Using complex surveys to estimate the $L_1$-median of a functional variable: application to electricity load curves

TL;DR

This paper introduces a robust $L_1$-median approach for estimating electricity load profiles using complex survey sampling, improving accuracy over traditional mean profiles especially in the presence of outliers.

Contribution

It develops new survey sampling estimators for the $L_1$-median of functional data, including stratified and auxiliary information-based methods, with demonstrated improvements.

Findings

Stratification based on linearized variables enhances estimator accuracy.

Proposed estimators outperform simple random sampling methods.

Application to electricity load curves shows robustness against outliers.

Abstract

Mean profiles are widely used as indicators of the electricity consumption habits of customers. Currently, in \'Electricit\'e De France (EDF), class load profiles are estimated using point-wise mean function. Unfortunately, it is well known that the mean is highly sensitive to the presence of outliers, such as one or more consumers with unusually high-levels of consumption. In this paper, we propose an alternative to the mean profile: the $L_1$-median profile which is more robust. When dealing with large datasets of functional data (load curves for example), survey sampling approaches are useful for estimating the median profile avoiding storing the whole data. We propose here estimators of the median trajectory using several sampling strategies and estimators. A comparison between them is illustrated by means of a test population. We develop a stratification based on the linearized variable which substantially improves the accuracy of the estimator compared to simple random sampling without replacement. We suggest also an improved estimator that takes into account auxiliary information. Some potential areas for future research are also highlighted.