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Structured penalties for functional linear models—partially empirical eigenvectors for regression

Randolph, Timothy W. and Harezlak, Jaroslaw and Feng, Ziding (2012) Structured penalties for functional linear models—partially empirical eigenvectors for regression. Electronic Journal of Statistics, 6. pp. 323-353. ISSN 1935-7524

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Article URL: http://dx.doi.org/10.1214/12-EJS676

Abstract

One of the challenges with functional data is incorporating geometric structure, or local correlation, into the analysis. This structure is inherent in the output from an increasing number of biomedical technologies, and a functional linear model is often used to estimate the relationship between the predictor functions and scalar responses. Common approaches to the problem of estimating a coefficient function typically involve two stages: regularization and estimation. Regularization is usually done via dimension reduction, projecting onto a predefined span of basis functions or a reduced set of eigenvectors (principal components). In contrast, we present a unified approach that directly incorporates geometric structure into the estimation process by exploiting the joint eigenproperties of the predictors and a linear penalty operator. In this sense, the components in the regression are ‘partially empirical’ and the framework is provided by the generalized singular value decomposition (GSVD). The form of the penalized estimation is not new, but the GSVD clarifies the process and informs the choice of penalty by making explicit the joint influence of the penalty and predictors on the bias, variance and performance of the estimated coefficient function. Laboratory spectroscopy data and simulations are used to illustrate the concepts.

Item Type: Article
Additional Information: For IMS statement of Creative Commons copyright license, visit: http://imstat.org/ejs/copyright.html
DOI: doi:10.1214/12-EJS676
NIHMSID: NIHMS368253
Grant Numbers: R01-CA126205, P01-CA053996, U01-CA086368, R01- NS036524, U01-MH083545
Subjects: Research Methodologies > Mathematics and statistics
Depositing User: Library Staff
Date Deposited: 03 Apr 2012 19:24
Last Modified: 04 Apr 2012 00:03
URI: http://authors.fhcrc.org/id/eprint/558

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