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. function does. ~~

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.

Usage

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

Value

A 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

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 is normalized to a maximum of 1.0.
...: Other arguments (currently unused)

Author

Michael Friendly

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)
	})