Skip to main content Accessibility help
×
Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-10-30T18:50:02.551Z Has data issue: false hasContentIssue false

6 - Regularization and Variable Selection for Parametric Models

Published online by Cambridge University Press:  05 June 2012

Gerhard Tutz
Affiliation:
Ludwig-Maximilians-Universität Munchen
Get access

Summary

In several chapters we discussed parametric regression modeling for a moderate number of explanatory variables based on maximum likelihood methods. In some areas of application, however, the number of explanatory variables may be very high. For example, in genetics, where binary regression is a frequently used tool, the number of predictors may be even larger than the number of predictors. In this “p > n problem” maximum likelihood and similar estimators are bound to fail. Typical data of this type are microarray data, where the expressions of thousands of predictors (genes) are observed and only some hundred samples are available. For example, the dataset considered by Golub et al. (1999a), which constitutes a milestone in the classification of cancer, consists of gene expression intensities for 7129 genes of 38 leukemia patients, from which 27 were diagnosed with acute lymphoblastic leukemia and the remaining patients acute myeloid leukemia.

In high-dimensional problems the reduction of the predictor space is the most important issue. A reduction technique with a long history is stepwise variable selection. However, stepwise variable selection as a discrete process is extremely variable. The results of a variable selection procedure may be determined by small changes in the data. The effect is often poor performance (see, e.g., Frank and Friedman, 1993). Moreover, it is challenging to investigate the sampling properties of stepwise variable selection procedures.

An alternative to stepwise subset selection is regularization methods. Ridge regression is a familiar regularization method that adds a simple penalty term to the log-likelihood and thereby shrinks estimates toward zero.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×