Depth-based reconstruction method for incomplete functional data

TL;DR

This paper introduces a non-parametric depth-based method for reconstructing incomplete functional data, outperforming traditional covariance-based methods especially with non-stationary data or scarce complete samples.

Contribution

A novel depth-based approach for functional data reconstruction that is effective with non-stationary covariance and limited complete data, unlike existing covariance-dependent methods.

Findings

Outperforms covariance-based methods with non-stationary data

Effective even when all functions are incomplete

Performs well on real temperature and mortality datasets

Abstract

The problem of estimating missing fragments of curves from a functional sample has been widely considered in the literature. However, a majority of the reconstruction methods rely on estimating the covariance matrix or the components of its eigendecomposition, a task that may be difficult. In particular, the accuracy of the estimation might be affected by the complexity of the covariance function and the poor availability of complete functional data. We introduce a non-parametric alternative based on a novel concept of depth for partially observed functional data. Our simulations point out that the available methods are unbeatable when the covariance function is stationary, and there is a large proportion of complete data. However, our approach was superior when considering non-stationary covariance functions or when the proportion of complete functions is scarce. Moreover, even in the most severe case of having all the functions incomplete, our method performs well meanwhile the competitors are unable. The methodology is illustrated with two real data sets: the Spanish daily temperatures observed in different weather stations and the age-specific mortality by prefectures in Japan.