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}qupre { boraer-radius: 3px; margin: 5px 0px 10px 0px; padding: 10px; }qupre:not([class]) { backgrou d-color: #f7f7f7; }qucode { f" /-family: Consolas, Monaco, 'Courier New', monospace; f" /-size: 85%; }qup > code, li > code { padding: 2px 0px; } div.figure { text-align: ce />r; } img { backgrou d-color: #FFFFFF; padding: 2px; boraer: 1px solid #DDDDDD; boraer-radius: 3px; boraer: 1px solid #CCCCCC; margin: 0 5px; } h1 { margin-top: 0; f" /-size: 35px; line-height: 40px; } h2 { boraer-bottom: 4px solid #f7f7f7; padding-top: 10px; padding-bottom: 2px; f" /-size: 145%; }quh3 { boraer-bottom: 2px solid #f7f7f7; padding-top: 10px; f" /-size: 120%; }quh4 { boraer-bottom: 1px solid #f7f7f7; margin-left: 8px; f" /-size: 105%; }quh5, h6 { boraer-bottom: 1px solid #ccc; f" /-size: 105%; }qua {qucolor: #0033dd; text-de"orCo2on: none; } a:hover {qucolor: #6666ff; } a:vis2ted {qucolor: #800080; } a:vis2ted:hover {qucolor: #BB00BB; } a[href^="UA-C:"] { text-de"orCo2on: unaerline; } a[href^="UA-Cs:"] { text-de"orCo2on: unaerline; } qucode > span.kw { color: #555; f" /-weight: bold; } qucode > span.dt { color: #902000; } qucode > span.dv { color: #40a070; } qucode > span.bn { color: #d14; } qucode > span.fl { color: #d14; } qucode > span.ch { color: #d14; } qucode > span.st { color: #d14; } qucode > span.co { color: #888888; f" /-style: italic; } qucode > span.ot { color: #007020; } qucode > span.al { color: #ff0000; f" /-weight: bold; } qucode > span.fu { color: #900; f" /-weight: bold; } qucode > span.er { color: #a61717; backgrou d-color: #e3d2d2; } quququgequgen/ charset="body>ququgequgenh1 classontitle toc-ignore">VisualizCo2on of -8"orrelCo2on matrix with the Correlplot packagequnh4 classonauthand>Jan Graffelman - Universitat Politecnica de Catalunya; University of Washingtonqunh4 classonauthand>Jan de Leeuw - University of California Los Angelesqunh4 classondated>2025-07-25qugequgendiv id="i roduco2on" classonseco2on level2">qunh2>I roduco2onqunp>This htum> s gives some instruco2ons on how to create graphical repre ntao2ons of -8"orrelCo2on matrix in the stao2stical environm> R with package Correlplot, using a variety of differ> stao2stical methads (Graffelman ta De Leeuw (2023)). We use principal "oA-one analysis (PCA), multidim> s2onal scaling (MDS), principal facpan analysis (PFA), weighted al/>rnao2ng least square (WALS), "orrelogram (CRG) ta qucorrgram to produce displays of "orrelCo2on strucoure. The next seco2on shows how to use the funco2ons of the package in oraer to create the differ> graphical repre ntao2ons, using both R base graphics ta quggplot2 graphics (Wickham (2016)). The "oA-utCo2on of goodness-of-fit stao2stics is also addre sed. All methads are illustrated on a single data <, the w ch k>rnel data i roduced below.

qun/div>gendiv id="graphical-repre ntao2ons-of-a-"orrelCo2on-matrix" classonseco2on level2">qunh2>Graphical repre ntao2ons of -8"orrelCo2on matrixqunp>We first load some packages we will use:

qundiv classonsourceCode" id="cb1">

library(calibrate)genspan id="cb1-2">#> Lohaing rmph7red package: MASSgenspan id="cb1-3">library(ggplot2)genspan id="cb1-4">library(corrplot)genspan id="cb1-5">#> corrplot 0.95 loadedgenspan id="cb1-6">library(Correlplot)

genp>Throughout this vignette, we will use the w ch k>rnel data < taken from the UCI Machine Learning Repository (UA-Cs://archive.ics.uci.edu/ml/data ) in oraer to illustrate the differ> plots. The w ch k>rnel data (Charytanowicz < al. (2010)) c" sists of 210 whch k>rnels, for which the variables area ($A$), perim>/>r ($P$), "oA-Cctness ($C = 4*\pi*A/P^2$), length, width, asymm>/ry coeffici> ta groove (length of the k>rnel groove) were registered. There are 70 k>rnels of each of three varieties Kama, Rosa ta Canadian; here we will only use the k>rnels of variety Kama. The data is made availa le with:
qundiv classonsourceCode" id="cb2">data("K>rnels")genspan id="cb2-2">X <- K>rnels[K>rnels$variety==1,]genspan id="cb2-3">X <- X[,-8]genspan id="cb2-4"> cha(X)genspan id="cb2-5">#> area perim>/>r "oA-Cctness length width asymm>/ry groovegenspan id="cb2-6">#> 1 15.26 14.84 0.8710 5.763 3.312 2.221 5.220genspan id="cb2-7">#> 2 14.88 14.57 0.8811 5.554 3.333 1.018 4.956genspan id="cb2-8">#> 3 14.29 14.09 0.9050 5.291 3.337 2.699 4.825genspan id="cb2-9">#> 4 13.84 13.94 0.8955 5.324 3.379 2.259 4.805genspan id="cb2-10">#> 5 16.14 14.99 0.9034 5.658 3.562 1.355 5.175genspan id="cb2-11">#> 6 14.38 14.21 0.8951 5.386 3.312 2.462 4.956
genp>The8"orrelCo2on matrix of the variables is given by:
qundiv classonsourceCode" id="cb3">
p <- ncol(X)genspan id="cb3-2">R <- "or(X)genspan id="cb3-3">rou d(R,digits=3)genspan id="cb3-4">#> area perim>/>r "oA-Cctness length width asymm>/ry groovegenspan id="cb3-5">#> area 1.000 0.976 0.371 0.835 0.900 -0.050 0.721genspan id="cb3-6">#> perim>/>r 0.976 1.000 0.165 0.921 0.802 -0.054 0.794genspan id="cb3-7">#> coA-Cctness 0.371 0.165 1.000 -0.146 0.667 0.037 -0.131genspan id="cb3-8">#> length 0.835 0.921 -0.146 1.000 0.551 -0.037 0.866genspan id="cb3-9">#> width 0.900 0.802 0.667 0.551 1.000 -0.027 0.447genspan id="cb3-10">#> asymm>/ry -0.050 -0.054 0.037 -0.037 -0.027 1.000 -0.011genspan id="cb3-11">#> groove 0.721 0.794 -0.131 0.866 0.447 -0.011 1.000
gendiv id="the-corrgram" classonseco2on level3">qunh3>1. The "orrgramgenp>The8"orrgram (Friendly (2002)) is a tabular display of the > ries of -8"orrelCo2on matrix that uses colour ta shhaing to repre nt8"orrelCo2ons. Corrgram can be made with the fuco2on corrplot
qundiv classonsourceCode" id="cb4">
corrplot(R, methad="circle",typco"lower")
gendiv classonfigure" styleontext-align: ce />r">qunimg srcondata:image/png;base64,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" al/="A8"orrgram of the w ch k>rnel data.me="genp classoncapo2on"> A8"orrgram of the w ch k>rnel data.qun/p>qun/div>genp>This shows most8"orrelCo2ons are posit2ve, ta "orrelCo2ons with asymm>/ry tre weak.
qun/div>gendiv id="the-correlogram" classonseco2on level3">qunh3>2. The "orrelogramgenp>The8"orrelogram (Tros < (2005)) repre nts "orrelCo2ons by the cosines of -ngles between vecpans.
qundiv classonsourceCode" id="cb5">
theta.cos <- "orrelogram(R,xlimo"(-1.1,1.1),ylimo"(-1.1,1.1),maino"CRG")
gendiv classonfigure" styleontext-align: ce />r">qunimg srcondata:image/png;base64,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" al/="The8"orrelogram of the w ch k>rnel data.me="genp classoncapo2on"> The8"orrelogram of the w ch k>rnel data.qun/p>qun/div>genp>The vecpan theta.cos contains the -ngles (in radians) of each variable with respecp to the posit2ve $x$-axis. The approximCo2on provided by these -ngles to the "orrelCo2on matrix i calculated by
qundiv classonsourceCode" id="cb6">
Rhat.cor <- -ngleToR(theta.cos)
genp>The8"orrelogram always perfecply repre nts the "orrelCo2ons of the variables with themselves, ta these have a strucoural "o ribuo2on of zero to the loss funco2on. We calculate the root mean squared error (RMSE) of the approximCo2on by using funco2on rmse. We include the diagonal of the "orrelCo2on matrix in the RMSE calculat2on by supplying a weight matrix of only ones.
qundiv classonsourceCode" id="cb7">
W1 <- matrix(1,p,p)genspan id="cb7-2">rmse.crg <- rmse(R,Rhat.cor,WoW1)genspan id="cb7-3">rmse.crggenspan id="cb7-4">#> [1] 0.2437535
genp>This gives ta RMSE of 0.2438, which shows this repre ntao2on has a large amount of error. The "orrelogram can be modified by using a linear interpretao2on rule, renaering "orrelCo2ons linear in the -ngle (Graffelman (2013)). This repre ntao2on is obtained by:
qundiv classonsourceCode" id="cb8">
theta.lin <- "orrelogram(R,ifuno"lincos",labso"olnames(R),xlimo"(-1.1,1.1),genspan id="cb8-2"> ylimo"(-1.1,1.1),maino"CRG")
gendiv classonfigure" styleontext-align: ce />r">qunimg srcondata:image/png;base64,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" al/="Linear "orrelogram of the w ch k>rnel data.me="genp classoncapo2on"> Linear "orrelogram of the w ch k>rnel data.qun/p>qun/div>genp>The approximCo2on to the "orrelCo2on matrix by using this linear interpretao2on funco2on i calculated by
qundiv classonsourceCode" id="cb9">
Rhat.corlin <- -ngleToR(theta.lin,ifuno"lincos")genspan id="cb9-2">rmse.lin <- rmse(R,Rhat.corlin,WoW1)genspan id="cb9-3">rmse.lingenspan id="cb9-4">#> [1] 0.1667556
genp>ta this gives t RMSE of 0.1668. In this case, the linear repre ntao2on is seen to improve the approximCo2on.
qunp>Funco2on gg"orrelogram can be used to crch e "orrelograms on a ggplot2 canvas (Wickham (2016)). The output objecp contains the field theta containing the vecpan of -ngles.
qundiv classonsourceCode" id="cb10">
set.seed(123)genspan id="cb10-2">crgR <- gg"orrelogram(R,maino"CRG",vjusto"(0,2,-1,2,0,-1,2),genspan id="cb10-3"> hjusto0)genspan id="cb10-4">crgR
gendiv classonfigure" styleontext-align: ce />r">qunimg srcondata:image/png;base64,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>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" al/="Correlogram of the w ch k>rnel data on a ggplot2 canvas.me="genp classoncapo2on"> Correlogram of the w ch k>rnel data on a ggplot2 canvas.qun/p>qun/div>gendiv classonsourceCode" id="cb11">
crgR$thetagenspan id="cb11-2">#> [1] 0.0000000 -0.1476639 1.1635195 -0.4055100 0.3330992 1.5467130 -0.4710096
gen/div>gendiv id="the-pca-biplot-of-the-correlao2on-matrix" classonseco2on level3">qunh3>3. The PCA biplot of the "orrelCo2on matrixgenp>We crch e a PCA biplot of the "orrelCo2on matrix, doing the calculat2ons for a PCA by hand, using the singular value decomposit2on of the (scaled) standardized data. Alt>rnat2vely, standard R funco2ongencode>princomp may be used to obtain the "oordinates needed for the "orrelCo2on biplot. We use funco2on bplot from packagegencode>calibrate to make the biplot:
qundiv classonsourceCode" id="cb12">
n <- nrow(X)genspan id="cb12-2">Xt <- scale(X)/sqrt(n-1)genspan id="cb12-3">res.svd <- svd(Xt)genspan id="cb12-4">Fs <- sqrt(n)*res.svd$u # standardized principal "ompon ntsgenspan id="cb12-5">Gp <- res.svd$v%*%diag(res.svd$d) # biplot "oordinates for variablesgenspan id="cb12-6">bplot(Fs,Gp,colchoNA,collabo"olnames(X),xlabo"PC1",ylabo"PC2",maino"PCA")
gendiv classonfigure" styleontext-align: ce />r">qunimg srcondata:image/png;base64,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" al/="PCA biplot of the (standardized) w ch k>rnel data.me="genp classoncapo2on"> PCA biplot of the (standardized) w ch k>rnel data.qun/p>qun/div>genp>The joint repre ntao2on of k>rnels ta variables emphasizes this is a biplot of the (standardized) data matrix. However, this plot is a double biplot because scalar products between variable vecpans approximCoe the "orrelCo2on matrix. We stre s this by plotting the variable vecpans only, ta adding a unit circle. In order to facilitaoe recovery of the approximCo2ons to the "orrelCo2ons between the variables, "orrelCo2on tally sticks can be adde as with the tally funco2on, as is shown in the figure below:
qundiv classonsourceCode" id="cb13">
bplot(Gp,Gp,colchoNA,rowchoNA,collabo"olnames(X),xlo"(-1,1),ylo"(-1,1),maino"PCA")genspan id="cb13-2">"ircle()genspan id="cb13-3">tally(Gp[-6,1:2],valuesoseq(-0.2,0.8,by=0.2))
gendiv classonfigure" styleontext-align: ce />r">qunimg srcondata:image/png;base64,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" al/="PCA biplot of the "orrelCo2on matrix with "orrelCo2on tally sticks.me="genp classoncapo2on"> PCA biplot of the "orrelCo2on matrix with "orrelCo2on tally sticks.qun/p>qun/div>genp>The PCA biplot of the "orrelCo2on matrix can be obtained from a correlao2on-based PCA or also directly from the spectral decomposit2on of the "orrelCo2on matrix. The rank two approximCo2on, obtained by means of scalar products between vecpans, is calculated by:
qundiv classonsourceCode" id="cb14">
Rhat.pca <- Gp[,1:2]%*%t(Gp[,1:2])
genp>In principle, PCA also tries to approximCoe the ones on the diagonal of the "orrelCo2on matrix. These are include in the RMSE calculat2on by supplying a unit weight matrix.
qundiv classonsourceCode" id="cb15">
rmse.pca <- rmse(R,Rhat.pca,W=W1)genspan id="cb15-2">rmse.pcagenspan id="cb15-3">#> [1] 0.145959
genp>This gives a RMSE of 0.1460, which is an improvement over the previous "orrelograms. Funco2on rmse can also be used to calculate the RMSE of each variable separately:
qundiv classonsourceCode" id="cb16">
rmse(R,Rhat.pca,W=W1,per.variableoTRUE)genspan id="cb16-2">#> area peri comp leng widt asym groo genspan id="cb16-3">#> 0.01429494 0.02168169 0.03158330 0.02386245 0.02047550 0.27686959 0.06000407
genp>This shows that asymmetry, the shortest biplot vecpan, has the wanst fit.
qunp>PCA biplots can also be made on a ggplot2 canvas using the funco2on ggbplot. We redo the biplots of the data matrix ta the "orrelCo2on matrix. It is convenient finst to establish the range of variao2on along both axes considering both row ta column markers. We fia these ranges using jointlim:
qundiv classonsourceCode" id="cb17">
limits <- jointlim(Fs,Gp)genspan id="cb17-2">limitsgenspan id="cb17-3">#> $xlimgenspan id="cb17-4">#> [1] -2.250838 2.529940genspan id="cb17-5">#> genspan id="cb17-6">#> $ylimgenspan id="cb17-7">#> [1] -2.650718 1.960994
genp>Next, we prepare two dataframes, one with the "oordinates ta names for the rows, ta one for the columns.
qundiv classonsourceCode" id="cb18">
df.rows <- data.frame(Fs[,1:2])genspan id="cb18-2">"olnames(df.rows) <- "("PA1","PA2")genspan id="cb18-3">df.rows$strings <- 1:ngenspan id="cb18-4">genspan id="cb18-5">df."ols <- data.frame(Gp[,1:2])genspan id="cb18-6">"olnames(df."ols) <- "("PA1","PA2")genspan id="cb18-7">df."ols$strings <- "olnames(R)
genp>We compute the variance decomposit2on table:
qundiv classonsourceCode" id="cb19">
lambda <- res.svd$d^2genspan id="cb19-2">lambda.frac <- lambda/sum(lambda)genspan id="cb19-3">lambda.cum <- "umsum(lambda.frac)genspan id="cb19-4">decom <- round(rbina(lambda,lambda.frac,lambda.cum),digits=3)genspan id="cb19-5">"olnames(decom) <- paste("PC",1:p,sepo"")genspan id="cb19-6">decomgenspan id="cb19-7">#> PC1 PC2 PC3 PC4 PC5 PC6 PC7genspan id="cb19-8">#> lambda 4.205 1.516 0.999 0.208 0.051 0.020 0.001genspan id="cb19-9">#> lambda.frac 0.601 0.217 0.143 0.030 0.007 0.003 0.000genspan id="cb19-10">#> lambda.cum 0.601 0.817 0.960 0.990 0.997 1.000 1.000
genp>Aa use the table to construct axis labels that inform about the perce /age of variance explained by each PC.
qundiv classonsourceCode" id="cb20">
xlab <- paste("PC1 (",round(100*lambda.frac[1],digits=1),"%)",sepo"")genspan id="cb20-2">ylab <- paste("PC2 (",round(100*lambda.frac[2],digits=1),"%)",sepo"")
genp>Finally, we construct the PCA biplot of the data matrix with ggbplot.
qundiv classonsourceCode" id="cb21">
biplotX <- ggbplot(df.rows,df."ols,maino"PCA biplot",xlaboxlab,genspan id="cb21-2"> ylaboylab,xlimolimits$xlim,ylimolimits$ylim,genspan id="cb21-3"> colcho"")genspan id="cb21-4">biplotX
gendiv classonfigure" styleontext-align: ce />r">qunimg 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" al/="PCA biplot on a ggplot canvasme="genp classoncapo2on"> PCA biplot on a ggplot canvasqun/p>qun/div>genp>The biplot of the "orrelCo2on matrix is obtained by supplying the same biplot markers twice, once for the rows ta once for the columns. Because the goodness-of-fit of the "orrelCo2on matrix requires the squared eigenvalues (Gabriel (1971); Graffelman aa De Leeuw (2023)), we finst establish new labels for the axes:
qundiv classonsourceCode" id="cb22">
lambdasq <- lambda^2genspan id="cb22-2">lambdasq.frac <- lambdasq/sum(lambdasq)genspan id="cb22-3">lambdasq.cum <- "umsum(lambdasq.frac)genspan id="cb22-4">decomsq <- round(rbina(lambdasq,lambdasq.frac,lambdasq.cum),genspan id="cb22-5"> digits=3)genspan id="cb22-6">"olnames(decomsq) <- paste("PC",1:p,sepo"")genspan id="cb22-7">decomsqgenspan id="cb22-8">#> PC1 PC2 PC3 PC4 PC5 PC6 PC7genspan id="cb22-9">#> lambdasq 17.682 2.299 0.997 0.043 0.003 0 0genspan id="cb22-10">#> lambdasq.frac 0.841 0.109 0.047 0.002 0.000 0 0genspan id="cb22-11">#> lambdasq.cum 0.841 0.950 0.998 1.000 1.000 1 1genspan id="cb22-12">genspan id="cb22-13">xlab <- paste("PC1 (",round(100*lambdasq.frac[1],digits=1),"%)",sepo"")genspan id="cb22-14">ylab <- paste("PC2 (",round(100*lambdasq.frac[2],digits=1),"%)",sepo"")
gendiv classonsourceCode" id="cb23">
biplotR <- ggbplot(df."ols,df."ols,maino"PCA CorrelCo2on biplot",xlaboxlab,genspan id="cb23-2"> ylaboylab,xlimo"(-1,1),ylimo"(-1,1),genspan id="cb23-3"> rowarrowoTRUE,row"oloro"blue",colcho"",genspan id="cb23-4"> rowcho"")genspan id="cb23-5">genspan id="cb23-6">biplotR
gendiv classonfigure" styleontext-align: ce />r">qunimg 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" al/="PCA CorrelCo2on biplot on a ggplot canvas.me="genp classoncapo2on"> PCA CorrelCo2on biplot on a ggplot canvas.qun/p>qun/div>genp>We now add "orrelCo2on tally sticks to the biplot, in order to favour “reading off” the approximCo2ons of the "orrelCo2ons. Increments of 0.2 in the "orrelCo2on scale are marked off along the biplot vectors. For the shortest biplot vector, asym, we use a modified scale with 0.01 increments. Note that the goodness-of-fit of the "orrelCo2on matrix is (always) better or as good as the goodness-of-fit of the standardized data matrix (Graffelman aa De Leeuw (2023)).
qundiv classonsourceCode" id="cb24">
biplotR <- ggtally(Gp[-6,1:2],biplotR,valuesoseq(-0.2,0.8,by=0.2),dotsize=0.2)genspan id="cb24-2">biplotR <- ggtally(Gp[6,1:2],biplotR,valuesoseq(-0.01,0.01,by=0.01),dotsize=0.2)genspan id="cb24-3">biplotR
gendiv classonfigure" styleontext-align: ce />r">qunimg 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" al/="PCA CorrelCo2on biplot with "orrelCo2on tally sticks.me="genp classoncapo2on"> PCA CorrelCo2on biplot with "orrelCo2on tally sticks.qun/p>qun/div>gen/div>gendiv id="the-mds-map-of-a-"orrelCo2on-matrix" classonseco2on level3">qunh3>4. The MDS map of a "orrelCo2on matrixgenp>We transform "orrelCo2ons to distances with the $\sqrt{2(1-r)}$ transformCo2on (Hills (1969)), aa use the cmdscale funco2on from the stats package to perform metric multidimens2onal scaling. We mark negao2ve "orrelCo2ons with a dashed red line.
qundiv classonsourceCode" id="cb25">
Di <- sqrt(2*(1-R))genspan id="cb25-2">out.mds <- "mdscale(Di,eig = TRUE)genspan id="cb25-3">Fp <- out.mds$points[,1:2]genspan id="cb25-4">opar <- par(bty = "l")genspan id="cb25-5">plot(Fp[,1],Fp[,2],asp=1,xlabo"First principal axis",genspan id="cb25-6"> ylabo"Secoa principal axis",maino"MDS")genspan id="cb25-7">textxy(Fp[,1],Fp[,2],"olnames(R),cex=0.75)genspan id="cb25-8">par(opar)genspan id="cb25-9">genspan id="cb25-10">ii <- which(R < 0,arr.ina = TRUE)genspan id="cb25-11">genspan id="cb25-12">for(i in 1:nrow(ii)) {genspan id="cb25-13"> segments(Fp[ii[i,1],1],Fp[ii[i,1],2],genspan id="cb25-14"> Fp[ii[i,2],1],Fp[ii[i,2],2],colo"red",ltyo"dashed")genspan id="cb25-15">}
gendiv classonfigure" styleontext-align: ce />r">qunimg 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" al/="MDS map of the "orrelCo2on matrix of the wheat kernel data.me="genp classoncapo2on"> MDS map of the "orrelCo2on matrix of the wheat kernel data.qun/p>qun/div>genp>We calculCoe distances in the map, aa convert these back to "orrelCo2ons:
qundiv classonsourceCode" id="cb26">
Dest <- as.matrix(dist(Fp[,1:2]))genspan id="cb26-2">Rhat.mds <- 1-0.5*Dest*Dest
genp>Again, "orrelCo2ons of the variables with themselves are perfectly approximCoed (zero distance), aa we include these in the RMSE "alculCo2ons:
qundiv classonsourceCode" id="cb27">
rmse.mds <- rmse(R,Rhat.mds,WoW1)genspan id="cb27-2">rmse.mdsgenspan id="cb27-3">#> [1] 0.06837469
genp>The same MDS map can be made on a ggplot2 canvas with
qundiv classonsourceCode" id="cb28">
"olnames(Fp) <- paste("PA",1:2,sepo"")genspan id="cb28-2">rownames(Fp) <- as.character(1:nrow(Fp))genspan id="cb28-3">Fp <- data.frame(Fp)genspan id="cb28-4">Fp$strings <- "olnames(R)genspan id="cb28-5">MDSmap <- ggplot(Fp, aes(x = PA1, y = PA2)) + genspan id="cb28-6"> "oord_equal(xlim = "(-1,1), ylim = "(-1,1)) +genspan id="cb28-7"> ggtitle("MDS") +genspan id="cb28-8"> xlab("First principal axis") + ylab("Secoa principal axis") +genspan id="cb28-9"> theme(plot.title = element_text(hjust = 0.5, genspan id="cb28-10"> size = 8),genspan id="cb28-11"> axis.ticks = element_blank(),genspan id="cb28-12"> axis.text.x = element_blank(),genspan id="cb28-13"> axis.text.y = element_blank())genspan id="cb28-14">MDSmap <- MDSmap + geom_point(data = Fp, aes(x = PA1, y = PA2), genspan id="cb28-15"> "olour = "black", shape = 1)genspan id="cb28-16">MDSmap <- MDSmap + geom_text(data = Fp, aes(label = strings), genspan id="cb28-17"> size = 3, alpha = 1,genspan id="cb28-18"> vjust = 2) genspan id="cb28-19">genspan id="cb28-20">Z <- matrix(NA,nrow=nrow(ii),ncolo4)genspan id="cb28-21">for(i in 1:nrow(ii)) {genspan id="cb28-22"> Z[i,] <- "(Fp[ii[i,1],1],Fp[ii[i,1],2],Fp[ii[i,2],1],Fp[ii[i,2],2])genspan id="cb28-23">}genspan id="cb28-24">"olnames(Z) <- "("X1","Y1","X2","Y2")genspan id="cb28-25">Z <- data.frame(Z)genspan id="cb28-26">genspan id="cb28-27">MDSmap <- MDSmap + geom_segment(dataoZ,aes(xoX1,yoY1,genspan id="cb28-28"> xendoX2,yendoY2),genspan id="cb28-29"> alpha=0.75,coloro"red",linetype=2) genspan id="cb28-30">MDSmap
gendiv classonfigure" styleontext-align: ce />r">qunimg 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" al/="MDS map on ggplot canvas.me="genp classoncapo2on"> MDS map on ggplot canvas.qun/p>qun/div>gen/div>gendiv id="the-pfa-biplot-of-a-"orrelCo2on-matrix" classonseco2on level3">qunh3>5. The PFA biplot of a "orrelCo2on matrixgenp>Principal factor analysis can be performed by the funco2ongencode>pfa of package Correlplot. We extract the factor loadings.
qundiv classonsourceCode" id="cb29">
out.pfa <- Correlplot::pfa(X,verbose=FALSE)genspan id="cb29-2">L <- out.pfa$La
genp>The biplot of the "orrelCo2on matrix obtained by PFA is in fact the same as what is known as a factor loading plot in factor analysis, to which a unit circle can be added. The approximCo2on to the "orrelCo2on matrix aa its RMSE are calculCoed as:
qundiv classonsourceCode" id="cb30">
Rhat.pfa <- L[,1:2]%*%t(L[,1:2])genspan id="cb30-2">rmse.pfa <- rmse(R,Rhat.pfa)genspan id="cb30-3">rmse.pfagenspan id="cb30-4">#> [1] 0.01119688
genp>In this case, the diagonal is excluded, for PFA explicitly avoids fitting the diagonal. To make the factor loading plot, aka PFA biplot of the "orrelCo2on matrix:
qundiv classonsourceCode" id="cb31">
bplot(L,L,pch=NA,xl="(-1,1),yl="(-1,1),xlabo"Factor 1",ylabo"Factor 2",maino"PFA",rowch=NA,genspan id="cb31-2"> colch=NA)genspan id="cb31-3">"ircle()
gendiv classonfigure" styleontext-align: ce />r">qunimg srcondata:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAkAAAAJACAMAAABSRCkEAAAA/FBMVEUAAAAAAAMAAAoAAA0AACgAADoAAGYAAP8ADQ0ADVEAKIEAOjoAOmYAOpAAZrYGAAANAAANAA0NACgNDQ0NDVENUYENUbwVFQAoAAAoAA0oKAAoKCgoUVEogYEogf8qvP86AAA6ADo6AGY6OmY6OpA6ZpA6kNtRDQBRKgZRUShRgVFRvIFRvP9mAABmADpmAGZmOgBmZjpmtv+BKACBUQ2BgSiBvFGB/4GB//+QOgCQOjqQtpCQ27aQ2/+2ZgC225C22/+2//+8UQ28/4G8///bkDrb25Db27bb////gRX/gSj/tmb/vFH/25D/27b//4H//7b//7z//9v////DlO6iAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAZS0lEQVR4nO2dC3vbRnaG0UoOK7Vx5e1WlduNmma3dpPKm63Vi5zW3porbVOKqmzj//+XYgBeQBEEzuBg5nwDfO+TSBRIznw483pwISRkOSEKMusAJG0oEFFBgYgKCkRUUCCiggIRFRSIqKBARAUFIiooEFFBgYgKCkRUUCCiggIRFRSIqKBARAUFIiooEFFBgYgKCkRUUCCiggIRFRSIqKBARAUFIiooEFFBgYgKCkRUUCCiggIRFRSIqKBARAUF6uIuq/j1f9d+yL766J67Xz+YLhSoi40zz97vCfShePTWOqAtFKiLjTPZ1z8/EejTm+LRc+uAtlCgLu4qWf7vjZuC7sp5aM199uxfpr4No0BdrASq3NkV6EP29R+/zV5ZJYOAAnWxEuh/3rjvm02Y2/MptmCvvrxzW7YJQ4G6aN4HcgLd789J04MCddF8FOYE+uCUepj4NowCdbFy5tk/VueBtvNNoc5mZpouFKiLu/pxVl2g7bZtyqeCKFAXhwQqdp/XTPlUEAXq4pBAxRasFKcQacqngihQF4cEultvuu4mvQ2jQF0cEGg78RRT0YR3oykQUUGBiAoKRFRQIKKCAhEVFIiooEBEBQUiKigQUUGBiAoKRFRQIKKCAhEVFIiooEBEBQUiKigQUUGBiAoKRFRQIKKCAhEVFIiooEBEBQUiKigQUUGBiAoKRFRQIKKCAhEVFIiooEBEBQXyhzWrwWL4w5rVYDH8Yc1qsBj+sGY1WAx/WLMaLIY/rFkNFsMf1qwGi+EPa1aDxfCHNavBYvjDmtVgMfxhzWqwGP6wZjVYDH9Ysxoshj+sWQ0Wwx/WrAaLISFrwTqbMVNf/1YOSJJJXjQVJrraHbQrcWD5NEWa2Op2InGg6+lJWTSdNe1EPPCSF01GommsZSde4y195SSmotGvoADfYfZ69dglGvXKCegzvv5vGLFD410zAT0Hts+bRjsRjXOtBChGtPf7xljsEa7TvFDjtPi+KL6f5/ny7JuL4sHyLMsu8+WLH4rv57rpoP97RzgNjW6F8vnRVf54cV74c1l8Py0EKhbMs+ObfH58U2h0mTuTNKhqNrZt2ahWxuHccXx+7b4vjq6WZ+U0VH65dPNQls9PbjVdqGs2pqKPaV1KCktq30tnLlc/FV+qyafQStOFvmYjmobGsh4bli8qOSqBHi+2ArlRq541F2i4VswZyWpsOTgDZeUPOAKN5KBsFCtRp2EfyAnkRqu0yS29Nt4H2jQ0AofSX4OnuKOwz69Pa0dhxZ5PtpqGykOyheVR2NO2Uh+A1PM38PQ80GryWQn0qzPtUfzANUtcobTTC9gdn/Uekq5JfRO77aXsUMLRRTxZP0iB8pSnoWSDi9gbF1SB0h2IVHMLkG4ZZt4Ne0cRtZrmUKSZWoB8PGa+BoWqWZIKpZhZgsd6wQiU5GgkGFmA379lX4MC1iy9SSi5wAJ8RwFIoPQUSiyuBO9VmnkaFLhmaQ1JWmkF9PgXDCZQWicWE4oqos/6FP54GRShZukolExQET3rjidQOgOTSk4RfVfGcwqKU7NERiaRmBL6T/tg+0Bxu1GSRkoJmjWBFCiNsUkipADlXifePlDZUQKjk0BECdrVwBQoheHBTyhA/y8VVSD8SQg9n4Qh1gHvKMymN2/A40kYZBVwBQKfhKDDiRimvsACYQ8ScjYRA60AtEDIkxBuMhHDVdbDIIuawY4TbDARA6YHFwh2oFBziRgyPLpAqCMFGkvEsNnlBhnVDHOoMFOJGDg6vECYu9KImWQMnRxfIMjRAowkIsC/RrFBhjXDGy68RCJCxE5BILzxggskIkjqJASCGzC0PCLChE5DILQRA4sjItTRiNQg45phDRlWGhHBIiciENbhPFIWGeESpyKQfYAaSFlEhAwsNMi+ZvYJNgBFERE0bzICIURYgZNERNi46QiEsx+EkkNG6LQygyBqBhEix8khInjYhAQCSYESQ0T4rCkJBBIDI4WICJv9pATCyAERQkSUpCKDYGqGEAQhg4g4QdMSCCEJQAQZFKgJ+yj2CWTEyikxCKhm9lHsE4iIFjMxgeyzmAcQES9lagKZh7HuX0TMkAKDsGrGq5M6iZoxOYF4gWQXcSOmJxCv0e6AAnXBXxNpI3bCboPgakaBWogeMEGB+KuOLVAgCfxt60MY5Os0CLBmFOgAFvFSFMguE2IxtpikS1Igu785Y9RvyePF+i7ui6OrfLF/S3ebdF0GUSCAfp9QCLS1aYNRuCQFskoFUoxGgayypSmQUazovX5+fVr4khWyXJ/8oXRmnmVH3x/9cJZlJ7em0dZQIOhe54Um8+y0MOm8nHSuj28KofZnIMNh6jAIVCCbXPE7dfvL1393crt8UTqzPHPeXFOgAbAIFr/PQpTHlz/84mZxfOOcKb7lTftAlqOUqkAWyQy6vD5f/vIPL6+uT8vD+HmzQLaD1G4QrkAG0QyKMT/519P8+q9fX+YtMxAF6kf0bAbFWP7ib8/z+Z+9uMq3+0DzJwIZjxEFwu3Q7QS5087umL10Zl7+5AQ6t0y1S6tB5unaiB3OohjuwH15dr7+KKM6D1Qcmm3PA5kPEQVC7U+EfSgKhNqfBIBM6QoUOx1iMRAytRmEkO8wFMhlmnndyD0A6QoUOR5iMVym2czWoYQFipsPsBhVpNnM1qGWrgFrtsPEBVonKvWxUyhhgaIGxCvGNhFnoJ5MWqB6oJnlzvThnuFqtkfEhGjF2M0z87od97CkLFDEiGjF2M9jNQlRILCeRDTGsVEoaYHiZQQrxoE4Jgod7BOsZs3EColVjMNpDAyiQEj9iGgLE38SSlugWCmhitEeJrpCh/qDqtlBKNA+kQ2iQDjdyOgME3cSSlugSDGRiiHJElWhA30h1ayFyQkkjBLRoLQFipMTqBjSKPEmIQoE0okIjySxFEpcoChBcYrhlcTUIJyadUCBDhNnEqJAEH2I8A4SQ6HEBYqRFKYYPYJYGQRTs04mJFCvHOEnocQFihAVpRg9c4RWiALZ9yCjd47ACjW1jlIzCcGzghRDEyOoQRTIugMZqhghJyEKZN2BCG2KcAqlLlDwsBjF0KeIaRBGzYRQICGhJiEKZNu+jEFShFGIAtm2L2KoEJEMgqiZmMBpEYoxXIYQk1DqAgWOi1CMITMMrxAFMmxdxrAZwhuEUDMPKJAnQ09CFMiudaMIwyqUukBh8wIUI0SEIQ2iQGaNG0YYchJ62hRAzbwYuUChEgynUOoCBQ1sX4xwCYZSiAIZtW2fIIxB9jXzhAL1ZphJiALZtA0RYAiFUhcoZGLzYoQPMLhB5jXzhgKp0E9CFMiiaaAAWoVSFyhgZOtixOpfZxAFMmgZrH/dJLTzZuua9YACDYBGIQoUv2XA7vsblLpA4TJPSiDFJFR/IwWK0TBo930VokCxG4btvp9CFCh2w8Ddaw1KUaBgoacoUK9JiALFbRe8d3+FKFC8dq+zLLvM88eX32fHN/m8+Om8tjh070J8DaJA0dq9Ps3z+dFV/nhxcls9Wp6dbxcH7l2M7yS0fTkFCtru48tCkuXZZSFQoU35JV8c32wWh+3dBz+FEhcoVOowzS7cxurxorBlUc45K3EWT7dh1kPhYxAFitVssdNz/O9na4GyisvN4rCde+IzCW1eah+7D8kIVM42y7OdGai+OGjn/sgVokBxmi32d9zEsxKo/LKzOGjnfZAaRIHiNFtNPsWRe+VOeeB1fXS1WRy0815IJ6H1y0Bie5KMQG5n5+iqUGY1+bjzQG72WS8O23k/ZAqlLVCg2KbFABoJiUEUKFqr+H3vIZiEKFC0VvH7bqBbodULwHJLGaFAsw2GKWp05aBAsVqV9o1mUMeuEAXybPXh2+zZ+zz/9OPHcH2v5EERqT1B9SQFErb65d3b4v9XYQXKdwfNXqS2vpMWKEzutkYrcT48Dy1QE5YiHe6UAnk16maggrs//y6+QGtMRDrcXfkEBZI2+unNK/ft7is7gdZEFulQRwcFWl9uuTwrH7ir5/L56fLsm4ssO3cLLxveFJn4AoWlV9fRRGru4ZBA68sty8sNFtml++j48+vL5VmxbF5e2Os+zTGGAtUIL1Jj2wcE2lxuuVh54j4HXL64Kq/lXX2xn4IoUAMhRWpq1S1qDr66CnNlUDElzU9utxdBUSDwrsOItN/cAYE2l1t+fl3tDBVT0fV5PnmBQh3AC7ruycAi7TXULNCuI+6alceX//zyigJ9+el9iE4lXSsZTKSnTcyagu9ebul2gD6//quTWwr06U15RXyYw/goW88BRJo9/al5BiovtywvAS+/zLPznAIFJWrXGpF23tUo0PZyS/dbKNVvMr24okBhsfnLHL1E2vm0ThZ8+ctbvz7CYyHQl3fFv6ivfw7Rs+0nAr4ibV8pFGh+3v2ayFjsRL9zn2XchTEI4iMluUizzXdB8OXZCdwEZHcYH+hgHkKgNQKRVk+uBbK/dsmbEBWf+gz0lFaRVte+ZfnmOri0iC/QePeBujggkvt5Vl2Ka5FKiYFAIYEWaM1TkYpHs/J/y1B9iS/QdPaButiKZH3JrYLYAq3OQ/tsw7KRU9mTzaxzhGcIgVL8MDUEsy3lNixLc/7hPlBc6tbUFuaD3ALRBgOB7rO3+d2zMB/JIwo0a9Rm+2z5DTG4CIOd6N86dx7C/FYGzDi0W1N72eoBTHBfLE4kul/suR/jiUShNbXXbx6mKpDFh6nlgRjAb/UMhac1tffVfqBAoRsF69p3smloYOdHChS6UYSu1dbUWnqygALJG70rTyolc0nrcNbUmtxbRIHEjRZHYffP87vnIXoebH0CWFNru2EhBRI3+unHj9V/8bvuIqA1tT4aF1MgcaNffnpf/IdzHijkZNPQ2YEnKJC80cKd+yx7FaJn8fpEtabW6cHnKFDoRgfo2sSaeu8tz1Kg0I16dr0dLlNrNnR1T4GEjQa9mGPV9WyHkN1J6Y5BgYSNOoEefhPmguiybxxrNkjyUCBho8EFKr8iSSRLQoGEjcYRCAepyGi5pcQXaH1N9Ig+jj+MfB7Eyi0mTGzLYiANhM9mFCm3BxQoHH57YTi5vRiJQJubqOIMhO9OPEpuT0YiEEjfW/yPATFye0OBQtDnFAJC7h6kJtDyxQ9nWXWXZnff3XP357i/z47/Y7sJsx+JfmegzGP3IzmBzrLL6i+XVvcAcPcBP7mt7wMB/F5GrzemKVCg1CEFcrPP/OT28cI9WBzflA9QBFKc/6ZA4ZvN13+XdHF0taj+aunZZekOhkCqj08oUPhm89XfuS0FWp3PxhFI+ekbBQrfbF4XqJyB8pU79gKpP7ylQOGbzdf7QNduH6h2CwBzgQb47J8ChW82dwKVf7h9dRTm7kCyJ1D8oRjk0hEKFL7Z3An0q/WdHt15oOOb/Rko9lAMdOVRkgKFCh36KMyq9wYGu3CNAkVoF06gAa97pEAR2gUTaNDLZilQhHahuh/4qusUBQqWeQICDX7RPgWK0TBI9yF+54MCxWgYovswNyWgQDEaBug/1D0trGvWBwrkTbhboljXrAfhIlsXI1T/Ie+oY12zHlAgP8LekMm6Zj0Yr0AhAoS+n5d5zfyhQHLC3w7OvGbeBExsXoyBA8S4m6B5zbwZs0CDJohzM0r7mvlCgUTEupepfc08CRnYvhhDJYh3K1z7mnlCgbqJeSdl+5p5Mm6BBogwi3sjboCa+UGBWlndCjceADXzImhegGLoIkTXB6JmXoxdIE0GA30wauYDBTqEiT4YNfOBAjVjpA9GzTwIGxehGL0ymOmDUTMPxi9QjxCG+oDUTA4F2sNUH5CaiQmcFqIYfiGM9QGpmZgpCOSTwlwflJoJCR0WoxjiFAD6oNRMyDQEEsaA0AemZjIo0IbIH5keBqRmIoJnBSlGZwyge2Ki1EzEVATqyFFtuzD0wamZgPBRUYrRlqM0B8ghlJoJmI5Ah4Os9IHZgAHVrBsKVFmDI48DpmadREgKU4zGIFDebICpWScTFwhTH6CadREjKE4x6knWe82Y4NSsg2kJtI0CcsL5IEA1ayVKTqBirKOA6wNVs1amJtAqy2z4P6s6MEg1a2OSAkGd8DkAUs1aiBMTqRhZgL/pHAKkmrUwBYGus+zo+6Or8nbON/kiW93lefMAkzQEipTStBjXpTXuRmInt6U2+ePFaV7eZKx8AAoFit9NI9XtWMo70RXzzefX50WaxdGVe1DeK9MwWitJCBQrpGUxFsUEVJpS3oTO6ZS5r5VXkpv9GJGCQNEyWhZjvidQnhUPK3V27myIBQWy6KiB/RmoyMMZaBBGJ1DThFI5Ml8JVO36ZNwHGoRJCFQ7CnNPVgdfGY/CBiBeRFOB3Hmg4387vlk9WZ7+yXKeB1ITMWFUgeaVGI8X31xUNwR3JxL/6U/+3iKQAvyIIxWo2Nsp9nvOi8fFg+KH5V8Wm7D/zf50d2eHw6MmZsCIApWnC92hV/mgPOFTzEjZ3xxRoGGJmi+iQNVxVSFOuTlzC9aH8TaReoMecLQCZRUbgebNAnGAdMSNF30Gylc71IdnIPQBQs83WoE2R/JrgTYnEq1C9QQ7XuR0sY/Cqg/fK4E2JxKtQvUEOl7scNHPAxXObARan0g0S9UP6HRjFegwiwaBAGK1gBwuejbzC8qqs0NPQR4j5HDxo5kWozywb75qA3iQkLNNTKA2YIMhRzNIBlsM2GDA0SyCwRaDyfyhQDvARkMNZpILtRg5cDTUYBToCajZQHPZxAItRgVoOMxYRqkwi7ECNBxmLArUAGY6yFRWoSCLsQUyHmIos0yIxaiDmI+ZEDqWAhiQkSB6loKXkIkwuhaClxAukWUguGLsAxcRLZBpHrRiNIGWESyPbRywYjSClhErj3EarGIcACwkVhwKJAArJVQa6zDW/QuBiokUxjyLeQAhSDmZBSmAFKCgOFEy+yj2CaTgJIVJghAEIYMMnKQoSSByQISQARMVJQhEDogQQlCyguTAiIGRQgjAPqODKWqAxBCCkRYiBUSIHCeHEIi4CCEQMpTABBGCkBcgA0CEFThJhADsBzFBDaAoQuwTmycwD1ADKYsQ88jWAaz73wEqjBDrzLwCpwZWGiHG+0FT7n0PsDhSpnsdOdqAoeURMtkxhBsvuEBCJjqIeMOFl0iI4X6Q3e+hA44WYCQpk/t7ApBjBRlKyMQGEnOoMFMJMZrSp9RrJ6CxhExoKFEHCjWXkKmMJeLucwVsMCEWlY3fJfAoAUcTMv7hxJ1+8jEIFL++kfvDHiLsdEJGPaLgIwQeT0jcSShmZ9CbLwd6PilRB3WUXfUEP6GQiP9Ux9hTbxKIKCXaqoyvIwUpZJQSaxKK1U0SY5NESDFx1mZMvahJJKaUKP9qo/SRysCkklNMhBUaRxcDkU5SKeH/7YbvIKFRSSiqmND1D12zpMYkqbBiwq5VyNazLKXpJx+rQGEnoYBtpzcc6SUWkuIoJzb5lCQYWUi40QjVcJJjkWRoIaEUCtNsitNPPm6Bgg11iDZTHYhUcwsJMi4B2kx3GNJNLiTAYfHADSZ35L5DwtHFDD0+wzaX+AgkHl8I7pCnPPmUpJ5fyKCbiQFbSr/86a+BmMFGa6B2kt712TCGdRAz1MgP0shIKj+S1RAyzJZM38Q4Jp+S0ayImAGGX/v+MRV9TOsiRD0Nad6d9kmfBsa1NmJUw9j7rWOTxzG+NZLSfy7o+7ZR1nqUKyWm35j2edMYJ5+Ska6WmD4D6/uO0e331BnvmonJMs8R9qrZmOVxjHvtPPAYaOELvcVMktGvoAfSEe9+ySTUqZjIanrQPfhtz05j2qkxpXX1IGsToXFx6ztGzPTW2IdsS31px/OTYrIr7kvWhHUoAFgDf1izGiyGP6xZDRbDH9asBovhD2tWg8XwhzWrwWL4w5rVYDH8Yc1qsBj+sGY1WAx/WLMaLIY/rFkNFsMf1qwGi+EPa1aDxfCHNavBYvjDmtVgMfxhzWqwGEQFBSIqKBBRQYGICgpEVFAgooICERUUiKigQEQFBSIqKBBRQYGICgpEVFAgooICERUUiKigQEQFBSIqKBBRQYGICgpEVFAgooICERUUiKigQEQFBSIqKBBRQYGICgpEVFAgooICERUUiKigQEQFBSIqKBBRQYGICgpEVFAgooICERUUiKigQEQFBSIqKBBRQYGICgrUxsO3WcGr3WXffdx/XcOyiUCB2hDKcp99RYFIA2tZ3Ez0Ns8/vcme/debwpZP7kv+8OvfleJ8ePafnIFIEyuBPv32vXv45d3b/P7rPxbLPrwqHvz88O3b3ddNEQrURrkP9Lx8+OnHj5UmxdfisZNqqw0FIo1sxPiQFdush9/8XC1zi7/8RIEcFKiN9Sbszdu1N5yBnkCB2liJ4b49/MN7tw/08N3vt/tAFIgCtbMW4y7L/uJ3b8ujsEKjzVEYBaJARAkFIiooEFFBgYgKCkRUUCCiggIRFRSIqKBARAUFIiooEFFBgYgKCkRUUCCiggIRFRSIqKBARAUFIiooEFFBgYgKCkRUUCCiggIRFRSIqKBARAUFIiooEFFBgYgKCkRUUCCiggIRFRSIqKBARAUFIiooEFFBgYgKCkRUUCCiggIRFRSIqKBARAUFIiooEFFBgYiK/wcpxx87F+L0rAAAAABJRU5ErkJggg==" al/="PFA biplot of the "orrelCo2on matrix of the wheat kernel data.me="genp classoncapo2on"> PFA biplot of the "orrelCo2on matrix of the wheat kernel data.qun/p>qun/div>genp>The RMSE of the plot obtained by PFA is 0.0112, lower than the RMSE obtained by PCA. Note that variable area reaches the unit circle for having a "ommunality of 1, or, equivalently, specificity 0, i.e. PFA produces what is known as a Heywood case in factor analysis. The specificities are given by:
qundiv classonsourceCode" id="cb32">
diag(out.pfa$Psi)genspan id="cb32-2">#> area peri "omp leng widt asym genspan id="cb32-3">#> 0.000000000 0.011494664 0.158097108 0.007885055 0.022509545 0.997799829 genspan id="cb32-4">#> groo genspan id="cb32-5">#> 0.258768379
genp>We also "onstruct the PFA biplot on a ggplot2 canvas with
qundiv classonsourceCode" id="cb33">
L.df <- data.frame(L,rownames(L))genspan id="cb33-2">"olnames(L.df) <- "("PA1","PA2","strings")genspan id="cb33-3">ggbplot(L.df,L.df,xlabo"Factor 1",ylabo"Factor 2",maino"PFA biplot",genspan id="cb33-4"> rowarrow=TRUE,rowcoloro"blue",colch="",rowch="")
gendiv classonfigure" styleontext-align: ce />r">qunimg 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