# Simultaneous nonparametric regression in RADWT dictionaries

## TL;DR

This paper introduces a novel nonparametric regression method for multichannel signals using RADWT with different Q-factors, enabling sparse representations of oscillatory components and joint analysis across channels.

## Contribution

The paper develops a new RADWT-based nonparametric regression technique with grouped lasso, providing asymptotic optimality and effective joint analysis of multichannel signals.

## Key findings

- Method performs well in synthetic scenarios.
- Effective in joint detection of EEG sleep events.
- Achieves asymptotic optimality under certain conditions.

## Abstract

A new technique for nonparametric regression of multichannel signals is presented. The technique is based on the use of the Rational-Dilation Wavelet Transform (RADWT), equipped with a tunable Q-factor able to provide sparse representations of functions with different oscillations persistence. In particular, two different frames are obtained by two RADWT with different Q-factors that give sparse representations of functions with low and high resonance. It is assumed that the signals are measured simultaneously on several independent channels and that they share the low resonance component and the spectral characteristics of the high resonance component. Then, a regression analysis is performed by means of the grouped lasso penalty. Furthermore, a result of asymptotic optimality of the estimator is presented using reasonable assumptions and exploiting recent results on group-lasso like procedures. Numerical experiments show the performance of the proposed method in different synthetic scenarios as well as in a real case example for the analysis and joint detection of sleep spindles and K-complex events for multiple electroencephalogram (EEG) signals.