### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 50-60 |

Number of pages | 11 |

Journal | American Statistician |

Volume | 66 |

Issue number | 1 |

DOIs | |

State | Published - 2012 |

Externally published | Yes |

### Fingerprint

### Keywords

- Projection principle
- Regularization
- Representation theorem
- Ridge regression
- Smoothing splines

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*American Statistician*,

*66*(1), 50-60. https://doi.org/10.1080/00031305.2012.678196

**Reproducing kernel Hilbert spaces for penalized regression : A tutorial.** / Nosedal-Sanchez, Alvaro; Storlie, Curtis; Lee, Thomas C M; Christensen, Ronald.

Research output: Contribution to journal › Article

*American Statistician*, vol. 66, no. 1, pp. 50-60. https://doi.org/10.1080/00031305.2012.678196

}

TY - JOUR

T1 - Reproducing kernel Hilbert spaces for penalized regression

T2 - A tutorial

AU - Nosedal-Sanchez, Alvaro

AU - Storlie, Curtis

AU - Lee, Thomas C M

AU - Christensen, Ronald

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

UR - http://www.scopus.com/inward/record.url?scp=84862649398&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84862649398&partnerID=8YFLogxK

U2 - 10.1080/00031305.2012.678196

DO - 10.1080/00031305.2012.678196

M3 - Article

VL - 66

SP - 50

EP - 60

JO - American Statistician

JF - American Statistician

SN - 0003-1305

IS - 1

ER -