This function is Deprecated; its Shrinkage PATH is both longer and more complicated than that of eff.ridge(). Specifically, the unr.ridge() PATH is a piece-wise Linear-Spline containing (p-1) "interior" KNOTs. None of these KNOTs usually correspond to the overall optimal m-Extent of Shrinkage [that minimizes MSE Risk under Normal-theory]. In sharp contrast, the eff.ridge() PATH contains exactly one "interior" KNOT when p > 1, and the overall Optimal m-Extent of Shrinkage does coinside with this KNOT! A vertical dashed-line marks this Optimal m-Extent on all 5-types of ridge TRACE displays.
unr.ridge(form, data, rscale = 1, steps = 8, delmax = 0.999999)A regression formula [y~x1+x2+...+xp] suitable for use with lm().
data.frame containing observations on all variables in the formula.
One of three possible choices (0, 1 or 2) for "rescaling" of variables (after being "centered") to remove all "non-essential" ill-conditioning: 0 implies no rescaling; 1 implies divide each variable by its standard error; 2 implies rescale as in option 1 but re-express answers as in option 0.
Number of equally spaced values per unit change along the horizontal M-extent-of-shrinkage axis for estimates to be calculated and displayed in TRACES (default = 8.)
Maximum allowed value for Shrinkage delta-factors that is strictly less than 1. (default = 0.999999, which prints as 1 when rounded to fewer than 6 decimal places.)
An output list object of class unr.ridge:
Name of the data.frame object specified as the second argument.
The regression formula is the first argument.
Number of regression x-predictor variables.
Number of complete observations after removal of all missing values.
Numerical value of R-squared: proportion of variance explained.
Numerical value of the residual mean square estimate of error.
Listing of 5 summary statistics for each of p-Principal Axes.
Variable re-scaling code of 0, 1 or 2 used in calculations.
The data.frame containing all variables listed in the formula.
Orthogonal Matrix of Direction Cosines for Principal Axes.
Matrix of shrinkage-ridge regression coefficient estimates.
Matrix of MSE risk estimates for fitted coefficients.
Matrix of excess MSE eigenvalues (ordinary least squares minus ridge.)
Matrix of direction cosines for the estimated inferior direction, if any.
Matrix of shrinkage pattern multiplicative delta-factors.
Listing of criteria for maximum likelihood selection of an m-Extent for Shrinkage.
Listing of summary statistics for all M-extents-of-shrinkage.
Unrestricted m-Extent of Shrinkage corresponding to k* == 1 on TRACE displays.
Minimum MSE Risk estimate.
Most Likely Observed Extent of Shrinkage: best multiple of (1/steps) <= p.
Minimum Observed Value of Normal-theory -2*log(Likelihood-Ratio).
Most Likely to be Optimal-values for Shrinkage Delta-factors [1:p].
Ill-conditioned and/or nearly multi-collinear regression models are unlikely to produce Ordinary Least Squares (OLS) regression coefficient estimates that are very close, numerically, to their unknown true values. Specifically, OLS estimates can have unreasonable relative magnitudes or "wrong" numerical signs when the number of x-Variables is >= 2. Shrunken (Generalized Ridge Regression) estimates chosen to maximize their likelihood of reducing Mean Squared Error (MSE) Risk (expected Squared Error Loss) can be more stable and reasonable, numerically. On the other hand, because only OLS estimates are guaranteed to be minimax when risk is matrix valued (truly multivariate), no guarantee of an actual reduction in MSE Risk is necessarily associated with Shrinkage.
Thompson JR. (1968) Some shrinkage techniques for estimating the mean. Journal of the American Statistical Association 63, 113-122. (The ``cubic'' estimator.)
Obenchain RL. (1978) Good and Optimal Ridge Estimators. Annals of Statistics 6, 1111-1121. <doi:10.1214/aos/1176344314>
Obenchain RL. (2020) Ridge TRACE Diagnostics. https://arxiv.org/abs/2005.14291
Obenchain RL. (2021) RXshrink_in_R.PDF RXshrink package vignette-like document, Version 2.0. http://localcontrolstatistics.org
mofk, kofm, correct.signs, MLtrue
and RXpredict.
# NOT RUN {
data(longley2)
form <- Employed~GNP+GNP.deflator+Unemployed+Armed.Forces+Population+Year
rxunrobj <- unr.ridge(form, data=longley2)
rxunrobj # print shrinkage summary statistics...
plot(rxunrobj) # 5 TRACEs on 1 plot...
str(rxunrobj)
# }
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