Type:	Package
Title:	Influence Measures and Diagnostic Plots for Multivariate Linear Models
Version:	0.9.0
Date:	2022-09-10
Maintainer:	Michael Friendly <friendly@yorku.ca>
Description:	Computes regression deletion diagnostics for multivariate linear models and provides some associated diagnostic plots. The diagnostic measures include hat-values (leverages), generalized Cook's distance, and generalized squared 'studentized' residuals. Several types of plots to detect influential observations are provided.
Depends:	car, heplots
Suggests:	knitr, rmarkdown, ggplot2, tibble, patchwork, rgl, dplyr
LazyData:	TRUE
VignetteBuilder:	knitr
Encoding:	UTF-8
License:	GPL-2
Language:	en-US
URL:	https://github.com/friendly/mvinfluence
BugReports:	https://github.com/friendly/mvinfluence/issues
Packaged:	2022-09-20 16:49:20 UTC; friendly
RoxygenNote:	7.2.1
NeedsCompilation:	no
Author:	Michael Friendly [aut, cre]
Repository:	CRAN
Date/Publication:	2022-09-20 17:10:02 UTC

Influence Measures and Diagnostic Plots for Multivariate Linear Models

Description

Functions in this package compute regression deletion diagnostics for multivariate linear models following methods proposed by Barrett & Ling (1992) and provide some associated diagnostic plots.

Details

The design goal for this package is that, as an extension of standard methods for univariate linear models, you should be able to fit a linear model with a multivariate response,

  mymlm <- lm( cbind(y1, y2, y3) ~ x1 + x2 + x3, data=mydata)

and then get useful diagnostics and plots with

  influence(mymlm)
  hatvalues(mymlm)
  influencePlot(mymlm, ...)  

The diagnostic measures include hat-values (leverages), generalized Cook's distance and generalized squared 'studentized' residuals. Several types of plots to detect influential observations are provided.

In addition, the functions provide diagnostics for deletion of subsets of observations of size m>1. This case is theoretically interesting because sometimes pairs (m=2) of influential observations can mask each other, sometimes they can have joint influence far exceeding their individual effects, as well as other interesting phenomena described by Lawrence (1995). Associated methods for the case m>1 are still under development in this package.

The main function in the package is the S3 method, influence.mlm, a simple wrapper for mlm.influence, which does the actual computations. This design was dictated by that used in the stats package, which provides the generic method influence and methods influence.lm and influence.glm. The car package extends this to include influence.lme for models fit by lme.

The following sections describe the notation and measures used in the calculations.

Notation

Let \mathbf{X} be the model matrix in the multivariate linear model, \mathbf{Y}_{n \times p} = \mathbf{X}_{n \times r} \mathbf{\beta}_{r \times p} + \mathbf{E}_{n \times p}. The usual least squares estimate of \mathbf{\beta} is given by \mathbf{B} = (\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T} \mathbf{Y}.

Then let

• \mathbf{X}_I be the submatrix of \mathbf{X} whose m rows are indexed by I,

• \mathbf{X}_{(I)} is the complement, the submatrix of \mathbf{X} with the m rows in I deleted,

Matrices \mathbf{Y}_I, \mathbf{Y}_{(I)} are defined similarly.

In the calculation of regression coefficients, \mathbf{B}_{(I)} = (\mathbf{X}_{(I)}^{T} \mathbf{X}_{(I)})^{-1} \mathbf{X}_{(I)}^{T} \mathbf{Y}_{I} are the estimated coefficients when the cases indexed by I have been removed. The corresponding residuals are \mathbf{E}_{(I)} = \mathbf{Y}_{(I)} - \mathbf{X}_{(I)} \mathbf{B}_{(I)}.

Measures

The influence measures defined by Barrett & Ling (1992) are functions of two matrices \mathbf{H}_I and \mathbf{Q}_I defined as follows:

• For the full data set, the “hat matrix”, \mathbf{H}, is given by \mathbf{H} = \mathbf{X} (\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T} ,

• \mathbf{H}_I is m \times m the submatrix of \mathbf{H} corresponding to the index set I, \mathbf{H}_I = \mathbf{X} (\mathbf{X}_I^{T} \mathbf{X}_I)^{-1} \mathbf{X}^{T} ,

• \mathbf{Q} is the analog of \mathbf{H} defined for the residual matrix \mathbf{E}, that is, \mathbf{Q} = \mathbf{E} (\mathbf{E}^{T} \mathbf{E})^{-1} \mathbf{E}^{T} , with corresponding submatrix \mathbf{Q}_I = \mathbf{E} (\mathbf{E}_I^{T} \mathbf{E}_I)^{-1} \mathbf{E}^{T} ,

Cook's distance

In these terms, Cook's distance is defined for a univariate response by

D_I = (\mathbf{b} - \mathbf{b}_{(I)})^T (\mathbf{X}^T \mathbf{X}) (\mathbf{b} - \mathbf{b}_{(I)}) / p s^2 \; ,

a measure of the squared distance between the coefficients \mathbf{b} for the full data set and those \mathbf{b}_{(I)} obtained when the cases in I are deleted.

In the multivariate case, Cook's distance is obtained by replacing the vector of coefficients \mathbf{b} by \mathrm{vec} (\mathbf{B}), the result of stringing out the coefficients for all responses in a single n \times p-length vector.

D_I = \frac{1}{p} [\mathrm{vec} (\mathbf{B} - \mathbf{B}_{(I)})]^T (S_{-1} \otimes \mathbf{X}^T \mathbf{X}) \mathrm{vec} (\mathbf{B} - \mathbf{B}_{(I)}) \; ,

where \otimes is the Kronecker (direct) product and \mathbf{S} = \mathbf{E}^T \mathbf{E} / (n-p) is the covariance matrix of the residuals.

Leverage and residual components

For a univariate response, and when m = 1, Cook's distance can be re-written as a product of leverage and residual components as

D_i = \left(\frac{n-p}{p} \right) \frac{h_{ii}}{(1 - h_{ii})^2 q_{ii} } \;.

Then we can define a leverage component L_i and residual component R_i as

L_i = \frac{h_{ii}}{1 - h_{ii}} \quad\quad R_i = \frac{q_{ii}}{1 - h_{ii}} \;.

R_i is the studentized residual, and D_i \propto L_i \times R_i.

In the general, multivariate case there are analogous matrix expressions for \mathbf{L} and \mathbf{R}. When m > 1, the quantities \mathbf{H}_I, \mathbf{Q}_I, \mathbf{L}_I, and \mathbf{R}_I are m \times m matrices. Where scalar quantities are needed, the package functions apply a function, FUN, either det() or tr() to calculate a measure of “size”, as in

  H <- sapply(x$H, FUN)
  Q <- sapply(x$Q, FUN)
  L <- sapply(x$L, FUN)
  R <- sapply(x$R, FUN)

References

Barrett, B. E. and Ling, R. F. (1992). General Classes of Influence Measures for Multivariate Regression. Journal of the American Statistical Association, 87(417), 184-191.

Barrett, B. E. (2003). Understanding Influence in Multivariate Regression. Communications in Statistics – Theory and Methods, 32, 3, 667-680.

A. J. Lawrence (1995). Deletion Influence and Masking in Regression. Journal of the Royal Statistical Society. Series B (Methodological) , 57, 1, 181-189.

Fertilizer Data

Description

A small data set on the use of fertilizer (x) in relation to the amount of grain (y1) and straw (y2) produced.

Format

A data frame with 8 observations on the following 3 variables.

grain

amount of grain produced

straw

amount of straw produced

fertilizer

amount of fertilizer applied

Details

The first observation is an obvious outlier and influential observation.

Source

Anderson, T. W. (1984). An Introduction to Multivariate Statistical Analysis, New York: Wiley, p. 369.

References

Hossain, A. and Naik, D. N. (1989). Detection of influential observations in multivariate regression. Journal of Applied Statistics, 16 (1), 25-37.

Examples

data(Fertilizer)

# simple plots
plot(Fertilizer, col=c('red', rep("blue",7)), 
     cex=c(2,rep(1.2,7)), 
     pch=as.character(1:8))

# A biplot shows the data in 2D. It gives another view of how case 1 stands out in data space
biplot(prcomp(Fertilizer))

# fit the mlm
mod <- lm(cbind(grain, straw) ~ fertilizer, data=Fertilizer)
Anova(mod)

# influence plots (m=1)
influencePlot(mod)
influencePlot(mod, type='LR')
influencePlot(mod, type='stres')

General Classes of Influence Measures

Description

These functions implement the general classes of influence measures for multivariate regression models defined in Barrett and Ling (1992), Eqn 2.3, 2.4, as shown in their Table 1.

Usage

Jtr(H, Q, a, b, f)

Jdet(H, Q, a, b, f)

COOKD(H, Q, n, p, r, m)

DFFITS(H, Q, n, p, r, m)

COVRATIO(H, Q, n, p, r, m)

Arguments

Details

There are two classes of functions, denoted J_I^{det} and J_I^{tr}, with parameters n, p, q of the data, m of the subset size and a and b which define powers of terms in the formulas, typically in the set -2, -1, 0.

They are defined in terms of the submatrices for a deleted index subset I,

H_I = X_I (X^T X)^{-1} X_I

Q_I = E_I (E^T E)^{-1} E_I

corresponding to the hat and residual matrices in univariate models.

For subset size m = 1 these evaluate to scalar equivalents of hat values and studentized residuals.

For subset size m > 1 these are m \times m matrices and functions in the J^{det} class use |H_I| and |Q_I|, while those in the J^{tr} class use tr(H_I) and tr(Q_I).

The functions COOKD, COVRATIO, and DFFITS implement some of the standard influence measures in these terms for the general cases of multivariate linear models and deletion of subsets of size m>1, but they have not yet been incorporated into our main functions mlm.influence and influence.mlm.

Value

The scalar result of the computation.

Author(s)

Michael Friendly

References

Barrett, B. E. and Ling, R. F. (1992). General Classes of Influence Measures for Multivariate Regression. Journal of the American Statistical Association, 87(417), 184-191.

Convert an inflmlm object to a data frame

Description

This function is used internally in the package to convert the result of mlm.influence() to a data frame. It is not normally called by the user.

Usage

## S3 method for class 'inflmlm'
as.data.frame(x, ..., FUN = det, funnames = TRUE)

Arguments

Value

A data frame containing the influence statistics

Examples

# none

Cook's distance for a MLM

Description

The functions cooks.distance.mlm and hatvalues.mlm are designed as extractor functions for regression deletion diagnostics for multivariate linear models following Barrett & Ling (1992). These are close analogs of methods for univariate and generalized linear models handled by the influence.measures in the stats package.

Usage

## S3 method for class 'mlm'
cooks.distance(model, infl = mlm.influence(model, do.coef = FALSE), ...)

Arguments

Details

In addition, the functions provide diagnostics for deletion of subsets of observations of size m>1.

Value

A vector of Cook's distances

References

Barrett, B. E. and Ling, R. F. (1992). General Classes of Influence Measures for Multivariate Regression. Journal of the American Statistical Association, 87(417), 184-191.

Examples

data(Rohwer, package="heplots")
Rohwer2 <- subset(Rohwer, subset=group==2)
rownames(Rohwer2)<- 1:nrow(Rohwer2)
Rohwer.mod <- lm(cbind(SAT, PPVT, Raven) ~ n+s+ns+na+ss, data=Rohwer2)

hatvalues(Rohwer.mod)
cooks.distance(Rohwer.mod)

Hatvalues for a MLM

Description

The functions cooks.distance.mlm and hatvalues.mlm are designed as extractor functions for regression deletion diagnostics for multivariate linear models following Barrett & Ling (1992). These are close analogs of methods for univariate and generalized linear models handled by the influence.measures in the stats package.

Usage

## S3 method for class 'mlm'
hatvalues(model, m = 1, infl, ...)

Arguments

Details

Hat values are a component of influence diagnostics, measuring the leverage or outlyingness of observations in the space of the predictor variables.

The usual case considers observations one at a time (m=1), where the hatvalue is proportional to the squared Mahalanobis distance, D^2 of each observation from the centroid of all observations. This function extends that definition to calculate a comparable quantity for subsets of size m>1.

Value

A vector of hatvalues

References

Barrett, B. E. and Ling, R. F. (1992). General Classes of Influence Measures for Multivariate Regression. Journal of the American Statistical Association, 87(417), 184-191.

See Also

cooks.distance.mlm

Examples

data(Rohwer, package="heplots")
Rohwer2 <- subset(Rohwer, subset=group==2)
rownames(Rohwer2)<- 1:nrow(Rohwer2)
Rohwer.mod <- lm(cbind(SAT, PPVT, Raven) ~ n+s+ns+na+ss, data=Rohwer2)

options(digits=3)
hatvalues(Rohwer.mod)
cooks.distance(Rohwer.mod)

Influence Index Plots for Multivariate Linear Models

Description

Provides index plots of some diagnostic measures for a multivariate linear model: Cook's distance, a generalized (squared) studentized residual, hat-values (leverages), and Mahalanobis squared distances of the residuals.

Usage

## S3 method for class 'mlm'
infIndexPlot(
  model,
  infl = mlm.influence(model, do.coef = FALSE),
  FUN = det,
  vars = c("Cook", "Studentized", "hat", "DSQ"),
  main = paste("Diagnostic Plots for", deparse(substitute(model))),
  pch = 19,
  labels,
  id.method = "y",
  id.n = if (id.method[1] == "identify") Inf else 0,
  id.cex = 1,
  id.col = palette()[1],
  id.location = "lr",
  grid = TRUE,
  ...
)

Arguments

Details

This function produces index plots of the various influence measures calculated by influence.mlm, and in addition, the measure based on the Mahalanobis squared distances of the residuals from the origin.

Value

None. Used for its side effect of producing a graph.

Author(s)

Michael Friendly; borrows code from car::infIndexPlot

References

Barrett, B. E. and Ling, R. F. (1992). General Classes of Influence Measures for Multivariate Regression. Journal of the American Statistical Association, 87(417), 184-191.

Barrett, B. E. (2003). Understanding Influence in Multivariate Regression Communications in Statistics - Theory and Methods, 32, 667-680.

See Also

influencePlot.mlm, Mahalanobis, infIndexPlot,

Examples

# iris data
data(iris)
iris.mod <- lm(as.matrix(iris[,1:4]) ~ Species, data=iris)
infIndexPlot(iris.mod, col=iris$Species, id.n=3)

# Sake data
data(Sake, package="heplots")
Sake.mod <- lm(cbind(taste,smell) ~ ., data=Sake)
infIndexPlot(Sake.mod, id.n=3)

# Rohwer data
data(Rohwer, package="heplots")
Rohwer2 <- subset(Rohwer, subset=group==2)
rownames(Rohwer2)<- 1:nrow(Rohwer2)
rohwer.mlm <- lm(cbind(SAT, PPVT, Raven) ~ n + s + ns + na + ss, data=Rohwer2)
infIndexPlot(rohwer.mlm, id.n=3)

Regression Deletion Diagnostics for Multivariate Linear Models

Description

This collection of functions is designed to compute regression deletion diagnostics for multivariate linear models following Barrett & Ling (1992) that are close analogs of methods for univariate and generalized linear models handled by the influence.measures in the stats package.

Usage

## S3 method for class 'mlm'
influence(model, do.coef = TRUE, m = 1, ...)

Arguments

Details

In addition, the functions provide diagnostics for deletion of subsets of observations of size m>1.

influence.mlm is a simple wrapper for the computational function, mlm.influence designed to provide an S3 method for class "mlm" objects.

There are still infelicities in the methods for the m>1 case in the current implementation. In particular, for m>1, you must call influence.mlm directly, rather than using the S3 generic influence().

Value

influence.mlm returns an S3 object of class inflmlm, a list with the following components

Author(s)

Michael Friendly

References

Barrett, B. E. and Ling, R. F. (1992). General Classes of Influence Measures for Multivariate Regression. Journal of the American Statistical Association, 87(417), 184-191.

See Also

influencePlot.mlm, mlm.influence

Examples

# Rohwer data
data(Rohwer, package="heplots")
Rohwer2 <- subset(Rohwer, subset=group==2)
rownames(Rohwer2)<- 1:nrow(Rohwer2)
Rohwer.mod <- lm(cbind(SAT, PPVT, Raven) ~ n+s+ns+na+ss, data=Rohwer2)

# m=1 diagnostics
influence(Rohwer.mod) |> head()

# try an m=2 case
## res2 <- influence.mlm(Rohwer.mod, m=2, do.coef=FALSE)
## res2.df <- as.data.frame(res2)
## head(res2.df)
## scatterplotMatrix(log(res2.df))


influencePlot(Rohwer.mod, id.n=4, type="cookd")


# Sake data
data(Sake, package="heplots")
Sake.mod <- lm(cbind(taste,smell) ~ ., data=Sake)
influence(Sake.mod)
influencePlot(Sake.mod, id.n=3, type="cookd")

Influence Plots for Multivariate Linear Models

Description

This function creates various types of “bubble” plots of influence measures with the areas of the circles representing the observations proportional to generalized Cook's distances.

Usage

## S3 method for class 'mlm'
influencePlot(
  model,
  scale = 12,
  type = c("stres", "LR", "cookd"),
  infl = mlm.influence(model, do.coef = FALSE),
  FUN = det,
  fill = TRUE,
  fill.col = "red",
  fill.alpha.max = 0.5,
  labels,
  id.method = "noteworthy",
  id.n = if (id.method[1] == "identify") Inf else 0,
  id.cex = 1,
  id.col = palette()[1],
  ref.col = "gray",
  ref.lty = 2,
  ref.lab = TRUE,
  ...
)

Arguments

Details

type="stres" plots squared (internally) Studentized residuals against hat values; type="cookd" plots Cook's distance against hat values; type="LR" plots residual components against leverage components, with the attractive property that contours of constant Cook's distance fall on diagonal lines with slope = -1. Adjacent reference lines represent multiples of influence.

The id.method="noteworthy" setting also requires setting id.n>0 to have any effect. Using id.method="noteworthy", and id.n>0, the number of points labeled is the union of the largest id.n values on each of L, R, and CookD.

Value

If points are identified, returns a data frame with the hat values, Studentized residuals and Cook's distance of the identified points. If no points are identified, nothing is returned. This function is primarily used for its side-effect of drawing a plot.

Author(s)

Michael Friendly

References

Barrett, B. E. and Ling, R. F. (1992). General Classes of Influence Measures for Multivariate Regression. Journal of the American Statistical Association, 87(417), 184-191.

Barrett, B. E. (2003). Understanding Influence in Multivariate Regression Communications in Statistics - Theory and Methods, 32, 667-680.

McCulloch, C. E. & Meeter, D. (1983). Discussion of "Outliers..." by R. J. Beckman and R. D. Cook. Technometrics, 25, 152-155

See Also

mlm.influence, lrPlot

influencePlot in the car package

Examples

data(Rohwer, package="heplots")
Rohwer2 <- subset(Rohwer, subset=group==2)
Rohwer.mod <- lm(cbind(SAT, PPVT, Raven) ~ n+s+ns+na+ss, data=Rohwer2)

influencePlot(Rohwer.mod, id.n=4, type="stres")
influencePlot(Rohwer.mod, id.n=4, type="LR")
influencePlot(Rohwer.mod, id.n=4, type="cookd")

# Sake data
data(Sake, package="heplots")
	Sake.mod <- lm(cbind(taste,smell) ~ ., data=Sake)
	influencePlot(Sake.mod, id.n=3, type="stres")
	influencePlot(Sake.mod, id.n=3, type="LR")
	influencePlot(Sake.mod, id.n=3, type="cookd")

# Adopted data	
data(Adopted, package="heplots")
Adopted.mod <- lm(cbind(Age2IQ, Age4IQ, Age8IQ, Age13IQ) ~ AMED + BMIQ, data=Adopted)
influencePlot(Adopted.mod, id.n=3)
influencePlot(Adopted.mod, id.n=3, type="LR", ylim=c(-4,-1.5))

Regression LR Influence Plot

Description

This function creates a “bubble” plot of functions, R = log(Studentized residuals^2) by L = log(H/p*(1-H)) of the hat values, with the areas of the circles representing the observations proportional to Cook's distances.

Usage

lrPlot(model, ...)

## S3 method for class 'lm'
lrPlot(
  model,
  scale = 12,
  xlab = "log Leverage factor [log H/p*(1-H)]",
  ylab = "log (Studentized Residual^2)",
  xlim = NULL,
  ylim,
  labels,
  id.method = "noteworthy",
  id.n = if (id.method[1] == "identify") Inf else 0,
  id.cex = 1,
  id.col = palette()[1],
  ref = c("h", "v", "d", "c"),
  ref.col = "gray",
  ref.lty = 2,
  ref.lab = TRUE,
  ...
)

Arguments

Details

This plot, suggested by McCulloch & Meeter (1983) has the attractive property that contours of equal Cook's distance are diagonal lines with slope = -1. Various reference lines are drawn on the plot corresponding to twice and three times the average hat value, a “large” squared studentized residual and contours of Cook's distance.

The id.method="noteworthy" setting also requires setting id.n>0 to have any effect. Using id.method="noteworthy", and id.n>0, the number of points labeled is the union of the largest id.n values on each of L, R, and CookD.

Value

If points are identified, returns a data frame with the hat values, Studentized residuals and Cook's distance of the identified points. If no points are identified, nothing is returned. This function is primarily used for its side-effect of drawing a plot.

Author(s)

Michael Friendly

References

A. J. Lawrence (1995). Deletion Influence and Masking in Regression Journal of the Royal Statistical Society. Series B (Methodological) , Vol. 57, No. 1, pp. 181-189.

McCulloch, C. E. & Meeter, D. (1983). Discussion of "Outliers..." by R. J. Beckman and R. D. Cook. Technometrics, 25, 152-155.

See Also

influencePlot.mlm influencePlot in the car package for other methods

Examples

# artificial example from Lawrence (1995)
x <- c( 0, 0, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 18, 18 )
y <- c( 0, 6, 6, 7, 6, 7, 6, 7, 6,  7,  6,  7,  7,  18 )
DF <- data.frame(x,y, row.names=LETTERS[1:length(x)])
DF

with(DF, {
	plot(x,y, pch=16, cex=1.3)
	abline(lm(y~x), col="red", lwd=2)
	NB <- c(1,2,13,14)
	text(x[NB],y[NB], LETTERS[NB], pos=c(4,4,2,2))
	}
)

mod <- lm(y~x, data=DF)
# standard influence plot from car
influencePlot(mod, id.n=4)

# lrPlot version
lrPlot(mod, id.n=4)


library(car)
dmod <- lm(prestige ~ income + education, data = Duncan)
influencePlot(dmod, id.n=3)
lrPlot(dmod, id.n=3)

Calculate Regression Deletion Diagnostics for Multivariate Linear Models

Description

mlm.influence is the main computational function in this package. It is usually not called directly, but rather via its alias, influence.mlm, the S3 method for a mlm object.

Usage

mlm.influence(model, do.coef = TRUE, m = 1, ...)

Arguments

Details

The computations and methods for the m=1 case are straight-forward, as are the computations for the m>1 case. Associated methods for m>1 are still under development.

Value

mlm.influence returns an S3 object of class inflmlm, a list with the following components:

Author(s)

Michael Friendly

References

Barrett, B. E. and Ling, R. F. (1992). General Classes of Influence Measures for Multivariate Regression. Journal of the American Statistical Association, 87(417), 184-191.

Barrett, B. E. (2003). Understanding Influence in Multivariate Regression. Communications in Statistics – Theory and Methods, 32, 3, 667-680.

See Also

influencePlot.mlm

Examples

Rohwer2 <- subset(Rohwer, subset=group==2)
rownames(Rohwer2)<- 1:nrow(Rohwer2)
Rohwer.mod <- lm(cbind(SAT, PPVT, Raven) ~ n+s+ns+na+ss, data=Rohwer2)
Rohwer.mod
influence(Rohwer.mod)

# extract the most influential cases
influence(Rohwer.mod) |> 
    as.data.frame() |> 
    dplyr::arrange(dplyr::desc(CookD)) |> 
    head()

# Sake data
Sake.mod <- lm(cbind(taste,smell) ~ ., data=Sake)
influence(Sake.mod) |>
    as.data.frame() |> 
    dplyr::arrange(dplyr::desc(CookD)) |> head()

General Matrix Power

Description

Calculates the n-th power of a square matrix, where n can be a positive or negative integer or a fractional power.

Usage

mpower(A, n)

A %^% n

Arguments

Details

If n<0, the method is applied to A^{-1}. When n is an integer, the function uses the Russian peasant method, or repeated squaring for efficiency. Otherwise, it uses the spectral decomposition of A, \mathbf{A}^n = \mathbf{V} \mathbf{D}^n \mathbf{V}^{T} requiring a symmetric matrix.

Value

Returns the matrix A^n

Author(s)

Michael Friendly

References

https://en.wikipedia.org/wiki/Exponentiation_by_squaring

See Also

Packages corpcor and expm define similar functions.

Examples

M <- matrix(sample(1:9), 3,3)
mpower(M,2)
mpower(M,4)

# make a symmetric matrix
MM <- crossprod(M)
mpower(MM, -1)
Mhalf <- mpower(MM, 1/2)
all.equal(MM, Mhalf %*% Mhalf)

Print an inflmlm object

Description

Print an inflmlm object

Usage

## S3 method for class 'inflmlm'
print(x, digits = max(3, getOption("digits") - 4), FUN = det, ...)

Arguments

Value

Invisibly returns the object

Examples

# none

Matrix trace

Description

Calculates the trace of a matrix

Usage

tr(M)

Arguments

Details

For square, symmetric matrices, such as covariance matrices, the trace is sometimes used as a measure of size, e.g., in Pillai's trace criterion for a MLM.

Value

returns the sum of the diagonal elements of the matrix

Author(s)

Michael Friendly

Examples

M <- matrix(sample(1:9), 3,3)
tr(M)