Wavelet-based LASSO in functional linear quantile regression

ABSTRACT

In this paper, we develop an efficient wavelet-based regularized linear quantile regression framework for coefficient estimations, where the responses are scalars and the predictors include both scalars and function. The framework consists of two important parts: wavelet transformation and regularized linear quantile regression. Wavelet transform can be used to approximate functional data through representing it by finite wavelet coefficients and effectively capturing its local features. Quantile regression is robust for response outliers and heavy-tailed errors. In addition, comparing with other methods it provides a more complete picture of how responses change conditional on covariates. Meanwhile, regularization can remove small wavelet coefficients to achieve sparsity and efficiency. A novel algorithm, Alternating Direction Method of Multipliers (ADMM) is derived to solve the optimization problems. We conduct numerical studies to investigate the finite sample performance of our method and applied it on real data from ADHD studies.

KEYWORDS:

• Quantile regression
• wavelets
• LASSO
• functional data analysis
• ADMM
• ADHD

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Drs Kong, Jiang and Heo acknowledge the support from the National Sciences and Engineering Research Council of Canada (NSERC). Dr Kong's research is also partially supported by the Canadian Statistical Sciences Institute (CANSSI). Dr Zhou's research is partially supported by the China Postdoctoral Science Foundation (No. 2018T110422).