Hiroshima Mathematical Journal

Optimization of ridge parameters in multivariate generalized ridge regression by plug-in methods

Isamu Nagai, Hirokazu Yanagihara, and Kenichi Satoh

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Generalized ridge (GR) regression for an univariate linear model was proposed simultaneously with ridge regression by Hoerl and Kennard (1970). In this paper, we deal with a GR regression for a multivariate linear model, referred to as a multivariate GR (MGR) regression. From the viewpoint of reducing the mean squared error (MSE) of a predicted value, many authors have proposed several GR estimators consisting of ridge parameters optimized by non-iterative methods. By expanding their optimizations of ridge parameters to the multiple response case, we derive some MGR estimators with ridge parameters optimized by the plug-in method. We analytically compare obtained MGR estimators with existing MGR estimators, and numerical studies are also given for an illustration.

Article information

Hiroshima Math. J. Volume 42, Number 3 (2012), 301-324.

First available in Project Euclid: 11 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J07: Ridge regression; shrinkage estimators
Secondary: 62H12: Estimation

Generalized ridge regression Mallows’ $C_p$ statistic model selection multivariate linear regression model non-iterative estimation plug-in method


Nagai, Isamu; Yanagihara, Hirokazu; Satoh, Kenichi. Optimization of ridge parameters in multivariate generalized ridge regression by plug-in methods. Hiroshima Math. J. 42 (2012), no. 3, 301--324. https://projecteuclid.org/euclid.hmj/1355238371.

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  • A. C. Atkinson, A note on the generalized information criterion for choice of a model, Biometrika, 67 (1980), 413-418.
  • S. J. V. Dien, S. Iwatani, Y. Usuda and K. Matsui, Theoretical analysis of amino acid-producing Eschenrichia coli using a stoixhiometrix model and multivariate linear regression, J. Biosci. Bioeng., 102 (2006), 34–40.
  • Y. Fujikoshi and K. Satoh, Modified AIC and $C_p$ in multivariate linear regression, Biometrika, 84 (1997), 707–716.
  • M. Goldstein and A. F. M. Smith, Ridge-type estimators for regression analysis, J. Roy. Statist. Soc. Ser. B, 36 (1974), 284–291.
  • W. J. Hemmerle, An explicit solution for generalized ridge regression, Technometrics, 17 (1975), 309–314.
  • A. E. Hoerl and R. W. Kennard, Ridge regression: biased estimation for nonorthogonal problems, Technometrics, 12 (1970), 55–67.
  • T. Kubokawa, An approach to improving the James-Stein estimator, J. Multivariate Anal., 36 (1991), 121–126.
  • J. F. Lawless, Mean squared error properties of generalized ridge estimators, J. Amer. Statist. Assoc., 76 (1981), 462–466.
  • W. F. Lott, The optimal set of principal component restrictions on a least squares regression, Comm. Statist., 2 (1973), 449–464.
  • C. L. Mallows, Some comments on $C_p$, Technometrics, 15 (1973), 661–675.
  • C. L. Mallows, More comments on $C_p$, Technometrics, 37 (1995), 362–372.
  • C. S$\hat{\rm a}$rbu, C. Onisor, M. Posa, S. Kevresan and K. Kuhajda, Modeling and prediction (correction) of partition coefficients of bile acids and their derivatives by multivariate regression methods, Talanta, 75 (2008), 651–657.
  • R. Sax$\acute{\rm e}$n and J. Sundell, $^{137}$Cs in freshwater fish in Finland since 1986– a statistical analysis with multivariate linear regression models, J. Environ. Radioactiv., 87 (2006), 62–76.
  • M. Siotani, T. Hayakawa, and Y. Fujikoshi, Modern Multivariate Statistical Analysis: A Graduate Course and Handbook, American Sciences Press, Columbus, Ohio, 1985.
  • B. Skagerberg, J. MacGregor and C. Kiparissides, Multivariate data analysis applied to low-density polyethylene reactors, Chemometr. Intell. Lab. Syst., 14 (1992), 341–356.
  • R. S. Sparks, D. Coutsourides and L. Troskie, The multivariate $C_p$, Comm. Statist. A – Theory Methods, 12 (1983), 1775–1793.
  • M. S. Srivastava, Methods of Multivariate Statistics, John Wiley & Sons, New York, 2002.
  • N. H. Timm, Applied Multivariate Analysis, Springer-Verlag, New York, 2002.
  • S. G. Walker and C. J. Page, Generalized ridge regression and a generalization of the $C_p$ statistics, J. Appl. Statist., 28 (2001), 911–922.
  • H. Yanagihara and K. Satoh, An unbiased $C_p$ criterion for multivariate ridge regression, J. Multivariate Anal., 101 (2010), 1226–1238.
  • H. Yanagihara, I. Nagai and K. Satoh, A bias-corrected $C_p$ criterion for optimizing ridge parameters in multivariate generalized ridge regression, Japanese J. Appl. Statist., 38 (2009), 151–172 (in Japanese).
  • A. Yoshimoto, H. Yanagihara and Y. Ninomiya, Finding factors affecting a forest stand growth through multivariate linear modeling, J. Jpn. For. Res., 87 (2005), 504–512 (in Japanese).