Wavelet-based estimation in aggregated functional data with positive and correlated errors

TL;DR

This paper introduces Bayesian wavelet-based methods for estimating individual functional components from aggregated data with positive, correlated Gamma errors, effectively capturing local features like peaks and discontinuities.

Contribution

It presents a novel wavelet-based Bayesian approach for functional data decomposition under positive, correlated errors, extending existing methods to more complex error structures.

Findings

Accurately estimates local features such as peaks and discontinuities.

Performs well in simulations and real data applications.

Handles positive, correlated Gamma errors effectively.

Abstract

We consider the statistical problem of estimating constituent curves from observations of their aggregated curves, referred to as \textit{aggregated functional data}, in models with strictly positive random errors following a Gamma distribution and correlated errors structured through AR(1) and ARFIMA processes. This problem arises in several areas of knowledge, such as chemometrics, for example, when absorbance curves of the constituents of a given substance must be estimated from its aggregated absorbance curve according to the Beer--Lambert law. In this context, we propose Bayesian wavelet-based methods to estimate the component functions within a functional data analysis framework. This approach has the advantage of accurately estimating curves with important local features, such as discontinuities, peaks, and oscillations, due to the representation properties of functions in wavelet bases. We further evaluate the performance of the proposed method through computational simulations, as well as applications to real data.