Compute and display TRACEs for the p-paramater Shrinkage PATH passing through the (classical) Normal-theory Maximum Likelihhod (ML) point-estimate of the Beta coefficient vector. The m-Extent of overall Optimal Shrinkage corresponding to this solution is marked by a vertical dashed-line on all 5-types of unr.ridge TRACE displays.
unr.ridge(form, data, rscale = 1, steps = 8, delmax = 0.999999)A regression formula [y~x1+x2+...] 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 specified as the first argument.
Number of regression predictor variables.
Number of complete observations after removal of all missing values.
Numerical value of R-square goodness-of-fit statistic.
Numerical value of the residual mean square estimate of error.
Listing of principal statistics.
Variable re-scaling code of 0, 1 or 2 used in calculations.
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 M-extent-of-shrinkage.
Listing of summary statistics for all M-extents-of-shrinkage.
Unrestricted m-Extent of Shrinkage corresponding to k* == 1 on TRACE displays.
Most Likely Extent of Shrinkage Observed: 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 then tend to have "wrong" numerical signs and/or unreasonable relative magnitudes, while shrunken (generalized ridge) estimates chosen to maximize their likelihood of reducing Mean Squared Error (MSE) Risk (expected loss) can be much 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. (2005) Shrinkage Regression: ridge, BLUP, Bayes, spline and Stein. Electronic book-in-progress; 185+ pages. http://localcontrolstatistics.org
Obenchain RL. (2020) Ridge TRACE Diagnostics. https://arxiv.org/abs/2005.14291
Obenchain RL. (2020) The Unrestricted Shrinkage Path: Technical Details. "unrPath.pdf" http://localcontrolstatistics.org
Obenchain RL. (2020) RXshrink_in_R.PDF RXshrink package vignette-like file. http://localcontrolstatistics.org
mofk, kofm, correct.signs, MLtrue
and RXpredict.
# NOT RUN {
data(longley2)
form <- Employed~GNP+GNP.deflator+Unemployed+Armed.Forces+Population+Year
rxuobj <- unr.ridge(form, data=longley2)
rxuobj
plot(rxuobj)
str(rxuobj)
# }
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