TY - JOUR

T1 - Reproducing kernel Hilbert spaces for penalized regression

T2 - A tutorial

AU - Nosedal-Sanchez, Alvaro

AU - Storlie, Curtis B.

AU - Lee, Thomas C.M.

AU - Christensen, Ronald

N1 - Funding Information:
Alvaro Nosedal-Sanchez, Assistant Professor, Mathematics Department, Indiana University of Pennsylvania, Indiana, PA 15705 (E-mail: anosedal@iup.edu). Curtis B. Storlie, Technical Staff, Statistical Sciences Group, Los Alamos National Laboratory, Los Alamos, NM 87545 (E-mail: storlie@lanl.gov). Thomas C.M. Lee, Professor of Statistics, Department of Statistics, University of California, Davis, Davis, CA 95616 (E-mail: tcm-lee@ucdavis.edu). Ronald Christensen, Professor of Statistics, Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131 (E-mail: fletcher@stat.unm.edu). The authors are grateful to the referees and the associate editor for their comments, which led to a much improved version of the article. Raymond Wong also provided helpful comments on an earlier version of the article. Nosedal was partially supported by the National Council of Science and Technology of Mexico (CONACYT). Lee was partially supported by the National Science Foundation under grant DMS 1007520. Storlie’s work was funded by Los Alamos National Security (LANS), LLC, operator of the Los Alamos National Laboratory under contract no. DE-AC52-06NA25396 with the U.S. Department of Energy. This article is published under LA-UR-12-10169.

PY - 2012

Y1 - 2012

N2 - Penalized regression procedures have become very popular ways to estimate complicated functions. The smoothing spline, for example, is the solution of a minimization problem in a functional space. If such a minimization problem is posed on a reproducing kernel Hilbert space (RKHS), the solution is guaranteed to exist, is unique, and has a very simple form. There are excellent books and articles about RKHS and their applications in statistics; however, this existing literature is very dense. This article provides a friendly reference for a reader approaching this subject for the first time. It begins with a simple problem, a system of linear equations, and then gives an intuitive motivation for reproducing kernels. Armed with the intuition gained from our first examples, we take the reader from vector spaces to Ba-nach spaces and to RKHS. Finally, we present some statistical estimation problems that can be solved using the mathematical machinery discussed. After reading this tutorial, the reader will be ready to study more advanced texts and articles about the subject, such as those by Wahba or Gu. Online supplements are available for this article.

AB - Penalized regression procedures have become very popular ways to estimate complicated functions. The smoothing spline, for example, is the solution of a minimization problem in a functional space. If such a minimization problem is posed on a reproducing kernel Hilbert space (RKHS), the solution is guaranteed to exist, is unique, and has a very simple form. There are excellent books and articles about RKHS and their applications in statistics; however, this existing literature is very dense. This article provides a friendly reference for a reader approaching this subject for the first time. It begins with a simple problem, a system of linear equations, and then gives an intuitive motivation for reproducing kernels. Armed with the intuition gained from our first examples, we take the reader from vector spaces to Ba-nach spaces and to RKHS. Finally, we present some statistical estimation problems that can be solved using the mathematical machinery discussed. After reading this tutorial, the reader will be ready to study more advanced texts and articles about the subject, such as those by Wahba or Gu. Online supplements are available for this article.

KW - Projection principle

KW - Regularization

KW - Representation theorem

KW - Ridge regression

KW - Smoothing splines

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U2 - 10.1080/00031305.2012.678196

DO - 10.1080/00031305.2012.678196

M3 - Article

AN - SCOPUS:84862649398

SN - 0003-1305

VL - 66

SP - 50

EP - 60

JO - American Statistician

JF - American Statistician

IS - 1

ER -