} poly.xy <- orthobasis.poly(xy, degree = 2) s.value(xy, poly.xy, Sp = france.map, plegend.drawKey = FALSE) ``` PCAIV is then performed using the `pcaiv` function: ```{r} pcaiv.xy <- pcaiv(pca, poly.xy, scannf = FALSE, nf = 2) ``` Here, 32.4 % of the total variance (sum of eigenvalues of PCA) is explained by the second-degree polynomial (sum of eigenvalues of PCAIV). The first two dimensions account for 51.4 % and 35.2 %, respectively, of the explained variance. ```{r} sum(pcaiv.xy$eig)/sum(pca$eig) * 100 pcaiv.xy$eig/sum(pcaiv.xy$eig) * 100 ``` The outputs of PCAIV-POLY (coefficients of variables, maps of départements scores, etc.) are very similar to those obtained by BCA. They can be represented easily by the generic `plot` function: ```{r, fig.dim=c(5,5)} plot(pcaiv.xy) ``` ### Moran's eigenvector maps An alternative way to build spatial predictors is by the diagonalization of the spatial weighting matrix **W**. Moran's eigenvector maps [MEM, @SD163] are the $n-1$ eigenvectors of the doubly-centered matrix **W**. They are orthogonal vectors with a unit norm maximizing MC [@SD264]. MEM associated with high positive (or negative) eigenvalues have high positive (or negative) autocorrelation. MEM associated with eigenvalues with small absolute values correspond to low spatial autocorrelation, and are not suitable for defining spatial structures. We used the spatial weighting matrix defined above to construct MEM. The first ten MEM, corresponding to the highest levels of spatial autocorrelation, have been mapped: ```{r, fig.dim = c(6,6)} mem1 <- scores.listw(lw) s.value(xy, mem1[, 1:9], Sp = france.map, plegend.drawKey = FALSE) ``` We introduced the first ten MEM as spatial explanatory variables in PCAIV. We call this approach PCAIV-MEM. ```{r} pcaiv.mem <- pcaiv(pca, mem1[,1:10], scannf = FALSE) ``` Here, 44.1 % of the total variance (sum of eigenvalues of PCA) is explained by the first ten MEM (sum of eigenvalues of PCAIV). The first two dimensions account for 54.9 % and 26.3 %, respectively, of the explained variance. ```{r} sum(pcaiv.mem$eig)/sum(pca$eig) * 100 pcaiv.mem$eig/sum(pcaiv.mem$eig) * 100 ``` The outputs of PCAIV-MEM (coefficients of variables, maps of départements scores, etc.) are very similar to those obtained by BCA. They can be represented easily by the generic `plot` function: ```{r, fig.dim=c(5,5)} plot(pcaiv.mem) ``` ## Spatial graph and weighting matrix The MEM framework introduced the spatial information into multivariate analysis through the eigendecomposition of the spatial weighting matrix. Usually, we consider only a part of the information contained in this matrix because only a subset of MEM are used as regressors in PCAIV. In this section, we focus on multivariate methods that consider the spatial weighting matrix under its original form. @SD694 was the first to develop a multivariate analysis based on MC. His work considered only normed and centered variables (i.e., normed PCA) for the multivariate part and a binary symmetric connectivity matrix for the spatial aspect. @SD807 generalized Wartenberg's method by introducing a row-standardized spatial weighting matrix in the analysis of a statistical triplet. This approach is very general and allows us to define spatially-constrained versions of various methods (corresponding to different triplets) such as correspondence analysis or multiple correspondence analysis. MULTISPATI finds coefficients to obtain a linear combination of variables that maximizes a compromise between the classical multivariate analysis and a generalized version of Moran's coefficient. ```{r} ms <- multispati(pca, lw, scannf = FALSE) ``` The main outputs of MULTISPATI can be represented easily by the generic `plot` function: ```{r, fig.dim=c(5,5)} plot(ms) ``` The barplot of eigenvalues suggests two main spatial structures. Eigenvalues of MULTISPATI are the product between the variance and the spatial autocorrelation of the scores, while PCA maximizes only the variance. The differences between the two methods are computed by the `summary` function: ```{r} summary(ms) ``` Hence, there is a loss of variance compared to PCA (2.14 versus 2.017 for axis 1; 1.201 versus 1.177 for axis 2) but a gain of spatial autocorrelation (0.551 versus 0.637 for axis 1; 0.561 versus 0.59 for axis 2). Coefficients of variables allow to interpret the structures: ```{r} s.arrow(ms$c1, plabel.cex = 0.8) ``` The first axis opposes literacy to property crime, suicides and illegitimate births. The second axis is aligned mainly with personal crime and donations to the poor. The maps of the scores show that the spatial structures are very close to those identified by PCA. The similarity of results between PCA and its spatially optimized version confirm that the main structures of Guerry's data are spatial. Spatial autocorrelation can be seen as the link between one variable and the lagged vector. This interpretation is used to construct the Moran scatterplot and can be extended to the multivariate case in MULTISPATI by analyzing the link between scores and lagged scores: ```{r, fig.dim = c(4,4)} s.match(ms$li, ms$ls, plabel.cex = 0) s.match(ms$li[c(10, 41, 27), ], ms$ls[c(10, 41, 27), ], label = dep.names[c(10, 41, 27)], plabel.cex = 0.8, add = TRUE) ``` Each département can be represented on the factorial map by an arrow (the bottom corresponds to its score, the head corresponds to its lagged score. A short arrow reveals a local spatial similarity (between one plot and its neighbors) while a long arrow reveals a spatial discrepancy. This viewpoint can be interpreted as a multivariate extension of the local index of spatial association [@SD565]. For instance: * Aude has a very small arrow, indicating that this département is very similar to its neighbors. * Haute-Loire has a long horizontal arrow which reflects its high values for the variables Infants (31017), Suicides (163241) and Crime\_prop (18043) compared to the average values over its neighbors (27032.4, 60097.8 and 10540.8 for these three variables). * Finistère corresponds to an arrow with a long vertical length which is due to its high values compared to its neighbors for Donations (23945 versus 12563) and Crime\_pers (29872 versus 25962). The link between the scores and the lagged scores (averages of neighbors weighted by the spatial connection matrix) can be mapped in the geographical space. For the first two axes, we have: ```{r, fig.dim = c(6,3)} s.value(xy, ms$li, Sp = france.map) ``` # Conclusions Even if the methods presented are quite different in their theoretical and practical viewpoints, their applications to Guerry's dataset yield very similar results. We provided a quantitative measure of this similarity by computing Procrustes statistics [@SD516;SD161] between the scores of the départements onto the first two axes for the different analyses. All the values of the statistic are very high and significant; this confirms the high concordance between the outputs of the different methods. ```{r} mat <- matrix(NA, 4, 4) mat.names <- c("PCA", "BCA", "PCAIV-POLY", "PCAIV-MEM", "MULTISPATI") colnames(mat) <- mat.names[-5] rownames(mat) <- mat.names[-1] mat[1, 1] <- procuste.randtest(pca$li[, 1:2], bet$ls[, 1:2])$obs mat[2, 1] <- procuste.randtest(pca$li[, 1:2], pcaiv.xy$ls[, 1:2])$obs mat[3, 1] <- procuste.randtest(pca$li[, 1:2], pcaiv.mem$ls[, 1:2])$obs mat[4, 1] <- procuste.randtest(pca$li[, 1:2], ms$li[, 1:2])$obs mat[2, 2] <- procuste.randtest(bet$ls[, 1:2], pcaiv.xy$ls[, 1:2])$obs mat[3, 2] <- procuste.randtest(bet$ls[, 1:2], pcaiv.mem$ls[, 1:2])$obs mat[4, 2] <- procuste.randtest(bet$ls[, 1:2], ms$li[, 1:2])$obs mat[3, 3] <- procuste.randtest(pcaiv.xy$ls[, 1:2], pcaiv.mem$ls[, 1:2])$obs mat[4, 3] <- procuste.randtest(pcaiv.xy$ls[, 1:2], ms$li[, 1:2])$obs mat[4, 4] <- procuste.randtest(pcaiv.mem$ls[, 1:2], ms$li[, 1:2])$obs mat ``` # References Guerry/NEWS.md0000644000176200001440000000550614513000475012632 0ustar liggesusers## Version 1.8.3 (2023-10-13) A problem with the adegraphics package becoming archived triggered a warning on Guerry threatening the same fate. I did not reply in time, and so Guerry was archived. This re-submission fixes that and other issues found by win-builder. * Moved some examples to `demo/` * Edit example in `gfrance85.Rd` * Restore use of `adegraphics` for vignette `MultiSpat.Rmd` * The `demo/guerry_ggradar.R` used a non-CRAN package; this package has been removed from Suggests: and the example moved to `examples/`. ## Version 1.8.2 (2023-05-10) * Incorporate parallel coord plots and radar charts in vignette ## Version 1.8.1 (2023-05-05) * This is largely a maintenance release to correct a problem with latin1 encoding used in the DESCRIPTION and in man/ files. All non-ascii characters have been removed. * Add reference to Friendly (2022) * Moved a number of examples/ to demo/ ## Version 1.8.0 (2022-10-04) * Fixed problems related to retirement of `maptools` #4, deleting Suggests: maptools * Added a new vignette, `guerry-multivariate`, illustrating some multivariate methods * Extensively revised README to give a better sense of the package datasets and some plots. ## Version 1.7.4 * Restore MultiSpat vignette ## Version 1.7.3 * begin guerry-multivariate vignette * Fixed problem arising from new spdep when rgeos is not installed [thx: Roger Bivand] ## Version 1.7.1 * translate/update/restore the main vignette, `MultiSpat.html` by Stephane Drey ## Version 1.7.0 * Update links to Guerry supplementary stuff to resolve CRAN nits * move `sp` to `Imports:` to resolve CRAN nits * Removed Suggests: shapefiles * added a hexsticker to README.md ## Version 1.6-2 * Package development has been moved from R-forge to github, https://github.com/friendly/Guerry * Updated Guerry-package.Rd from `utils::promptPackage()` ## Version 1.6 (2014-09-23) * Removed MultiSpat vignette because it is no longer compatible with CRAN policies. The old PDF version will be made available on the R-Forge project page. * Removed Suggests: spacemakeR (not on CRAN); moved sp, shapefiles to Suggests: * Removed thinSpatialPoly, as this is now provided in maptools * Made some examples in gfrance.Rd \dontrun{} to reduce check time ## Version 1.5 (2011-11-08) * Added back MultiSpat.Rnw vignette with MultiSpat.tex disguised as MultiSpat.Rnw * Added NAMESPACE for R 2.14+ ## Version 1.4 (2010-02-15) * Remove temporarily MultiSpat.Rnw vignette ## Version 1.3 (2009-11-19) * Added thinnedSpatialPoly to calculate thinned maps * Added MultiSpat.Rnw vignette ## Version 1.2 (2009-11-12) * Added MultiSpat.pdf vignette (without .Rnw) ## Version 1.1 (2009-10-28) * Added Angeville data * Released to CRAN ## Version 1.0 (2009-10-20) * Initial version uploaded to R-Forge Guerry/MD50000644000176200001440000000524514516023372012050 0ustar liggesusersac41a1b61819b0990c771a9cd012c0a1 *DESCRIPTION 195963848221ae8485a2687d34977ed2 *NAMESPACE af3969ad56f304eea5496f7825bc8306 *NEWS.md 96b987a34603fd84c6812422456bf2a0 *build/partial.rdb 7a6c0748d63c87eabf11107a662d7331 *build/vignette.rds 4d3e3b1d1307b61102745af6c164195f *data/Angeville.RData 5c12af3e1e4d205ea46f96e4d0ce9813 *data/Guerry.RData 91bd93041f2bf418dd0424591dc4b5d9 *data/gfrance.RData f45237956849addb495f83a6cded42d6 *data/gfrance85.RData 38c0b190d2e04426f583f730508fa0c4 *data/propensity.RData 9d335b5da2c9d08bef69c1c0843b386e *demo/00Index 5efbccd5b4938134fe3bb71ad7e2b0e8 *demo/guerry_bivar.R 2325b820028904882212d8c181ccb746 *demo/guerry_density.R e9767bee7b6a9aecc81586b67c5916aa *demo/guerry_manova.R 97cc3976d97e9498554ed6276f558568 *demo/guerry_mra.R c2205213c9cb481eecca62e249e495e7 *inst/WORDLIST 53d06e0f28e9a4b15c6b705e5678162d *inst/doc/MultiSpat.R 236b43e2bd31825fdb9e47aeacc75cf9 *inst/doc/MultiSpat.Rmd b7be1409437a2e0583a996f652cf1b43 *inst/doc/MultiSpat.html 4232086974d295b93f9fb696a54fe72e *inst/doc/guerry-multivariate.R 45b35a13dec1249da4721432100db57f *inst/doc/guerry-multivariate.Rmd fdbd38bde258773d437a88675b788540 *inst/doc/guerry-multivariate.html 784aec924058f686ae71c7ee30f311b9 *man/Angeville.Rd d523f6c5eb0e61dfd0023f47cba886e2 *man/Guerry-package.Rd c52ff62c1bcf54b8d5926fe760530350 *man/Guerry.Rd 67c34055c5d8715634bd379e3b546c02 *man/figures/Guerry-logo.png 95147e3a56e181a7593ed2e9738f4c4f *man/figures/Guerry-vars.png 4d33f110f118c23e3fae7a5de6582afa *man/figures/README-ex-bivar1-1.png 8e5f86feea111438c100c645f661921d *man/figures/README-ex-bivar2-1.png 312a3bfbcfd21f20cd93f4993f8e9254 *man/figures/README-gfrance1-1.png caf191eef097cbf06db511508892f5f8 *man/figures/README-gfrance2-1.png dad456b45718ae97d2669064b7839af8 *man/figures/README-gfrance3-1.png 4e2eb16daddc9b15aa19d19581b932a3 *man/figures/README-gfrance4-1.png 6b9d1da64fd1c7641fc61b7fe2f9a1dd *man/figures/README-gfrance85-labels-1.png bcb56ad207d38925fd1efb67c58970ad *man/figures/ex-bivar1.png 41344262e0fac14ad1926648b58b2829 *man/figures/ex-bivar2.png 825a20bf7dd3d9eca569bb5c01b326bc *man/gfrance.Rd ab3c1d250c29f06e5b92a943fd112547 *man/gfrance85.Rd 3fa0344e30ec297cd23840f60107722e *man/partials/_ggradar.Rmd 7af59b0b943707b57d500cb05a03058b *man/propensity.Rd 236b43e2bd31825fdb9e47aeacc75cf9 *vignettes/MultiSpat.Rmd b8a19ad70279d557599217caead45ff4 *vignettes/figures/ggbiplot.png b4b6dd98e513ae94ccae98785799e2ee *vignettes/figures/mra-effect1.png b9cb4eda65e56c5f125c873cc346e781 *vignettes/figures/mra-effect2.png 45b35a13dec1249da4721432100db57f *vignettes/guerry-multivariate.Rmd 8136ef405a97754b841982699313c585 *vignettes/reference.bib b7abedc70441849d1c51c243ff902149 *vignettes/refs.bib Guerry/inst/0000755000176200001440000000000014515773424012520 5ustar liggesusersGuerry/inst/doc/0000755000176200001440000000000014515773424013265 5ustar liggesusersGuerry/inst/doc/guerry-multivariate.R0000644000176200001440000002054514515773423017436 0ustar liggesusers## ----setup, include = FALSE--------------------------------------------------- knitr::opts_chunk$set( collapse = TRUE, warning = FALSE, message = FALSE, # suppress package loading messages comment = "#>", fig.height = 5, fig.width = 5, dpi = 96 ) # packages to be cited here. Code at the end automatically updates packages.bib to.cite <- c("car", "effects", "ggplot2", "GGally", "ggbiplot") ## ----load--------------------------------------------------------------------- library(Guerry) # Guerry data library(car) # better scatterplots library(effects) # Effect Displays for Linear Models library(ggplot2) # Elegant Data Visualisations Using the Grammar of Graphics library(ggrepel) # better handling of text labels library(patchwork) # combine plots library(heplots) # Hypothesis-Error plots library(candisc) # Visualizing Generalized Canonical Discriminant Analysis library(dplyr) # A Grammar of Data Manipulation library(tidyr) # Tidy Messy Data data(Guerry) ## ----guerry-more-vars--------------------------------------------------------- names(Guerry)[-(1:9)] ## ----guerry-long-------------------------------------------------------------- data("Guerry", package="Guerry") guerry_long <- Guerry |> filter(!is.na(Region)) |> select(dept:Suicides) |> pivot_longer(cols = Crime_pers:Suicides, names_to = "variable", values_to = "value") guerry_long ## ----guerry-density1---------------------------------------------------------- ggplot(data = guerry_long, aes(x=value, fill=TRUE)) + geom_density(alpha=0.2) + geom_rug() + facet_wrap(~variable, scales="free") + theme_bw(base_size = 14) + theme(legend.position = "none", axis.ticks.y=element_blank(), axis.text.y=element_blank()) ## ----guerry-density2---------------------------------------------------------- col.region <- colors()[c(149, 254, 468, 552, 26)] # colors for region ggplot(data = guerry_long, aes(x=value, fill=Region)) + geom_density(alpha=0.2) + geom_rug() + facet_wrap(~variable, scales="free") + scale_fill_manual(values=col.region) + theme_bw(base_size = 14) + theme(legend.position = "bottom", axis.ticks.y=element_blank(), axis.text.y=element_blank()) ## ----lit-pers-scat0----------------------------------------------------------- ggplot(aes(x=Literacy, y=Crime_pers/1000), data=Guerry) + geom_point(size=2) ## ----lit-pers-scat------------------------------------------------------------ gdf <- Guerry[, c("Literacy", "Crime_pers", "Department")] gdf$dsq <- mahalanobis(gdf[,1:2], colMeans(gdf[,1:2]), cov(gdf[,1:2])) ggplot(aes(x=Literacy, y=Crime_pers/1000, label=Department), data=gdf) + geom_point(size=2) + stat_ellipse(level=0.68, color="blue", size=1.2) + stat_ellipse(level=0.95, color="gray", size=1, linetype=2) + geom_smooth(method="lm", formula=y~x, fill="lightblue") + geom_smooth(method="loess", formula=y~x, color="red", se=FALSE) + geom_label_repel(data = gdf[gdf$dsq > 4.6,]) + theme_bw() ## ----lit-prop-scat------------------------------------------------------------ gdf <- Guerry[, c("Literacy", "Crime_prop", "Department")] gdf$dsq <- mahalanobis(gdf[,1:2], colMeans(gdf[,1:2]), cov(gdf[,1:2])) ggplot(aes(x=Literacy, y=Crime_prop/1000, label=Department), data=gdf) + geom_point(size=2) + stat_ellipse(level=0.68, color="blue", size=1.2) + stat_ellipse(level=0.95, color="gray", size=1, linetype=2) + geom_smooth(method="lm", formula=y~x, fill="lightblue") + geom_smooth(method="loess", formula=y~x, color="red", se=FALSE) + geom_label_repel(data = gdf[gdf$dsq > 4.6,]) + theme_bw() ## ----spm1--------------------------------------------------------------------- par(mar=rep(2,4)) library(car) # Companion to Applied Regression scatterplotMatrix(Guerry[,4:9], ellipse=list(levels=0.68), smooth=FALSE) ## ----corrgram1---------------------------------------------------------------- par(mar=rep(1,4)+.1) library(corrgram) # Plot a Correlogram corrgram(Guerry[,4:9], upper=panel.pie) ## ----corrgram2---------------------------------------------------------------- par(mar=rep(1,4)+.1) corrgram(Guerry[,4:9], upper=panel.ellipse, order=TRUE, lwd=2) ## ----guerry.pca--------------------------------------------------------------- gdata <- Guerry |> select(Region, Crime_pers:Suicides) |> # keep only main variables filter(!is.na(Region)) # delete Corsica (Region==NA) guerry.pca <- gdata |> select(-Region) |> prcomp(scale = TRUE) print(guerry.pca, digits=3) ## ----biplot1------------------------------------------------------------------ # if(!require(ggbiplot)) remotes::install_github("vqv/ggbiplot") # library(ggbiplot) # A ggplot2 based biplot # ggbiplot(guerry.pca, groups=gdata$Region, # ellipse=TRUE, # var.scale = 3, varname.size = 5) + # theme_bw() + # labs(color="Region") + # theme(legend.position = c(0.1, 0.8)) ## ----ggbiplot----------------------------------------------------------------- knitr::include_graphics("figures/ggbiplot.png") ## ----biplot2, out.width="95%"------------------------------------------------- op <- par(mar=c(5,4,1,1)+.1) cols = colorspace::rainbow_hcl(5) covEllipses(guerry.pca$x, group=gdata$Region, pooled=FALSE, fill=TRUE, fill.alpha=0.1, col=cols, label.pos=c(3,0,1,1,3), cex=2, xlim=c(-4,4), ylim=c(-4,4), xlab = "Dimension 1 (35.7 %)", ylab = "Dimension 2 (20.0 %)", cex.lab=1.4 ) points(guerry.pca$x, pch=(15:19)[Guerry$Region], col=cols[Guerry$Region]) candisc::vectors(guerry.pca$rotation, scale=5, col="black", lwd=3, cex=1.4, pos = c(4,2,4,2,2,2), xpd=TRUE) abline(h=0, v=0, col=gray(.70)) ## ----mra-models--------------------------------------------------------------- crime.mod1 <- lm(Crime_pers ~ Region + Literacy + Donations + Infants + Suicides, data=Guerry) crime.mod2 <- lm(Crime_prop ~ Region + Literacy + Donations + Infants + Suicides, data=Guerry) ## ----mra-anova---------------------------------------------------------------- Anova(crime.mod1) Anova(crime.mod2) ## ----mra-effect1-code--------------------------------------------------------- # plot(predictorEffects(crime.mod1, ~ Region + Literacy + Infants + Suicides), # lwd=2, main="") ## ----mra-effect1-------------------------------------------------------------- knitr::include_graphics("figures/mra-effect1.png") ## ----mra-effect2-code--------------------------------------------------------- # plot(predictorEffects(crime.mod2, ~ Region + Literacy + Infants + Suicides), # lwd=2, main="") ## ----mra-effect2-------------------------------------------------------------- knitr::include_graphics("figures/mra-effect2.png") ## ----manova1------------------------------------------------------------------ crime.mod <- lm(cbind(Crime_pers, Crime_prop) ~ Region + Literacy + Donations + Infants + Suicides, data=Guerry) Anova(crime.mod) ## ----cqplot------------------------------------------------------------------- par(mar=c(3,3,1,1)+.1) labels <- paste0(Guerry$dept,":", Guerry$Department) cqplot(crime.mod, id.n=4, labels=labels) ## ----heplot------------------------------------------------------------------- par(mar = c(3,3,1,1)+.1) heplot(crime.mod, fill=TRUE, fill.alpha=0.05, cex=1.4, cex.lab=1.3 ) ## ----candisc------------------------------------------------------------------ crime.can <- candisc(crime.mod) crime.can ## ----hecan-------------------------------------------------------------------- par(mar = c(3,3,1,1)+.1) heplot(crime.can, fill=TRUE, fill.alpha=0.1, var.col = "black", var.cex = 1.3, cex=1.4, cex.lab=1.3) ## ----write-bib, echo = FALSE-------------------------------------------------- # write a packages.bib file of the packages (.packages()) that have been used here pkgs <- unique(c(to.cite, .packages())) knitr::write_bib(pkgs, file = here::here("vignettes", "packages.bib")) Guerry/inst/doc/guerry-multivariate.Rmd0000644000176200001440000005455514513000475017753 0ustar liggesusers--- title: "Guerry data: Multivariate Analysis" author: "Michael Friendly" date: "`r Sys.Date()`" package: Guerry output: rmarkdown::html_vignette: toc: true toc_depth: 2 bibliography: ["refs.bib", "packages.bib"] link-citations: yes vignette: > %\VignetteIndexEntry{Guerry data: Multivariate Analysis} %\VignetteKeywords{crime, literacy, suicide, France, multivariate analysis} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r setup, include = FALSE} knitr::opts_chunk$set( collapse = TRUE, warning = FALSE, message = FALSE, # suppress package loading messages comment = "#>", fig.height = 5, fig.width = 5, dpi = 96 ) # packages to be cited here. Code at the end automatically updates packages.bib to.cite <- c("car", "effects", "ggplot2", "GGally", "ggbiplot") ``` André-Michel Guerry's _Essai sur la Statistique Morale de la France_ [@Guerry:1833] collected data on crimes, suicide, literacy and other "moral statistics" for various départements in France. He provided the first real social data analysis, using graphics and maps to summarize this multivariate dataset. One of his main goals in this ground-breaking study was to determine if the prevalence of crime in France could be explained by other social variables. In 1833, the scatterplot had not yet been invented; the idea of a correlation or a regression was still 50 years in the future [@Galton:1886]. Guerry displayed his data in shaded choropleth maps and semi-graphic tables and argued how these could be seen as implying systematic, lawful relations among moral variables. In this analysis, we ignore the spatial context of the départements and focus on multivariate analyses of the the data set. # Load data and packages We will primarily use the following packages, so load them now. ```{r load} library(Guerry) # Guerry data library(car) # better scatterplots library(effects) # Effect Displays for Linear Models library(ggplot2) # Elegant Data Visualisations Using the Grammar of Graphics library(ggrepel) # better handling of text labels library(patchwork) # combine plots library(heplots) # Hypothesis-Error plots library(candisc) # Visualizing Generalized Canonical Discriminant Analysis library(dplyr) # A Grammar of Data Manipulation library(tidyr) # Tidy Messy Data data(Guerry) ``` ## `Guerry` data set Guerry's (1833) data consisted of six main moral variables shown in the table below. He wanted all of these to be recorded on aligned scales so that **larger** numbers consistently reflected "**morally better**". Thus, four of the variables are recorded in the inverse form, as "Population per ...". | Name | Description | |:------------|:-----------------------------------------------------| |`Crime_pers` | Population per crime against persons | |`Crime_prop` | Population per crime against property | |`Literacy` | Percent of military conscripts who can read and write| |`Donations` | Donations to the poor | |`Infants` | Population per illegitimate birth | |`Suicides` | Population per suicide | The `Guerry` data set also contains: * `dept` and `Department`, the French ID numbers and names for the 86 départements of metropolitan France in 1830, including Corsica. * `Region`: a factor with main levels "N", "S", "E", "W", "C". Corsica is coded as `NA`. * A collection of 14 other related variables from other sources at the same time. See `?Guerry` for their precise definitions. ```{r guerry-more-vars} names(Guerry)[-(1:9)] ``` Among these, as other aspects of criminal behavior, we see crime against parents, `Infanticide` and `Prostitutes`. `Clergy` and `Donations_clergy` are considered to be measures of moral rectitude, potentially counteracting crime. ## Guerry's questions The main questions that concerned Guerry were whether indicators of crime could be shown to be related to factors which might be considered to ameliorate crime. Among these, Guerry focused most on `Literacy` defined as the number of military conscripts who could do more than mark an "X" on their enrollment form. A related variable is `Instruction`, the rank recorded from Guerry's map; as defined, it is inversely related to `Literacy`. Other potential explanatory variables are: : `Donations` (a measure of donations to the poor), : `Donation_clergy` (a measure of donations to clergy) : `Clergy` (the rank of number of Catholic priests in active service, per population) # Multivariate visualization methods Visualization methods for multivariate data take an enormous variety of forms simply because more than two dimensions of data offer exponentially increasingly possibilities. It is useful to distinguish several broad categories: * __data plots__ : primarily plot the raw data, often with annotations to aid interpretation (regression lines and smooths, data ellipses, marginal distributions) * __model plots__ : primarily plot the results of a fitted model, considering that the fitted model may involve more variables than can be shown in a static 2D plot. Some examples are: Added variable plots, effect plots, coefficient plots, ... * __diagnostic plots__ : indicating potential problems with the fitted model. These include residual plots, influence plots, plots for testing homogeneity of variance and so forth. * __dimension reduction plots__ : plot representations of the data into a space of fewer dimensions than the number of variables in the data set. Simple examples include principal components analysis (PCA) and the related biplots, and multidimensional scaling (MDS) methods. # Data plots Data plots portray the data in a space where the coordinate axes are the observed variables. * 1D plots include line plots, histograms and density estimates * 2D plots are most often scatterplots, but contour plots or hex-binned plots are also useful when the sample size is large. * For higher dimensions, biplots, showing the data in principal components space, together with vectors representing the correlations among variables, are often the most useful. ## Density plots It is useful to examine the distributions of the variables and **density** plots are quite informative. I want to do this for each of the 6 main variables, so I'll use this trick of tidy data analysis with `ggplot2`: 1. Reshape the data from wide to long. This gives `guerry_long`, where the different variables are in a column labeled `variable` and the values are in `value`. ```{r guerry-long} data("Guerry", package="Guerry") guerry_long <- Guerry |> filter(!is.na(Region)) |> select(dept:Suicides) |> pivot_longer(cols = Crime_pers:Suicides, names_to = "variable", values_to = "value") guerry_long ``` 2. Plot the density, but make a different subplot by `facet_wrap(~ variable)`. These plots all have different scales for the X and Y (density) values, so it is important to use `scales="FREE"`. Moreover, I'm primarily interested in the **shape** of these distributions, so I suppress the Y axis tick marks and labels. ```{r guerry-density1} #| out.width = "100%" ggplot(data = guerry_long, aes(x=value, fill=TRUE)) + geom_density(alpha=0.2) + geom_rug() + facet_wrap(~variable, scales="free") + theme_bw(base_size = 14) + theme(legend.position = "none", axis.ticks.y=element_blank(), axis.text.y=element_blank()) ``` You can see that all variables are positively skewed, `Donations`, `Infants` and `Suicides` particularly so, but not so much as to cause alarm. It is also of interest to see whether and how these distributions differ according to `Region`. This is easy to do, using `aes(... fill=Region)` ```{r guerry-density2} #| out.width = "100%", #| fig.height = 6 col.region <- colors()[c(149, 254, 468, 552, 26)] # colors for region ggplot(data = guerry_long, aes(x=value, fill=Region)) + geom_density(alpha=0.2) + geom_rug() + facet_wrap(~variable, scales="free") + scale_fill_manual(values=col.region) + theme_bw(base_size = 14) + theme(legend.position = "bottom", axis.ticks.y=element_blank(), axis.text.y=element_blank()) ``` For some variables, like `Infants` and `Suicides` the differences do not seem particularly large. However, both crime variables and `Literacy` show marked differences across region. ## Bivariate relations Let's start with plots of crime (`Crime_pers` and `Crime_prop`) in relation to `Literacy`. A simple scatterplot is not very informative. All that can be seen is that there is not much of a relation between personal crime and literacy. ```{r lit-pers-scat0} ggplot(aes(x=Literacy, y=Crime_pers/1000), data=Guerry) + geom_point(size=2) ``` More useful scatterplots are annotated with additional statistical summaries to aid interpretation: * linear regression line, * smoothed non-parametric (loess) curve, to diagnose potential non-linear relations, * data ellipses, to highlight the overall trend and variability, * point labels for potentially outlying or influential points. I use `ggplot2` here. It provides most of these features, except that to label unusual points, I calculate the Mahalanobis squared distance of all points from the grand means. ```{r lit-pers-scat} gdf <- Guerry[, c("Literacy", "Crime_pers", "Department")] gdf$dsq <- mahalanobis(gdf[,1:2], colMeans(gdf[,1:2]), cov(gdf[,1:2])) ggplot(aes(x=Literacy, y=Crime_pers/1000, label=Department), data=gdf) + geom_point(size=2) + stat_ellipse(level=0.68, color="blue", size=1.2) + stat_ellipse(level=0.95, color="gray", size=1, linetype=2) + geom_smooth(method="lm", formula=y~x, fill="lightblue") + geom_smooth(method="loess", formula=y~x, color="red", se=FALSE) + geom_label_repel(data = gdf[gdf$dsq > 4.6,]) + theme_bw() ``` The flat (blue) regression line and the nearly circular data ellipses show that the correlation is nearly zero; the smoothed (red) curve indicates that there is no tendency for a nonlinear relation. Doing the same for crimes against property: ```{r lit-prop-scat} gdf <- Guerry[, c("Literacy", "Crime_prop", "Department")] gdf$dsq <- mahalanobis(gdf[,1:2], colMeans(gdf[,1:2]), cov(gdf[,1:2])) ggplot(aes(x=Literacy, y=Crime_prop/1000, label=Department), data=gdf) + geom_point(size=2) + stat_ellipse(level=0.68, color="blue", size=1.2) + stat_ellipse(level=0.95, color="gray", size=1, linetype=2) + geom_smooth(method="lm", formula=y~x, fill="lightblue") + geom_smooth(method="loess", formula=y~x, color="red", se=FALSE) + geom_label_repel(data = gdf[gdf$dsq > 4.6,]) + theme_bw() ``` So, somewhat surprisingly, increased literacy is associated with an increase in property crime (greater population per crime) as opposed to the situation with personal crime, which seems unrelated to literacy. Creuse again stands out as an unusual point, one that is likely to be influential in regression models. ## Reconnaisance plots Reconnaisance plots attempt to give a bird's-eye overview of a multivariate data set. For example, to see the relations among more than two variables we could turn to a scatterplot matrix or some other display to show all pairwise bivariate relations. For these, my preferred package is `car` [@R-car] with the `scatterplotMatrix` function. `GGally` [@R-GGally] works within the the `ggplot2` framework, but doesn't have the flexibility I'd like. ```{r spm1} #| out.width="100%", #| echo=-1 par(mar=rep(2,4)) library(car) # Companion to Applied Regression scatterplotMatrix(Guerry[,4:9], ellipse=list(levels=0.68), smooth=FALSE) ``` ### Corrgrams Sometimes, particularly with more variables than this, we want to see a more schematic overview. A _correlation diagram_ or "corrgram" [@Friendly:02:corrgram] is a graphic display of a correlation matrix, allowing different renderings of the correlation between each pair of variables: as a shaded box, a pie symbol, a schematic data ellipse, and other options. This is implemented in the `corrgram` package [@R-corrgram]. The panels in the upper and lower triangles can be rendered differently. ```{r corrgram1} #| echo=-1 par(mar=rep(1,4)+.1) library(corrgram) # Plot a Correlogram corrgram(Guerry[,4:9], upper=panel.pie) ``` Or, the data in each pairwise tile can be rendered with data ellipses and smoothed curves to show possible nonlinear relations. Another feature is that the rows/column variables can be permuted to put similar variables together, using the `order` option, which arranges the variables according to similarity of their correlations. ```{r corrgram2} #| echo=-1 par(mar=rep(1,4)+.1) corrgram(Guerry[,4:9], upper=panel.ellipse, order=TRUE, lwd=2) ``` Here, there are a number of pairwise plots that appear markedly nonlinear. For the main crime variables, the most nonlinear are that of personal crime vs. donations to the poor, and property crime vs. infants born out of wedlock and suicides. `Literacy` stands out here as having negative relations with all other variables. An alternative analysis might include: * converting the data to ranks. * considering transformations of some of the variables ## Biplots Rather than viewing the data in **data space**, a biplot shows the data in the **reduced-rank PCA space** that explains most of the variation of the observations. This is essentially a plot of the observation scores on the first principal component overlaid with vectors representing the variables projected into PCA space. First, we use `prcomp()` to carry out the PCA. We'd like to visualize the result in relation to `Region`, so delete Corsica where `Region` is missing. ```{r guerry.pca} gdata <- Guerry |> select(Region, Crime_pers:Suicides) |> # keep only main variables filter(!is.na(Region)) # delete Corsica (Region==NA) guerry.pca <- gdata |> select(-Region) |> prcomp(scale = TRUE) print(guerry.pca, digits=3) ``` In the `ggplot2` framework, biplots can be produced by the `ggbiplot` package [@R-ggbiplot], but this package is not on CRAN, so cannot be directly used in this vignette. Instead, the code below was run locally and the result included. ```{r biplot1} #| eval=FALSE if(!require(ggbiplot)) remotes::install_github("vqv/ggbiplot") library(ggbiplot) # A ggplot2 based biplot ggbiplot(guerry.pca, groups=gdata$Region, ellipse=TRUE, var.scale = 3, varname.size = 5) + theme_bw() + labs(color="Region") + theme(legend.position = c(0.1, 0.8)) ``` ```{r ggbiplot} knitr::include_graphics("figures/ggbiplot.png") ``` This is OK, but there are many features of such plots that cannot be customized (line widths, colors, ... ). I prefer those created using the `heplots` package. ```{r biplot2, out.width="95%"} op <- par(mar=c(5,4,1,1)+.1) cols = colorspace::rainbow_hcl(5) covEllipses(guerry.pca$x, group=gdata$Region, pooled=FALSE, fill=TRUE, fill.alpha=0.1, col=cols, label.pos=c(3,0,1,1,3), cex=2, xlim=c(-4,4), ylim=c(-4,4), xlab = "Dimension 1 (35.7 %)", ylab = "Dimension 2 (20.0 %)", cex.lab=1.4 ) points(guerry.pca$x, pch=(15:19)[Guerry$Region], col=cols[Guerry$Region]) candisc::vectors(guerry.pca$rotation, scale=5, col="black", lwd=3, cex=1.4, pos = c(4,2,4,2,2,2), xpd=TRUE) abline(h=0, v=0, col=gray(.70)) ``` An interpretation can be read from both the directions of the variable arrows and the relative positions of the ellipses representing the scatter of the component scores for the different regions. * The first component is largely aligned positively with `Literacy` and negatively with property crime, suicides and children born out of wedlock (`Infants`) * The second dimension reflects mainly the correlation of personal crime and donations to the poor. * The South region is generally lower on PC2, the West, generally higher. * The North stands out as being higher than the others on PC1; the West somewhat higher on PC2. # Models Here we illustrate: * Model based plots for linear regression models predicting personal crime and property crime * Multivariate analysis of variance (MANOVA) and HE plots for the joint relation of the crime variables to other predictors. ## Predicting crime: Univariate regression The simplest approach to predicting the crime variables would be to fit a separate multiple regression to each. ```{r mra-models} crime.mod1 <- lm(Crime_pers ~ Region + Literacy + Donations + Infants + Suicides, data=Guerry) crime.mod2 <- lm(Crime_prop ~ Region + Literacy + Donations + Infants + Suicides, data=Guerry) ``` Tests for the predictors are best obtained using `car::Anova()` which gives **partial** (Type II) tests, adjusting for other predictors, rather than the **sequential** (Type I) tests provided by `stats::anova()` ```{r mra-anova} Anova(crime.mod1) Anova(crime.mod2) ``` These are somewhat disappointing if you look only at the significance stars: Only `Region` is significant for personal crime and only `Suicides` for property crime. There is no evidence for the argument, supported by the liberal hygenicists of Guerry's time, that increased `Literacy` would reduce crime. For such models, we can understand the nature of the predicted effects using the `effects` package. The (marginal) effect for a given term gives the predicted values, averaging over all other terms in the model. ```{r mra-effect1-code} #| eval=FALSE plot(predictorEffects(crime.mod1, ~ Region + Literacy + Infants + Suicides), lwd=2, main="") ``` ```{r mra-effect1} #| echo = FALSE knitr::include_graphics("figures/mra-effect1.png") ``` Doing the same for property crime, we get: ```{r mra-effect2-code} #| eval=FALSE plot(predictorEffects(crime.mod2, ~ Region + Literacy + Infants + Suicides), lwd=2, main="") ``` ```{r mra-effect2} #| echo = FALSE knitr::include_graphics("figures/mra-effect2.png") ``` ## Predicting crime: Multivariate regression The two regression models can be fit together in a multivariate regression for both crime variables jointly. ```{r manova1} crime.mod <- lm(cbind(Crime_pers, Crime_prop) ~ Region + Literacy + Donations + Infants + Suicides, data=Guerry) Anova(crime.mod) ``` As a quick check on the assumption that the residuals are bivariate normally distributed and a check for outliers, a $\chi^2$ Q-Q plot graphs the squared Mahalanobis distances of the residuals against the corresponding $\chi^2_2$ quantiles these would have in a bivariate normal distribution. The data for Creuse stands out as a potential outlier. ```{r cqplot} #| echo=-1 par(mar=c(3,3,1,1)+.1) labels <- paste0(Guerry$dept,":", Guerry$Department) cqplot(crime.mod, id.n=4, labels=labels) ``` ### HE plots Hypothesis-Error (HE) plots [@Friendly:2007:heplots; @FoxFriendlyMonette:09:compstat; @heplots] provide a convenient graphical summary of hypothesis tests in multivariate linear model. They plot a data ellipse for the residuals in the model, representing the $\mathbf{E}$ matrix in the test statistics (Roy's maximum root test, Pillai and Hotelling trace criteria and Wilks' Lambda). Overlaid on this are $\mathbf{H}$ ellipses for each term in the model, representing the data ellipses for the fitted values. Using Roy's test, these have a convenient interpretation: a term is significant _iff_ the H ellipse projects anywhere outside the E ellipse. For a 1 df (regression) variable, the H ellipse collapses to a line. ```{r heplot} #| echo=-1 par(mar = c(3,3,1,1)+.1) heplot(crime.mod, fill=TRUE, fill.alpha=0.05, cex=1.4, cex.lab=1.3 ) ``` In this plot, the effect of `Suicides` is completely aligned with crimes against property. The effect of `Region` is positively correlated with both types of crime. The means for the regions show that the South of France is lower (worse) on personal crime; the other regions vary most in property crime, with the North being lower and the Center being higher. ### Canonical plots The HE plot displays these relations in **data space**. An alternative is provided by canonical discriminant analysis, which finds the weighted sums of the response variables leading to the largest test statistics for the terms, which can be visualized in **canonical space**. The analysis below reflects the effect of `Region` in relation to both crime variables. ```{r candisc} crime.can <- candisc(crime.mod) crime.can ``` The HE plot for this analysis is shown below. Variable vector represent the correlations of the crime variables with the canonical dimension. ```{r hecan} #| echo=-1 par(mar = c(3,3,1,1)+.1) heplot(crime.can, fill=TRUE, fill.alpha=0.1, var.col = "black", var.cex = 1.3, cex=1.4, cex.lab=1.3) ``` This gives a simple interpretation of the differences in `Region` on the crime variables. The first canonical dimension accounts for 83% of differences among the regions, and this is nearly perfectly aligned with personal crime, with the largest difference between the South and the other regions. The second canonical dimension, accounting for the remaining 17%, is perfectly aligned with property crime. On this dimension, the North stands out compared to the other regions. ```{r write-bib, echo = FALSE} # write a packages.bib file of the packages (.packages()) that have been used here pkgs <- unique(c(to.cite, .packages())) knitr::write_bib(pkgs, file = here::here("vignettes", "packages.bib")) ``` # References Guerry/inst/doc/MultiSpat.R0000644000176200001440000001542314515773415015337 0ustar liggesusers## ----setup, include = FALSE--------------------------------------------------- knitr::opts_chunk$set( collapse = TRUE, warning = FALSE, message = FALSE, # suppress package loading messages comment = "#>", fig.height = 3, fig.width = 3, fig.align = "center" ) ## ----------------------------------------------------------------------------- library(Guerry) # Guerry's data library(sp) # management of spatial data library(ade4) # multivariate analysis library(adegraphics) # graphical representation library(spdep) # spatial dependency library(adespatial) # multivariate spatial analysis ## ----names-------------------------------------------------------------------- names(gfrance85) ## ----components--------------------------------------------------------------- data(gfrance85) df <- data.frame(gfrance85)[, 7:12] # the 6 variables france.map <- as(gfrance85, "SpatialPolygons") # the map xy <- coordinates(gfrance85) # spatial coordinates dep.names <- data.frame(gfrance85)[, 6] # departement names region.names <- data.frame(gfrance85)[, 5] # region names col.region <- colors()[c(149, 254, 468, 552, 26)] # colors for region ## ----dudi-pca----------------------------------------------------------------- pca <- dudi.pca(df, scannf = FALSE, nf = 3) ## ----------------------------------------------------------------------------- biplot(pca, plabel.cex = 0.8) ## ----------------------------------------------------------------------------- pca$eig/sum(pca$eig) * 100 ## ----------------------------------------------------------------------------- s.corcircle(pca$co) ## ----fig.width = 4, fig.height = 4------------------------------------------- s.label(pca$li, ppoint.col = col.region[region.names], plabel.optim = TRUE, plabel.cex = 0.6) s.Spatial(france.map, col = col.region[region.names], plabel.cex = 0) s.class(xy, region.names, col = col.region, add = TRUE, ellipseSize = 0, starSize = 0) ## ----------------------------------------------------------------------------- nb <- poly2nb(gfrance85) lw <- nb2listw(nb, style = "W") ## ----------------------------------------------------------------------------- s.Spatial(france.map, nb = nb, plabel.cex = 0, pSp.border = "white") ## ----------------------------------------------------------------------------- moran.randtest(df, lw) ## ----fig.width = 4, fig.height=4---------------------------------------------- x <- df[, 3] x.lag <- lag.listw(lw, df[, 3]) moran.plot(x, lw) text(x[5], x.lag[5], dep.names[5], pos = 1, cex = 0.8) ## ----fig.dim = c(6,3)--------------------------------------------------------- moran.randtest(pca$li, lw) s.value(xy, pca$li[, 1:2], Sp = france.map, pSp.border = "white", symbol = "circle", pgrid.draw = FALSE) ## ----------------------------------------------------------------------------- bet <- bca(pca, region.names, scannf = FALSE, nf = 2) ## ----------------------------------------------------------------------------- bet$ratio ## ----fig.dim=c(5,5)----------------------------------------------------------- plot(bet) ## ----------------------------------------------------------------------------- barplot(bet$eig) bet$eig/sum(bet$eig) * 100 ## ----------------------------------------------------------------------------- s.arrow(bet$c1, plabel.cex = 0.8) ## ----fig.dim = c(4,4)--------------------------------------------------------- s.label(bet$ls, as.character(dep.names), ppoint.cex = 0, plabel.optim = TRUE, plabel.col = col.region[region.names], plabel.cex = 0.5) s.class(bet$ls, fac = region.names, col = col.region, ellipse = 0, add = TRUE) ## ----fig.dim = c(6,3)--------------------------------------------------------- s.value(xy, bet$ls, symbol = "circle", Sp = france.map, pSp.col = col.region[region.names], pSp.border = "transparent") ## ----fig.dim = c(6,4)--------------------------------------------------------- poly.xy <- orthobasis.poly(xy, degree = 2) s.value(xy, poly.xy, Sp = france.map, plegend.drawKey = FALSE) ## ----------------------------------------------------------------------------- pcaiv.xy <- pcaiv(pca, poly.xy, scannf = FALSE, nf = 2) ## ----------------------------------------------------------------------------- sum(pcaiv.xy$eig)/sum(pca$eig) * 100 pcaiv.xy$eig/sum(pcaiv.xy$eig) * 100 ## ----fig.dim=c(5,5)----------------------------------------------------------- plot(pcaiv.xy) ## ----fig.dim = c(6,6)--------------------------------------------------------- mem1 <- scores.listw(lw) s.value(xy, mem1[, 1:9], Sp = france.map, plegend.drawKey = FALSE) ## ----------------------------------------------------------------------------- pcaiv.mem <- pcaiv(pca, mem1[,1:10], scannf = FALSE) ## ----------------------------------------------------------------------------- sum(pcaiv.mem$eig)/sum(pca$eig) * 100 pcaiv.mem$eig/sum(pcaiv.mem$eig) * 100 ## ----fig.dim=c(5,5)----------------------------------------------------------- plot(pcaiv.mem) ## ----------------------------------------------------------------------------- ms <- multispati(pca, lw, scannf = FALSE) ## ----fig.dim=c(5,5)----------------------------------------------------------- plot(ms) ## ----------------------------------------------------------------------------- summary(ms) ## ----------------------------------------------------------------------------- s.arrow(ms$c1, plabel.cex = 0.8) ## ----fig.dim = c(4,4)--------------------------------------------------------- s.match(ms$li, ms$ls, plabel.cex = 0) s.match(ms$li[c(10, 41, 27), ], ms$ls[c(10, 41, 27), ], label = dep.names[c(10, 41, 27)], plabel.cex = 0.8, add = TRUE) ## ----fig.dim = c(6,3)--------------------------------------------------------- s.value(xy, ms$li, Sp = france.map) ## ----------------------------------------------------------------------------- mat <- matrix(NA, 4, 4) mat.names <- c("PCA", "BCA", "PCAIV-POLY", "PCAIV-MEM", "MULTISPATI") colnames(mat) <- mat.names[-5] rownames(mat) <- mat.names[-1] mat[1, 1] <- procuste.randtest(pca$li[, 1:2], bet$ls[, 1:2])$obs mat[2, 1] <- procuste.randtest(pca$li[, 1:2], pcaiv.xy$ls[, 1:2])$obs mat[3, 1] <- procuste.randtest(pca$li[, 1:2], pcaiv.mem$ls[, 1:2])$obs mat[4, 1] <- procuste.randtest(pca$li[, 1:2], ms$li[, 1:2])$obs mat[2, 2] <- procuste.randtest(bet$ls[, 1:2], pcaiv.xy$ls[, 1:2])$obs mat[3, 2] <- procuste.randtest(bet$ls[, 1:2], pcaiv.mem$ls[, 1:2])$obs mat[4, 2] <- procuste.randtest(bet$ls[, 1:2], ms$li[, 1:2])$obs mat[3, 3] <- procuste.randtest(pcaiv.xy$ls[, 1:2], pcaiv.mem$ls[, 1:2])$obs mat[4, 3] <- procuste.randtest(pcaiv.xy$ls[, 1:2], ms$li[, 1:2])$obs mat[4, 4] <- procuste.randtest(pcaiv.mem$ls[, 1:2], ms$li[, 1:2])$obs mat Guerry/inst/doc/MultiSpat.html0000644000176200001440000132717214515773415016112 0ustar liggesusers Guerry data: Spatial Multivariate Analysis

Guerry data: Spatial Multivariate Analysis

Stéphane Dray

2023-10-24

This vignette indicates how to perform the analyses described in Dray and Jombart (2011) of data derived from André-Michel Guerry’s (1833) Essai sur la Statistique Morale de la France. It illustrates some classical methods for analysis of multivariate spatial data that focus either on the multivariate aspect or on the spatial one, as well as some more modern methods that attempt to integrate geographical and multivariate aspects simultaneously.

Preliminary steps

Several packages are required to run the different analyses and should be loaded. The ade4 package for multivariate analysis is supplemented by adegraphics (for associated graphical methods) and adespatial for multivariate spatial analysis. For further information on this spatial analysis approach, see vignette(package="adespatial", "tutorial").

library(Guerry)       # Guerry's data
library(sp)           # management of spatial data
library(ade4)         # multivariate analysis
library(adegraphics)  # graphical representation
library(spdep)        # spatial dependency
library(adespatial)   # multivariate spatial analysis

Guerry gathered data on crimes, suicide, literacy and other moral statistics for various départements (i.e., counties) in France. He provided the first real social data analysis, using graphics and maps to summarize this georeferenced multivariate dataset. We use the dataset gfrance85 and consider six key quantitative variables (shown in the table below) for each of the 85 départements of France in 1830 (Corsica, an island and often an outlier, was excluded).

Data are recorded on aligned scales so that larger numbers consistently reflect “morally better”. Thus, four of the “bad” variables are recorded in the inverse form, as “Population per …”. With this scaling, it would be expected that all correlations be \(\ge 0\).

Name Description
Crime_pers Population per crime against persons
Crime_prop Population per crime against property
Literacy Percent of military conscripts who can read and write
Donations Donations to the poor
Infants Population per illegitimate birth
Suicides Population per suicide

The dataset gfrance85 is actually a SpatialPolygonsDataFrame object created with the sp package. It contains the polygon boundaries of the map of France in 1830, as well the variables in the Guerry data frame. As an ordinary data.frame, it has these components

names(gfrance85)
#>  [1] "CODE_DEPT"       "COUNT"           "AVE_ID_GEO"      "dept"           
#>  [5] "Region"          "Department"      "Crime_pers"      "Crime_prop"     
#>  [9] "Literacy"        "Donations"       "Infants"         "Suicides"       
#> [13] "MainCity"        "Wealth"          "Commerce"        "Clergy"         
#> [17] "Crime_parents"   "Infanticide"     "Donation_clergy" "Lottery"        
#> [21] "Desertion"       "Instruction"     "Prostitutes"     "Distance"       
#> [25] "Area"            "Pop1831"

To simplify analyses, we extract the components to be used below.

 data(gfrance85)
 df           <- data.frame(gfrance85)[, 7:12]    # the 6 variables
 france.map   <- as(gfrance85, "SpatialPolygons") # the map
 xy           <- coordinates(gfrance85)           # spatial coordinates
 dep.names    <- data.frame(gfrance85)[, 6]       # departement names
 region.names <- data.frame(gfrance85)[, 5]       # region names
 col.region   <- colors()[c(149, 254, 468, 552, 26)] # colors for region

Standard approaches

In this section, we focus on classical approaches that consider either the multivariate or the spatial aspect of the data.

Multivariate analysis

Here we consider \(p=6\) variables measured for \(n=85\) individuals (départements of France). As only quantitative variables have been recorded, principal component analysis (PCA, Hotelling 1933) is well adapted. PCA summarizes the data by maximizing simultaneously the variance of the projection of the individuals onto the principal axes and the sum of the squared correlations between the principal component and the variables.

pca <- dudi.pca(df, scannf = FALSE, nf = 3)

The biplot is simply obtained by

biplot(pca, plabel.cex = 0.8)

The first two PCA dimensions account for 35.7% and 20% ,respectively, of the total variance.

pca$eig/sum(pca$eig) * 100
#> [1] 35.674507 20.013665 18.367448 11.116102  9.144584  5.683694

Correlations between variables and principal components can be represented on a correlation circle. The first axis is negatively correlated to literacy and positively correlated to property crime, suicides and illegitimate births. The second axis is aligned mainly with personal crime and donations to the poor.

s.corcircle(pca$co)

Spatial information can be added on the factorial map representing the projections of départements on principal axes by coloring according to colors representing the different regions of France.

s.label(pca$li, ppoint.col = col.region[region.names], plabel.optim = TRUE, plabel.cex = 0.6)

s.Spatial(france.map, col = col.region[region.names], plabel.cex = 0)
s.class(xy, region.names, col = col.region, add = TRUE, ellipseSize = 0, starSize = 0)

For the first axis, the North and East are characterized by negative scores, corresponding to high levels of literacy and high rates of suicides, crimes against property and illegitimate births. The second axis mainly contrasts the West (high donations to the the poor and low levels of crime against persons) to the South.

Spatial autocorrelation

Spatial autocorrelation statistics, such as Moran (1948) Coefficient (MC) and Geary (1954) Ratio, aim to measure and analyze the degree of dependency among observations in a geographical context (Cliff and Ord 1973).

The spatial weighting matrix

The first step of spatial autocorrelation analysis is to define a spatial weighting matrix \(\mathbf{W}=[w_{ij}]\) . In the case of Guerry’s data, we simply defined a binary neighborhood where two départements are considered as neighbors if they share a common border. The spatial weighting matrix is then obtained after row-standardization (style = "W"):

nb <- poly2nb(gfrance85)
lw <- nb2listw(nb, style = "W")

We can represent this neighborhood on the geographical map:

s.Spatial(france.map, nb = nb, plabel.cex = 0, pSp.border = "white")

Moran’s Coefficient

Once the spatial weights have been defined, the spatial autocorrelation statistics can then be computed. Let us consider the \(n\)-by-1 vector \(\mathbf{x}=\left[ {x_1 \cdots x_n } \right]^\textrm{T}\) containing measurements of a quantitative variable for \(n\) spatial units. The usual formulation for Moran’s coefficient of spatial autocorrelation (Cliff and Ord 1973; Moran 1948) is \[\begin{equation} \label{eq1} MC(\mathbf{x})=\frac{n\sum\nolimits_{\left( 2 \right)} {w_{ij} (x_i -\bar {x})(x_j -\bar {x})} }{\sum\nolimits_{\left( 2 \right)} {w_{ij} } \sum\nolimits_{i=1}^n {(x_i -\bar {x})^2} }\mbox{ where }\sum\nolimits_{\left( 2 \right)} =\sum\limits_{i=1}^n {\sum\limits_{j=1}^n } \mbox{ with }i\ne j. \end{equation}\]

MC can be rewritten using matrix notation: \[\begin{equation} \label{eq2} MC(\mathbf{x})=\frac{n}{\mathbf{1}^\textrm{T}\mathbf{W1}}\frac{\mathbf{z}^\textrm{T}{\mathbf{Wz}}}{\mathbf{z}^\textrm{T}\mathbf{z}}, \end{equation}\] where \(\mathbf{z}=\left ( \mathbf{I}_n-\mathbf{1}_n\mathbf{1}_n^\textrm{T} /n \right )\mathbf{x}\) is the vector of centered values (\(z_i=x_i-\bar{x}\)) and \(\mathbf{1}_n\) is a vector of ones (of length \(n\)).

The significance of the observed value of MC can be tested by a Monte-Carlo procedure, in which locations are permuted to obtain a distribution of MC under the null hypothesis of random distribution. An observed value of MC that is greater than that expected at random indicates the clustering of similar values across space (positive spatial autocorrelation), while a significant negative value of MC indicates that neighboring values are more dissimilar than expected by chance (negative spatial autocorrelation).

We computed MC for the Guerry’s dataset. A positive and significant autocorrelation is identified for each of the six variables. Thus, the values of literacy are the most covariant in adjacent departments, while illegitimate births (Infants) covary least.

moran.randtest(df, lw)
#> class: krandtest lightkrandtest 
#> Monte-Carlo tests
#> Call: moran.randtest(x = df, listw = lw)
#> 
#> Number of tests:   6 
#> 
#> Adjustment method for multiple comparisons:   none 
#> Permutation number:   999 
#>         Test       Obs   Std.Obs   Alter Pvalue
#> 1 Crime_pers 0.4114597  5.797137 greater  0.001
#> 2 Crime_prop 0.2635533  4.160819 greater  0.002
#> 3   Literacy 0.7176053 10.357515 greater  0.001
#> 4  Donations 0.3533613  5.204476 greater  0.001
#> 5    Infants 0.2287241  3.628118 greater  0.002
#> 6   Suicides 0.4016812  6.081180 greater  0.001

Moran scatterplot

If the spatial weighting matrix is row-standardized, we can define the lag vector \(\mathbf{\tilde{z}} = \mathbf{Wz}\) (i.e., \(\tilde{z}_i = \sum\limits_{j=1}^n{w_{ij}x_j}\)) composed of the weighted (by the spatial weighting matrix) averages of the neighboring values. Thus, we have: \[\begin{equation} \label{eq3} MC(\mathbf{x})=\frac{\mathbf{z}^\textrm{T}{\mathbf{\tilde{z}}}}{\mathbf{z}^\textrm{T}\mathbf{z}}, \end{equation}\] since in this case \(\mathbf{1}^\textrm{T}\mathbf{W1}=n\). This shows clearly that MC measures the autocorrelation by giving an indication of the intensity of the linear association between the vector of observed values \(\mathbf{z}\) and the vector of weighted averages of neighboring values \(\mathbf{\tilde{z}}\). Anselin (1996) proposed to visualize MC in the form of a bivariate scatterplot of \(\mathbf{\tilde{z}}\) against \(\mathbf{z}\). A linear regression can be added to this Moran scatterplot, with slope equal to MC.

Considering the Literacy variable of Guerry’s data, the Moran scatterplot clearly shows strong autocorrelation. It also shows that the Hautes-Alpes département has a slightly outlying position characterized by a high value of Literacy compared to its neighbors.

x <- df[, 3]
x.lag <- lag.listw(lw, df[, 3])
moran.plot(x, lw)
text(x[5], x.lag[5], dep.names[5], pos = 1, cex = 0.8)

Indirect integration of multivariate and geographical aspects

The simplest approach considered a two-step procedure where the data are first summarized with multivariate analysis such as PCA. In a second step, univariate spatial statistics or mapping techniques are applied to PCA scores for each axis separately. One can also test for the presence of spatial autocorrelation for the first few scores of the analysis, with univariate autocorrelation statistics such as MC. We mapped scores of the départements for the first two axes of the PCA of Guerry’s data. Even if PCA maximizes only the variance of these scores, there is also a clear spatial structure, as the scores are highly autocorrelated. The map for the first axis corresponds closely to the split between la France éclairée (North-East characterized by an higher level of Literacy) and la France obscure.

moran.randtest(pca$li, lw)
#> class: krandtest lightkrandtest 
#> Monte-Carlo tests
#> Call: moran.randtest(x = pca$li, listw = lw)
#> 
#> Number of tests:   3 
#> 
#> Adjustment method for multiple comparisons:   none 
#> Permutation number:   999 
#>    Test       Obs  Std.Obs   Alter Pvalue
#> 1 Axis1 0.5506343 7.704990 greater  0.001
#> 2 Axis2 0.5614030 8.237265 greater  0.001
#> 3 Axis3 0.1805569 2.701054 greater  0.006
s.value(xy, pca$li[, 1:2], Sp = france.map, pSp.border = "white", symbol = "circle", pgrid.draw = FALSE)

Spatial multivariate analysis

Over the last decades, several approaches have been developed to consider both geographical and multivariate information simultaneously. The multivariate aspect is usually treated by techniques of dimensionality reduction similar to PCA. On the other hand, several alternatives have been proposed to integrate the spatial information.

Spatial partition

One alternative is to consider a spatial partition of the study area. In this case, the spatial information is coded as a categorical variable, and each category corresponds to a region of the whole study area. For instance, Guerry’s data contained a partition of France into 5 regions.

We used the between-class analysis (BCA, Dolédec and Chessel 1987), to investigate differences between regions. BCA maximizes the variance between groups.

 bet <- bca(pca, region.names, scannf = FALSE, nf = 2)

Here, 28.8 % of the total variance (sum of eigenvalues of PCA) corresponds to the between-regions variance (sum of the eigenvalues of BCA).

bet$ratio
#> [1] 0.288139

The main graphical outputs are obtained by the generic plot function:

plot(bet)

The barplot of eigenvalues indicates that two axes should be interpreted. The first two BCA dimensions account for 59 % and 30.2 %, respectively, of the between-regions variance.

 barplot(bet$eig)

 bet$eig/sum(bet$eig) * 100
#> [1] 58.995823 30.159802  7.417445  3.426930

The coefficients used to construct the linear combinations of variables are represented:

s.arrow(bet$c1, plabel.cex = 0.8)

The first axis opposed literacy to property crime, suicides and illegitimate births. The second axis is mainly aligned with personal crime and donations to the poor.

Projections of départements on the BCA axes can be represented on the factorial map:

s.label(bet$ls, as.character(dep.names), ppoint.cex = 0, plabel.optim = TRUE, plabel.col = col.region[region.names], plabel.cex = 0.5)
s.class(bet$ls, fac = region.names, col = col.region, ellipse = 0, add = TRUE)

The scores can be mapped to show the spatial aspects:

s.value(xy, bet$ls, symbol = "circle", Sp = france.map, pSp.col = col.region[region.names], pSp.border = "transparent")

The results are very close to those obtained by PCA: the first axis contrasted the North and the East (la France éclairée) to the other regions while the South is separated from the other regions by the second axis. The high variability of the region Centre is also noticeable. In contrast, the South is very homogeneous.

Spatial explanatory variables

Principal component analysis with respect to the instrumental variables (PCAIV, Rao 1964), and related methods, have been often used in community ecology to identify spatial relationships. The spatial information is introduced in the form of spatial predictors and the analysis maximized the “spatial variance” (i.e., the variance explained by spatial predictors). Note that BCA can also be considered as a particular case of PCAIV, where the explanatory variables are dummy variables indicating group membership.

Trend surface of geographic coordinates

Student (1914) proposed to express observed values in time series as a polynomial function of time, and mentioned that this could be done for spatial data as well. Borcard, Legendre, and Drapeau (1992) extended this approach to the spatial and multivariate case by introducing polynomial functions of geographic coordinates as predictors in PCAIV. We call this approach PCAIV-POLY.

The centroids of départements of France were used to construct a second-degree orthogonal polynomial.

poly.xy <- orthobasis.poly(xy, degree = 2)
s.value(xy, poly.xy, Sp = france.map, plegend.drawKey = FALSE)

PCAIV is then performed using the pcaiv function:

pcaiv.xy <- pcaiv(pca, poly.xy, scannf = FALSE, nf = 2)

Here, 32.4 % of the total variance (sum of eigenvalues of PCA) is explained by the second-degree polynomial (sum of eigenvalues of PCAIV). The first two dimensions account for 51.4 % and 35.2 %, respectively, of the explained variance.

sum(pcaiv.xy$eig)/sum(pca$eig) * 100
#> [1] 32.36025
pcaiv.xy$eig/sum(pcaiv.xy$eig) * 100
#> [1] 51.423264 35.152362  5.954475  4.799075  2.670825

The outputs of PCAIV-POLY (coefficients of variables, maps of départements scores, etc.) are very similar to those obtained by BCA. They can be represented easily by the generic plot function:

plot(pcaiv.xy)

Moran’s eigenvector maps

An alternative way to build spatial predictors is by the diagonalization of the spatial weighting matrix W. Moran’s eigenvector maps (MEM, Dray, Legendre, and Peres-Neto 2006) are the \(n-1\) eigenvectors of the doubly-centered matrix W. They are orthogonal vectors with a unit norm maximizing MC (Griffith 1996). MEM associated with high positive (or negative) eigenvalues have high positive (or negative) autocorrelation. MEM associated with eigenvalues with small absolute values correspond to low spatial autocorrelation, and are not suitable for defining spatial structures.

We used the spatial weighting matrix defined above to construct MEM. The first ten MEM, corresponding to the highest levels of spatial autocorrelation, have been mapped:

mem1 <- scores.listw(lw)
s.value(xy, mem1[, 1:9], Sp = france.map, plegend.drawKey = FALSE)

We introduced the first ten MEM as spatial explanatory variables in PCAIV. We call this approach PCAIV-MEM.

pcaiv.mem <- pcaiv(pca, mem1[,1:10], scannf = FALSE)

Here, 44.1 % of the total variance (sum of eigenvalues of PCA) is explained by the first ten MEM (sum of eigenvalues of PCAIV). The first two dimensions account for 54.9 % and 26.3 %, respectively, of the explained variance.

sum(pcaiv.mem$eig)/sum(pca$eig) * 100
#> [1] 44.11597
pcaiv.mem$eig/sum(pcaiv.mem$eig) * 100
#> [1] 54.928640 26.300082  9.042300  4.574408  3.532962  1.621607

The outputs of PCAIV-MEM (coefficients of variables, maps of départements scores, etc.) are very similar to those obtained by BCA. They can be represented easily by the generic plot function:

plot(pcaiv.mem)

Spatial graph and weighting matrix

The MEM framework introduced the spatial information into multivariate analysis through the eigendecomposition of the spatial weighting matrix. Usually, we consider only a part of the information contained in this matrix because only a subset of MEM are used as regressors in PCAIV. In this section, we focus on multivariate methods that consider the spatial weighting matrix under its original form.

Wartenberg (1985) was the first to develop a multivariate analysis based on MC. His work considered only normed and centered variables (i.e., normed PCA) for the multivariate part and a binary symmetric connectivity matrix for the spatial aspect. Dray, Saïd, and Débias (2008) generalized Wartenberg’s method by introducing a row-standardized spatial weighting matrix in the analysis of a statistical triplet. This approach is very general and allows us to define spatially-constrained versions of various methods (corresponding to different triplets) such as correspondence analysis or multiple correspondence analysis. MULTISPATI finds coefficients to obtain a linear combination of variables that maximizes a compromise between the classical multivariate analysis and a generalized version of Moran’s coefficient.

 ms <- multispati(pca, lw, scannf = FALSE)

The main outputs of MULTISPATI can be represented easily by the generic plot function:

plot(ms)

The barplot of eigenvalues suggests two main spatial structures. Eigenvalues of MULTISPATI are the product between the variance and the spatial autocorrelation of the scores, while PCA maximizes only the variance. The differences between the two methods are computed by the summary function:

summary(ms)
#> 
#> Multivariate Spatial Analysis
#> Call: multispati(dudi = pca, listw = lw, scannf = FALSE)
#> 
#> Scores from the initial duality diagram:
#>          var      cum     ratio     moran
#> RS1 2.140470 2.140470 0.3567451 0.5506343
#> RS2 1.200820 3.341290 0.5568817 0.5614030
#> RS3 1.102047 4.443337 0.7405562 0.1805569
#> 
#> Multispati eigenvalues decomposition:
#>           eig      var     moran
#> CS1 1.2858683 2.017171 0.6374612
#> CS2 0.6939887 1.176551 0.5898499

Hence, there is a loss of variance compared to PCA (2.14 versus 2.017 for axis 1; 1.201 versus 1.177 for axis 2) but a gain of spatial autocorrelation (0.551 versus 0.637 for axis 1; 0.561 versus 0.59 for axis 2).

Coefficients of variables allow to interpret the structures:

s.arrow(ms$c1, plabel.cex = 0.8)

The first axis opposes literacy to property crime, suicides and illegitimate births. The second axis is aligned mainly with personal crime and donations to the poor. The maps of the scores show that the spatial structures are very close to those identified by PCA. The similarity of results between PCA and its spatially optimized version confirm that the main structures of Guerry’s data are spatial.

Spatial autocorrelation can be seen as the link between one variable and the lagged vector. This interpretation is used to construct the Moran scatterplot and can be extended to the multivariate case in MULTISPATI by analyzing the link between scores and lagged scores:

s.match(ms$li, ms$ls, plabel.cex = 0)
s.match(ms$li[c(10, 41, 27), ], ms$ls[c(10, 41, 27), ], label = dep.names[c(10, 
     41, 27)], plabel.cex = 0.8, add = TRUE)

Each département can be represented on the factorial map by an arrow (the bottom corresponds to its score, the head corresponds to its lagged score. A short arrow reveals a local spatial similarity (between one plot and its neighbors) while a long arrow reveals a spatial discrepancy. This viewpoint can be interpreted as a multivariate extension of the local index of spatial association (Anselin 1995). For instance: * Aude has a very small arrow, indicating that this département is very similar to its neighbors. * Haute-Loire has a long horizontal arrow which reflects its high values for the variables Infants (31017), Suicides (163241) and Crime_prop (18043) compared to the average values over its neighbors (27032.4, 60097.8 and 10540.8 for these three variables). * Finistère corresponds to an arrow with a long vertical length which is due to its high values compared to its neighbors for Donations (23945 versus 12563) and Crime_pers (29872 versus 25962).

The link between the scores and the lagged scores (averages of neighbors weighted by the spatial connection matrix) can be mapped in the geographical space. For the first two axes, we have:

s.value(xy, ms$li, Sp = france.map)

Conclusions

Even if the methods presented are quite different in their theoretical and practical viewpoints, their applications to Guerry’s dataset yield very similar results. We provided a quantitative measure of this similarity by computing Procrustes statistics [Peres-Neto and Jackson (2001);SD161] between the scores of the départements onto the first two axes for the different analyses. All the values of the statistic are very high and significant; this confirms the high concordance between the outputs of the different methods.

mat <- matrix(NA, 4, 4)
mat.names <- c("PCA", "BCA", "PCAIV-POLY", "PCAIV-MEM", "MULTISPATI")
colnames(mat) <- mat.names[-5]
rownames(mat) <- mat.names[-1]

mat[1, 1] <- procuste.randtest(pca$li[, 1:2], bet$ls[, 1:2])$obs
mat[2, 1] <- procuste.randtest(pca$li[, 1:2], pcaiv.xy$ls[, 1:2])$obs
mat[3, 1] <- procuste.randtest(pca$li[, 1:2], pcaiv.mem$ls[, 1:2])$obs
mat[4, 1] <- procuste.randtest(pca$li[, 1:2], ms$li[, 1:2])$obs
mat[2, 2] <- procuste.randtest(bet$ls[, 1:2], pcaiv.xy$ls[, 1:2])$obs
mat[3, 2] <- procuste.randtest(bet$ls[, 1:2], pcaiv.mem$ls[, 1:2])$obs
mat[4, 2] <- procuste.randtest(bet$ls[, 1:2], ms$li[, 1:2])$obs
mat[3, 3] <- procuste.randtest(pcaiv.xy$ls[, 1:2], pcaiv.mem$ls[, 1:2])$obs
mat[4, 3] <- procuste.randtest(pcaiv.xy$ls[, 1:2], ms$li[, 1:2])$obs
mat[4, 4] <- procuste.randtest(pcaiv.mem$ls[, 1:2], ms$li[, 1:2])$obs

mat
#>                  PCA       BCA PCAIV-POLY PCAIV-MEM
#> BCA        0.9788594        NA         NA        NA
#> PCAIV-POLY 0.9791939 0.9897463         NA        NA
#> PCAIV-MEM  0.9885944 0.9936217  0.9954116        NA
#> MULTISPATI 0.9868799 0.9954034  0.9951133 0.9986471

References

Anselin, L. 1995. “Local Indicators of Spatial Association.” Geographical Analysis 27: 93–115.
———. 1996. “The Moran Scatterplot as an ESDA Tool to Assess Local Instability in Spatial Association.” In Spatial Analytical Perspectives on GIS, edited by M. M. Fischer, H. J. Scholten, and D. Unwin, 111–25. London: Taylor; Francis.
Borcard, D., P. Legendre, and P. Drapeau. 1992. “Partialling Out the Spatial Component of Ecological Variation.” Ecology 73: 1045–55.
Cliff, A. D., and J. K. Ord. 1973. Spatial Autocorrelation. London: Pion.
Dolédec, S., and D. Chessel. 1987. “Rythmes Saisonniers Et Composantes Stationnelles En Milieu Aquatique I- Description d’un Plan d’observations Complet Par Projection de Variables.” Acta Oecologica - Oecologia Generalis 8 (3): 403–26.
Dray, S., and T. Jombart. 2011. “Revisiting Guerry’s Data: Introducing Spatial Constraints in Multivariate Analysis.” Annals of Applied Statistics 5 (4): 2278–99.
Dray, S., P. Legendre, and P. R. Peres-Neto. 2006. “Spatial Modeling: A Comprehensive Framework for Principal Coordinate Analysis of Neighbor Matrices (PCNM).” Ecological Modelling 196: 483–93.
Dray, S., S. Saïd, and F. Débias. 2008. “Spatial Ordination of Vegetation Data Using a Generalization of Wartenberg’s Multivariate Spatial Correlation.” Journal of Vegetation Science 19: 45–56.
Geary, R. C. 1954. “The Contiguity Ratio and Statistical Mapping.” The Incorporated Statistician 5 (3): 115–45.
Griffith, D. A. 1996. “Spatial Autocorrelation and Eigenfunctions of the Geographic Weights Matrix Accompanying Geo-Referenced Data.” Canadian Geographer 40 (4): 351–67.
Guérry, A. M. 1833. Essai Sur La Statistique Morale de La France. Paris: Crochard.
Hotelling, H. 1933. “Analysis of a Complex of Statistical Variables into Principal Components.” Journal of Educational Psychology 24: 417–41.
Moran, P. A. P. 1948. “The Interpretation of Statistical Maps.” Journal of the Royal Statistical Society Series B-Methodological 10: 243–51.
Peres-Neto, P. R., and D. A. Jackson. 2001. “How Well Do Multivariate Data Sets Match? The Advantages of a Procrustean Superimposition Approach over the Mantel Test.” Oecologia 129: 169–78.
Rao, C. R. 1964. “The Use and Interpretation of Principal Component Analysis in Applied Research.” Sankhya A 26: 329–59.
Student, W. S. 1914. “The Elimination of Spurious Correlation Due to Position in Time or Space.” Biometrika 10: 179–80.
Wartenberg, D. 1985. “Multivariate Spatial Correlation: A Method for Exploratory Geographical Analysis.” Geographical Analysis 17 (4): 263–83.
Guerry/inst/doc/guerry-multivariate.html0000644000176200001440000113337014515773424020204 0ustar liggesusers Guerry data: Multivariate Analysis

Guerry data: Multivariate Analysis

Michael Friendly

2023-10-24

André-Michel Guerry’s Essai sur la Statistique Morale de la France (Guerry 1833) collected data on crimes, suicide, literacy and other “moral statistics” for various départements in France. He provided the first real social data analysis, using graphics and maps to summarize this multivariate dataset. One of his main goals in this ground-breaking study was to determine if the prevalence of crime in France could be explained by other social variables.

In 1833, the scatterplot had not yet been invented; the idea of a correlation or a regression was still 50 years in the future (Galton 1886). Guerry displayed his data in shaded choropleth maps and semi-graphic tables and argued how these could be seen as implying systematic, lawful relations among moral variables.

In this analysis, we ignore the spatial context of the départements and focus on multivariate analyses of the the data set.

Load data and packages

We will primarily use the following packages, so load them now.

library(Guerry)         # Guerry data
library(car)            # better scatterplots
library(effects)        # Effect Displays for Linear Models
library(ggplot2)        # Elegant Data Visualisations Using the Grammar of Graphics
library(ggrepel)        # better handling of text labels
library(patchwork)      # combine plots
library(heplots)        # Hypothesis-Error plots
library(candisc)        # Visualizing Generalized Canonical Discriminant Analysis
library(dplyr)          # A Grammar of Data Manipulation
library(tidyr)          # Tidy Messy Data
data(Guerry)

Guerry data set

Guerry’s (1833) data consisted of six main moral variables shown in the table below. He wanted all of these to be recorded on aligned scales so that larger numbers consistently reflected “morally better”. Thus, four of the variables are recorded in the inverse form, as “Population per …”.

Name Description
Crime_pers Population per crime against persons
Crime_prop Population per crime against property
Literacy Percent of military conscripts who can read and write
Donations Donations to the poor
Infants Population per illegitimate birth
Suicides Population per suicide

The Guerry data set also contains:

  • dept and Department, the French ID numbers and names for the 86 départements of metropolitan France in 1830, including Corsica.
  • Region: a factor with main levels “N”, “S”, “E”, “W”, “C”. Corsica is coded as NA.
  • A collection of 14 other related variables from other sources at the same time. See ?Guerry for their precise definitions.
names(Guerry)[-(1:9)]
#>  [1] "MainCity"        "Wealth"          "Commerce"        "Clergy"         
#>  [5] "Crime_parents"   "Infanticide"     "Donation_clergy" "Lottery"        
#>  [9] "Desertion"       "Instruction"     "Prostitutes"     "Distance"       
#> [13] "Area"            "Pop1831"

Among these, as other aspects of criminal behavior, we see crime against parents, Infanticide and Prostitutes. Clergy and Donations_clergy are considered to be measures of moral rectitude, potentially counteracting crime.

Guerry’s questions

The main questions that concerned Guerry were whether indicators of crime could be shown to be related to factors which might be considered to ameliorate crime. Among these, Guerry focused most on Literacy defined as the number of military conscripts who could do more than mark an “X” on their enrollment form. A related variable is Instruction, the rank recorded from Guerry’s map; as defined, it is inversely related to Literacy.

Other potential explanatory variables are:

Donations (a measure of donations to the poor),

Donation_clergy (a measure of donations to clergy)
Clergy (the rank of number of Catholic priests in active service, per population)

Multivariate visualization methods

Visualization methods for multivariate data take an enormous variety of forms simply because more than two dimensions of data offer exponentially increasingly possibilities. It is useful to distinguish several broad categories:

  • data plots : primarily plot the raw data, often with annotations to aid interpretation (regression lines and smooths, data ellipses, marginal distributions)

  • model plots : primarily plot the results of a fitted model, considering that the fitted model may involve more variables than can be shown in a static 2D plot. Some examples are: Added variable plots, effect plots, coefficient plots, …

  • diagnostic plots : indicating potential problems with the fitted model. These include residual plots, influence plots, plots for testing homogeneity of variance and so forth.

  • dimension reduction plots : plot representations of the data into a space of fewer dimensions than the number of variables in the data set. Simple examples include principal components analysis (PCA) and the related biplots, and multidimensional scaling (MDS) methods.

Data plots

Data plots portray the data in a space where the coordinate axes are the observed variables.

  • 1D plots include line plots, histograms and density estimates
  • 2D plots are most often scatterplots, but contour plots or hex-binned plots are also useful when the sample size is large.
  • For higher dimensions, biplots, showing the data in principal components space, together with vectors representing the correlations among variables, are often the most useful.

Density plots

It is useful to examine the distributions of the variables and density plots are quite informative. I want to do this for each of the 6 main variables, so I’ll use this trick of tidy data analysis with ggplot2:

  1. Reshape the data from wide to long. This gives guerry_long, where the different variables are in a column labeled variable and the values are in value.
data("Guerry", package="Guerry")
guerry_long <- Guerry |>
  filter(!is.na(Region)) |>
  select(dept:Suicides) |>
  pivot_longer(cols = Crime_pers:Suicides,
               names_to = "variable",
               values_to = "value")
guerry_long
#> # A tibble: 510 × 5
#>     dept Region Department variable   value
#>    <int> <fct>  <fct>      <chr>      <int>
#>  1     1 E      Ain        Crime_pers 28870
#>  2     1 E      Ain        Crime_prop 15890
#>  3     1 E      Ain        Literacy      37
#>  4     1 E      Ain        Donations   5098
#>  5     1 E      Ain        Infants    33120
#>  6     1 E      Ain        Suicides   35039
#>  7     2 N      Aisne      Crime_pers 26226
#>  8     2 N      Aisne      Crime_prop  5521
#>  9     2 N      Aisne      Literacy      51
#> 10     2 N      Aisne      Donations   8901
#> # ℹ 500 more rows
  1. Plot the density, but make a different subplot by facet_wrap(~ variable). These plots all have different scales for the X and Y (density) values, so it is important to use scales="FREE". Moreover, I’m primarily interested in the shape of these distributions, so I suppress the Y axis tick marks and labels.
ggplot(data = guerry_long,
       aes(x=value, fill=TRUE)) +
  geom_density(alpha=0.2) +
  geom_rug() +
  facet_wrap(~variable, scales="free") +
  theme_bw(base_size = 14) +
  theme(legend.position = "none",
        axis.ticks.y=element_blank(),
        axis.text.y=element_blank())

You can see that all variables are positively skewed, Donations, Infants and Suicides particularly so, but not so much as to cause alarm.

It is also of interest to see whether and how these distributions differ according to Region. This is easy to do, using aes(... fill=Region)

col.region   <- colors()[c(149, 254, 468, 552, 26)] # colors for region
ggplot(data = guerry_long,
       aes(x=value, fill=Region)) +
  geom_density(alpha=0.2) +
  geom_rug() +
  facet_wrap(~variable, scales="free") +
  scale_fill_manual(values=col.region) +
  theme_bw(base_size = 14) +
  theme(legend.position = "bottom",
        axis.ticks.y=element_blank(),
        axis.text.y=element_blank())

For some variables, like Infants and Suicides the differences do not seem particularly large. However, both crime variables and Literacy show marked differences across region.

Bivariate relations

Let’s start with plots of crime (Crime_pers and Crime_prop) in relation to Literacy. A simple scatterplot is not very informative. All that can be seen is that there is not much of a relation between personal crime and literacy.

ggplot(aes(x=Literacy, y=Crime_pers/1000), data=Guerry) +
  geom_point(size=2)

More useful scatterplots are annotated with additional statistical summaries to aid interpretation:

  • linear regression line,
  • smoothed non-parametric (loess) curve, to diagnose potential non-linear relations,
  • data ellipses, to highlight the overall trend and variability,
  • point labels for potentially outlying or influential points.

I use ggplot2 here. It provides most of these features, except that to label unusual points, I calculate the Mahalanobis squared distance of all points from the grand means.

gdf <- Guerry[, c("Literacy", "Crime_pers", "Department")]
gdf$dsq <- mahalanobis(gdf[,1:2], colMeans(gdf[,1:2]), cov(gdf[,1:2]))

ggplot(aes(x=Literacy, y=Crime_pers/1000, label=Department), data=gdf) +
  geom_point(size=2) +
  stat_ellipse(level=0.68, color="blue", size=1.2) +  
  stat_ellipse(level=0.95, color="gray", size=1, linetype=2) + 
  geom_smooth(method="lm", formula=y~x, fill="lightblue") +
  geom_smooth(method="loess", formula=y~x, color="red", se=FALSE) +
  geom_label_repel(data = gdf[gdf$dsq > 4.6,]) +
  theme_bw()

The flat (blue) regression line and the nearly circular data ellipses show that the correlation is nearly zero; the smoothed (red) curve indicates that there is no tendency for a nonlinear relation.

Doing the same for crimes against property:

gdf <- Guerry[, c("Literacy", "Crime_prop", "Department")]
gdf$dsq <- mahalanobis(gdf[,1:2], colMeans(gdf[,1:2]), cov(gdf[,1:2]))

ggplot(aes(x=Literacy, y=Crime_prop/1000, label=Department), data=gdf) +
  geom_point(size=2) +
  stat_ellipse(level=0.68, color="blue", size=1.2) +  
  stat_ellipse(level=0.95, color="gray", size=1, linetype=2) + 
  geom_smooth(method="lm", formula=y~x, fill="lightblue") +
  geom_smooth(method="loess", formula=y~x, color="red", se=FALSE) +
  geom_label_repel(data = gdf[gdf$dsq > 4.6,]) +
  theme_bw()

So, somewhat surprisingly, increased literacy is associated with an increase in property crime (greater population per crime) as opposed to the situation with personal crime, which seems unrelated to literacy. Creuse again stands out as an unusual point, one that is likely to be influential in regression models.

Reconnaisance plots

Reconnaisance plots attempt to give a bird’s-eye overview of a multivariate data set. For example, to see the relations among more than two variables we could turn to a scatterplot matrix or some other display to show all pairwise bivariate relations.

For these, my preferred package is car (John Fox, Weisberg, and Price 2023) with the scatterplotMatrix function. GGally (Schloerke et al. 2021) works within the the ggplot2 framework, but doesn’t have the flexibility I’d like.

library(car)          # Companion to Applied Regression
scatterplotMatrix(Guerry[,4:9],
                  ellipse=list(levels=0.68), 
                  smooth=FALSE)

Corrgrams

Sometimes, particularly with more variables than this, we want to see a more schematic overview.
A correlation diagram or “corrgram” (Friendly 2002) is a graphic display of a correlation matrix, allowing different renderings of the correlation between each pair of variables: as a shaded box, a pie symbol, a schematic data ellipse, and other options. This is implemented in the corrgram package (Wright 2021). The panels in the upper and lower triangles can be rendered differently.

library(corrgram)             # Plot a Correlogram
corrgram(Guerry[,4:9], upper=panel.pie)

Or, the data in each pairwise tile can be rendered with data ellipses and smoothed curves to show possible nonlinear relations.

Another feature is that the rows/column variables can be permuted to put similar variables together, using the order option, which arranges the variables according to similarity of their correlations.

corrgram(Guerry[,4:9], 
         upper=panel.ellipse, 
         order=TRUE,
         lwd=2)

Here, there are a number of pairwise plots that appear markedly nonlinear. For the main crime variables, the most nonlinear are that of personal crime vs. donations to the poor, and property crime vs. infants born out of wedlock and suicides. Literacy stands out here as having negative relations with all other variables.

An alternative analysis might include:

  • converting the data to ranks.
  • considering transformations of some of the variables

Biplots

Rather than viewing the data in data space, a biplot shows the data in the reduced-rank PCA space that explains most of the variation of the observations. This is essentially a plot of the observation scores on the first principal component overlaid with vectors representing the variables projected into PCA space.

First, we use prcomp() to carry out the PCA. We’d like to visualize the result in relation to Region, so delete Corsica where Region is missing.

gdata <- Guerry |>
  select(Region, Crime_pers:Suicides) |>   # keep only main variables
  filter(!is.na(Region))                   # delete Corsica (Region==NA)

guerry.pca <- gdata |>
  select(-Region) |>
  prcomp(scale = TRUE)

print(guerry.pca, digits=3)
#> Standard deviations (1, .., p=6):
#> [1] 1.463 1.096 1.050 0.817 0.741 0.584
#> 
#> Rotation (n x k) = (6 x 6):
#>                PC1     PC2     PC3      PC4     PC5     PC6
#> Crime_pers -0.0659  0.5906 -0.6732  0.13973 -0.0102 -0.4172
#> Crime_prop -0.5123 -0.0884 -0.4765 -0.09861  0.1381  0.6884
#> Literacy    0.5118 -0.1294 -0.2090  0.00797  0.8213  0.0560
#> Donations  -0.1062  0.6990  0.4134 -0.47298  0.2742  0.1741
#> Infants    -0.4513  0.1033  0.3238  0.73031  0.3776 -0.0696
#> Suicides   -0.5063 -0.3569 -0.0169 -0.46220  0.2976 -0.5602

In the ggplot2 framework, biplots can be produced by the ggbiplot package (Vu and Friendly 2023), but this package is not on CRAN, so cannot be directly used in this vignette. Instead, the code below was run locally and the result included.

if(!require(ggbiplot)) remotes::install_github("vqv/ggbiplot")
library(ggbiplot) # A ggplot2 based biplot
ggbiplot(guerry.pca, groups=gdata$Region, 
         ellipse=TRUE,
         var.scale = 3, varname.size = 5) + 
  theme_bw() + 
  labs(color="Region") +
  theme(legend.position = c(0.1, 0.8))
knitr::include_graphics("figures/ggbiplot.png")

This is OK, but there are many features of such plots that cannot be customized (line widths, colors, … ). I prefer those created using the heplots package.

op <- par(mar=c(5,4,1,1)+.1)
cols = colorspace::rainbow_hcl(5)
covEllipses(guerry.pca$x, 
            group=gdata$Region, 
            pooled=FALSE, 
            fill=TRUE, fill.alpha=0.1,
            col=cols, 
            label.pos=c(3,0,1,1,3), 
            cex=2,
            xlim=c(-4,4), ylim=c(-4,4),
            xlab = "Dimension 1 (35.7 %)", 
            ylab = "Dimension 2 (20.0 %)",
            cex.lab=1.4
            )
points(guerry.pca$x, pch=(15:19)[Guerry$Region], col=cols[Guerry$Region])

candisc::vectors(guerry.pca$rotation, scale=5,  
                 col="black", lwd=3, cex=1.4, 
                 pos = c(4,2,4,2,2,2),
                 xpd=TRUE)
abline(h=0, v=0, col=gray(.70))

An interpretation can be read from both the directions of the variable arrows and the relative positions of the ellipses representing the scatter of the component scores for the different regions.

  • The first component is largely aligned positively with Literacy and negatively with property crime, suicides and children born out of wedlock (Infants)
  • The second dimension reflects mainly the correlation of personal crime and donations to the poor.
  • The South region is generally lower on PC2, the West, generally higher.
  • The North stands out as being higher than the others on PC1; the West somewhat higher on PC2.

Models

Here we illustrate:

  • Model based plots for linear regression models predicting personal crime and property crime
  • Multivariate analysis of variance (MANOVA) and HE plots for the joint relation of the crime variables to other predictors.

Predicting crime: Univariate regression

The simplest approach to predicting the crime variables would be to fit a separate multiple regression to each.

crime.mod1 <- lm(Crime_pers ~  Region + Literacy + Donations +  Infants + Suicides, data=Guerry)
crime.mod2 <- lm(Crime_prop ~  Region + Literacy + Donations +  Infants + Suicides, data=Guerry)

Tests for the predictors are best obtained using car::Anova() which gives partial (Type II) tests, adjusting for other predictors, rather than the sequential (Type I) tests provided by stats::anova()

Anova(crime.mod1)
#> Anova Table (Type II tests)
#> 
#> Response: Crime_pers
#>               Sum Sq Df F value    Pr(>F)    
#> Region    1388267847  4  9.0398 5.005e-06 ***
#> Literacy    77140249  1  2.0092    0.1604    
#> Donations   54505520  1  1.4197    0.2372    
#> Infants       102152  1  0.0027    0.9590    
#> Suicides      205432  1  0.0054    0.9419    
#> Residuals 2917886368 76                      
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova(crime.mod2)
#> Anova Table (Type II tests)
#> 
#> Response: Crime_prop
#>              Sum Sq Df F value    Pr(>F)    
#> Region     52269436  4  2.0939 0.0898250 .  
#> Literacy   13366819  1  2.1419 0.1474514    
#> Donations   9218353  1  1.4771 0.2279870    
#> Infants     7577617  1  1.2142 0.2739759    
#> Suicides  100890796  1 16.1665 0.0001355 ***
#> Residuals 474296314 76                      
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

These are somewhat disappointing if you look only at the significance stars: Only Region is significant for personal crime and only Suicides for property crime. There is no evidence for the argument, supported by the liberal hygenicists of Guerry’s time, that increased Literacy would reduce crime.

For such models, we can understand the nature of the predicted effects using the effects package. The (marginal) effect for a given term gives the predicted values, averaging over all other terms in the model.

plot(predictorEffects(crime.mod1, ~ Region + Literacy + Infants + Suicides), 
     lwd=2, main="")

Doing the same for property crime, we get:

plot(predictorEffects(crime.mod2, ~ Region + Literacy + Infants + Suicides), 
     lwd=2, main="")

Predicting crime: Multivariate regression

The two regression models can be fit together in a multivariate regression for both crime variables jointly.

crime.mod <- lm(cbind(Crime_pers, Crime_prop) ~ 
                Region + Literacy + Donations +  Infants + Suicides, data=Guerry)
Anova(crime.mod)
#> 
#> Type II MANOVA Tests: Pillai test statistic
#>           Df test stat approx F num Df den Df    Pr(>F)    
#> Region     4   0.42933   5.1936      8    152 9.563e-06 ***
#> Literacy   1   0.03707   1.4434      2     75 0.2425951    
#> Donations  1   0.02615   1.0071      2     75 0.3701736    
#> Infants    1   0.01833   0.7001      2     75 0.4997450    
#> Suicides   1   0.20772   9.8315      2     75 0.0001615 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

As a quick check on the assumption that the residuals are bivariate normally distributed and a check for outliers, a \(\chi^2\) Q-Q plot graphs the squared Mahalanobis distances of the residuals against the corresponding \(\chi^2_2\) quantiles these would have in a bivariate normal distribution. The data for Creuse stands out as a potential outlier.

labels <- paste0(Guerry$dept,":", Guerry$Department)
cqplot(crime.mod, id.n=4, labels=labels)

HE plots

Hypothesis-Error (HE) plots (Friendly 2007; J. Fox, Friendly, and Monette 2009; Friendly, Fox, and Georges Monette 2022) provide a convenient graphical summary of hypothesis tests in multivariate linear model. They plot a data ellipse for the residuals in the model, representing the \(\mathbf{E}\) matrix in the test statistics (Roy’s maximum root test, Pillai and Hotelling trace criteria and Wilks’ Lambda). Overlaid on this are \(\mathbf{H}\) ellipses for each term in the model, representing the data ellipses for the fitted values. Using Roy’s test, these have a convenient interpretation: a term is significant iff the H ellipse projects anywhere outside the E ellipse. For a 1 df (regression) variable, the H ellipse collapses to a line.

heplot(crime.mod, 
       fill=TRUE, fill.alpha=0.05, 
       cex=1.4, cex.lab=1.3 )

In this plot, the effect of Suicides is completely aligned with crimes against property. The effect of Region is positively correlated with both types of crime. The means for the regions show that the South of France is lower (worse) on personal crime; the other regions vary most in property crime, with the North being lower and the Center being higher.

Canonical plots

The HE plot displays these relations in data space. An alternative is provided by canonical discriminant analysis, which finds the weighted sums of the response variables leading to the largest test statistics for the terms, which can be visualized in canonical space.

The analysis below reflects the effect of Region in relation to both crime variables.

crime.can <- candisc(crime.mod)
crime.can
#> 
#> Canonical Discriminant Analysis for Region:
#> 
#>     CanRsq Eigenvalue Difference Percent Cumulative
#> 1 0.337068    0.50845    0.40681   83.34      83.34
#> 2 0.092267    0.10164    0.40681   16.66     100.00
#> 
#> Test of H0: The canonical correlations in the 
#> current row and all that follow are zero
#> 
#>   LR test stat approx F numDF denDF   Pr(> F)    
#> 1      0.60177   5.7097     8   158 2.209e-06 ***
#> 2      0.90773   2.7105     3    80   0.05051 .  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The HE plot for this analysis is shown below. Variable vector represent the correlations of the crime variables with the canonical dimension.

heplot(crime.can, fill=TRUE, fill.alpha=0.1,
       var.col = "black", 
       var.cex = 1.3,
       cex=1.4, cex.lab=1.3)

#> Vector scale factor set to  3.10537

This gives a simple interpretation of the differences in Region on the crime variables. The first canonical dimension accounts for 83% of differences among the regions, and this is nearly perfectly aligned with personal crime, with the largest difference between the South and the other regions. The second canonical dimension, accounting for the remaining 17%, is perfectly aligned with property crime. On this dimension, the North stands out compared to the other regions.

References

Fox, J., M. Friendly, and G. Monette. 2009. “Visualizing Hypothesis Tests in Multivariate Linear Models: The Heplots Package for R.” Computational Statistics 24 (2): 233–46. https://datavis.ca/papers/FoxFriendlyMonette-2009.pdf.
Fox, John, Sanford Weisberg, and Brad Price. 2023. Car: Companion to Applied Regression. https://r-forge.r-project.org/projects/car/.
Friendly, Michael. 2002. “Corrgrams: Exploratory Displays for Correlation Matrices.” The American Statistician 56 (4): 316–24. http://datavis.ca/papers/corrgram.pdf.
———. 2007. “HE Plots for Multivariate General Linear Models.” Journal of Computational and Graphical Statistics 16 (4): 421–44. http://datavis.ca/papers/jcgs-heplots.pdf.
Friendly, Michael, John Fox, and and Georges Monette. 2022. heplots: Visualizing Tests in Multivariate Linear Models. https://CRAN.R-project.org/package=heplots.
Galton, Francis. 1886. “Regression Towards Mediocrity in Hereditary Stature.” Journal of the Anthropological Institute 15: 246–63.
Guerry, André-Michel. 1833. Essai Sur La Statistique Morale de La France. Paris: Crochard.
Schloerke, Barret, Di Cook, Joseph Larmarange, Francois Briatte, Moritz Marbach, Edwin Thoen, Amos Elberg, and Jason Crowley. 2021. GGally: Extension to Ggplot2. https://ggobi.github.io/ggally/.
Vu, Vincent, and Michael Friendly. 2023. Ggbiplot: A Ggplot2 Based Biplot. https://github.com/friendly/ggbiplot.
Wright, Kevin. 2021. Corrgram: Plot a Correlogram. https://kwstat.github.io/corrgram/.
Guerry/inst/doc/MultiSpat.Rmd0000644000176200001440000005474414513000475015654 0ustar liggesusers--- title: "Guerry data: Spatial Multivariate Analysis" author: "Stéphane Dray" date: "`r Sys.Date()`" output: rmarkdown::html_vignette: toc: true toc_depth: 2 bibliography: reference.bib link-citations: yes vignette: > %\VignetteIndexEntry{Guerry data: Spatial Multivariate Analysis} %\VignetteKeywords{crime, literacy, suicide, France, spatial multivariate analysis} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r setup, include = FALSE} knitr::opts_chunk$set( collapse = TRUE, warning = FALSE, message = FALSE, # suppress package loading messages comment = "#>", fig.height = 3, fig.width = 3, fig.align = "center" ) ``` This vignette indicates how to perform the analyses described in @SD966 of data derived from André-Michel Guerry's [-@SD955] *Essai sur la Statistique Morale de la France*. It illustrates some classical methods for analysis of multivariate spatial data that focus *either* on the multivariate aspect or on the spatial one, as well as some more modern methods that attempt to integrate geographical and multivariate aspects *simultaneously*. # Preliminary steps Several packages are required to run the different analyses and should be loaded. The `ade4` package for multivariate analysis is supplemented by `adegraphics` (for associated graphical methods) and `adespatial` for multivariate spatial analysis. For further information on this spatial analysis approach, see `vignette(package="adespatial", "tutorial")`. ```{r} library(Guerry) # Guerry's data library(sp) # management of spatial data library(ade4) # multivariate analysis library(adegraphics) # graphical representation library(spdep) # spatial dependency library(adespatial) # multivariate spatial analysis ``` Guerry gathered data on crimes, suicide, literacy and other moral statistics for various départements (i.e., counties) in France. He provided the first real social data analysis, using graphics and maps to summarize this georeferenced multivariate dataset. We use the dataset `gfrance85` and consider six key quantitative variables (shown in the table below) for each of the 85 départements of France in 1830 (Corsica, an island and often an outlier, was excluded). Data are recorded on aligned scales so that **larger numbers** consistently reflect "morally better". Thus, four of the "bad" variables are recorded in the inverse form, as "Population per ...". With this scaling, it would be expected that all correlations be $\ge 0$. | Name | Description | |:------------|:------------| |`Crime_pers` | Population per crime against persons| |`Crime_prop` | Population per crime against property| |`Literacy` | Percent of military conscripts who can read and write| |`Donations` | Donations to the poor| |`Infants` | Population per illegitimate birth| |`Suicides` | Population per suicide| The dataset `gfrance85` is actually a `SpatialPolygonsDataFrame` object created with the `sp` package. It contains the polygon boundaries of the map of France in 1830, as well the variables in the `Guerry` data frame. As an ordinary data.frame, it has these components ```{r names} names(gfrance85) ``` To simplify analyses, we extract the components to be used below. ```{r components} data(gfrance85) df <- data.frame(gfrance85)[, 7:12] # the 6 variables france.map <- as(gfrance85, "SpatialPolygons") # the map xy <- coordinates(gfrance85) # spatial coordinates dep.names <- data.frame(gfrance85)[, 6] # departement names region.names <- data.frame(gfrance85)[, 5] # region names col.region <- colors()[c(149, 254, 468, 552, 26)] # colors for region ``` # Standard approaches In this section, we focus on classical approaches that consider either the multivariate or the spatial aspect of the data. ## Multivariate analysis Here we consider $p=6$ variables measured for $n=85$ individuals (départements of France). As only quantitative variables have been recorded, principal component analysis [PCA, @SD308] is well adapted. PCA summarizes the data by maximizing simultaneously the variance of the projection of the individuals onto the principal axes and the sum of the squared correlations between the principal component and the variables. ```{r dudi-pca} pca <- dudi.pca(df, scannf = FALSE, nf = 3) ``` The biplot is simply obtained by ```{r} biplot(pca, plabel.cex = 0.8) ``` The first two PCA dimensions account for 35.7% and 20% ,respectively, of the total variance. ```{r} pca$eig/sum(pca$eig) * 100 ``` Correlations between variables and principal components can be represented on a correlation circle. The first axis is negatively correlated to literacy and positively correlated to property crime, suicides and illegitimate births. The second axis is aligned mainly with personal crime and donations to the poor. ```{r} s.corcircle(pca$co) ``` Spatial information can be added on the factorial map representing the projections of départements on principal axes by coloring according to colors representing the different regions of France. ```{r, fig.width = 4, fig.height = 4} s.label(pca$li, ppoint.col = col.region[region.names], plabel.optim = TRUE, plabel.cex = 0.6) s.Spatial(france.map, col = col.region[region.names], plabel.cex = 0) s.class(xy, region.names, col = col.region, add = TRUE, ellipseSize = 0, starSize = 0) ``` For the first axis, the North and East are characterized by negative scores, corresponding to high levels of literacy and high rates of suicides, crimes against property and illegitimate births. The second axis mainly contrasts the West (high donations to the the poor and low levels of crime against persons) to the South. ## Spatial autocorrelation Spatial autocorrelation statistics, such as @SD455 Coefficient (MC) and @SD223 Ratio, aim to measure and analyze the degree of dependency among observations in a geographical context [@SD577]. ### The spatial weighting matrix The first step of spatial autocorrelation analysis is to define a spatial weighting matrix $\mathbf{W}=[w_{ij}]$ . In the case of Guerry's data, we simply defined a binary neighborhood where two départements are considered as neighbors if they share a common border. The spatial weighting matrix is then obtained after row-standardization (`style = "W"`): ```{r} nb <- poly2nb(gfrance85) lw <- nb2listw(nb, style = "W") ``` We can represent this neighborhood on the geographical map: ```{r} s.Spatial(france.map, nb = nb, plabel.cex = 0, pSp.border = "white") ``` ### Moran's Coefficient Once the spatial weights have been defined, the spatial autocorrelation statistics can then be computed. Let us consider the $n$-by-1 vector $\mathbf{x}=\left[ {x_1 \cdots x_n } \right]^\textrm{T}$ containing measurements of a quantitative variable for $n$ spatial units. The usual formulation for Moran's coefficient of spatial autocorrelation [@SD577;@SD455] is \begin{equation} \label{eq1} MC(\mathbf{x})=\frac{n\sum\nolimits_{\left( 2 \right)} {w_{ij} (x_i -\bar {x})(x_j -\bar {x})} }{\sum\nolimits_{\left( 2 \right)} {w_{ij} } \sum\nolimits_{i=1}^n {(x_i -\bar {x})^2} }\mbox{ where }\sum\nolimits_{\left( 2 \right)} =\sum\limits_{i=1}^n {\sum\limits_{j=1}^n } \mbox{ with }i\ne j. \end{equation} MC can be rewritten using matrix notation: \begin{equation} \label{eq2} MC(\mathbf{x})=\frac{n}{\mathbf{1}^\textrm{T}\mathbf{W1}}\frac{\mathbf{z}^\textrm{T}{\mathbf{Wz}}}{\mathbf{z}^\textrm{T}\mathbf{z}}, \end{equation} where $\mathbf{z}=\left ( \mathbf{I}_n-\mathbf{1}_n\mathbf{1}_n^\textrm{T} /n \right )\mathbf{x}$ is the vector of centered values ($z_i=x_i-\bar{x}$) and $\mathbf{1}_n$ is a vector of ones (of length $n$). The significance of the observed value of MC can be tested by a Monte-Carlo procedure, in which locations are permuted to obtain a distribution of MC under the null hypothesis of random distribution. An observed value of MC that is greater than that expected at random indicates the clustering of similar values across space (positive spatial autocorrelation), while a significant negative value of MC indicates that neighboring values are more dissimilar than expected by chance (negative spatial autocorrelation). We computed MC for the Guerry's dataset. A positive and significant autocorrelation is identified for each of the six variables. Thus, the values of literacy are the most covariant in adjacent departments, while illegitimate births (Infants) covary least. ```{r} moran.randtest(df, lw) ``` ### Moran scatterplot If the spatial weighting matrix is row-standardized, we can define the lag vector $\mathbf{\tilde{z}} = \mathbf{Wz}$ (i.e., $\tilde{z}_i = \sum\limits_{j=1}^n{w_{ij}x_j}$) composed of the weighted (by the spatial weighting matrix) averages of the neighboring values. Thus, we have: \begin{equation} \label{eq3} MC(\mathbf{x})=\frac{\mathbf{z}^\textrm{T}{\mathbf{\tilde{z}}}}{\mathbf{z}^\textrm{T}\mathbf{z}}, \end{equation} since in this case $\mathbf{1}^\textrm{T}\mathbf{W1}=n$. This shows clearly that MC measures the autocorrelation by giving an indication of the intensity of the linear association between the vector of observed values $\mathbf{z}$ and the vector of weighted averages of neighboring values $\mathbf{\tilde{z}}$. @SD566 proposed to visualize MC in the form of a bivariate scatterplot of $\mathbf{\tilde{z}}$ against $\mathbf{z}$. A linear regression can be added to this *Moran scatterplot*, with slope equal to MC. Considering the Literacy variable of Guerry's data, the Moran scatterplot clearly shows strong autocorrelation. It also shows that the Hautes-Alpes département has a slightly outlying position characterized by a high value of Literacy compared to its neighbors. ```{r, fig.width = 4, fig.height=4} x <- df[, 3] x.lag <- lag.listw(lw, df[, 3]) moran.plot(x, lw) text(x[5], x.lag[5], dep.names[5], pos = 1, cex = 0.8) ``` ## Indirect integration of multivariate and geographical aspects The simplest approach considered a two-step procedure where the data are first summarized with multivariate analysis such as PCA. In a second step, univariate spatial statistics or mapping techniques are applied to PCA scores for each axis separately. One can also test for the presence of spatial autocorrelation for the first few scores of the analysis, with univariate autocorrelation statistics such as MC. We mapped scores of the départements for the first two axes of the PCA of Guerry's data. Even if PCA maximizes only the variance of these scores, there is also a clear spatial structure, as the scores are highly autocorrelated. The map for the first axis corresponds closely to the split between *la France éclairée* (North-East characterized by an higher level of Literacy) and *la France obscure*. ```{r, fig.dim = c(6,3)} moran.randtest(pca$li, lw) s.value(xy, pca$li[, 1:2], Sp = france.map, pSp.border = "white", symbol = "circle", pgrid.draw = FALSE) ``` # Spatial multivariate analysis Over the last decades, several approaches have been developed to consider both geographical and multivariate information simultaneously. The multivariate aspect is usually treated by techniques of dimensionality reduction similar to PCA. On the other hand, several alternatives have been proposed to integrate the spatial information. ## Spatial partition One alternative is to consider a spatial partition of the study area. In this case, the spatial information is coded as a categorical variable, and each category corresponds to a region of the whole study area. For instance, Guerry's data contained a partition of France into 5 regions. We used the between-class analysis [BCA, @SD148], to investigate differences between regions. BCA maximizes the variance between groups. ```{r} bet <- bca(pca, region.names, scannf = FALSE, nf = 2) ``` Here, 28.8 % of the total variance (sum of eigenvalues of PCA) corresponds to the between-regions variance (sum of the eigenvalues of BCA). ```{r} bet$ratio ``` The main graphical outputs are obtained by the generic `plot` function: ```{r, fig.dim=c(5,5)} plot(bet) ``` The barplot of eigenvalues indicates that two axes should be interpreted. The first two BCA dimensions account for 59 % and 30.2 %, respectively, of the between-regions variance. ```{r} barplot(bet$eig) bet$eig/sum(bet$eig) * 100 ``` The coefficients used to construct the linear combinations of variables are represented: ```{r} s.arrow(bet$c1, plabel.cex = 0.8) ``` The first axis opposed literacy to property crime, suicides and illegitimate births. The second axis is mainly aligned with personal crime and donations to the poor. Projections of départements on the BCA axes can be represented on the factorial map: ```{r, fig.dim = c(4,4)} s.label(bet$ls, as.character(dep.names), ppoint.cex = 0, plabel.optim = TRUE, plabel.col = col.region[region.names], plabel.cex = 0.5) s.class(bet$ls, fac = region.names, col = col.region, ellipse = 0, add = TRUE) ``` The scores can be mapped to show the spatial aspects: ```{r, fig.dim = c(6,3)} s.value(xy, bet$ls, symbol = "circle", Sp = france.map, pSp.col = col.region[region.names], pSp.border = "transparent") ``` The results are very close to those obtained by PCA: the first axis contrasted the North and the East (*la France éclairée*) to the other regions while the South is separated from the other regions by the second axis. The high variability of the region Centre is also noticeable. In contrast, the South is very homogeneous. ## Spatial explanatory variables Principal component analysis with respect to the instrumental variables [PCAIV, @SD540], and related methods, have been often used in community ecology to identify spatial relationships. The spatial information is introduced in the form of spatial predictors and the analysis maximized the "spatial variance" (i.e., the variance explained by spatial predictors). Note that BCA can also be considered as a particular case of PCAIV, where the explanatory variables are dummy variables indicating group membership. ### Trend surface of geographic coordinates @SD626 proposed to express observed values in time series as a polynomial function of time, and mentioned that this could be done for spatial data as well. @SD59 extended this approach to the spatial and multivariate case by introducing polynomial functions of geographic coordinates as predictors in PCAIV. We call this approach PCAIV-POLY. The centroids of départements of France were used to construct a second-degree orthogonal polynomial. ```{r, fig.dim = c(6,4)} poly.xy <- orthobasis.poly(xy, degree = 2) s.value(xy, poly.xy, Sp = france.map, plegend.drawKey = FALSE) ``` PCAIV is then performed using the `pcaiv` function: ```{r} pcaiv.xy <- pcaiv(pca, poly.xy, scannf = FALSE, nf = 2) ``` Here, 32.4 % of the total variance (sum of eigenvalues of PCA) is explained by the second-degree polynomial (sum of eigenvalues of PCAIV). The first two dimensions account for 51.4 % and 35.2 %, respectively, of the explained variance. ```{r} sum(pcaiv.xy$eig)/sum(pca$eig) * 100 pcaiv.xy$eig/sum(pcaiv.xy$eig) * 100 ``` The outputs of PCAIV-POLY (coefficients of variables, maps of départements scores, etc.) are very similar to those obtained by BCA. They can be represented easily by the generic `plot` function: ```{r, fig.dim=c(5,5)} plot(pcaiv.xy) ``` ### Moran's eigenvector maps An alternative way to build spatial predictors is by the diagonalization of the spatial weighting matrix **W**. Moran's eigenvector maps [MEM, @SD163] are the $n-1$ eigenvectors of the doubly-centered matrix **W**. They are orthogonal vectors with a unit norm maximizing MC [@SD264]. MEM associated with high positive (or negative) eigenvalues have high positive (or negative) autocorrelation. MEM associated with eigenvalues with small absolute values correspond to low spatial autocorrelation, and are not suitable for defining spatial structures. We used the spatial weighting matrix defined above to construct MEM. The first ten MEM, corresponding to the highest levels of spatial autocorrelation, have been mapped: ```{r, fig.dim = c(6,6)} mem1 <- scores.listw(lw) s.value(xy, mem1[, 1:9], Sp = france.map, plegend.drawKey = FALSE) ``` We introduced the first ten MEM as spatial explanatory variables in PCAIV. We call this approach PCAIV-MEM. ```{r} pcaiv.mem <- pcaiv(pca, mem1[,1:10], scannf = FALSE) ``` Here, 44.1 % of the total variance (sum of eigenvalues of PCA) is explained by the first ten MEM (sum of eigenvalues of PCAIV). The first two dimensions account for 54.9 % and 26.3 %, respectively, of the explained variance. ```{r} sum(pcaiv.mem$eig)/sum(pca$eig) * 100 pcaiv.mem$eig/sum(pcaiv.mem$eig) * 100 ``` The outputs of PCAIV-MEM (coefficients of variables, maps of départements scores, etc.) are very similar to those obtained by BCA. They can be represented easily by the generic `plot` function: ```{r, fig.dim=c(5,5)} plot(pcaiv.mem) ``` ## Spatial graph and weighting matrix The MEM framework introduced the spatial information into multivariate analysis through the eigendecomposition of the spatial weighting matrix. Usually, we consider only a part of the information contained in this matrix because only a subset of MEM are used as regressors in PCAIV. In this section, we focus on multivariate methods that consider the spatial weighting matrix under its original form. @SD694 was the first to develop a multivariate analysis based on MC. His work considered only normed and centered variables (i.e., normed PCA) for the multivariate part and a binary symmetric connectivity matrix for the spatial aspect. @SD807 generalized Wartenberg's method by introducing a row-standardized spatial weighting matrix in the analysis of a statistical triplet. This approach is very general and allows us to define spatially-constrained versions of various methods (corresponding to different triplets) such as correspondence analysis or multiple correspondence analysis. MULTISPATI finds coefficients to obtain a linear combination of variables that maximizes a compromise between the classical multivariate analysis and a generalized version of Moran's coefficient. ```{r} ms <- multispati(pca, lw, scannf = FALSE) ``` The main outputs of MULTISPATI can be represented easily by the generic `plot` function: ```{r, fig.dim=c(5,5)} plot(ms) ``` The barplot of eigenvalues suggests two main spatial structures. Eigenvalues of MULTISPATI are the product between the variance and the spatial autocorrelation of the scores, while PCA maximizes only the variance. The differences between the two methods are computed by the `summary` function: ```{r} summary(ms) ``` Hence, there is a loss of variance compared to PCA (2.14 versus 2.017 for axis 1; 1.201 versus 1.177 for axis 2) but a gain of spatial autocorrelation (0.551 versus 0.637 for axis 1; 0.561 versus 0.59 for axis 2). Coefficients of variables allow to interpret the structures: ```{r} s.arrow(ms$c1, plabel.cex = 0.8) ``` The first axis opposes literacy to property crime, suicides and illegitimate births. The second axis is aligned mainly with personal crime and donations to the poor. The maps of the scores show that the spatial structures are very close to those identified by PCA. The similarity of results between PCA and its spatially optimized version confirm that the main structures of Guerry's data are spatial. Spatial autocorrelation can be seen as the link between one variable and the lagged vector. This interpretation is used to construct the Moran scatterplot and can be extended to the multivariate case in MULTISPATI by analyzing the link between scores and lagged scores: ```{r, fig.dim = c(4,4)} s.match(ms$li, ms$ls, plabel.cex = 0) s.match(ms$li[c(10, 41, 27), ], ms$ls[c(10, 41, 27), ], label = dep.names[c(10, 41, 27)], plabel.cex = 0.8, add = TRUE) ``` Each département can be represented on the factorial map by an arrow (the bottom corresponds to its score, the head corresponds to its lagged score. A short arrow reveals a local spatial similarity (between one plot and its neighbors) while a long arrow reveals a spatial discrepancy. This viewpoint can be interpreted as a multivariate extension of the local index of spatial association [@SD565]. For instance: * Aude has a very small arrow, indicating that this département is very similar to its neighbors. * Haute-Loire has a long horizontal arrow which reflects its high values for the variables Infants (31017), Suicides (163241) and Crime\_prop (18043) compared to the average values over its neighbors (27032.4, 60097.8 and 10540.8 for these three variables). * Finistère corresponds to an arrow with a long vertical length which is due to its high values compared to its neighbors for Donations (23945 versus 12563) and Crime\_pers (29872 versus 25962). The link between the scores and the lagged scores (averages of neighbors weighted by the spatial connection matrix) can be mapped in the geographical space. For the first two axes, we have: ```{r, fig.dim = c(6,3)} s.value(xy, ms$li, Sp = france.map) ``` # Conclusions Even if the methods presented are quite different in their theoretical and practical viewpoints, their applications to Guerry's dataset yield very similar results. We provided a quantitative measure of this similarity by computing Procrustes statistics [@SD516;SD161] between the scores of the départements onto the first two axes for the different analyses. All the values of the statistic are very high and significant; this confirms the high concordance between the outputs of the different methods. ```{r} mat <- matrix(NA, 4, 4) mat.names <- c("PCA", "BCA", "PCAIV-POLY", "PCAIV-MEM", "MULTISPATI") colnames(mat) <- mat.names[-5] rownames(mat) <- mat.names[-1] mat[1, 1] <- procuste.randtest(pca$li[, 1:2], bet$ls[, 1:2])$obs mat[2, 1] <- procuste.randtest(pca$li[, 1:2], pcaiv.xy$ls[, 1:2])$obs mat[3, 1] <- procuste.randtest(pca$li[, 1:2], pcaiv.mem$ls[, 1:2])$obs mat[4, 1] <- procuste.randtest(pca$li[, 1:2], ms$li[, 1:2])$obs mat[2, 2] <- procuste.randtest(bet$ls[, 1:2], pcaiv.xy$ls[, 1:2])$obs mat[3, 2] <- procuste.randtest(bet$ls[, 1:2], pcaiv.mem$ls[, 1:2])$obs mat[4, 2] <- procuste.randtest(bet$ls[, 1:2], ms$li[, 1:2])$obs mat[3, 3] <- procuste.randtest(pcaiv.xy$ls[, 1:2], pcaiv.mem$ls[, 1:2])$obs mat[4, 3] <- procuste.randtest(pcaiv.xy$ls[, 1:2], ms$li[, 1:2])$obs mat[4, 4] <- procuste.randtest(pcaiv.mem$ls[, 1:2], ms$li[, 1:2])$obs mat ``` # References Guerry/inst/WORDLIST0000644000176200001440000000152314502054541013677 0ustar liggesusersAlmanach Andr André Angeville Angeville's Angleterre Bailliere Brunsdon Compte Comptes Corrgrams Creuse Crochard DOI Darfour De Doufour Duchatelet Dupin Essai GISRUK Guerry's Guerry’s Hotelling Jombart Lewiston Lifecycle MDS Maynooth Mellen Multivariable NUI PROJ Pillai Reconnaisance Reinking Reinking's SpatialPolygonsDataFrame Statistique Stéphane Whitt Wilks aise avec centroid centroids choropleth compar comptes corrgram criminelle d'Angeville d'apres d'autorisation dans de des df du départements d’ d’Angeville eprint et etat fils fran francaise geo georeferenced geospatial guerre hygenicists l'Angleterre l'administration l'age le les loess lois mathbf ministere ministre officiel ordunn rendus sp statistique sur variables’ ville vivios