In linear regression with functional predictors and scalar responses, it may be advantageous, particularly if the function is thought to contain features at many scales, to restrict the coefficient function to the span of a wavelet basis, thereby converting the problem into one of variable selection. If the coefficient function is sparsely represented in the wavelet domain, we may employ the well-known LASSO to select a relatively small number of nonzero wavelet coefficients. This is a natural approach to take but to date, the properties of such an estimator have not been studied. In this article we describe the wavelet-based LASSO approach to regressing scalars on functions and investigate both its asymptotic convergence and its finite-sample performance through both simulation and real-data application.We compare the performance of this approach with existing methods and find that the wavelet-based LASSO performs relatively well, particularly when the true coefficient function is spiky. Source code to implement the method and datasets used in the study are provided as supplementary materials available online.
Bibliographical noteFunding Information:
The authors thank the editor, the associate editor, and the reviewers for their helpful suggestions and comments that greatly improve the manuscript. They also thank Prof. Hervé Cardot for kindly providing his computer program and Prof. Richard P. Sloan for his generous support through this project. Yihong Zhao was partly supported by the National Cancer Institute grant 5T32CA09529, R. Todd Ogden’s research was supported in part by the NIH grant 5R01EB009744, and Philip T. Reiss was supported in part by the National Science Foundation grant DMS-0907017 and the National Institutes of Health grant 5R01EB009744.
- Functional data analysis
- Penalized linear regression
- Variable selection
- Wavelet regression
ASJC Scopus subject areas
- Statistics and Probability
- Discrete Mathematics and Combinatorics
- Statistics, Probability and Uncertainty