rstats2.lmridge {lmridge}

Ordinary Ridge Regression Statistics 2

Description

The rstats2 function computes the ordinary ridge related statistics such as Ck, \sigma^2, ridge degrees of freedom, effective degrees of freedom (EDF), and prediction residual error sum of squares PRESS statistics for scalar or vector value of biasing parameter K (See Allen, 1974 <doi:10.2307/1267500>; Lee, 1979; Hoerl and Kennard, 1970 <doi:10.2307/1267351>).

Usage

   rstats2(x, ...)
   ## S3 method for class 'lmridge'
rstats2(x, ...)
   ## S3 method for class 'rstats2'
print(x, digits = max(5,getOption("digits") - 5), ...)

Arguments

x

For the rstats2 method, an object of class "lmridge", i.e., a fitted model.

digits

Minimum number of significant digits to be used.

...

Not presently used in this implementation.

Details

The rstats2 function computes the ridge regression related different statistics which may help in selecting the optimal value of biasing parameter K. If value of K is zero then these statistics are equivalent to the relevant OLS statistics.

Value

Following are ridge related statistics computed for given scalar or vector value of biasing parameter K provided as argument to lmridge or lmridgeEst function.

CK

Ck similar to Mallows Cp statistics for given biasing parameter K.

dfridge

DF of ridge for given biasing parameter K, i.e., Trace[Hat_{R,k}].

EP

Effective number of Parameters for given biasing parameter K, i.e., Trace[2Hat_{R,k}-Hat_{R,k}t(Hat_{R,k})].

redf

Residual effective degrees of freedom for given biasing parameter K from Hastie and Tibshirani, (1990), i.e., n-Trace[2Hat_{R,k}-Hat_{R,k}t(Hat_{R,k})].

EF

Effectiveness index for given biasing parameter K, also called the ratio of reduction in total variance in the total squared bias by the ridge regression, i.e., EF=\frac{\sigma^2 trace(X'X)^{-1}-\sigma^2 trace(VIF_R)}{Bias^2(\hat{\beta}_R)}.

ISRM

Quantification of concept of stable region proposed by Vinod and Ullah, 1981, i.e., ISRM_k=\sum_{j=1}^p (\frac{p(\frac{\lambda_j}{\lambda_j+k})^2}{\sum_{j=1}^p \frac{\lambda_j}{(\lambda_j+k)^2} \lambda_j}-1)^2.

m

m-scale for given value of biasing parameter proposed by Vinod (1976) alternative to plotting of the ridge coefficients, i.e., p-\sum_{j-1}^p \frac{\lambda_j}{\lambda_j+k}.

PRESS

PRESS statistics for ridge regression introduced by Allen, 1971, 1974, i.e., PRESS_k=\sum_{i=1}^n e^2_{i,-i} for scalar or vector value of biasing parameter K.

Author(s)

Muhammad Imdad Ullah, Muhammad Aslam

References

Allen, D. M. (1971). Mean Square Error of Prediction as a Criterion for Selecting Variables. Technometrics, 13, 469-475. doi:10.1080/00401706.1971.10488811.

Allen, D. M. (1974). The Relationship between Variable Selection and Data Augmentation and Method for Prediction. Technometrics, 16, 125-127. doi:10.1080/00401706.1974.10489157.

Cule, E. and De lorio, M. (2012). A semi-Automatic method to guide the choice of ridge parameter in ridge regression. arXiv:1205.0686v1 [stat.AP].

Hastie, T. and Tibshirani, R. (1990). Generalized Additive Models. Chapman & Hall.

Hoerl, A. E., Kennard, R. W., and Baldwin, K. F. (1975). Ridge Regression: Some Simulation. Communication in Statistics, 4, 105-123. doi:10.1080/03610927508827232.

Hoerl, A. E. and Kennard, R. W., (1970). Ridge Regression: Biased Estimation of Nonorthogonal Problems. Technometrics, 12, 55-67. doi:10.1080/00401706.1970.10488634.

Imdad, M. U. Addressing Linear Regression Models with Correlated Regressors: Some Package Development in R (Doctoral Thesis, Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan), 2017.

Kalivas, J. H., and Palmer, J. (2014). Characterizing Multivariate Calibration Tradeoffs (Bias, Variance, Selectivity, and Sensitivity) to Select Model Tuning Parameters. Journal of Chemometrics, 28(5), 347–357. doi:10.1002/cem.2555.

Lee, W. F. (1979). Model Estimation Using Ridge Regression with the Variane Normalization Criterion. Master thesis, Department of Educational Foundation Memorial University of Newfoundland.

See Also

Ridge related statistics rstats1, ridge model fitting lmridge

Examples

data(Hald)
mod <- lmridge(y~., data=as.data.frame(Hald), K = seq(0,0.2, 0.001) )

rstats2(mod)