ridge: Ridge Regression Estimates
In genridge: Generalized Ridge Trace Plots for Ridge Regression

Description Usage Arguments Details Value Author(s) References See Also Examples

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.

ridge(y, ...)

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

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

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

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

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

`y`	A numeric vector containing the response variable. NAs not allowed.
`X`	A matrix of predictor variables. NA's not allowed. Should not include a column of 1's for the intercept
`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.
`x, object`	An object of class `ridge`
`...`	Other arguments, passed down to methods
`digits`	For the `print` method, the number of digits to print.

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 is supplied the equivalent values of lambda. 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.

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

If svd==TRUE, 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.

Michael Friendly

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.

lm.ridge, simple.ridge for other implementations of ridge regression

traceplot, plot.ridge, pairs.ridge, plot3d.ridge, for 1D, 2D, 3D plotting methods

pca.ridge, biplot.ridge, biplot.pcaridge for views in PCA/SVD space

precision.ridge for measures of shrinkage and precision

# 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)
traceplot(lridge)
traceplot(lridge, X="df")
pairs(lridge, radius=0.5)


if (require("ElemStatLearn")) {
	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.ridge(mridge)
traceplot(mpridge, X="df")
biplot(mpridge, radius=0.25)