precision: Measures of Precision and Shrinkage for Ridge Regression

Description

Calculates measures of precision based on the size of the estimated covariance matrices of the parameters and shrinkage of the parameters in a ridge regression model.

Usage

precision(object, ...)

## S3 method for class 'ridge':
precision(object, det.fun=c("log","root"), normalize=TRUE, ...)
## S3 method for class 'lm':
precision(object, det.fun=c("log","root"), normalize=TRUE, ...)

Arguments

Value

A data.frame with the following columnslambdaThe ridge constantdfThe equivalent effective degrees of freedomdetThe det.fun function of the determinant of the covariance matrixtraceThe trace of the covariance matrixmax.eigMaximum eigen value of the covariance matrixnorm.betaThe root mean square of the estimated coefficients, possibly normalized

Details

Three measures of (inverse) precision based on the size of the covariance matrix of the parameters are calculated. Let $V_k$ be the covariance matrix for a given ridge constant, and let $\lambda_i , i= 1, \dots p$ be its eigenvalues

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

Examples

Run this code

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)
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)
	})