Author: Jan Graffelman - Universitat Politecnica de Catalunya; University of Washington
Date: 2026-04-01

# Visualization of a correlation matrix with the Correlplot package

#### Jan Graffelman - Universitat Politecnica de Catalunya; University of Washington

#### Jan de Leeuw - University of California Los Angeles

#### 2026-04-01

## Introduction

This documents gives some instructions on how to create graphical representations of correlation matrices (within sets and between sets of variables) in the statistical environment R with package `Correlplot`, using a variety of different statistical methods (Graffelman and De Leeuw (2023)). For the within-set correlation matrix, we use principal component analysis (PCA), multidimensional scaling (MDS), principal factor analysis (PFA), weighted alternating least squares (WALS), correlograms (CRG) and corrgrams to produce displays of correlation structure. The next section shows how to use the functions of the package in order to create the different graphical representations, using both R base graphics and `ggplot2` graphics (Wickham (2016)). The computation of goodness-of-fit statistics is also addressed. All these methods are illustrated on a single data set, the wheat kernel data introduced below. A separate section is dedicated to the representation of the between-set correlation matrix by means of canonical correlation analysis (CCA) and various adjustments of this matrix by means of an alternating generalised least squares (AGLS) algorithm (Graffelman (2026)). The canonical methods are illustrated with the psychology-academic achievement data.

## Graphical representations of a within-set correlation matrix

We first load some packages we will use:

    library(calibrate)
    #> Loading required package: MASS
    library(ggplot2)
    library(corrplot)
    #> corrplot 0.95 loaded
    library(Correlplot)

Throughout this section, we will use the wheat kernel data set taken from the UCI Machine Learning Repository (https://archive.ics.uci.edu/ml/datasets/seeds) in order to illustrate the different plots. The wheat kernel data (Charytanowicz et al. (2010)) consists of 210 wheat kernels, for which the variables area (\(A\)), perimeter (\(P\)), compactness (\(C = 4*\pi*A/P^2\)), length, width, asymmetry coefficient and groove (length of the kernel groove) were registered. There are 70 kernels of each of three varieties Kama, Rosa and Canadian; here we will only use the kernels of variety Kama. The data is made available with:

    data("Kernels")
    X <- Kernels[Kernels$variety==1,]
    X <- X[,-8]
    head(X)
    #>    area perimeter compactness length width asymmetry groove
    #> 1 15.26     14.84      0.8710  5.763 3.312     2.221  5.220
    #> 2 14.88     14.57      0.8811  5.554 3.333     1.018  4.956
    #> 3 14.29     14.09      0.9050  5.291 3.337     2.699  4.825
    #> 4 13.84     13.94      0.8955  5.324 3.379     2.259  4.805
    #> 5 16.14     14.99      0.9034  5.658 3.562     1.355  5.175
    #> 6 14.38     14.21      0.8951  5.386 3.312     2.462  4.956

The correlation matrix of the variables is given by:

    p <- ncol(X)
    R <- cor(X)
    round(R,digits=3)
    #>               area perimeter compactness length  width asymmetry groove
    #> area         1.000     0.976       0.371  0.835  0.900    -0.050  0.721
    #> perimeter    0.976     1.000       0.165  0.921  0.802    -0.054  0.794
    #> compactness  0.371     0.165       1.000 -0.146  0.667     0.037 -0.131
    #> length       0.835     0.921      -0.146  1.000  0.551    -0.037  0.866
    #> width        0.900     0.802       0.667  0.551  1.000    -0.027  0.447
    #> asymmetry   -0.050    -0.054       0.037 -0.037 -0.027     1.000 -0.011
    #> groove       0.721     0.794      -0.131  0.866  0.447    -0.011  1.000

### 1. The corrgram

The corrgram (Friendly (2002)) is a tabular display of the entries of a correlation matrix that uses colour and shading to represent correlations. Corrgrams can be made with the fuction `corrplot`

    corrplot(R, method="circle",type="lower")

A corrgram of the wheat kernel data.

This shows most correlations are positive, and correlations with asymmetry are weak.

### 2. The correlogram

The correlogram (Trosset (2005)) represents correlations by the cosines of angles between vectors.

    theta.cos <- correlogram(R,xlim=c(-1.1,1.1),ylim=c(-1.1,1.1),
                             main="Correlogram")

The correlogram of the wheat kernel data.

The vector `theta.cos` contains the angles (in radians) of each variable with respect to the positive \(x\)-axis. The approximation provided by these angles to the correlation matrix is calculated by

    Rhat.cor <- angleToR(theta.cos)

The correlogram always perfectly represents the correlations of the variables with themselves, and these have a structural contribution of zero to the loss function. We calculate the root mean squared error (RMSE) of the approximation by using function `rmse`. We include the diagonal of the correlation matrix in the RMSE calculation by supplying a weight matrix of only ones.

    W1 <- matrix(1,p,p)
    rmse.crg <- rmse(R,Rhat.cor,W=W1)
    rmse.crg
    #> [1] 0.2437535

This gives and RMSE of 0.2438, which shows this representation has a large amount of error. The correlogram can be modified by using a linear interpretation rule, rendering correlations linear in the angle (Graffelman (2013)). This representation is obtained by:

    theta.lin <- correlogram(R,ifun="lincos",labs=colnames(R),xlim=c(-1.1,1.1),
                             ylim=c(-1.1,1.1),main="Linear Correlogram")

Linear correlogram of the wheat kernel data.

The approximation to the correlation matrix by using this linear interpretation function is calculated by

    Rhat.corlin <- angleToR(theta.lin,ifun="lincos")
    rmse.lin <- rmse(R,Rhat.corlin,W=W1)
    rmse.lin
    #> [1] 0.1667556

and this gives a RMSE of 0.1668. In this case, the linear representation is seen to improve the approximation.

Function `ggcorrelogram` can be used to create correlograms on a `ggplot2` canvas (Wickham (2016)). The output object contains the field `theta` containing the vector of angles.

    set.seed(123)
    crgR <- ggcorrelogram(R,main="Correlogram",vjust=c(0,2,-1,2,0,-1,2),
                         hjust=0)
    crgR

Correlogram of the wheat kernel data on a ggplot2 canvas.

    crgR$theta
    #> [1]  0.0000000 -0.1476639  1.1635195 -0.4055100  0.3330992  1.5467130 -0.4710096

### 3. The PCA biplot of the correlation matrix

We create a PCA biplot of the correlation matrix, doing the calculations for a PCA by hand, using the singular value decomposition of the (scaled) standardized data. Alternatively, standard R function `princomp` may be used to obtain the coordinates needed for the correlation biplot. We use function `bplot` from package `calibrate` to make the biplot:

    n <- nrow(X)
    Xt <- scale(X)/sqrt(n-1)
    res.svd <- svd(Xt)
    Fs <- sqrt(n)*res.svd$u # standardized principal components
    Gp <- res.svd$v%*%diag(res.svd$d) # biplot coordinates for variables
    bplot(Fs,Gp,colch=NA,collab=colnames(X),xlab="PC1",ylab="PC2",main="PCA")

PCA biplot of the (standardized) wheat kernel data.

The joint representation of kernels and variables emphasizes this is a biplot of the (standardized) data matrix. However, this plot is a double biplot because scalar products between variable vectors approximate the correlation matrix. We stress this by plotting the variable vectors only, and adding a unit circle. In order to facilitate recovery of the approximations to the correlations between the variables, correlation tally sticks can be added as with the `tally` function, as is shown in the figure below:

    bplot(Gp,Gp,colch=NA,rowch=NA,collab=colnames(X),xl=c(-1,1),yl=c(-1,1),main="PCA")
    circle()
    tally(Gp[-6,1:2],values=seq(-0.2,0.8,by=0.2))

PCA biplot of the correlation matrix with correlation tally sticks.

The PCA biplot of the correlation matrix can be obtained from a correlation-based PCA or also directly from the spectral decomposition of the correlation matrix. The rank two approximation, obtained by means of scalar products between vectors, is calculated by:

    Rhat.pca <- Gp[,1:2]%*%t(Gp[,1:2])

In principle, PCA also tries to approximate the ones on the diagonal of the correlation matrix. These are included in the RMSE calculation by supplying a unit weight matrix.

    rmse.pca <- rmse(R,Rhat.pca,W=W1)
    rmse.pca
    #> [1] 0.145959

This gives a RMSE of 0.1460, which is an improvement over the previous correlograms. Function `rmse` can also be used to calculate the RMSE of each variable separately:

    rmse(R,Rhat.pca,W=W1,per.variable=TRUE)
    #>       area       peri       comp       leng       widt       asym       groo
    #> 0.01429494 0.02168169 0.03158330 0.02386245 0.02047550 0.27686959 0.06000407

This shows that asymmetry, the shortest biplot vector, has the worst fit.

PCA biplots can also be made on a `ggplot2` canvas using the function `ggbplot`. We redo the biplots of the data matrix and the correlation matrix. It is convenient first to establish the range of variation along both axes considering both row and column markers. We find these ranges using `jointlim`:

    limits <- jointlim(Fs,Gp)
    limits
    #> $xlim
    #> [1] -2.250838  2.529940
    #>
    #> $ylim
    #> [1] -2.650718  1.960994

Next, we prepare two dataframes, one with the coordinates and names for the rows, and one for the columns.

    df.rows <- data.frame(Fs[,1:2])
    colnames(df.rows) <- c("PA1","PA2")
    df.rows$strings <- 1:n

    df.cols <- data.frame(Gp[,1:2])
    colnames(df.cols) <- c("PA1","PA2")
    df.cols$strings <- colnames(R)

We compute the variance decomposition table:

    lambda      <- res.svd$d^2
    lambda.frac <- lambda/sum(lambda)
    lambda.cum  <- cumsum(lambda.frac)
    decom <- round(rbind(lambda,lambda.frac,lambda.cum),digits=3)
    colnames(decom) <- paste("PC",1:p,sep="")
    decom
    #>               PC1   PC2   PC3   PC4   PC5   PC6   PC7
    #> lambda      4.205 1.516 0.999 0.208 0.051 0.020 0.001
    #> lambda.frac 0.601 0.217 0.143 0.030 0.007 0.003 0.000
    #> lambda.cum  0.601 0.817 0.960 0.990 0.997 1.000 1.000

And use the table to construct axis labels that inform about the percentage of variance explained by each PC.

    xlab <- paste("PC1 (",round(100*lambda.frac[1],digits=1),"%)",sep="")
    ylab <- paste("PC2 (",round(100*lambda.frac[2],digits=1),"%)",sep="")

Finally, we construct the PCA biplot of the data matrix with `ggbplot`.

    biplotX <- ggbplot(df.rows,df.cols,main="PCA biplot",xlab=xlab,
                  ylab=ylab,xlim=limits$xlim,ylim=limits$ylim,
                  colch="",size=2)
    biplotX

PCA biplot on a ggplot canvas

The biplot of the correlation matrix is obtained, because of its symmetry, by supplying the same biplot markers twice, once for the rows and once for the columns. Because the goodness-of-fit of the correlation matrix requires the squared eigenvalues (Gabriel (1971); Graffelman and De Leeuw (2023)), we first establish new labels for the axes:

    lambdasq      <- lambda^2
    lambdasq.frac <- lambdasq/sum(lambdasq)
    lambdasq.cum  <- cumsum(lambdasq.frac)
    decomsq <- round(rbind(lambdasq,lambdasq.frac,lambdasq.cum),
                     digits=3)
    colnames(decomsq) <- paste("PC",1:p,sep="")
    decomsq
    #>                  PC1   PC2   PC3   PC4   PC5 PC6 PC7
    #> lambdasq      17.682 2.299 0.997 0.043 0.003   0   0
    #> lambdasq.frac  0.841 0.109 0.047 0.002 0.000   0   0
    #> lambdasq.cum   0.841 0.950 0.998 1.000 1.000   1   1

    xlab <- paste("PC1 (",round(100*lambdasq.frac[1],digits=1),"%)",sep="")
    ylab <- paste("PC2 (",round(100*lambdasq.frac[2],digits=1),"%)",sep="")

    biplotR <- ggbplot(df.cols,df.cols,
                       main="PCA Correlation biplot",
                       xlab=xlab,ylab=ylab,
                       xlim=c(-1,1),ylim=c(-1,1),
                       rowarrow=TRUE,rowcolor="blue",
                       colch="",rowch="",size=3)

    biplotR

PCA Correlation biplot on a ggplot canvas.

We now add correlation tally sticks to the biplot, in order to favour “reading off” the approximations of the correlations. Increments of 0.2 in the correlation 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 correlation matrix is (always) better or as good as the goodness-of-fit of the standardized data matrix (Graffelman and De Leeuw (2023)).

    biplotR <- ggtally(biplotR,Gp,Gp,R,ind=c(1:5,7),
                       values=seq(-0.2,0.8,by=0.2),dotsize=1.0,
                       verbose=FALSE)
    biplotR <- ggtally(biplotR,Gp,Gp,R,ind=6,
                       values=seq(-0.01,0.01,by=0.01),
                       dotsize=0.2)
    biplotR

PCA Correlation biplot with correlation tally sticks.

### 4. The MDS map of a correlation matrix

We transform correlations to distances with the \(\sqrt{2(1-r)}\) transformation (Hills (1969)), and use the `cmdscale` function from the `stats` package to perform metric multidimensional scaling. We mark negative correlations with a dashed red line.

    Di <- sqrt(2*(1-R))
    out.mds <- cmdscale(Di,eig = TRUE)
    Fp <- out.mds$points[,1:2]
    opar <- par(bty = "l")
    plot(Fp[,1],Fp[,2],asp=1,xlab="First principal axis",
         ylab="Second principal axis",main="MDS")
    textxy(Fp[,1],Fp[,2],colnames(R),cex=0.75)
    par(opar)

    ii <- which(R < 0,arr.ind = TRUE)

    for(i in 1:nrow(ii)) {
      segments(Fp[ii[i,1],1],Fp[ii[i,1],2],
               Fp[ii[i,2],1],Fp[ii[i,2],2],col="red",lty="dashed")
    }

MDS map of the correlation matrix of the wheat kernel data.

We calculate distances in the map, and convert these back to correlations:

    Dest <- as.matrix(dist(Fp[,1:2]))
    Rhat.mds <- 1-0.5*Dest*Dest

Again, correlations of the variables with themselves are perfectly approximated (zero distance), and we include these in the RMSE calculations:

    rmse.mds <- rmse(R,Rhat.mds,W=W1)
    rmse.mds
    #> [1] 0.06837469

The fit by distances offered by MDS is better than the approximation by scalar products obtained in PCA. The same MDS map can be made on a `ggplot2` canvas with

    colnames(Fp) <- paste("PA",1:2,sep="")
    rownames(Fp) <- as.character(1:nrow(Fp))
    Fp <- data.frame(Fp)
    Fp$strings <- colnames(R)
    MDSmap <- ggplot(Fp, aes(x = PA1, y = PA2)) +
                coord_equal(xlim = c(-1,1), ylim = c(-1,1)) +
                ggtitle("MDS") +
                xlab("First principal axis") + ylab("Second principal axis") +
                theme(plot.title = element_text(hjust = 0.5,
                                                size = 8),
                      axis.ticks = element_blank(),
                      axis.text.x = element_blank(),
                      axis.text.y = element_blank())
    MDSmap <- MDSmap + geom_point(data = Fp, aes(x = PA1, y = PA2),
                            colour = "black", shape = 1)
    MDSmap <- MDSmap + geom_text(data = Fp, aes(label = strings),
                            size = 3, alpha = 1,
                          vjust = 2)

    Z <- matrix(NA,nrow=nrow(ii),ncol=4)
    for(i in 1:nrow(ii)) {
      Z[i,] <- c(Fp[ii[i,1],1],Fp[ii[i,1],2],Fp[ii[i,2],1],Fp[ii[i,2],2])
    }
    colnames(Z) <- c("X1","Y1","X2","Y2")
    Z <- data.frame(Z)

    MDSmap <- MDSmap + geom_segment(data=Z,aes(x=X1,y=Y1,
                                               xend=X2,yend=Y2),
                             alpha=0.75,color="red",linetype=2)
    MDSmap

MDS map on ggplot canvas.

### 5. The PFA biplot of a correlation matrix

Principal factor analysis can be performed by the function `pfa` of package `Correlplot`. We extract the factor loadings.

    out.pfa <- Correlplot::pfa(X,verbose=FALSE)
    L <- out.pfa$La

The biplot of the correlation 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 approximation to the correlation matrix and its RMSE are calculated as:

    Rhat.pfa <- L[,1:2]%*%t(L[,1:2])
    rmse.pfa <- rmse(R,Rhat.pfa)
    rmse.pfa
    #> [1] 0.01119688

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 correlation matrix:

    bplot(L,L,pch=NA,xl=c(-1,1),yl=c(-1,1),xlab="Factor 1",ylab="Factor 2",main="PFA",rowch=NA,
          colch=NA)
    circle()

PFA biplot of the correlation matrix of the wheat kernel data.

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 communality 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:

    diag(out.pfa$Psi)
    #>        area        peri        comp        leng        widt        asym
    #> 0.000000000 0.011494664 0.158097108 0.007885055 0.022509545 0.997799829
    #>        groo
    #> 0.258768379

We also construct the PFA biplot on a `ggplot2` canvas with

    L.df <- data.frame(L,rownames(L))
    colnames(L.df) <- c("PA1","PA2","strings")
    ggbplot(L.df,L.df,xlab="Factor 1",ylab="Factor 2",main="PFA biplot",
            rowarrow=TRUE,rowcolor="blue",colch="",rowch="",size=3)

PFA biplot on a ggplot canvas.

### 6. The WALS biplot of a correlation matrix

The correlation matrix can also be factored using weighted alternating least squares, avoiding the fit of the ones on the diagonal of the correlation matrix by assigning them weight 0, using function `ipSymLS` (De Leeuw (2006)).

    Wdiag0 <- matrix(1,nrow(R),nrow(R))
    diag(Wdiag0) <- 0
    Fp.als <- ipSymLS(R,w=Wdiag0,eps=1e-15)

    bplot(Fp.als,Fp.als,rowch=NA,colch=NA,collab=colnames(R),
          xl=c(-1.1,1.1),yl=c(-1.1,1.1),main="WALS")
    circle()

WALS biplot of the correlation matrix of the wheat kernel data.

Weighted alternating least squares has, in contrast to PFA, no restriction on the vector length. If the vector lengths in the biplot are calculated, then variable area is seen to just move out of the unit circle.

    Rhat.wals <- Fp.als%*%t(Fp.als)
    sqrt(diag(Rhat.wals))
    #> [1] 1.00124368 0.99394213 0.91345321 0.99646265 0.99026217 0.04686397 0.86124152
    rmse.als <- rmse(R,Rhat.wals)
    rmse.als
    #> [1] 0.01118619

The RMSE of the approximation obtained by WALS is 0.011186, slightly below the RMSE of PFA. The WALS low rank approximation to the correlation matrix can also be obtained by the more generic function `wAddPCA`, which allows for non-symmetric matrices and adjustments (see the next section). In order to get uniquely defined biplot vectors for each variable, corresponding to symmetric input, an eigendecomposition is applied to the fitted correlation matrix.

    out.wals <- wAddPCA(R, Wdiag0, add = "nul", verboseout = FALSE, epsout = 1e-10)
    Rhat.wals <- out.wals$a%*%t(out.wals$b)
    out.eig <- eigen(Rhat.wals)
    Fp.adj <- out.eig$vectors[,1:2]%*%diag(sqrt(out.eig$values[1:2]))
    rmse.als <- rmse(R,Rhat.wals)
    rmse.als
    #> [1] 0.01118619

We also make the WALS biplot on a `ggplot2` canvas with

    Fp.als.df <- data.frame(Fp.als,colnames(R))
    colnames(Fp.als.df) <- c("PA1","PA2","strings")
    ggbplot(Fp.als.df,Fp.als.df,xlab="Dimension 1",
            ylab="Dimension 2",main="WALS biplot",
            rowarrow=TRUE,rowcolor="blue",colch="",rowch="",
            size=3)

WALS biplot of the correlation matrix of the wheat kernel data.

### 7. The WALS biplot using a scalar adjustment of the correlation matrix

A scalar adjustment can be employed to improve the approximation of the correlation matrix, and the adjusted correlation matrix is factored for a biplot representation. That means we seek the factorization

\[ {\mathbf R} - \delta {\mathbf J} = {\mathbf R}_a = {\mathbf G} {\mathbf G}', \]

where both \(\delta\) and \(\mathbf G\) are chosen such that the (weighted) residual sum of squares is minimal. This problem is solved by using function `wAddPCA`.

    out.wals <- wAddPCA(R, Wdiag0, add = "one", verboseout = FALSE, epsout = 1e-10)
    delta <- out.wals$delta[1,1]
    Rhat <- out.wals$a%*%t(out.wals$b)
    out.eig <- eigen(Rhat)
    Fp.adj <- out.eig$vectors[,1:2]%*%diag(sqrt(out.eig$values[1:2]))

The optimal adjustment \(\delta\) is 0.071. The corresponding biplot is shown below.

    bplot(Fp.adj,Fp.adj,rowch=NA,colch=NA,collab=colnames(R),
          xl=c(-1.2,1.2),yl=c(-1.2,1.2),main="WALS adjusted")
    circle()

WALS biplot of the correlation matrix of the wheat kernel data, with the use of an additive adjustment.

Note that, when calculating the fitted correlation matrix, adjustment \(\delta\) is added back. The fitted correlation matrix and its RMSE are now calculated as:

    Rhat.adj <- Fp.adj%*%t(Fp.adj)+delta
    rmse.adj <- rmse(R,Rhat.adj)
    rmse.adj
    #> [1] 0.005560242

This gives RMSE 0.0056. This is smaller than the RMSE obtained by WALS without adjustment. We summarize the values of the RMSE of all methods in a table below:

    rmsevector <- c(rmse.crg,rmse.lin,rmse.pca,rmse.mds,rmse.pfa,rmse.als,rmse.adj)
    methods <- c("CRG (cos)","CRG (lin)","PCA","MDS",
    "PFA","WALS R","WALS Radj")
    results <- data.frame(methods,rmsevector)
    results <- results[order(rmsevector),]
    results
    #>     methods  rmsevector
    #> 7 WALS Radj 0.005560242
    #> 6    WALS R 0.011186190
    #> 5       PFA 0.011196885
    #> 4       MDS 0.068374694
    #> 3       PCA 0.145959040
    #> 2 CRG (lin) 0.166755585
    #> 1 CRG (cos) 0.243753461

A summary of RMSE calculations for four methods (PCA and WALS, both with and without \(\delta\) adjustment) can be obtained by the functions `FitRwithPCAandWALS` and `rmsePCAandWALS`; the first applies the four methods to the correlation matrix, and the latter calculates the RMSE statistics for the four approximations. The bottom line of the table produced by `rmsePCAandWALS` gives the overall RMSE for each methods.

    output <- FitRwithPCAandWALS(R,eps=1e-15)
    rmsePCAandWALS(R,output)
    #>         PCA  PCA-A   WALS WALS-A
    #> area 0.0143 0.0542 0.0081 0.0043
    #> peri 0.0217 0.0405 0.0088 0.0064
    #> comp 0.0316 0.0590 0.0110 0.0054
    #> leng 0.0239 0.0485 0.0067 0.0045
    #> widt 0.0205 0.0626 0.0076 0.0068
    #> asym 0.2769 0.1182 0.0181 0.0049
    #> groo 0.0600 0.0675 0.0135 0.0061
    #> All  0.1460 0.0706 0.0112 0.0056

The scalar adjustment changes the interpretation of the origin, and it is convenient to show the zero correlation point on each biplot vector. We do so by creating tally sticks, redoing the biplot on a `ggplot2` canvas. The origin now represents correlation 0.07 for all variables.

    Fp.adj.df <- data.frame(Fp.adj,colnames(R))
    colnames(Fp.adj) <- c("PA1","PA2")
    colnames(Fp.adj.df) <- c("PA1","PA2","strings")
    WALSbiplot.adj <- ggbplot(Fp.adj.df,Fp.adj.df,
                              xlab="Dimension 1",ylab="Dimension 2",
                              main="WALS adjusted",rowarrow=TRUE,
                              rowcolor = "blue",rowch="",colch="",
                              size=3)
    WALSbiplot.adj <- ggtally(WALSbiplot.adj,Fp.adj[,1:2],
                              Fp.adj[,1:2],R,ind=c(1:5,7),
                              adj=-out.wals$delta[1,1],
                              values=seq(-0.2,0.8,by=0.2),
                              dotsize=1.0)
    WALSbiplot.adj

WALS biplot of the correlation matrix of the wheat kernel data, with additive adjustment and tally sticks.

### 8. The WALS biplot using column adjustment of the correlation matrix

We generate an asymmetric approximation to the correlation matrix using a column adjustment, based on the decomposition

\[ {\mathbf R} = \delta {\mathbf 1} {\mathbf 1}' + {\mathbf 1} {\mathbf q}' + {\mathbf G} {\mathbf G}' + {\mathbf E}. \]

This decomposition can be obtained with function `FitDeltaQSym`.

    out.walsq.sym <- FitRDeltaQSym(R,Wdiag0,eps=1e-8)
    Gq.sym <- out.walsq.sym$G
    rownames(Gq.sym) <- colnames(R)
    Rhat.wsym <- out.walsq.sym$Rhat
    rmse.wsym <- rmse(R,Rhat.wsym)
    rmse.wsym
    #> [1] 0.00540002

This approximation has an RMSE of 0.0054, a very minor improvement in comparison with using a single scalar adjustment. The corresponding biplot can be obtained with

    Gq.sym.df <- data.frame(Gq.sym)
    Gq.sym.df$strings <- colnames(R)
    colnames(Gq.sym.df) <- c("PA1","PA2","strings")
    ll <- 1.5
    WALSbiplotq.sym <- ggbplot(Gq.sym.df,Gq.sym.df,
                               main="WALS-q-sym biplot",
                               xlab="Dimension 1",
                               ylab="Dimension 2",
                               ylim=c(-ll,ll),xlim=c(-ll,ll),circle=TRUE,
                               rowarrow=TRUE,rowcolor="blue",rowch="",
                               colch="",size=3)
    for(i in c(1:5,7)) {

       WALSbiplotq.sym <- ggtally(WALSbiplotq.sym,Gq.sym[,1:2],
                                  Gq.sym[,1:2],R,ind=i,
                                  adj=-out.walsq.sym$delta-out.walsq.sym$q[i],
                  values=seq(-0.2,1,by=0.2),dotsize=1.0)
    }
    WALSbiplotq.sym

WALS biplot of the correlation matrix with column adjustment and tally sticks.

This biplot is similar in appearance to the previous biplot that uses a single scalar adjustment only. However, in principle it no longer has a unique origin, as the origin represents a (slightly) different correlation for each variable:

    out.walsq.sym$delta + out.walsq.sym$q
    #>       area       peri       comp       leng       widt       asym       groo
    #> 0.06640039 0.06234622 0.06582835 0.06548268 0.06719080 0.06612880 0.06899342

For this data, a single scalar adjustment is preferable for simplicity.

## Graphical representations of a between-set correlation matrix

Canonical correlation analysis (CCA) is a classic tool to analyse the correlations between two sets of variables, the \(X\) set and the \(Y\) set. CCA allows the construction of a biplot of the between-set correlation matrix \(R_{xy}\). We again use adjustments by scalar, row and/or column effects to improve the approximation obtained by CCA. We use a data set consisting of three psychological variables (locus, self and motivation), four acadamic achievement variables (read, write, math, science) and gender (female) to illustrate the use of CCA and the various adjustments fitted with an alternating generalised least squares algorithm. We first calculate the between-set and within-set correlation matrices.

    data(achievement)
    X <- achievement[,1:3]
    Y <- achievement[,4:ncol(achievement)]
    Rxy  <- cor(X,Y)
    Rxx  <- cor(X)
    Ryy  <- cor(Y)
    round(Rxy,3)
    #>             read write  math science female
    #> locus      0.374 0.359 0.337   0.325  0.113
    #> self       0.061 0.019 0.054   0.070 -0.126
    #> motivation 0.211 0.254 0.195   0.116  0.098
    Rxx.inv <- solve(Rxx)
    Ryy.inv <- solve(Ryy)

Next, we fit all models (CCA plus all adjusted version of CCA with scalar, row and/or column adjustment) and compare their loss and RMSE.

    Results <- FitAllModelsRxy(Rxy,Rxx,Ryy,verbose=FALSE,
                           eps=1e-08,ndim=2)
    #> Algorithm converged in 449 iterations.
    #> Algorithm converged in 22 iterations.
    #> Algorithm converged in 15 iterations.
    #> Algorithm converged in 18 iterations.
    print(round(Results,6))
    #>           LOSS  GLSRMSE  OLSRMSE
    #> cca   0.010814 0.026850 0.018816
    #> delta 0.000409 0.005220 0.003104
    #> row   0.000408 0.005216 0.003038
    #> col   0.000000 0.000015 0.000017
    #> r&c   0.000000 0.000028 0.000023

We first run CCA using the `canocor` function from the `calibrate` package, obtain the canonical correlations and the biplot coordinates.

    out.cca <- canocor(X,Y)
    Fp <- out.cca$Fp
    Gs <- out.cca$Gs
    Gp <- out.cca$Gp
    Fs <- out.cca$Fs
    diag(out.cca$ccor)
    #> [1] 0.4640861 0.1675091 0.1039911

We calculate the root-mean-squared error of a two-dimensional approximation to the between-set correlation matrix with function `rmse.rxy`.

    Rhat.cca <- tcrossprod(Fp[,1:2],Gs[,1:2])
    rmse.cca <- round(rmse.rxy(Rxy,Rhat.cca,R=Rxx.inv,C=Ryy.inv),4)
    rmse.cca
    #> [1] 0.0269

We set up title, axis labels, and dataframes with biplot markers for the rows and the columns, and make a biplot of the between-set correlation matrix.

    header <- paste("CCA (",toString(rmse.cca),")",sep="")

    fra1 <- 100*out.cca$fitRxy[2,1]
    fra2 <- 100*out.cca$fitRxy[2,2]

    xlab <- paste("First canonical axis (",
                  round(fra1,digits=1),"%)",sep="")
    ylab <- paste("Second canonical axis (",
                  round(fra2,digits=1),"%)",sep="")

    df.row <- data.frame(Fp[,1:2])
    colnames(df.row) <- c("PA1","PA2")
    df.row$strings <- rownames(Rxy)
    df.row$ve <- c(  2, 1, -0.5)
    df.row$ho <- c(  1, 0, 1)

    df.col <- data.frame(Gs[,1:2])
    colnames(df.col) <- c("PA1","PA2")
    df.col$strings <- colnames(Rxy)
    df.col$ve <- c(   1.15, -1, -1,    1,  0)
    df.col$ho <- c(   0.5, 0.75, 0.5, -0.5, -0.5)

    biplotRxy <- ggbplot(df.row,df.col,main=header,xlab=xlab,ylab=ylab,
                      rowch="",colch="",rowarrow = TRUE, size = 3,
                      linewidth = 0.1)
    biplotRxy

CCA biplot of the between-set correlation matrix.

We illustrate the construction of a calibrated correlation scale for variable science, (`ind=4`) marking off 0.01, 0.05 and 0.10 increments, and showing the approximations of its correlations with perpendiculars.

    biplotRxy <- ggtally(biplotRxy,Fp,Gs,Rxy,ind=4,
                  values=round(seq(-0.2,0.8,by=0.01),2),dotsize=0.1,
                  W=Rxx.inv,dotcolour = "grey",linewidth = 0.1,
                  dp=TRUE)
    biplotRxy <- ggtally(biplotRxy,Fp,Gs,Rxy,ind=4,
                  values=round(seq(-0.2,0.8,by=0.05),2),dotsize=0.2,
                  W=Rxx.inv,dotcolour = "dimgrey",linewidth = 0.1,
                  dp=TRUE)
    biplotRxy <- ggtally(biplotRxy,Fp,Gs,Rxy,ind=4,
                  values=round(seq(-0.2,0.8,by=0.1),2),dotsize=1.0,
                  W=Rxx.inv,dotcolour = "black",linewidth = 0.1,
                  dp=TRUE)
    biplotRxy

CCA biplot of the between-set correlation matrix with calibration of variable science.

The output of `FitAllModelsRxy` above shows adjustment with a single scalar brings the RMSE below 0.01. We fit the model with a scalar adjustment (`adjust="delta"`), and obtain the approximation and its RMSE.

    out.delta <- FitRxy(Rxy,Rxx.inv,Ryy.inv,
                      adjust="delta",eps=1e-08,
                      verbose=FALSE)
    #> Algorithm converged in 449 iterations.
    out.delta$delta
    #> [1] -0.268789
    round(out.delta$y,3)
    #>             read write  math science female
    #> locus      0.373 0.358 0.343   0.324  0.112
    #> self       0.062 0.021 0.045   0.071 -0.123
    #> motivation 0.211 0.255 0.190   0.117  0.100
    rmse.delta <- rmse.rxy(Rxy,out.delta$y,R=Rxx.inv,C=Ryy.inv)
    rmse.delta
    #> [1] 0.005220467

We construct the biplot according to the adjusted analysis.

    header <- paste("CCA-delta (",round(rmse.delta,4),")",sep="")

    xlab <- "First adjusted canonical axis"
    ylab <- "Second adjusted canonical axis"

    Gc    <- out.delta$Gc
    Fc    <- out.delta$Fc
    delta <- out.delta$delta

    df.col <- data.frame(out.delta$Gc[,1:2])
    df.row <- data.frame(out.delta$Fc[,1:2])
    colnames(df.col) <- c("PA1","PA2")
    colnames(df.row) <- c("PA1","PA2")
    df.col$strings <- colnames(Rxy)
    df.row$strings <- rownames(Rxy)

    df.col$ve <- c(   1.15, -1, -1,    1,  0)
    df.col$ho <- c(   0.5, 0.75, 0.5, -0.5, -0.5)

    df.row$ve <- c(  2, 1, 1)
    df.row$ho <- c(  1, 0, 1)

    biplotRxy.delta <- ggbplot(df.row,df.col,main=header,
                               xlab=xlab,ylab=ylab,
                               rowarrow = TRUE, rowch="", colch="",
                               size=3,linewidth = 0.1)
    biplotRxy.delta

CCA biplot of the between-set correlation matrix with delta adjustment.

To emphasise the change in interpretation of the origin, we, calibrate all biplot vectors marking the positive part of the correlation scale in blue and the negative part in red.

    biplotRxy.delta <- ggtally(biplotRxy.delta,Fc,Gc,Rxy,
                               linewidth=0.1,ind=1,
                               values=round(seq(-0.4,0.4,by=0.10),2),
                               dotsize=1.0,W=Rxx.inv,adj=delta,
                               dotcolour = "black",dp=FALSE)

    biplotRxy.delta <- ggtally(biplotRxy.delta,Fc,Gc,Rxy,
                               linewidth=0.1,ind=2,
                               values=round(seq(-0.4,0.3,by=0.10),2),
                               dotsize=1.0,W=Rxx.inv,adj=delta,
                               dotcolour = "black",dp=FALSE)

    biplotRxy.delta <- ggtally(biplotRxy.delta,Fc,Gc,Rxy,
                               linewidth=0.1,ind=3,
                               values=round(seq(-0.4,0.4,by=0.10),2),
                               dotsize=1.0,W=Rxx.inv,adj=delta,
                               dotcolour = "black",dp=FALSE)

    biplotRxy.delta <- ggtally(biplotRxy.delta,Fc,Gc,Rxy,
                               linewidth=0.1,ind=4,
                               values=round(seq(-0.4,0.6,by=0.10),2),
                               dotsize=1.0,W=Rxx.inv,adj=delta,
                               dotcolour = "black",dp=TRUE)

    biplotRxy.delta <- ggtally(biplotRxy.delta,Fc,Gc,Rxy,
                               linewidth=0.1,ind=5,
                               values=round(seq(-0.4,0.2,by=0.10),2),
                               dotsize=1.0,W=Rxx.inv,adj=delta,
                               dotcolour = "black",dp=FALSE)

As an alternative to the RMSE, the goodness-of-fit of the two-dimensional approximation can be calculated as

    TSS   <- tr(Rxx.inv%*%(Rxy)%*%Ryy.inv%*%t(Rxy))
    ESS   <- tr(Rxx.inv%*%out.delta$y%*%Ryy.inv%*%t(out.delta$y))
    gof2D <- ESS/TSS
    gof2D
    #> [1] 0.9980534
    fitstring <- paste("(",toString(round(100*gof2D,1)),"%)",sep="")

    biplotRxy.delta <- biplotRxy.delta + annotate("text", x = 0.9, y = 0.9,
                              label = fitstring, size = 2)
    biplotRxy.delta

Adjustment of the columns reduces the RMSE to almost zero (0.000015). The iteration history (`verbose=TRUE`), the approximation of \(R_{xy}\), the RMSE and the estimations of the column adjustments can be obtained as follows.

    out.col <- FitRxy(Rxy,Rxx.inv,Ryy.inv,
                      adjust="col",eps=1e-08,verbose=TRUE)
    #> 3 X-variables and  5 Y-variables.
    #> itel  1 loss.old  0.0108141590 loss.new  0.0046605043
    #> itel  2 loss.old  0.0046605043 loss.new  0.0018029505
    #> itel  3 loss.old  0.0018029505 loss.new  0.0006702195
    #> itel  4 loss.old  0.0006702195 loss.new  0.0002455383
    #> itel  5 loss.old  0.0002455383 loss.new  0.0000894779
    #> itel  6 loss.old  0.0000894779 loss.new  0.0000325442
    #> itel  7 loss.old  0.0000325442 loss.new  0.0000118284
    #> itel  8 loss.old  0.0000118284 loss.new  0.0000042980
    #> itel  9 loss.old  0.0000042980 loss.new  0.0000015616
    #> itel  10 loss.old  0.0000015616 loss.new  0.0000005674
    #> itel  11 loss.old  0.0000005674 loss.new  0.0000002061
    #> itel  12 loss.old  0.0000002061 loss.new  0.0000000749
    #> itel  13 loss.old  0.0000000749 loss.new  0.0000000272
    #> itel  14 loss.old  0.0000000272 loss.new  0.0000000099
    #> itel  15 loss.old  0.0000000099 loss.new  0.0000000036
    #> Algorithm converged in 15 iterations.
    out.col$y
    #>                  read      write       math    science      female
    #> locus      0.37356822 0.35887972 0.33727196 0.32462839  0.11340611
    #> self       0.06063429 0.01942905 0.05357759 0.06981648 -0.12591984
    #> motivation 0.21058993 0.25423008 0.19499482 0.11566035  0.09813196
    rmse.col <- out.col$rmse.approximation
    rmse.col
    #> [1] 1.547339e-05
    ce <- out.col$ce
    ce
    #>        read       write        math     science      female
    #>  0.03217693  0.03821020  0.03154600 -0.00584184 -0.07241002

The column adjustments are relatively small, and similar for read, write and math. For simplicity, the previous adjustment by a single scalar may be preferred, which already has its RMSE close to zero (0.0052). We nevertheless show how a biplot for the column-adjusted analysis can be obtained. For calibration, the adjustment argument `adj` of function `ggtally` should now be set to the estimated column adjustment for each corresponding variable.

    header <- paste("CCA-col (",round(rmse.col,6),")",sep="")

    xlab <- "First adjusted canonical axis"
    ylab <- "Second adjusted canonical axis"

    Gc    <- out.col$Gc
    Fc    <- out.col$Fc

    df.col <- data.frame(out.col$Gc[,1:2])
    df.row <- data.frame(out.col$Fc[,1:2])
    colnames(df.col) <- c("PA1","PA2")
    colnames(df.row) <- c("PA1","PA2")
    df.col$strings <- colnames(Rxy)
    df.row$strings <- rownames(Rxy)

    df.col$ve <- c(   1.15, -1, -1,    1,  0)
    df.col$ho <- c(   0.5, 0.75, 0.5, -0.5, -0.5)

    df.row$ve <- c(  2, 1, 1)
    df.row$ho <- c(  1, 0, 1)

    biplotRxy.col <- ggbplot(df.row,df.col,main=header,
                               xlab=xlab,ylab=ylab,
                               rowarrow = TRUE, rowch="", colch="",
                               size=3,linewidth = 0.1)

    biplotRxy.col <- ggtally(biplotRxy.col,Fc,Gc,Rxy,
                               linewidth=0.1,ind=1,
                               values=round(seq(-0.2,0.6,by=0.10),2),
                               dotsize=1.0,W=Rxx.inv,adj=ce[1],
                               dotcolour = "black",dp=FALSE)

    biplotRxy.col <- ggtally(biplotRxy.col,Fc,Gc,Rxy,
                               linewidth=0.1,ind=2,
                               values=round(seq(-0.2,0.6,by=0.10),2),
                               dotsize=1.0,W=Rxx.inv,adj=ce[2],
                               dotcolour = "black",dp=FALSE)

    biplotRxy.col <- ggtally(biplotRxy.col,Fc,Gc,Rxy,
                               linewidth=0.1,ind=3,
                               values=round(seq(-0.2,0.4,by=0.10),2),
                               dotsize=1.0,W=Rxx.inv,adj=ce[3],
                               dotcolour = "black",dp=FALSE)

    biplotRxy.col <- ggtally(biplotRxy.col,Fc,Gc,Rxy,
                               linewidth=0.1,ind=4,
                               values=round(seq(-0.2,0.6,by=0.10),2),
                               dotsize=1.0,W=Rxx.inv,adj=ce[4],
                               dotcolour = "black",dp=TRUE)

    biplotRxy.col <- ggtally(biplotRxy.col,Fc,Gc,Rxy,
                               linewidth=0.1,ind=5,
                               values=round(seq(-0.2,0.4,by=0.10),2),
                               dotsize=1.0,W=Rxx.inv,adj=ce[5],
                               dotcolour = "black",dp=FALSE)

    ESS   <- tr(Rxx.inv%*%out.col$y%*%Ryy.inv%*%t(out.col$y))
    gof2D <- ESS/TSS
    gof2D
    #> [1] 0.9999522
    fitstring <- paste("(",toString(round(100*gof2D,3)),"%)",sep="")

    biplotRxy.col <- biplotRxy.col + annotate("text", x = 0.9, y = 0.9,
                              label = fitstring, size = 2)
    biplotRxy.col

CCA biplot of the between-set correlation matrix with column adjustment.

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