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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://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/gfrance85.Rd0000644000176200001440000000424214053175241014357 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) 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)] if (require(spdep)) { lw <- nb2listw(poly2nb(gfrance85)) # neighbors list } # 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=0.4) par(op) } \keyword{datasets} \keyword{spatial} Guerry/man/Guerry-package.Rd0000644000176200001440000000735314124372565015460 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). \doi{10.1214/10-aoas356}. 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/}. 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'apr?s les comptes de l'administration de la justice criminelle en Angleterre et en France, etc.} Paris: J.-B. Bailli?re et fils. } %~~ Optionally other standard keywords, one per line, from file KEYWORDS in ~~ %~~ the R documentation directory ~~ \keyword{spatial} %\examples{ %#~~ simple examples of the most important functions ~~ %} Guerry/man/Guerry.Rd0000644000176200001440000001627714124362261014064 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{ André-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 général, 1825-1830)} \item{\code{Crime_prop}}{Population per Crime against property. Source: A2 (Compte général, 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 général, 1827-1830)} \item{\code{MainCity}}{Size of principal city ('1:Sm', '2:Med', '3:Lg'), used as a surrogate for poulation 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 général) } \item{\code{Infanticide}}{Infanticides per capita. A ranked ratio of number of infanticides to population-- Average for the years 1825-1830. Source: A1 (Compte général) } \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 disertion, 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 francais}, 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. and Jombart, T. (2009). A Revisit Of Guerry's Data: Introducing Spatial Constraints In Multivariate Analysis. Unpublished manuscript. Brunsdon, C. and Dykes, J. (2007). Geographically weighted visualization: interactive graphics for scale-varying exploratory analysis. 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{http://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.Rd0000644000176200001440000000666314053175241014514 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{ data(Angeville) ## maybe str(Angeville) ; plot(Angeville) ... } \keyword{datasets} Guerry/DESCRIPTION0000644000176200001440000000265114125066266013251 0ustar liggesusersPackage: Guerry Type: Package Title: Maps, Data and Methods Related to Guerry (1833) "Moral Statistics of France" Version: 1.7.4 Date: 2021-09-27 Authors@R: c(person(given = "Michael", family = "Friendly", role=c("aut", "cre"), email="friendly@yorku.ca"), person(given = "Stephane", family = "Dray", role="aut", email="stephane.dray@univ-lyon1.fr"), person(given = "Roger", family = "Bivand", role="ctb") ) Author: Michael Friendly [aut, cre], Stephane Dray [aut], Roger Bivand [ctb] Maintainer: Michael Friendly Encoding: latin1 Language: en-US Depends: R (>= 2.10) Suggests: knitr, spdep, ade4, adegraphics, adespatial, maptools, RColorBrewer, corrgram, car, rmarkdown, ggplot2, ggrepel, heplots, patchwork, candisc, colorspace Imports: sp 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: 2021-09-28 13:52:26 UTC; friendly Repository: CRAN Date/Publication: 2021-09-29 13:40:05 UTC Guerry/build/0000755000176200001440000000000014124617031012625 5ustar liggesusersGuerry/build/vignette.rds0000644000176200001440000000042014124617031015160 0ustar liggesuserseQMk0 u$ 9۠d1] c;-~:X'mK/8K9\AkI+aѾxj(yIc!dЂ-ڈyWieoF9 !4oA˸d#mwOr 4  lB "Dk;}¥O//9EO?Mt)LV ;tllտUYӜGW)qL.ۿv$sT9s?BjE/~5U /My9d/' QS4 2.ذW(s)/gρ0̯~> _c2™Jqa.QeNMͨem{Qq욥*FvVlέK;4TuCU'ʃY-J(&QJ4\i+Id-@GvE#TpQPﷀϥryJEiËd-l| A|:`#ovZ= aįt[(ՁgdO! &A|zfC!xG* z=$V%OBU3:iĢX 4Kݮa k2 㹲 `!>jŏ_gyPV5\<#("ֵ,菮t|w> ?ǭBn3D,?h;6g+Eu@sd1ufi(F֩[4FrNT#Eħە52#սӍ@漩9tb]W=LIQw ͓՘K+5ݨ2 1+b~QAD|3P+Ru94p5IXU-sIxZIux,M ݥFɞл:݋*Jli5y&p>TLIegˠ8cP!$,gˈ/ǖec۱v5ɶ(ctuaU%uK|%jRw~FR7߈-ӷ`aN8zŶW1]SJ҄fJ4YZr@&Ы8/QJ*y෈˩7'X{D(Ƞ3v#DҨ8nWDpb—]}' w?-YxxTޙV \"+,*{I]Lx[hTnL.?.5,6 OCl;ܟ #<ÈX?MI Ljkʉ9rdvRm;5e EQtM=bg`Éir8D:Zrb,pz ?u8|K223=hg9(|41'qstCþɝ(QmM,sR<)jIQF >[%C2cu|LccwӄVx a+ nj:SA.W}䥷'۪=y;pѓj8yn:97Q:IdA62%C c]Nl,և,GO}zLz/C|:4I9"Ejk_ZSD1,!6xgJ~؉[f#Uıؘa20$ g?-Dog 5<$ӝNdo2wOB!t x=LEЙ5}/fs!>1-4&ёSYT=we܃ZC >S̈́ODtyE?Hy]e`e@PW]WdKqt6G3dQqVՉ\җ7ozZa2Aw,)#E[?K !kObjxdK6*Uu,BW9ydaɺpfz+6%D`B[79sb>}޶ L1/bj,|1 `(MD7GbalDA(Gқӿ&3SS3٩ Rؽ_~JWG& 8ϩx܈u^ h3>GZKn?/R#g?~QqJ/Lkǡq&wk$1©{*kc?coG-@p )*0CacgX}S=U'}c\i]%mrIJ(t./"Ep}H2 B܊{gHڇ>̳\K#!7Or)`1| ׇc/tܛ}8qѣ6# oz6vdm&l) WFU7eKv~zf*ckňqxD8̡ǝ b>ħl05GeMJ \ڶAa/ Ol)#WM'T LQezr@|#Lm_C>f$bnd#zu ~xG9gYG6^t=iOtߵHke M5į%f馅QLZUW⯏Zdp W?r"\E>b󀮿 jԲhmy5cE̒2qy1ǗV2ʐʺWr]nR=8L2sdIMNA-2&#v "N߬E>bH'/;TG 7>|8H~:'A|v]n+ٴ_x$ohmr4^,"6Kn3E~L Kab}:\%vg]#VEsCU1OӉξG=7]Kg@I(2yCJ(E/&> "P>◺_+@ m65dRRyb6zy|E|g%OL6ܾŝ A9W +Cg,]S\{^6PsCUЙf)վUXME4ӀbČ{}2` < reoqS2ԅViyM sJ}d~@kު:N#s_la[wiE%P[zԨܔZ84 \ 5;[{qz XDɽ,le vLyA?w [U7_ ; s\67qoש̽,++洬i̲N:w_G?_?`YNyۆkm ܊dKٜ:9]TLj D@./&^튴Yjn:`/Zlu[VnTHSa5k6^X-t߾({8r 9%üe 2$3T^ooM1sW#S1e!7!\oΊ񭿝ޝMp-fCtzo CD _ODdڈ-ɘ:$ :Jp8A o{EIcjnyxz8PzWCZ䃉D|[#'Z/VW77EC{}ȇd@GZLBCm~ %(M$$q&QY9f}/٪F q-qO=u Z|T{fVzW .E cv/PHAVGuerry/vignettes/0000755000176200001440000000000014124617031013536 5ustar liggesusersGuerry/vignettes/reference.bib0000644000176200001440000001155614063162341016163 0ustar liggesusers@incollection{SD566, author = {Anselin, L.}, title = {The {M}oran scatterplot as an {ESDA} tool to assess local instability in spatial association}, booktitle = {Spatial analytical perspectives on GIS}, publisher = {Taylor and Francis}, year = {1996}, editor = {Fischer, M.M. and Scholten, H.J. and Unwin, D.}, pages = {111-125}, address = {London} } @article{SD565, author = {Anselin, L.}, title = {Local indicators of spatial association}, journal = {Geographical Analysis}, year = {1995}, volume = {27}, pages = {93-115}, endnotereftype = {Journal Article} } @article{SD59, author = {Borcard, D. and Legendre, P. and Drapeau, P.}, title = {Partialling out the spatial component of ecological variation}, journal = {Ecology}, year = {1992}, volume = {73}, pages = {1045-1055} } @book{SD577, title = {Spatial autocorrelation}, publisher = {Pion}, year = {1973}, author = {Cliff, A.D. and Ord, J.K.}, address = {London} } @article{SD148, author = {Dolédec, S. and Chessel, D.}, title = {Rythmes saisonniers et composantes stationnelles en milieu aquatique {I}- {D}escription d'un plan d'observations complet par projection de variables}, journal = {Acta Oecologica - Oecologia Generalis}, year = {1987}, volume = {8}, pages = {403-426}, number = {3} } @article{SD161, author = {Dray, S. and Chessel, D. and Thioulouse, J.}, title = {Procrustean co-inertia analysis for the linking of multivariate data sets}, journal = {Ecoscience}, year = {2003}, volume = {10}, pages = {110-119}, number = {1} } @article{SD966, author = {Dray, S. and Jombart, T.}, title = {Revisiting Guerry's data: introducing spatial constraints in multivariate analysis}, journal = {Annals of Applied Statistics}, year = {2011}, volume = {5}, pages = {2278-2299}, number = {4} } @article{SD163, author = {Dray, S. and Legendre, P. and Peres-Neto, P.R.}, title = {Spatial modeling: a comprehensive framework for principal coordinate analysis of neighbor matrices ({PCNM})}, journal = {Ecological Modelling}, year = {2006}, volume = {196}, pages = {483-493} } @article{SD807, author = {Dray, S. and Saïd, S. and Débias, F.}, title = {Spatial ordination of vegetation data using a generalization of {W}artenberg's multivariate spatial correlation}, journal = {Journal of Vegetation Science}, year = {2008}, volume = {19}, pages = {45-56} } @article{SD922, author = {Friendly, M.}, title = {A.-{M}. {G}uerry's moral statistics of {F}rance: challenges for multivariable spatial analysis}, journal = {Statistical Science}, year = {2007}, volume = {22}, pages = {368-399} } @article{SD223, author = {Geary, R.C.}, title = {The contiguity ratio and statistical mapping}, journal = {The incorporated Statistician}, year = {1954}, volume = {5}, pages = {115-145}, number = {3} } @article{SD264, author = {Griffith, D. A.}, title = {Spatial autocorrelation and eigenfunctions of the geographic weights matrix accompanying geo-referenced data}, journal = {Canadian Geographer}, year = {1996}, volume = {40}, pages = {351-367}, number = {4} } @book{SD955, title = {Essai sur la Statistique Morale de la France}, publisher = {Crochard}, year = {1833}, author = {Guérry, A.M.}, address = {Paris} } @article{SD308, author = {Hotelling, H.}, title = {Analysis of a complex of statistical variables into principal components}, journal = {Journal of Educational Psychology}, year = {1933}, volume = {24}, pages = {417-441} } @article{SD455, author = {Moran, P.A.P.}, title = {The interpretation of statistical maps}, journal = {Journal of the Royal Statistical Society Series B-Methodological}, year = {1948}, volume = {10}, pages = {243-251} } @article{SD516, author = {Peres-Neto, P.R. and Jackson, D.A.}, title = {How well do multivariate data sets match? {T}he advantages of a {P}rocrustean superimposition approach over the {M}antel test}, journal = {Oecologia}, year = {2001}, volume = {129}, pages = {169-178} } @article{SD540, author = {Rao, C.R.}, title = {The use and interpretation of principal component analysis in applied research}, journal = {Sankhya A}, year = {1964}, volume = {26}, pages = {329-359} } @article{SD626, author = {Student, W.S.}, title = {The elimination of spurious correlation due to position in time or space}, journal = {Biometrika}, year = {1914}, volume = {10}, pages = {179-180} } @article{SD694, author = {Wartenberg, D.}, title = {Multivariate spatial correlation: a method for exploratory geographical analysis}, journal = {Geographical Analysis}, year = {1985}, volume = {17}, pages = {263-283}, number = {4} } Guerry/vignettes/MultiSpat.Rmd0000644000176200001440000005444114124100247016130 0ustar liggesusers--- title: "Guerry data: Spatial Multivariate Analysis" author: "Stéphane Dray" date: "`r Sys.Date()`" output: rmarkdown::html_vignette bibliography: reference.bib 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. ```{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(gfrance85) ``` To simplify analyses, we extract the components to be used below. ```{r} 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} 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/NEWS0000644000176200001440000000311714124350437012233 0ustar liggesusersVersion 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/MD50000644000176200001440000000201614125066266012046 0ustar liggesusersaf4580bbffb413e17ccad47ec1ab9576 *DESCRIPTION 195963848221ae8485a2687d34977ed2 *NAMESPACE 112cd20fd3f84e74bb04985019e66589 *NEWS 1f3d336e042b913fb53a0957828d0e07 *build/partial.rdb 72824d4e53a968c82e60202a8bef02a7 *build/vignette.rds 4d3e3b1d1307b61102745af6c164195f *data/Angeville.RData 5c12af3e1e4d205ea46f96e4d0ce9813 *data/Guerry.RData 91bd93041f2bf418dd0424591dc4b5d9 *data/gfrance.RData f45237956849addb495f83a6cded42d6 *data/gfrance85.RData 77681528c11458ab0d5ef1566bde5f47 *inst/WORDFILE 2234a49a9894b0add189101b621f8dda *inst/doc/MultiSpat.R 29c7e9eb842c5e8c1fc542b31a72364c *inst/doc/MultiSpat.Rmd df4a5bdef78f82854467455267ff2580 *inst/doc/MultiSpat.html 0543a444f9982c2ee4e186e2b1b9d2c6 *man/Angeville.Rd d0380be47d4fa6d32736a209b8d9cf97 *man/Guerry-package.Rd 614528e0cbf9dee1ffd8294c6ab52822 *man/Guerry.Rd 1f732c25f06d44de9f98b17977b6eb5d *man/gfrance.Rd e795de3c57238650dd45aadfee3e34d3 *man/gfrance85.Rd 29c7e9eb842c5e8c1fc542b31a72364c *vignettes/MultiSpat.Rmd 3370ed77dc47aeebe758872617f1cb52 *vignettes/reference.bib Guerry/inst/0000755000176200001440000000000014124617031012503 5ustar liggesusersGuerry/inst/WORDFILE0000644000176200001440000000053614124362244013650 0ustar liggesusersAlmanach Alpes André Angeville Angleterre Aude autocorrelated BCA cdots centroid centroids Centre choropleth covary Crochard d'Angeville département départements diagonalization Duchatelet Dupin éclairée eigendecomposition général github Guerry Guerry's geospatial Jombart PCAIV Reinking Statistique Stéphane Guerry/inst/doc/0000755000176200001440000000000014124617031013250 5ustar liggesusersGuerry/inst/doc/MultiSpat.R0000644000176200001440000001542314124617031015322 0ustar liggesusers## ----setup, include = FALSE--------------------------------------------------- knitr::opts_chunk$set( collapse = TRUE, warning = FALSE, message = FALSE, # suppress package loading messages comment = "#>", fig.height = 3, fig.width = 3, fig.align = "center" ) ## ----------------------------------------------------------------------------- library(Guerry) # Guerry's data library(sp) # management of spatial data library(ade4) # multivariate analysis library(adegraphics) # graphical representation library(spdep) # spatial dependency library(adespatial) # multivariate spatial analysis ## ----------------------------------------------------------------------------- names(gfrance85) ## ----------------------------------------------------------------------------- data(gfrance85) df <- data.frame(gfrance85)[, 7:12] # the 6 variables france.map <- as(gfrance85, "SpatialPolygons") # the map xy <- coordinates(gfrance85) # spatial coordinates dep.names <- data.frame(gfrance85)[, 6] # departement names region.names <- data.frame(gfrance85)[, 5] # region names col.region <- colors()[c(149, 254, 468, 552, 26)] # colors for region ## ----------------------------------------------------------------------------- pca <- dudi.pca(df, scannf = FALSE, nf = 3) ## ----------------------------------------------------------------------------- biplot(pca, plabel.cex = 0.8) ## ----------------------------------------------------------------------------- pca$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.html0000644000176200001440000130334714124617031016073 0ustar liggesusers Guerry data: Spatial Multivariate Analysis

Guerry data: Spatial Multivariate Analysis

Stéphane Dray

2021-09-28

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.

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

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

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.

The biplot is simply obtained by

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

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.

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.

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"):

We can represent this neighborhood on the geographical map:

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 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.

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.

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.

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

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

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.

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

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:

The scores can be mapped to show the spatial aspects:

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.

PCAIV is then performed using the pcaiv function:

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.

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:

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:

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

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.

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:

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.

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

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:

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:

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:

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:

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.

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/MultiSpat.Rmd0000644000176200001440000005444114124100247015642 0ustar liggesusers--- title: "Guerry data: Spatial Multivariate Analysis" author: "Stéphane Dray" date: "`r Sys.Date()`" output: rmarkdown::html_vignette bibliography: reference.bib 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. ```{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(gfrance85) ``` To simplify analyses, we extract the components to be used below. ```{r} 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} 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