[Paper Review] Sparse Wavelet Estimation in Quantile Regression with Multiple Functional Predictors

Dengdeng Yu, Li Zhang+3 more|arXiv (Cornell University)|Jun 7, 2017

Sparse and Compressive Sensing Techniques46 references3 citations

TL;DR

This paper proposes a sparse wavelet-based quantile regression method for scalar responses with multiple functional predictors, using a sparse group lasso penalty to simultaneously select important functional predictors and capture shared structure among them. The method employs wavelet basis expansion for efficient local feature detection and solves the estimation problem via second-order cone programming and ADMM, achieving optimal convergence rates and prediction error bounds in theory and strong performance in simulations and ADHD-200 fMRI data.

ABSTRACT

In this manuscript, we study quantile regression in partial functional linear model where response is scalar and predictors include both scalars and multiple functions. Wavelet basis are adopted to better approximate functional slopes while effectively detect local features. The sparse group lasso penalty is imposed to select important functional predictors while capture shared information among them. The estimation problem can be reformulated into a standard second-order cone program and then solved by an interior point method. We also give a novel algorithm by using alternating direction method of multipliers (ADMM) which was recently employed by many researchers in solving penalized quantile regression problems. The asymptotic properties such as the convergence rate and prediction error bound have been established. Simulations and a real data from ADHD-200 fMRI data are investigated to show the superiority of our proposed method.

Motivation & Objective

• Address the need for robust, high-dimensional functional quantile regression in the presence of multiple functional predictors and scalar covariates.
• Overcome limitations of existing methods by incorporating wavelet basis for localized functional feature detection and computational efficiency.
• Simultaneously perform variable selection across multiple functional predictors and sparsity within each predictor using the sparse group lasso penalty.
• Establish theoretical guarantees including convergence rate and prediction error bounds under regularity conditions.
• Demonstrate the method’s effectiveness on real fMRI data from the ADHD-200 project, particularly in identifying relevant brain regions (ROIs) at different quantiles.

Proposed method

• Represent functional coefficients using a common wavelet basis to enable shared information capture across multiple functional predictors.
• Formulate the estimation problem as a penalized quantile regression with a sparse group lasso penalty combining lasso and group lasso penalties.
• Reformulate the optimization problem into a standard second-order cone program solvable via interior point methods.
• Develop an efficient ADMM algorithm for solving the penalized quantile regression problem, enabling scalability to high-dimensional settings.
• Use the Knight identity to linearize the check function in quantile regression for theoretical analysis and derivation of asymptotic properties.
• Employ wavelet thresholding and basis expansion to achieve sparse, localized, and adaptive estimation of functional coefficients.

Experimental results

Research questions

• RQ1 Can wavelet basis representation improve the estimation accuracy and local feature detection in quantile regression with multiple functional predictors?
• RQ2 Does the sparse group lasso penalty effectively select important functional predictors while capturing shared structure among them in a quantile regression framework?
• RQ3 What are the theoretical convergence rates and prediction error bounds for the proposed estimator under regularity conditions?
• RQ4 How does the proposed method compare to existing methods in terms of variable selection and prediction performance on real fMRI data?
• RQ5 Can the method identify biologically meaningful brain regions (ROIs) associated with ADHD index across different quantile levels?

Key findings

• The proposed method achieves optimal convergence rates for the functional coefficient estimator, with the rate depending on the sparsity and smoothness of the true coefficient functions.
• The prediction error bound is established under regularity conditions, showing that the estimator converges to the true coefficient at a rate that depends on sample size and the number of wavelet coefficients.
• The ADMM algorithm converges efficiently and enables scalable computation, making the method applicable to high-dimensional functional data such as fMRI.
• Simulations show that the method outperforms competing approaches in terms of variable selection accuracy and prediction error, especially under sparse and correlated functional predictors.
• In the ADHD-200 fMRI application, the method successfully identifies a small subset of relevant brain ROIs (e.g., prefrontal and parietal regions) associated with ADHD severity at different quantiles.
• The sparse group lasso penalty effectively balances group-level sparsity (selecting entire ROIs) and within-ROI sparsity (selecting relevant wavelet coefficients), improving interpretability and performance.