Approximation of a correlation matrix with column adjustment and symmetric low rank factorization

Description

Program FitRDeltaQSym calculates a low rank factorization for a correlation matrix. It adjusts for column effects, and the approximation is therefore asymmetric.

Usage

FitRDeltaQSym(R, W = NULL, nd = 2, eps = 1e-10, delta = 0, q = colMeans(R),
              itmax.inner = 1000, itmax.outer = 1000, verbose = FALSE)

Arguments

Details

Program FitRDeltaQSym implements an iterative algorithm for the low rank factorization of the correlation matrix. It decomposes the correlation matrix as R = delta J + 1 q' + G G' + E. The approximation of R is ultimately asymmetric, but the low rank factorization used for biplotting (G G') is symmetric.

Value

A list object with fields:

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

References

Graffelman, J. and De Leeuw, J. (2023) Improved approximation and visualization of the correlation matrix. The American Statistician pp. 1–20. Available online as latest article doi:10.1080/00031305.2023.2186952

See Also

wAddPCA,ipSymLS,Keller

Examples

data(HeartAttack)
X <- HeartAttack[,1:7]
X[,7] <- log(X[,7])
colnames(X)[7] <- "logPR"
R <- cor(X)
W <- matrix(1, 7, 7)
diag(W) <- 0
out.sym <- FitRDeltaQSym(R, W, eps=1e-6) 
Rhat <- out.sym$Rhat

Calculate a low-rank approximation to the correlation matrix with four methods

Description

Function FitRwithPCAandWALS uses principal component analysis (PCA) and weighted alternating least squares (WALS) to calculate different low-rank approximations to the correlation matrix.

Usage

FitRwithPCAandWALS(R, nd = 2, itmaxout = 10000, itmaxin = 10000, eps = 1e-08)

Arguments

Details

Four methods are run succesively: standard PCA; PCA with an additive adjustment; WALS avoiding the fit of the diagonal; WALS avoiding the fit of the diagonal and with an additive adjustment.

Value

A list object with fields:

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

References

Graffelman, J. and De Leeuw, J. (2023) Improved approximation and visualization of the correlation matrix. The American Statistician pp. 1–20. Available online as latest article doi:10.1080/00031305.2023.2186952

See Also

wAddPCA

Examples

data(HeartAttack)
X <- HeartAttack[,1:7]
X[,7] <- log(X[,7])
colnames(X)[7] <- "logPR"
R <- cor(X)
## Not run: 
out <- FitRwithPCAandWALS(R)

## End(Not run)

Myocardial infarction or Heart attack data

Description

Data set consisting of 101 observations of patients who suffered a heart attack.

Usage

data("HeartAttack")

Format

A data frame with 101 observations on the following 8 variables.

Pulse

Pulse

CI

Cardiac index

SI

Systolic index

DBP

Diastolic blood pressure

PA

Pulmonary artery pressure

VP

Ventricular pressure

PR

Pulmonary resistance

Status

Deceased or survived

Source

Table 18.1, (Saporta 1990, pp. 452–454)

References

Saporta, G. (1990) Probabilites analyse des donnees et statistique. Paris, Editions technip

Examples

data(HeartAttack)
str(HeartAttack)

Program Keller calculates a rank p approximation to a correlation matrix according to Keller's method.

Description

Keller's method is based on iterated eigenvalue decompositions that are used to adjust the diagonal of the correlation matrix.

Usage

Keller(R, eps = 1e-06, nd = 2, itmax = 10)

Arguments

Value

A matrix containing the approximation to the correlation matrix-

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

References

Keller, J.B. (1962) Factorization of Matrices by Least-Squares. Biometrika, 49(1 and 2) pp. 239–242.

Graffelman, J. and De Leeuw, J. (2023) Improved approximation and visualization of the correlation matrix. The American Statistician pp. 1–20. Available online as latest article doi:10.1080/00031305.2023.2186952

See Also

ipSymLS

Examples

data(Kernels)
R <- cor(Kernels)
Rhat <- Keller(R)

Wheat kernel data

Description

Wheat kernel data set taken from the UCI Machine Learning Repository

Usage

data("Kernels")

Format

A data frame with 210 observations on the following 8 variables.

area

Area of the kernel

perimeter

Perimeter of the kernel

compactness

Compactness (C = 4*pi*A/P^2)

length

Length of the kernel

width

Width of the kernel

asymmetry

Asymmetry coefficient

groove

Length of the groove of the kernel

variety

Variety (1=Kama, 2=Rosa, 3=Canadian)

Source

https://archive.ics.uci.edu/ml/datasets/seeds

References

M. Charytanowicz, J. Niewczas, P. Kulczycki, P.A. Kowalski, S. Lukasik, S. Zak, A Complete Gradient Clustering Algorithm for Features Analysis of X-ray Images. in: Information Technologies in Biomedicine, Ewa Pietka, Jacek Kawa (eds.), Springer-Verlag, Berlin-Heidelberg, 2010, pp. 15-24.

Examples

data(Kernels)

Heights of mothers and daughters

Description

Heights of 1375 mothers and daughters (in cm) in the UK in 1893-1898.

Usage

data(PearsonLee)

Format

dataframe with Mheight and Dheight

Source

Weisberg, Chapter 1

References

Weisberg, S. (2005) Applied Linear Regression, John Wiley & Sons, New Jersey

Characteristics of aircraft

Description

Four variables registered for 21 types of aircraft.

Usage

data("aircraft")

Format

A data frame with 21 observations on the following 4 variables.

SPR

specific power

RGF

flight range factor

PLF

payload

SLF

sustained load factor

Source

Gower and Hand, Table 2.1

References

Gower, J.C. and Hand, D.J. (1996) Biplots, Chapman & Hall, London

Examples

data(aircraft)
str(aircraft)

Correlations between characteristics of aircraft

Description

Correlations between SPR (specific power), RGF (flight range factor), PLF (payload) and SLF (sustained load factor) for 21 types of aircraft.

Usage

data(aircraftR)

Format

a matrix containing the correlations

Source

Gower and Hand, Table 2.1

References

Gower, J.C. and Hand, D.J. (1996) Biplots, Chapman & Hall, London

Convert angles to correlations.

Description

Function angleToR converts a vector of angles (in radians) to an estimate of the correlation matrix, given an interpretation function.

Usage

angleToR(x, ifun = "cos")

Arguments

Value

A correlation matrix

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

References

Graffelman, J. (2012) Linear-angle correlation plots: new graphs for revealing correlation structure. Journal of Computational and Graphical Statistics. 22(1): 92-106.

See Also

cos,lincos

Examples

angles <- c(0,pi/3)
R <- angleToR(angles)
print(R)

Correlations for 10 generated variables

Description

A 10 by 10 artificial correlation matrix

Usage

data(artificialR)

Format

A matrix of correlations

Source

Trosset (2005), Table 1.

References

Trosset, M.W. (2005) Visualizing correlation. Journal of Computational and Graphical Statistics, 14(1), pp. 1–19.

Correlation matrix of characteristics of Australian athletes

Description

Correlation matrix of 12 characteristics of Austration athletes (Sex, Height, Weight, Lean Body Mass, RCC, WCC, Hc, Hg, Ferr, BMI, SSF, Bfat)

Usage

data(athletesR)

Format

A matrix of correlations

Source

Weisberg (2005), file ais.txt

References

Weisberg, S. (2005) Applied Linear Regression. Third edition, John Wiley & Sons, New Jersey.

Swiss banknote data

Description

The Swiss banknote data consist of six measures taken on 200 banknotes, of which 100 are counterfeits, and 100 are normal.

Usage

data("banknotes")

Format

A data frame with 200 observations on the following 7 variables.

Length

Banknote length

Left

Left width

Right

Right width

Bottom

Bottom margin

Top

Top margin

Diagonal

Length of the diagonal of the image

Counterfeit

0 = normal, 1 = counterfeit

References

Weisberg, S. (2005) Applied Linear Regression. Third edition. John Wiley & Sons, New Jersey.

Examples

data(banknotes)

Correlation matrix for boys of the Berkeley Guidance Study

Description

Correlation matrix for sex, height and weight at age 2, 9 and 18 and somatotype

Usage

data(berkeleyR)

Format

A matrix of correlations

Source

Weisberg (2005), file BGSBoys.txt

References

Weisberg, S. (2005) Applied Linear Regression. Third edition, John Wiley & Sons, New Jersey.

Correlation matrix for height and length

Description

Correlation between nave height and total length

Usage

data(cathedralsR)

Format

A matrix of correlations

Source

Weisberg (2005), file cathedral.txt

References

Weisberg, S. (2005) Applied Linear Regression. Third edition, John Wiley & Sons, New Jersey.

Plot a correlogram

Description

correlogram plots a correlogram for a correlation matrix.

Usage

correlogram(R,labs=colnames(R),ifun="cos",cex=1,main="",ntrials=50,
            xlim=c(-1.2,1.2),ylim=c(-1.2,1.2),pos=NULL,...)

Arguments

Details

correlogram makes a correlogram on the basis of a set of angles. All angles are given w.r.t the positive x-axis. Variables are represented by unit vectors emanating from the origin.

Value

A vector of angles

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

References

Trosset, M.W. (2005) Visualizing correlation. Journal of Computational and Graphical Statistics 14(1), pp. 1–19

See Also

fit_angles,nlminb

Examples

X <- matrix(rnorm(90),ncol=3)
R <- cor(X)
angles <- correlogram(R)

Correlations between educational and demographic variables

Description

Correlations between infant mortality, educational and demographic variables (infd, phys, dens, agds, lit, hied, gnp)

Usage

data(countriesR)

Format

A matrix of correlations

Source

Chatterjee and Hadi (1988)

References

Chatterjee, S. and Hadi, A.S. (1988), Sensitivity Analysis in Regression. Wiley, New York.

Fit angles to a correlation matrix

Description

fit_angles finds a set of optimal angles for representing a particular correlation matrix by angles between vectors

Usage

fit_angles(R, ifun = "cos", ntrials = 10, verbose = FALSE)

Arguments

Value

a vector of angles (in radians)

Author(s)

anonymous

References

Trosset, M.W. (2005) Visualizing correlation. Journal of Computational and Graphical Statistics 14(1), pp. 1–19

See Also

nlminb

Examples

X <- matrix(rnorm(90),ncol=3)
R <- cor(X)
angles <- fit_angles(R)
print(angles)

Correlations between thirtheen fysiological variables

Description

Correlations of 13 fysiological variables (sys, dia, p.p., pul, cort, u.v., tot/100, adr/100, nor/100, adr/tot, tot/hr, adr/hr, nor/hr) obtained from 48 medical students

Usage

data(fysiologyR)

Format

A matrix of correlations

Source

Hills (1969), Table 1.

References

Hills, M (1969) On looking at large correlation matices Biometrika 56(2): pp. 249.

Create a biplot with ggplot2

Description

Function ggbiplot creates a biplot of a matrix with ggplot2 graphics.

Usage

ggbplot(A, B, main = "", circle = TRUE, xlab = "", ylab = "", main.size = 8,
xlim = c(-1, 1), ylim = c(-1, 1), rowcolor = "red", rowch = 1, colcolor = "blue",
colch = 1, rowarrow = FALSE, colarrow = TRUE)

Arguments

Details

Dataframes A and B must consists of three columns labeled "PA1", "PA2" (coordinates of the first and second principal axis) and a column "strings" with the labels for the coordinates.

Dataframe B is optional. If it is not specified, a biplot with a single set of markers is constructed, for which the row settings must be specified.

Value

A ggplot2 object

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

References

Graffelman, J. and De Leeuw, J. (2023) On the visualisation of the correlation matrix. Available online. doi:10.48550/arXiv.2211.13150

See Also

bplot,ggtally,biplot

Examples

data("HeartAttack")
X <- as.matrix(HeartAttack[,1:7])
n <- nrow(X)
Xt <- scale(X)/sqrt(n-1)
res.svd <- svd(Xt)
Fs <- sqrt(n)*res.svd$u # standardized principal components
Gp <- crossprod(t(res.svd$v),diag(res.svd$d)) # biplot coordinates for variables
rows.df <- data.frame(Fs[,1:2],as.character(1:n))
colnames(rows.df) <- c("PA1","PA2","strings")
cols.df <- data.frame(Gp[,1:2],colnames(X))
colnames(cols.df) <- c("PA1","PA2","strings")
ggbplot(rows.df,cols.df,xlab="PA1",ylab="PA2",main="PCA")

Create a correlogram as a ggplot object.

Description

Function ggcorrelogram creates a correlogram of a correlation matrix using ggplot graphics.

Usage

ggcorrelogram(R, labs = colnames(R), ifun = "cos", cex = 1, main = "", ntrials = 50,
              xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2), hjust = 1, vjust = 2, size = 2,
	      main.size = 8)

Arguments

Details

ggcorrelogram makes a correlogram on the basis of a set of angles. All angles are given w.r.t the positive x-axis. Variables are represented by unit vectors emanating from the origin.

Value

A ggplot object. Field theta of the output contains the angles for the variables.

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

References

Trosset, M.W. (2005) Visualizing correlation. Journal of Computational and Graphical Statistics 14(1), pp. 1–19

See Also

correlogram,fit_angles,nlminb

Examples

 set.seed(123)
 X <- matrix(rnorm(90),ncol=3)
 R <- cor(X)
 angles <- ggcorrelogram(R)

Create a correlation tally stick on a biplot vector

Description

Function ggtally puts a series of dots along a biplot vector of a correlation matrix, so marking the change in correlation along the vector with specified values.

Usage

ggtally(G, p1, adj = 0, values = seq(-1, 1, by = 0.2), dotsize = 0.1, dotcolour = "black")

Arguments

Details

Any set of values for the correlation to be marked off can be used, though a standard scale with 0.2 increments is recommmended.

Value

A ggplot2 object with the updated biplot

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

References

Graffelman, J. and De Leeuw, J. (2023) On the visualisation of the correlation matrix. Available online. doi:10.48550/arXiv.2211.13150

See Also

ggbplot

Examples

library(calibrate)
data(goblets)
R <- cor(goblets)
out.sd <- eigen(R)
V  <- out.sd$vectors[,1:2]
Dl <- diag(out.sd$values[1:2])
Gp <- crossprod(t(V),sqrt(Dl))
pca.df <- data.frame(Gp)
pca.df$strings <- colnames(R)
colnames(pca.df) <- c("PA1","PA2","strings")
p1 <- ggbplot(pca.df,pca.df,main="PCA correlation biplot",xlab="",ylab="",rowarrow=TRUE,
              rowcolor="blue",rowch="",colch="")
p1 <- ggtally(Gp,p1,values=seq(-0.2,0.6,by=0.2),dotsize=0.1)

Correlations between size measurements of archeological goblets

Description

Correlations between 6 size measurements of archeological goblets

Usage

data(gobletsR)

Format

A matrix of correlations

Source

Manly (1989)

References

Manly, B.F.J. (1989) Multivariate statistical methods: a primer. Chapman and Hall, London.

Function for obtaining a weighted least squares low-rank approximation of a symmetric matrix

Description

Function ipSymLS implements an alternating least squares algorithm that uses both decomposition and block relaxation to find the optimal positive semidefinite approxation of given rank p to a known symmetric matrix of order n.

Usage

ipSymLS(target, w = matrix(1, dim(target)[1], dim(target)[2]), ndim = 2,
        init = FALSE, itmax = 100, eps = 1e-06, verbose = FALSE)

Arguments

Value

A matrix with the coordinates for the variables

Author(s)

deleeuw@stat.ucla.edu

References

De Leeuw, J. (2006) A decomposition method for weighted least squares low-rank approximation of symmetric matrices. Department of Statistics, UCLA. Retrieved from https://escholarship.org/uc/item/1wh197mh

Graffelman, J. and De Leeuw, J. (2023) Improved approximation and visualization of the correlation matrix. The American Statistician pp. 1–20. Available online as latest article doi:10.1080/00031305.2023.2186952

Examples

data(banknotes)
R <- cor(banknotes)
W <- matrix(1,nrow(R),nrow(R))
diag(W) <- 0
Fp.als <- ipSymLS(R,w=W,verbose=TRUE,eps=1e-15)
Rhat.als <- Fp.als%*%t(Fp.als)

Establish limits for x and y axis

Description

jointlim computes a sensible range for x and y axis if two sets of points are to be plotted simultaneously

Usage

jointlim(X, Y)

Arguments

Value

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

Examples

X <- matrix(runif(20),ncol=2)
Y <- matrix(runif(20),ncol=2)
print(jointlim(X,Y)$xlim)

Linang plot

Description

linangplot produces a plot of two variables, such that the correlation between the two variables is linear in the angle.

Usage

linangplot(x, y, tmx = NULL, tmy = NULL, ...)

Arguments

Value

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

See Also

plotcorrelogram

Examples

x <- runif(10)
y <- rnorm(10)
linangplot(x,y)

Linearized cosine function

Description

Function lincos linearizes the cosine function over the interval [0,2pi]. The function returns -2/pi*x + 1 over [0,pi] and 2/pi*x - 3 over [pi,2pi]

Usage

lincos(x)

Arguments

Value

a real number in [-1,1].

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

References

Graffelman, J. (2012) Linear-angle correlation plots: new graphs for revealing correlation structure. Journal of Computational and Graphical Statistics. 22(1): 92-106.

See Also

cos

Examples

angle <- pi
y <- lincos(angle)
print(y)

Principal Coordinate Analysis

Description

pco is a program for Principal Coordinate Analysis.

Usage

pco(Dis)

Arguments

Details

The program pco does a principal coordinates analysis of a dissimilarity (or distance) matrix (Dij) where the diagonal elements, Dii, are zero.

Note that when we dispose of a similarity matrix rather that a distance matrix, a transformation is needed before calling coorprincipal. For instance, if Sij is a similarity matrix, Dij might be obtained as Dij = 1 - Sij/diag(Sij)

Goodness of fit calculations need to be revised such as to deal (in different ways) with negative eigenvalues.

Value

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

See Also

cmdscale

Examples

citynames <- c("Aberystwyth","Brighton","Carlisle","Dover","Exeter","Glasgow","Hull",
"Inverness","Leeds","London","Newcastle", "Norwich")    
A <-matrix(c(
0,244,218,284,197,312,215,469,166,212,253,270,
244,0,350,77,167,444,221,583,242,53,325,168,
218,350,0,369,347,94,150,251,116,298,57,284,
284,77,369,0,242,463,236,598,257,72,340,164,
197,167,347,242,0,441,279,598,269,170,359,277,
312,444,94,463,441,0,245,169,210,392,143,378,
215,221,150,236,279,245,0,380,55,168,117,143,
469,583,251,598,598,169,380,0,349,531,264,514,
166,242,116,257,269,210,55,349,0,190,91,173,
212,53,298,72,170,392,168,531,190,0,273,111,
253,325,57,340,359,143,117,264,91,273,0,256,
270,168,284,164,277,378,143,514,173,111,256,0),ncol=12)
rownames(A) <- citynames
colnames(A) <- citynames
out <- pco(A)
plot(out$PC[,2],-out$PC[,1],pch=19,asp=1)
textxy(out$PC[,2],-out$PC[,1],rownames(A))

Principal factor analysis

Description

Program pfa performs (iterative) principal factor analysis, which is based on the computation of eigenvalues of the reduced correlation matrix.

Usage

pfa(X, option = "data", m = 2, initial.communality = "R2", crit = 0.001, verbose = FALSE)

Arguments

Value

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

References

Mardia, K.V., Kent, J.T. and Bibby, J.M. (1979) Multivariate analysis.

Rencher, A.C. (1995) Methods of multivriate analysis.

Satorra, A. and Neudecker, H. (1998) Least-Squares Approximation of off-Diagonal Elements of a Variance Matrix in the Context of Factor Analysis. Econometric Theory 14(1) pp. 156–157.

See Also

princomp

Examples

   X <- matrix(rnorm(100),ncol=2)
   out.pfa <- pfa(X)
#  based on a correlation matrix
   R <- cor(X)
   out.pfa <- pfa(R,option="cor")

Correlations between sources of protein

Description

Correlations between sources of protein for a number of countries (Red meat, White meat, Eggs, Milk, Fish, Cereals, Starchy food, Nuts, Fruits and vegetables.

Usage

data(proteinR)

Format

A matrix of correlations

Source

Manly (1989)

References

Manly, B.F.J. (1989) Multivariate statistical methods: a primer. Chapman and Hall, London.

Correlations between sources of protein

Description

Correlations between sources of protein for a number of countries (Red meat, White meat, Eggs, Milk, Fish, Cereals, Starchy food, Nuts, Fruits and vegetables.

Usage

data(proteinR)

Format

A matrix of correlations

Source

Manly (1989)

References

Manly, B.F.J. (1989) Multivariate statistical methods: a primer. Chapman and Hall, London.

Correlations between national track records for men

Description

Correlations between national track records for men (100m,200m,400m,800m,1500m,5000m,10.000m and Marathon

Usage

data(recordsR)

Format

A matrix of correlations

Source

Johnson and Wichern, Table 8.6

References

Johnson, R.A. and Wichern, D.W. (2002) Applied Multivariate Statistical Analysis. Fifth edition. New Jersey: Prentice Hall.

Calculate the root mean squared error

Description

Program rmse calculates the RMSE for a matrix approximation.

Usage

rmse(R, Rhat, W = matrix(1, nrow(R), ncol(R)) - diag(nrow(R)),
     verbose = FALSE, per.variable = FALSE)

Arguments

Details

By default, function rmse assumes a symmetric correlation matrix as input, together with its approximation. The approximation does not need to be symmetric. Weight matrix W has to be symmetric. By default, the diagonal is excluded from RMSE calcuations (W = J - I). To include it, specify W = J, that is set W = matrix(1, nrow(R), ncol(R))

Value

the calculated rmse

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

References

Graffelman, J. and De Leeuw, J. (2023) Improved approximation and visualization of the correlation matrix. The American Statistician pp. 1–20. doi:10.1080/00031305.2023.2186952

Examples

data(banknotes)
X <- as.matrix(banknotes[,1:6])
p <- ncol(X)
J <- matrix(1,p,p)
R <- cor(X)
out.sd <- eigen(R)
V <- out.sd$vectors
Dl <- diag(out.sd$values)
V2 <- V[,1:2]
D2 <- Dl[1:2,1:2]
Rhat <- V2%*%D2%*%t(V2)
rmse(R,Rhat,W=J)

Generate a table of root mean square error (RMSE) statistics for principal component analysis (PCA) and weighted alternating least squares (WALS).

Description

Function rmsePCAandWALS creates table with the RMSE for each variable, for a low-rank approximation to the correlation matrix obtained by PCA or WALS.

Usage

rmsePCAandWALS(R, output, digits = 4, omit.diagonals = c(FALSE,FALSE,TRUE,TRUE))

Arguments

Value

A matrix with one row per variable and four columns for RMSE statistics.

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

References

Graffelman, J. and De Leeuw, J. (2023) Improved approximation and visualization of the correlation matrix. The American Statistician pp. 1–20. doi:10.1080/00031305.2023.2186952

See Also

FitRwithPCAandWALS

Examples

data(HeartAttack)
X <- HeartAttack[,1:7]
X[,7] <- log(X[,7])
colnames(X)[7] <- "logPR"
R <- cor(X)
## Not run: 
out <- FitRwithPCAandWALS(R)
Results <- rmsePCAandWALS(R,out)

## End(Not run)

Correlations between three variables

Description

Danish data from 1953-1977 giving the correlations between nesting storks, human birth rate and per capita electricity consumption.

Usage

data(storksR)

Format

A matrix of correlations

Source

Gabriel and Odoroff, Table 1.

References

Gabriel, K. R. and Odoroff, C. L. (1990) Biplots in biomedical research. Statistics in Medicine 9(5): pp. 469-485.

Marks for 5 student exams

Description

Matrix of marks for five exams, two with closed books and three with open books (Mechanics (C), Vectors (C), Algebra (O), Analysis (O) and Statistics (O)).

Usage

data(students)

Format

A data matrix

Source

Mardia et al., Table 1.2.1

References

Mardia, K.V., Kent, J.T. and Bibby, J.M. (1979) Multivariate Analysis, Academic Press London.

Correlations between marks for 5 exams

Description

Correlation matrix of marks for five exams, two with closed books and three with open books (Mechanics (C), Vectors (C), Algebra (O), Analysis (O) and Statistics (O)).

Usage

data(studentsR)

Format

A matrix of correlations

Source

Mardia et al., Table 1.2.1

References

Mardia, K.V., Kent, J.T. and Bibby, J.M. (1979) Multivariate Analysis, Academic Press London.

Create a tally on a biplot vector

Description

Function tally marks of a set of dots on a biplot vector. It is thought for biplot vectors representing correlations, such that their correlation scale becomes visible, without doing a full calibration with tick marks and tick mark labels.

Usage

tally(G, adj = 0, values = seq(-1, 1, by = 0.2), pch = 19, dotcolor = "black", cex = 0.5,
      color.negative = "red", color.positive = "blue")

Arguments

Value

NULL

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

References

Graffelman, J. and De Leeuw, J. (2023) Improved approximation and visualization of the correlation matrix. The American Statistician pp. 1–20. doi:10.1080/00031305.2023.2186952

See Also

bplot, calibrate

Examples

data(goblets)
R <- cor(goblets)
results <- eigen(R)
V  <- results$vectors
Dl <- diag(results$values)
#
# Calculate correlation biplot coordinates
#
G  <- crossprod(t(V[,1:2]),sqrt(Dl[1:2,1:2]))
#
# Make the biplot
#
bplot(G,G,rowch=NA,colch=NA,collab=colnames(R),
      xl=c(-1.1,1.1),yl=c(-1.1,1.1))
#
# Create a correlation tally stick for variable X1
#
tally(G[1,])

Compute the trace of a matrix

Description

tr computes the trace of a matrix.

Usage

tr(X)

Arguments

Value

the trace (a scalar)

Author(s)

Jan Graffelman (jan.graffelman@upc.edu)

Examples

X <- matrix(runif(25),ncol=5)
print(X)
print(tr(X))

Low-rank matrix approximation by weighted alternating least squares

Description

Function wAddPCA calculates a weighted least squares approximation of low rank to a given matrix.

Usage

wAddPCA(x, w = matrix(1, nrow(x), ncol(x)), p = 2, add = "all", bnd = "opt",
        itmaxout = 1000, itmaxin = 1000, epsout = 1e-06, epsin = 1e-06,
	verboseout = TRUE, verbosein = FALSE)

Arguments

Value

A list object with fields:

Author(s)

jan@deleeuwpdx.net

References

Graffelman, J. and De Leeuw, J. (2023) Improved approximation and visualization of the correlation matrix. The American Statistician pp. 1–20. Available online as latest article doi:10.1080/00031305.2023.2186952

https://jansweb.netlify

See Also

ipSymLS

Examples

data(HeartAttack)
X <- HeartAttack[,1:7]
X[,7] <- log(X[,7])
colnames(X)[7] <- "logPR"
R <- cor(X)
W <- matrix(1, 7, 7)
diag(W) <- 0
Wals.out <- wAddPCA(R, W, add = "nul", verboseout = FALSE) 
Rhat <- Wals.out$y