Help for package genridge

Type:

Package

Title:

Generalized Ridge Trace Plots for Ridge Regression

Version:

0.8.0

Date:

2024-11-30

Maintainer:

Michael Friendly <friendly@yorku.ca>

Depends:

R (≥ 3.5.0), car

Imports:

rgl, colorspace, splines

Suggests:

MASS, bestglm, vcdExtra

Description:

The genridge package introduces generalizations of the standard univariate ridge trace plot used in ridge regression and related methods. These graphical methods show both bias (actually, shrinkage) and precision, by plotting the covariance ellipsoids of the estimated coefficients, rather than just the estimates themselves. 2D and 3D plotting methods are provided, both in the space of the predictor variables and in the transformed space of the PCA/SVD of the predictors.

License:

GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]

LazyLoad:

yes

LazyData:

yes

Language:

en-US

BugReports:

https://github.com/friendly/genridge/issues

URL:

https://github.com/friendly/genridge, https://friendly.github.io/genridge/

RoxygenNote:

7.3.2

Encoding:

UTF-8

NeedsCompilation:

Packaged:

2024-12-02 14:41:12 UTC; friendly

Author:

Michael Friendly

[aut, cre]

Repository:

CRAN

Date/Publication:

2024-12-02 15:00:02 UTC

Generalized ridge trace plots for ridge regression

Description

The genridge package introduces generalizations of the standard univariate ridge trace plot used in ridge regression and related methods (Friendly, 2012). These graphical methods show both bias (actually, shrinkage) and precision, by plotting the covariance ellipsoids of the estimated coefficients, rather than just the estimates themselves. 2D and 3D plotting methods are provided, both in the space of the predictor variables and in the transformed space of the PCA/SVD of the predictors.

Details

This package provides computational support for the graphical methods described in Friendly (2013). Ridge regression models may be fit using the function ridge, which incorporates features of lm.ridge. In particular, the shrinkage factors in ridge regression may be specified either in terms of the constant added to the diagonal of X^T X matrix (lambda), or the equivalent number of degrees of freedom.

More importantly, the ridge function also calculates and returns the associated covariance matrices of each of the ridge estimates, allowing precision to be studied and displayed graphically.

This provides the support for the main plotting functions in the package:

plot.ridge: Bivariate ridge trace plots

pairs.ridge: All pairwise bivariate ridge trace plots

plot3d.ridge: 3D ridge trace plots

traceplot: Traditional univariate ridge trace plots

In addition, the function pca.ridge transforms the coefficients and covariance matrices of a ridge object from predictor space to the equivalent, but more interesting space of the PCA of X^T X or the SVD of X. The main plotting functions also work for these objects, of class c("ridge", "pcaridge").

Finally, the functions precision and vif.ridge provide other useful measures and plots.

Author(s)

Michael Friendly

Maintainer: Michael Friendly <friendly@yorku.ca>

References

Friendly, M. (2013). The Generalized Ridge Trace Plot: Visualizing Bias and Precision. Journal of Computational and Graphical Statistics, 22(1), 50-68, doi:10.1080/10618600.2012.681237, https://www.datavis.ca/papers/genridge-jcgs.pdf

Arthur E. Hoerl and Robert W. Kennard (1970). Ridge Regression: Biased Estimation for Nonorthogonal Problems, Technometrics, 12(1), pp. 55-67.

Arthur E. Hoerl and Robert W. Kennard (1970). Ridge Regression: Applications to Nonorthogonal Problems Technometrics, 12(1), pp. 69-82.

Examples


# see examples for ridge, etc.

Acetylene Data

Description

The data consist of measures of yield of a chemical manufacturing process for acetylene in relation to numeric parameters.

Format

A data frame with 16 observations on the following 4 variables.

yield: conversion percentage yield of acetylene
temp: reactor temperature (celsius)
ratio: H2 to N-heptone ratio
time: contact time (sec)

Details

Marquardt and Snee (1975) used these data to illustrate ridge regression in a model containing quadratic and interaction terms, particularly the need to center and standardize variables appearing in high-order terms.

Typical models for these data include the interaction of temp:ratio, and a squared term in temp

Source

SAS documentation example for PROC REG, Ridge Regression for Acetylene Data.

References

Marquardt, D.W., and Snee, R.D. (1975), "Ridge Regression in Practice," The American Statistician, 29, 3-20.

Marquardt, D.W. (1980), "A Critique of Some Ridge Regression Methods: Comment," Journal of the American Statistical Association, Vol. 75, No. 369 (Mar., 1980), pp. 87-91

Examples


data(Acetylene)

# naive model, not using centering
amod0 <- lm(yield ~ temp + ratio + time + I(time^2) + temp:time, data=Acetylene)

y <- Acetylene[,"yield"]
X0 <- model.matrix(amod0)[,-1]

lambda <- c(0, 0.0005, 0.001, 0.002, 0.005, 0.01)
aridge0 <- ridge(y, X0, lambda=lambda)

traceplot(aridge0)
traceplot(aridge0, X="df")
pairs(aridge0, radius=0.2)

Detroit Homicide Data for 1961-1973

Description

The data set Detroit was used extensively in the book by Miller (2002) on subset regression. The data are unusual in that a subset of three predictors can be found which gives a very much better fit to the data than the subsets found from the Efroymson stepwise algorithm, or from forward selection or backward elimination. They are also unusual in that, as time series data, the assumption of independence is patently violated, and the data suffer from problems of high collinearity.

As well, ridge regression reveals somewhat paradoxical paths of shrinkage in univariate ridge trace plots, that are more comprehensible in multivariate views.

Format

A data frame with 13 observations on the following 14 variables.

Police: Full-time police per 100,000 population
Unemp: Percent unemployed in the population
MfgWrk: Number of manufacturing workers in thousands
GunLic: Number of handgun licences per 100,000 population
GunReg: Number of handgun registrations per 100,000 population
HClear: Percent of homicides cleared by arrests
WhMale: Number of white males in the population
NmfgWrk: Number of non-manufacturing workers in thousands
GovWrk: Number of government workers in thousands
HrEarn: Average hourly earnings
WkEarn: Average weekly earnings
Accident: Death rate in accidents per 100,000 population
Assaults: Number of assaults per 100,000 population
Homicide: Number of homicides per 100,000 of population

Details

The data were originally collected and discussed by Fisher (1976) but the complete dataset first appeared in Gunst and Mason (1980, Appendix A). Miller (2002) discusses this dataset throughout his book, but doesn't state clearly which variables he used as predictors and which is the dependent variable. (Homicide was the dependent variable, and the predictors were Police ... WkEarn.) The data were obtained from StatLib.

A similar version of this data set, with different variable names appears in the bestglm package.

Source

https://lib.stat.cmu.edu/datasets/detroit

References

Fisher, J.C. (1976). Homicide in Detroit: The Role of Firearms. Criminology, 14, 387–400.

Gunst, R.F. and Mason, R.L. (1980). Regression analysis and its application: A data-oriented approach. Marcel Dekker.

Miller, A. J. (2002). Subset Selection in Regression. 2nd Ed. Chapman & Hall/CRC. Boca Raton.

Examples


data(Detroit)

# Work with a subset of predictors, from Miller (2002, Table 3.14),
# the "best" 6 variable model
#    Variables: Police, Unemp, GunLic, HClear, WhMale, WkEarn
# Scale these for comparison with other methods

Det <- as.data.frame(scale(Detroit[,c(1,2,4,6,7,11)]))
Det <- cbind(Det, Homicide=Detroit[,"Homicide"])

# use the formula interface; specify ridge constants in terms
# of equivalent degrees of freedom
dridge <- ridge(Homicide ~ ., data=Det, df=seq(6,4,-.5))

# univariate trace plots are seemingly paradoxical in that
# some coefficients "shrink" *away* from 0
traceplot(dridge, X="df")
vif(dridge)
pairs(dridge, radius=0.5)


plot3d(dridge, radius=0.5, labels=dridge$df)

# transform to PCA/SVD space
dpridge <- pca(dridge)

# not so paradoxical in PCA space
traceplot(dpridge, X="df")
biplot(dpridge, radius=0.5, labels=dpridge$df)

# show PCA vectors in variable space
biplot(dridge, radius=0.5, labels=dridge$df)

Hospital manpower data

Description

The hospital manpower data, taken from Myers (1990), table 3.8, are a well-known example of highly collinear data to which ridge regression and various shrinkage and selection methods are often applied.

The data consist of measures taken at 17 U.S. Naval Hospitals and the goal is to predict the required monthly man hours for staffing purposes.

Format

A data frame with 17 observations on the following 6 variables.

Hours: monthly man hours (response variable)
Load: average daily patient load
Xray: monthly X-ray exposures
BedDays: monthly occupied bed days
AreaPop: eligible population in the area in thousands
Stay: average length of patient's stay in days

Details

Myers (1990) indicates his source was "Procedures and Analysis for Staffing Standards Development: Data/Regression Analysis Handbook", Navy Manpower and Material Analysis Center, San Diego, 1979.

Source

Raymond H. Myers (1990). Classical and Modern Regression with Applications, 2nd ed., PWS-Kent, pp. 130-133.

References

Donald R. Jensen and Donald E. Ramirez (2012). Variations on Ridge Traces in Regression, Communications in Statistics - Simulation and Computation, 41 (2), 265-278.

Examples


data(Manpower)
mmod <- lm(Hours ~ ., data=Manpower)
vif(mmod)
# ridge regression models, specified in terms of equivalent df
mridge <- ridge(Hours ~ ., data=Manpower, df=seq(5, 3.75, -.25))
vif(mridge)

# univariate ridge trace plots
traceplot(mridge)
traceplot(mridge, X="df")

# bivariate ridge trace plots
plot(mridge, radius=0.25, labels=mridge$df)
pairs(mridge, radius=0.25)


# 3D views
# ellipsoids for Load, Xray & BedDays are nearly 2D
plot3d(mridge, radius=0.2, labels=mridge$df)
# variables in model selected by AIC & BIC
plot3d(mridge, variables=c(2,3,5), radius=0.2, labels=mridge$df)

# plots in PCA/SVD space
mpridge <- pca(mridge)
traceplot(mpridge, X="df")
biplot(mpridge, radius=0.25)

Biplot of Ridge Regression Trace Plot in SVD Space

Description

biplot.pcaridge supplements the standard display of the covariance ellipsoids for a ridge regression problem in PCA/SVD space with labeled arrows showing the contributions of the original variables to the dimensions plotted.

Usage

## S3 method for class 'pcaridge'
biplot(
  x,
  variables = (p - 1):p,
  labels = NULL,
  asp = 1,
  origin,
  scale,
  var.lab = rownames(V),
  var.lwd = 1,
  var.col = "black",
  var.cex = 1,
  xlab,
  ylab,
  prefix = "Dim ",
  suffix = TRUE,
  ...
)

Arguments

x

A pcaridge object computed by pca.ridge or a ridge object.

variables

The dimensions or variables to be shown in the the plot. By default, the last two dimensions, corresponding to the smallest singular values, are plotted for class("pcaridge") objects or the first two variables for class("ridge") objects.

labels

A vector of character strings or expressions used as labels for the ellipses. Use labels=NULL to suppress these.

asp

Aspect ratio for the plot. The default value, asp=1 helps ensure that lengths and angles are preserved in these plots. Use asp=NA to override this.

origin

The origin for the variable vectors in this plot, a vector of length 2. If not specified, the function calculates an origin to make the variable vectors approximately centered in the plot window.

scale

The scale factor for variable vectors in this plot. If not specified, the function calculates a scale factor to make the variable vectors approximately fill the plot window.

var.lab

Labels for variable vectors. The default is the names of the predictor variables.

var.lwd, var.col, var.cex

Line width, color and character size used to draw and label the arrows representing the variables in this plot.

xlab, ylab

Labels for the plot dimensions. If not specified, prefix and suffix are used to construct informative dimension labels.

prefix

Prefix for labels of the plot dimensions.

suffix

Suffix for labels of the plot dimensions. If suffix=TRUE the percent of variance accounted for by each dimension is added to the axis label.

...

Other arguments, passed to plot.pcaridge

Details

The biplot view showing the dimensions corresponding to the two smallest singular values is particularly useful for understanding how the predictors contribute to shrinkage in ridge regression.

This is only a biplot in the loose sense that results are shown in two spaces simultaneously – the transformed PCA/SVD space of the original predictors, and vectors representing the predictors projected into this space.

biplot.ridge is a similar extension of plot.ridge, adding vectors showing the relation of the PCA/SVD dimensions to the plotted variables.

class("ridge") objects use the transpose of the right singular vectors, t(x$svd.V) for the dimension weights plotted as vectors.

Value

None

Author(s)

Michael Friendly, with contributions by Uwe Ligges

References

Examples


longley.y <- longley[, "Employed"]
longley.X <- data.matrix(longley[, c(2:6,1)])

lambda <- c(0, 0.005, 0.01, 0.02, 0.04, 0.08)
lridge <- ridge(longley.y, longley.X, lambda=lambda)

plridge <- pca(lridge)

plot(plridge, radius=0.5)

# same, with variable vectors
biplot(plridge, radius=0.5)
# add some other options
biplot(plridge, radius=0.5, var.col="brown", var.lwd=2, var.cex=1.2, prefix="Dimension ")

# biplots for ridge objects, showing PCA vectors
plot(lridge, radius=0.5)
biplot(lridge, radius=0.5)
biplot(lridge, radius=0.5, asp=NA)

Enhanced Contour Plots

Description

This is an enhancement to contour, written as a wrapper for that function. It creates a contour plot, or adds contour lines to an existing plot, allowing the contours to be filled and returning the list of contour lines.

Usage

contourf(
  x = seq(0, 1, length.out = nrow(z)),
  y = seq(0, 1, length.out = ncol(z)),
  z,
  nlevels = 10,
  levels = pretty(zlim, nlevels),
  zlim = range(z, finite = TRUE),
  col = par("fg"),
  color.palette = colorRampPalette(c("white", col)),
  fill.col = color.palette(nlevels + 1),
  fill.alpha = 0.5,
  add = FALSE,
  ...
)

Arguments

x, y

locations of grid lines at which the values in z are measured. These must be in ascending order. By default, equally spaced values from 0 to 1 are used. If x is a list, its components x$x and x$y are used for x and y, respectively. If the list has component x$z this is used for z.

z

a matrix containing the values to be plotted (NAs are allowed). Note that x can be used instead of z for convenience.

nlevels

number of contour levels desired iff levels is not supplied

levels

numeric vector of levels at which to draw contour lines

zlim

z-limits for the plot. x-limits and y-limits can be passed through ...

col

color for the lines drawn

color.palette

a color palette function to be used to assign fill colors in the plot

fill.col

a call to the color.palette function or an an explicit set of colors to be used in the plot. Use fill.col=NULL to suppress the filled polygons. a vector of fill colors corresponding to levels. By default, a set of possibly transparent colors is calculated ranging from white to col, using transparency given by fill.alpha

fill.alpha

transparency value for fill.col, either a hex character string, or a numeric value between 0 and 1. Use fill.alpha=NA to suppress transparency.

add

logical. If TRUE, add to a current plot.

...

additional arguments passed to contour, including all arguments of contour.default not mentioned above, as well as additional graphical parameters passed by contour.default to more basic functions.

Value

Returns invisibly the list of contours lines, with components levels, x, y. See contourLines.

Author(s)

Michael Friendly

Examples


x <- 10*1:nrow(volcano)
y <- 10*1:ncol(volcano)
contourf(x,y,volcano, col="blue")
contourf(x,y,volcano, col="blue", nlevels=6)

# return value, unfilled, other graphic parameters
res <- contourf(x,y,volcano, col="blue", fill.col=NULL, lwd=2)
# levels used in the plot
sapply(res, function(x) x[[1]])

Diabetes Progression

Description

These data consist of observations on 442 patients, with the response of interest being a quantitative measure of disease progression one year after baseline.

There are ten baseline variables: age, sex, body-mass index (bmi), average blood pressure (map) and six blood serum measurements.

Usage

data("diab")

Format

A data frame with 442 observations on the following 11 variables.

prog: disease progression, a numeric vector
age: age, a numeric vector
sex: integer, a numeric vector
bmi: body mass index, a numeric vector
map: mean arterial blood pressure, a numeric vector
tc: blood serum TC, a numeric vector
ldl: blood serum low-density lipoprotein ("bad cholersterol"), a numeric vector
hdl: blood serum high-density lipoprotein ("good cholersterol"), a numeric vector
tch: blood serum TCH, a numeric vector
ltg: blood serum lamotrigine, a numeric vector
glu: blood serum glucose, a numeric vector

Details

Efron & Hastie describe their analysis using the centered predictor variables standardized to unit L2 norm. ridge does not (yet) provide this scaling.

Source

The dataset was taken from the web site for Efron & Hastie (2021), Computer Age Statistical Inference, https://hastie.su.domains/CASI_files/DATA/diabetes.csv.

References

Efron, B., Hastie, T., Johnstone, I., & Tibshirani, R. (2004). Least Angle Regression. The Annals of Statistics, 32(2), 407-499. doi:10.1214/009053604000000067

Efron, B., & Hastie, T. (2021). Computer Age Statistical Inference, Student Edition: Algorithms, Evidence, and Data Science, Cambridge University Press. doi:10.1017/9781108914062

Examples

data(diab)
## maybe str(diab) ; plot(diab) ...

Scatterplot Matrix of Bivariate Ridge Trace Plots

Description

Displays all possible pairs of bivariate ridge trace plots for a given set of predictors.

Usage

## S3 method for class 'ridge'
pairs(
  x,
  variables,
  radius = 1,
  lwd = 1,
  lty = 1,
  col = c("black", "red", "darkgreen", "blue", "darkcyan", "magenta", "brown",
    "darkgray"),
  center.pch = 16,
  center.cex = 1.25,
  digits = getOption("digits") - 3,
  diag.cex = 2,
  diag.panel = panel.label,
  fill = FALSE,
  fill.alpha = 0.3,
  ...
)

Arguments

x

A ridge object, as fit by ridge

variables

Predictors in the model to be displayed in the plot: an integer or character vector, giving the indices or names of the variables.

radius

Radius of the ellipse-generating circle for the covariance ellipsoids.

lwd, lty

Line width and line type for the covariance ellipsoids. Recycled as necessary.

col

A numeric or character vector giving the colors used to plot the covariance ellipsoids. Recycled as necessary.

center.pch

Plotting character used to show the bivariate ridge estimates. Recycled as necessary.

center.cex

Size of the plotting character for the bivariate ridge estimates

digits

Number of digits to be displayed as the (min, max) values in the diagonal panels

diag.cex

Character size for predictor labels in diagonal panels

diag.panel

Function to draw diagonal panels. Not yet implemented: just uses internal panel.label to write the variable name and ranges.

fill

Logical vector: Should the covariance ellipsoids be filled? Recycled as necessary.

fill.alpha

Numeric vector: alpha transparency value(s) for filled ellipsoids. Recycled as necessary.

...

Other arguments passed down

Value

None. Used for its side effect of plotting.

Author(s)

Michael Friendly

References

Examples


longley.y <- longley[, "Employed"]
longley.X <- data.matrix(longley[, c(2:6,1)])

lambda <- c(0, 0.005, 0.01, 0.02, 0.04, 0.08)
lridge <- ridge(longley.y, longley.X, lambda=lambda)

pairs(lridge, radius=0.5, diag.cex=1.75)

data(prostate)
py <- prostate[, "lpsa"]
pX <- data.matrix(prostate[, 1:8])
pridge <- ridge(py, pX, df=8:1)

pairs(pridge)

Transform Ridge Estimates to PCA Space

Description

The function pca.ridge transforms a ridge object from parameter space, where the estimated coefficients are \beta_k with covariance matrices \Sigma_k, to the principal component space defined by the right singular vectors, V, of the singular value decomposition of the scaled predictor matrix, X.

In this space, the transformed coefficients are V \beta_k, with covariance matrices

V \Sigma_k V^T

This transformation provides alternative views of ridge estimates in low-rank approximations. In particular, it allows one to see where the effects of collinearity typically reside — in the smallest PCA dimensions.

Usage

pca(x, ...)

Arguments

x

A ridge object, as fit by ridge

...

Other arguments passed down. Not presently used in this implementation.

Value

An object of class c("ridge", "pcaridge"), with the same components as the original ridge object.

Author(s)

Michael Friendly

References

Examples


longley.y <- longley[, "Employed"]
longley.X <- data.matrix(longley[, c(2:6,1)])

lambda <- c(0, 0.005, 0.01, 0.02, 0.04, 0.08)
lridge <- ridge(longley.y, longley.X, lambda=lambda)

plridge <- pca(lridge)
traceplot(plridge)
pairs(plridge)
# view in space of smallest singular values
plot(plridge, variables=5:6)

Plot Shrinkage vs. Variance for Ridge Precision

Description

This function uses the results of precision to plot a measure of shrinkage of the coefficients in ridge regression against a selected measure of their estimated sampling variance, so as to provide a direct visualization of the tradeoff between bias and precision.

Usage

## S3 method for class 'precision'
plot(
  x,
  xvar = "norm.beta",
  yvar = c("det", "trace", "max.eig"),
  labels = c("lambda", "df"),
  label.cex = 1.25,
  label.prefix,
  criteria = NULL,
  pch = 16,
  cex = 1.5,
  col,
  main = NULL,
  xlab,
  ylab,
  ...
)

Arguments

x

A data frame of class "precision" resulting from precision called on a "ridge" object. Named x only to conform with the plot generic.

xvar

The character name of the column to be used for the horizontal axis. Typically, this is the normalized sum of squares of the coefficients ("norm.beta") used as a measure of shrinkage / bias.

yvar

The character name of the column to be used for the vertical axis. One of c("det", "trace", "max.eig"). See precision for definitions of these measures.

labels

The character name of the column to be used for point labels. One of c("lambda", "df").

label.cex

Character size for point labels.

label.prefix

Character or expression prefix for the point labels.

criteria

The vector of optimal shrinkage criteria from the ridge call to be added as points in the plot.

pch

Plotting character for points

cex

Character size for points

col

Point colors

main

Plot title

xlab

Label for horizontal axis

ylab

Label for vertical axis

...

Other arguments passed to plot.

Value

Returns nothing. Used for the side effect of plotting.

Author(s)

Michael Friendly

Examples

lambda <- c(0, 0.001, 0.005, 0.01, 0.02, 0.04, 0.08)
lridge <- ridge(Employed ~ GNP + Unemployed + Armed.Forces + 
                  Population + Year + GNP.deflator, 
                data=longley, lambda=lambda)

criteria <- lridge$criteria |> print()

pridge <- precision(lridge) |> print()

plot(pridge)
# also show optimal criteria
plot(pridge, criteria = criteria)

# use degrees of freedom as point labels 
plot(pridge, labels = "df")
plot(pridge, labels = "df", label.prefix="df:")
# show the trace measure
plot(pridge, yvar="trace")

Bivariate Ridge Trace Plots

Description

The bivariate ridge trace plot displays 2D projections of the covariance ellipsoids for a set of ridge regression estimates indexed by a ridge tuning constant.

The centers of these ellipses show the bias induced for each parameter, and also how the change in the ridge estimate for one parameter is related to changes for other parameters.

The size and shapes of the covariance ellipses show directly the effect on precision of the estimates as a function of the ridge tuning constant.

plot.pcaridge does these bivariate ridge trace plots for "pcaridge" objects, defaulting to plotting the two smallest components.

Usage

## S3 method for class 'ridge'
plot(
  x,
  variables = 1:2,
  radius = 1,
  which.lambda = 1:length(x$lambda),
  labels = lambda,
  pos = 3,
  cex = 1.2,
  lwd = 2,
  lty = 1,
  xlim,
  ylim,
  col = c("black", "red", "darkgreen", "blue", "darkcyan", "magenta", "brown",
    "darkgray"),
  center.pch = 16,
  center.cex = 1.5,
  fill = FALSE,
  fill.alpha = 0.3,
  ref = TRUE,
  ref.col = gray(0.7),
  ...
)

## S3 method for class 'pcaridge'
plot(x, variables = (p - 1):p, labels = NULL, ...)

Arguments

x

A ridge object, as fit by ridge

variables

Predictors in the model to be displayed in the plot: an integer or character vector of length 2, giving the indices or names of the variables. Defaults to the first two predictors for ridge objects or the last two dimensions for pcaridge objects.

radius

Radius of the ellipse-generating circle for the covariance ellipsoids. The default, radius=1 gives a standard “unit” ellipsoid. Typically, values radius<1 gives less cluttered displays.

which.lambda

A vector of indices used to select the values of lambda for which ellipses are plotted. The default is to plot ellipses for all values of lambda in the ridge object.

labels

A vector of character strings or expressions used as labels for the ellipses. Use labels=NULL to suppress these.

pos, cex

Scalars or vectors of positions (relative to the ellipse centers) and character size used to label the ellipses

lwd, lty

Line width and line type for the covariance ellipsoids. Recycled as necessary.

xlim, ylim

X, Y limits for the plot, each a vector of length 2. If missing, the range of the covariance ellipsoids is used.

col

A numeric or character vector giving the colors used to plot the covariance ellipsoids. Recycled as necessary.

center.pch

Plotting character used to show the bivariate ridge estimates. Recycled as necessary.

center.cex

Size of the plotting character for the bivariate ridge estimates

fill

Logical vector: Should the covariance ellipsoids be filled? Recycled as necessary.

fill.alpha

Numeric vector: alpha transparency value(s) in the range (0, 1) for filled ellipsoids. Recycled as necessary.

ref

Logical: whether to draw horizontal and vertical reference lines at 0.

ref.col

Color of reference lines.

...

Other arguments passed down to plot.default, e.g., xlab, ylab, and other graphic parameters.

Value

None. Used for its side effect of plotting.

Author(s)

Michael Friendly

References

Examples


longley.y <- longley[, "Employed"]
longley.X <- data.matrix(longley[, c(2:6,1)])

lambda <- c(0, 0.005, 0.01, 0.02, 0.04, 0.08)
lambdaf <- c("", ".005", ".01", ".02", ".04", ".08")
lridge <- ridge(longley.y, longley.X, lambda=lambda)

op <- par(mfrow=c(2,2), mar=c(4, 4, 1, 1)+ 0.1)
for (i in 2:5) {
	plot(lridge, variables=c(1,i), radius=0.5, cex.lab=1.5)
	text(lridge$coef[1,1], lridge$coef[1,i], expression(~widehat(beta)^OLS), 
	     cex=1.5, pos=4, offset=.1)
	if (i==2) text(lridge$coef[-1,1:2], lambdaf[-1], pos=3, cex=1.25)
}
par(op)

data(prostate)
py <- prostate[, "lpsa"]
pX <- data.matrix(prostate[, 1:8])
pridge <- ridge(py, pX, df=8:1)

plot(pridge)
plot(pridge, fill=c(TRUE, rep(FALSE,7)))

3D Ridge Trace Plots

Description

The 3D ridge trace plot displays 3D projections of the covariance ellipsoids for a set of ridge regression estimates indexed by a ridge tuning constant.

The centers of these ellipses show the bias induced for each parameter, and also how the change in the ridge estimate for one parameter is related to changes for other parameters.

The size and shapes of the covariance ellipsoids show directly the effect on precision of the estimates as a function of the ridge tuning constant.

plot3d.ridge and plot3d.pcaridge differ only in the defaults for the variables plotted.

Usage

plot3d(x, ...)

## S3 method for class 'pcaridge'
plot3d(x, variables = (p - 2):p, ...)

## S3 method for class 'ridge'
plot3d(
  x,
  variables = 1:3,
  radius = 1,
  which.lambda = 1:length(x$lambda),
  lwd = 1,
  lty = 1,
  xlim,
  ylim,
  zlim,
  xlab,
  ylab,
  zlab,
  col = c("black", "red", "darkgreen", "blue", "darkcyan", "magenta", "brown",
    "darkgray"),
  labels = lambda,
  ref = TRUE,
  ref.col = gray(0.7),
  segments = 40,
  shade = TRUE,
  shade.alpha = 0.1,
  wire = FALSE,
  aspect = 1,
  add = FALSE,
  ...
)

Arguments

x

A ridge object, as fit by ridge or a pcaridge object as transformed by pca.ridge

...

Other arguments passed down

variables

Predictors in the model to be displayed in the plot: an integer or character vector of length 3, giving the indices or names of the variables. Defaults to the first three predictors for ridge objects or the last three dimensions for pcaridge objects.

radius

Radius of the ellipse-generating circle for the covariance ellipsoids. The default, radius=1 gives a standard “unit” ellipsoid. Typically, radius<1 gives less cluttered displays.

which.lambda

A vector of indices used to select the values of lambda for which ellipsoids are plotted. The default is to plot ellipsoids for all values of lambda in the ridge object.

lwd, lty

Line width and line type for the covariance ellipsoids. Recycled as necessary.

xlim, ylim, zlim

X, Y, Z limits for the plot, each a vector of length 2. If missing, the range of the covariance ellipsoids is used.

xlab, ylab, zlab

Labels for the X, Y, Z variables in the plot. If missing, the names of the predictors given in variables is used.

col

A numeric or character vector giving the colors used to plot the covariance ellipsoids. Recycled as necessary.

labels

A numeric or character vector giving the labels to be drawn at the centers of the covariance ellipsoids.

ref

Logical: whether to draw horizontal and vertical reference lines at 0. This is not yet implemented.

ref.col

Color of reference lines.

segments

Number of line segments used in drawing each dimension of a covariance ellipsoid.

shade

a logical scalar or vector, indicating whether the ellipsoids should be rendered with shade3d. Recycled as necessary.

shade.alpha

a numeric value in the range [0,1], or a vector of such values, giving the alpha transparency for ellipsoids rendered with shade=TRUE.

wire

a logical scalar or vector, indicating whether the ellipsoids should be rendered with wire3d. Recycled as necessary.

aspect

a scalar or vector of length 3, or the character string "iso", indicating the ratios of the x, y, and z axes of the bounding box. The default, aspect=1 makes the bounding box display as a cube approximately filling the display. See aspect3d for details.

add

if TRUE, add to the current rgl plot; the default is FALSE.

Value

None. Used for its side-effect of plotting

Note

This is an initial implementation. The details and arguments are subject to change.

Author(s)

Michael Friendly

References

Examples


lmod <- lm(Employed ~ GNP + Unemployed + Armed.Forces + Population + 
                      Year + GNP.deflator, data=longley)
longley.y <- longley[, "Employed"]
longley.X <- model.matrix(lmod)[,-1]

lambda <- c(0, 0.005, 0.01, 0.02, 0.04, 0.08)
lambdaf <- c("0", ".005", ".01", ".02", ".04", ".08")
lridge <- ridge(longley.y, longley.X, lambda=lambda)


plot3d(lridge, var=c(1,4,5), radius=0.5)

# view in SVD/PCA space
plridge <- pca(lridge)
plot3d(plridge, radius=0.5)

Measures of Precision and Shrinkage for Ridge Regression

Description

The goal of precision is to allow you to study the relationship between shrinkage of ridge regression coefficients and their precision directly by calculating measures of each.

Three measures of (inverse) precision based on the “size” of the covariance matrix of the parameters are calculated. Let V_k \equiv \text{Var}(\mathbf{\beta}_k) be the covariance matrix for a given ridge constant, and let \lambda_i , i= 1, \dots p be its eigenvalues. Then the variance (= 1/precision) measures are:

"det": \log | V_k | = \log \prod \lambda (with det.fun = "log", the default) or |V_k|^{1/p} =(\prod \lambda)^{1/p} (with det.fun = "root") measures the linearized volume of the covariance ellipsoid and corresponds conceptually to Wilks' Lambda criterion
"trace": \text{trace}( V_k ) = \sum \lambda corresponds conceptually to Pillai's trace criterion
"max.eig": \lambda_1 = \max (\lambda) corresponds to Roy's largest root criterion.

Two measures of shrinkage are also calculated:

norm.beta: the root mean square of the coefficient vector \lVert\mathbf{\beta}_k \rVert, normalized to a maximum of 1.0 if normalize == TRUE (the default).
norm.diff: the root mean square of the difference from the OLS estimate \lVert \mathbf{\beta}_{\text{OLS}} - \mathbf{\beta}_k \rVert. This measure is inversely related to norm.beta

A plot method, plot.precision facilitates making graphs of these quantities.

Usage

precision(object, det.fun, normalize, ...)

Arguments

object

An object of class ridge or lm

det.fun

Function to be applied to the determinants of the covariance matrices, one of c("log","root").

normalize

If TRUE the length of the coefficient vector \mathbf{\beta}_k is normalized to a maximum of 1.0.

...

Other arguments (currently unused)

Value

An object of class c("precision", "data.frame") with the following columns:

lambda

The ridge constant

df

The equivalent effective degrees of freedom

det

The det.fun function of the determinant of the covariance matrix

trace

The trace of the covariance matrix

max.eig

Maximum eigen value of the covariance matrix

norm.beta

The root mean square of the estimated coefficients, possibly normalized

norm.diff

The root mean square of the difference between the OLS solution (lambda = 0) and ridge solutions

Note

Models fit by lm and ridge use a different scaling for the predictors, so the results of precision for an lm model will not correspond to those for ridge with ridge constant = 0.

Author(s)

Michael Friendly

Examples


longley.y <- longley[, "Employed"]
longley.X <- data.matrix(longley[, c(2:6,1)])

lambda <- c(0, 0.005, 0.01, 0.02, 0.04, 0.08)
lridge <- ridge(longley.y, longley.X, lambda=lambda)

# same, using formula interface
lridge <- ridge(Employed ~ GNP + Unemployed + Armed.Forces + Population + Year + GNP.deflator, 
		data=longley, lambda=lambda)

clr <- c("black", rainbow(length(lambda)-1, start=.6, end=.1))
coef(lridge)

(pdat <- precision(lridge))
# plot log |Var(b)| vs. length(beta)
with(pdat, {
	plot(norm.beta, det, type="b", 
	cex.lab=1.25, pch=16, cex=1.5, col=clr, lwd=2,
	xlab='shrinkage: ||b|| / max(||b||)',
	ylab='variance: log |Var(b)|')
	text(norm.beta, det, lambda, cex=1.25, pos=c(rep(2,length(lambda)-1),4))
	text(min(norm.beta), max(det), "Variance vs. Shrinkage", cex=1.5, pos=4)
	})

# plot trace[Var(b)] vs. length(beta)
with(pdat, {
	plot(norm.beta, trace, type="b",
	cex.lab=1.25, pch=16, cex=1.5, col=clr, lwd=2,
	xlab='shrinkage: ||b|| / max(||b||)',
	ylab='variance: trace [Var(b)]')
	text(norm.beta, trace, lambda, cex=1.25, pos=c(2, rep(4,length(lambda)-1)))
#	text(min(norm.beta), max(det), "Variance vs. Shrinkage", cex=1.5, pos=4)
	})

Prostate Cancer Data

Description

Data to examine the correlation between the level of prostate-specific antigen and a number of clinical measures in men who were about to receive a radical prostatectomy.

Format

A data frame with 97 observations on the following 10 variables.

lcavol: log cancer volume
lweight: log prostate weight
age: in years
lbph: log of the amount of benign prostatic hyperplasia
svi: seminal vesicle invasion
lcp: log of capsular penetration
gleason: a numeric vector
pgg45: percent of Gleason score 4 or 5
lpsa: response
train: a logical vector

Details

This data set came originally from the (now defunct) ElemStatLearn package.

The last column indicates which 67 observations were used as the "training set" and which 30 as the test set, as described on page 48 in the book.

Note

There was an error in this dataset in earlier versions of the package, as indicated in a footnote on page 3 of the second edition of the book. As of version 2012.04-0 this was corrected.

Source

Stamey, T., Kabalin, J., McNeal, J., Johnstone, I., Freiha, F., Redwine, E. and Yang, N (1989) Prostate specific antigen in the diagnosis and treatment of adenocarcinoma of the prostate II. Radical prostatectomy treated patients, Journal of Urology, 16: 1076–1083.

Examples


data(prostate)
str( prostate )
cor( prostate[,1:8] )
prostate <- prostate[, -10]

prostate.mod <- lm(lpsa ~ ., data=prostate)
vif(prostate.mod)

py <- prostate[, "lpsa"]
pX <- data.matrix(prostate[, 1:8])
pridge <- ridge(py, pX, df=8:1)
pridge

# univariate ridge trace plots
traceplot(pridge)
traceplot(pridge, X="df")

# bivariate ridge trace plots
plot(pridge)
pairs(pridge)

Ridge Regression Estimates

Description

The function ridge fits linear models by ridge regression, returning an object of class ridge designed to be used with the plotting methods in this package.

It is also designed to facilitate an alternative representation of the effects of shrinkage in the space of uncorrelated (PCA/SVD) components of the predictors.

The standard formulation of ridge regression is that it regularizes the estimates of coefficients by adding small positive constants \lambda to the diagonal elements of \mathbf{X}^\top\mathbf{X} in the least squares solution to achieve a more favorable tradeoff between bias and variance (inverse of precision) of the coefficients.

\widehat{\mathbf{\beta}}^{\text{RR}}_k = (\mathbf{X}^\top \mathbf{X} + \lambda \mathbf{I})^{-1} \mathbf{X}^\top \mathbf{y}

Ridge regression shrinkage can be parameterized in several ways.

If a vector of lambda values is supplied, these are used directly in the ridge regression computations.
Otherwise, if a vector df can be supplied the equivalent values for effective degrees of freedom corresponding to shrinkage, going down from the number of predictors in the model.

In either case, both lambda and df are returned in the ridge object, but the rownames of the coefficients are given in terms of lambda.

coef extracts the estimated coefficients for each value of the shrinkage factor

vcov extracts the estimated p \times p covariance matrices of the coefficients for each value of the shrinkage factor.

best extracts the optimal shrinkage values according to several criteria: HKB: Hoerl et al. (1975); LW: Lawless & Wang (1976); GCV: Golub et al. (1975)

Usage

ridge(y, ...)

## S3 method for class 'formula'
ridge(formula, data, lambda = 0, df, svd = TRUE, contrasts = NULL, ...)

## Default S3 method:
ridge(y, X, lambda = 0, df, svd = TRUE, ...)

## S3 method for class 'ridge'
coef(object, ...)

## S3 method for class 'ridge'
print(x, digits = max(5, getOption("digits") - 5), ...)

## S3 method for class 'ridge'
vcov(object, ...)

best(object, ...)

## S3 method for class 'ridge'
best(object, ...)

Arguments

y

A numeric vector containing the response variable. NAs not allowed.

...

Other arguments, passed down to methods

formula

For the formula method, a two-sided formula.

data

For the formula method, data frame within which to evaluate the formula.

lambda

A scalar or vector of ridge constants. A value of 0 corresponds to ordinary least squares.

df

A scalar or vector of effective degrees of freedom corresponding to lambda

svd

If TRUE the SVD of the centered and scaled X matrix is returned in the ridge object.

contrasts

a list of contrasts to be used for some or all of factor terms in the formula. See the contrasts.arg of model.matrix.default.

X

A matrix of predictor variables. NA's not allowed. Should not include a column of 1's for the intercept.

x, object

An object of class ridge

digits

For the print method, the number of digits to print.

Details

If an intercept is present in the model, its coefficient is not penalized. (If you want to penalize an intercept, put in your own constant term and remove the intercept.)

The predictors are centered, but not (yet) scaled in this implementation.

A number of the methods in the package assume that lambda is a vector of shrinkage constants increasing from lambda[1] = 0, or equivalently, a vector of df decreasing from p.

Value

A list with the following components:

lambda

The vector of ridge constants

df

The vector of effective degrees of freedom corresponding to lambda

coef

The matrix of estimated ridge regression coefficients

scales

scalings used on the X matrix

kHKB

HKB estimate of the ridge constant

kLW

L-W estimate of the ridge constant

GCV

vector of GCV values

kGCV

value of lambda with the minimum GCV

criteria

Collects the criteria kHKB, kLW, and kGCV in a named vector

If svd==TRUE (the default), the following are also included:

svd.D

Singular values of the svd of the scaled X matrix

svd.U

Left singular vectors of the svd of the scaled X matrix. Rows correspond to observations and columns to dimensions.

svd.V

Right singular vectors of the svd of the scaled X matrix. Rows correspond to variables and columns to dimensions.

A data.frame with one row for each of the HKB, LW, and GCV criteria

Author(s)

Michael Friendly

References

Hoerl, A. E., Kennard, R. W., and Baldwin, K. F. (1975), "Ridge Regression: Some Simulations," Communications in Statistics, 4, 105-123.

Lawless, J.F., and Wang, P. (1976), "A Simulation Study of Ridge and Other Regression Estimators," Communications in Statistics, 5, 307-323.

Golub G.H., Heath M., Wahba G. (1979) Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics, 21:215–223. doi:10.2307/1268518

Examples



#\donttest{
# Longley data, using number Employed as response
longley.y <- longley[, "Employed"]
longley.X <- data.matrix(longley[, c(2:6,1)])

lambda <- c(0, 0.005, 0.01, 0.02, 0.04, 0.08)
lridge <- ridge(longley.y, longley.X, lambda=lambda)

# same, using formula interface
lridge <- ridge(Employed ~ GNP + Unemployed + Armed.Forces + Population + Year + GNP.deflator, 
		data=longley, lambda=lambda)


coef(lridge)

# standard trace plot
traceplot(lridge)
# plot vs. equivalent df
traceplot(lridge, X="df")
pairs(lridge, radius=0.5)
#}


data(prostate)
py <- prostate[, "lpsa"]
pX <- data.matrix(prostate[, 1:8])
pridge <- ridge(py, pX, df=8:1)
pridge

plot(pridge)
pairs(pridge)
traceplot(pridge)
traceplot(pridge, X="df")


# Hospital manpower data from Table 3.8 of Myers (1990) 
data(Manpower)
str(Manpower)

mmod <- lm(Hours ~ ., data=Manpower)
vif(mmod)
# ridge regression models, specified in terms of equivalent df
mridge <- ridge(Hours ~ ., data=Manpower, df=seq(5, 3.75, -.25))
vif(mridge)

# univariate ridge trace plots
traceplot(mridge)
traceplot(mridge, X="df")


# bivariate ridge trace plots
plot(mridge, radius=0.25, labels=mridge$df)
pairs(mridge, radius=0.25)

# 3D views
# ellipsoids for Load, Xray & BedDays are nearly 2D
plot3d(mridge, radius=0.2, labels=mridge$df)
# variables in model selected by AIC & BIC
plot3d(mridge, variables=c(2,3,5), radius=0.2, labels=mridge$df)

# plots in PCA/SVD space
mpridge <- pca(mridge)
traceplot(mpridge, X="df")
biplot(mpridge, radius=0.25)

Univariate Ridge Trace Plots

Description

The traceplot function extends and simplifies the univariate ridge trace plots for ridge regression provided in the plot method for lm.ridge

Usage

traceplot(
  x,
  X = c("lambda", "df"),
  col = c("black", "red", "darkgreen", "blue", "darkcyan", "magenta", "brown",
    "darkgray"),
  pch = c(15:18, 7, 9, 12, 13),
  xlab,
  ylab = "Coefficient",
  xlim,
  ylim,
  ...
)

Arguments

x

A ridge object, as fit by ridge

X

What to plot as the horizontal coordinate, one of c("lambda", "df")

col

A numeric or character vector giving the colors used to plot the ridge trace curves. Recycled as necessary.

pch

Vector of plotting characters used to plot the ridge trace curves. Recycled as necessary.

xlab

Label for horizontal axis

ylab

Label for vertical axis

xlim, ylim

x, y limits for the plot. You may need to adjust these to allow for the variable labels.

...

Other arguments passed to matplot

Details

For ease of interpretation, the variables are labeled at the side of the plot (left, right) where the coefficient estimates are expected to be most widely spread. If xlim is not specified, the range of the X variable is extended slightly to accommodate the variable names.

Value

None. Used for its side effect of plotting.

Author(s)

Michael Friendly

References

Hoerl, A. E. and Kennard R. W. (1970). "Ridge Regression: Applications to Nonorthogonal Problems", Technometrics, 12(1), 69-82.

Examples


longley.y <- longley[, "Employed"]
longley.X <- data.matrix(longley[, c(2:6,1)])

lambda <- c(0, 0.005, 0.01, 0.02, 0.04, 0.08)
lridge <- ridge(longley.y, longley.X, lambda=lambda)

traceplot(lridge)
#abline(v=lridge$kLW, lty=3)
#abline(v=lridge$kHKB, lty=3)
#text(lridge$kLW, -3, "LW")
#text(lridge$kHKB, -3, "HKB")

traceplot(lridge, X="df")

Make Colors Transparent

Description

Takes a vector of colors (as color names or rgb hex values) and adds a specified alpha transparency to each.

Usage

trans.colors(col, alpha = 0.5, names = NULL)

Arguments

col

A character vector of colors, either as color names or rgb hex values

alpha

alpha transparency value(s) to apply to each color (0 means fully transparent and 1 means opaque)

names

optional character vector of names for the colors

Details

Colors (col) and alpha need not be of the same length. The shorter one is replicated to make them of the same length.

Value

A vector of color values of the form "#rrggbbaa"

Author(s)

Michael Friendly

Examples


trans.colors(palette(), alpha=0.5)

# alpha can be vectorized
trans.colors(palette(), alpha=seq(0, 1, length=length(palette())))

# lengths need not match: shorter one is repeated as necessary
trans.colors(palette(), alpha=c(.1, .2))

trans.colors(colors()[1:20])

# single color, with various alphas
trans.colors("red", alpha=seq(0,1, length=5))
# assign names
trans.colors("red", alpha=seq(0,1, length=5), names=paste("red", 1:5, sep=""))

Variance Inflation Factors for Ridge Regression

Description

The function vif.ridge calculates variance inflation factors for the predictors in a set of ridge regression models indexed by the tuning/shrinkage factor, returning one row for each value of the \lambda parameter.

Variance inflation factors are calculated using the simplified formulation in Fox & Monette (1992).

The plot.vif.ridge method plots variance inflation factors for a "vif.ridge" object in a similar style to what is provided by traceplot. That is, it plots the VIF for each coefficient in the model against either the ridge \lambda tuning constant or it's equivalent effective degrees of freedom.

Usage

## S3 method for class 'ridge'
vif(mod, ...)

## S3 method for class 'vif.ridge'
print(x, digits = max(4, getOption("digits") - 5), ...)

## S3 method for class 'vif.ridge'
plot(
  x,
  X = c("lambda", "df"),
  Y = c("vif", "sqrt"),
  col = c("black", "red", "darkgreen", "blue", "darkcyan", "magenta", "brown",
    "darkgray"),
  pch = c(15:18, 7, 9, 12, 13),
  xlab,
  ylab,
  xlim,
  ylim,
  ...
)

Arguments

mod

A "ridge" object computed by ridge

...

Other arguments passed to methods

x

A ridge object, as fit by ridge

digits

Number of digits to display in the print method

X

What to plot as the horizontal coordinate, one of c("lambda", "df")

Y

What to plot as the vertical coordinate, one of c("vif", "sqrt"), where the latter plots \sqrt{VIF}.

col

A numeric or character vector giving the colors used to plot the ridge trace curves. Recycled as necessary.

pch

Vector of plotting characters used to plot the ridge trace curves. Recycled as necessary.

xlab

Label for horizontal axis

ylab

Label for vertical axis

xlim, ylim

x, y limits for the plot. You may need to adjust these to allow for the variable labels.

Value

vif returns a "vif.ridge" object, which is a list of four components

vif

a data frame of the same size and shape as coef{mod}. The columns correspond to the predictors in the model and the rows correspond to the values of lambda in ridge estimation.

lambda

the vector of ridge constants from the original call to ridge

df

the vector of effective degrees of freedom corresponding to lambda

criteria

the optimal values of lambda

Author(s)

Michael Friendly

References

Fox, J. and Monette, G. (1992). Generalized collinearity diagnostics. JASA, 87, 178-183, doi:10.1080/01621459.1992.10475190.

Examples


data(longley)
lmod <- lm(Employed ~ GNP + Unemployed + Armed.Forces + Population + 
                      Year + GNP.deflator, data=longley)
vif(lmod)

lambda <- c(0, 0.005, 0.01, 0.02, 0.04, 0.08)
lridge <- ridge(Employed ~ GNP + Unemployed + Armed.Forces + 
                           Population + Year + GNP.deflator, 
		             data=longley, lambda=lambda)

coef(lridge)

# get VIFs for the shrunk estimates
vridge <- vif(lridge)
vridge
names(vridge)



# plot VIFs
pch <- c(15:18, 7, 9)
clr <- c("black", rainbow(5, start=.6, end=.1))

plot(vridge, 
     col=clr, pch=pch, cex = 1.2,
     xlim = c(-0.02, 0.08))

plot(vridge, X = "df",
     col=clr, pch=pch, cex = 1.2,
     xlim = c(4, 6.5))

# Better to plot sqrt(VIF). Plot against degrees of freedom
plot(vridge, X = "df", Y="sqrt",
     col=clr, pch=pch, cex = 1.2,
     xlim = c(4, 6.5))

Generalized ridge trace plots for ridge regression

Description

Details

Author(s)

References

See Also

Examples

Acetylene Data

Description

Format

Details

Source

References

Examples

Detroit Homicide Data for 1961-1973

Description

Format

Details

Source

References

Examples

Hospital manpower data

Description

Format

Details

Source

References

See Also

Examples

Biplot of Ridge Regression Trace Plot in SVD Space

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Enhanced Contour Plots

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Diabetes Progression

Description

Usage

Format

Details

Source

References

Examples

Scatterplot Matrix of Bivariate Ridge Trace Plots

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Transform Ridge Estimates to PCA Space

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Plot Shrinkage vs. Variance for Ridge Precision

Description

Usage

Arguments

Value

Author(s)

See Also