Comparison of different sample designs and construction of confidence bands to estimate the mean of functional data: An illustration on electricity consumption

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Hervé Cardot, Alain Dessertaine, Camelia Goga, Étienne Josserand and Pauline Lardin¹

Abstract

When the study variables are functional and storage capacities are limited or transmission costs are high, using survey techniques to select a portion of the observations of the population is an interesting alternative to using signal compression techniques. In this context of functional data, our focus in this study is on estimating the mean electricity consumption curve over a one-week period. We compare different estimation strategies that take account of a piece of auxiliary information such as the mean consumption for the previous period. The first strategy consists in using a simple random sampling design without replacement, then incorporating the auxiliary information into the estimator by introducing a functional linear model. The second approach consists in incorporating the auxiliary information into the sampling designs by considering unequal probability designs, such as stratified and $\pi ps$. designs. We then address the issue of constructing confidence bands for these estimators of the mean. When effective estimators of the covariance function are available and the mean estimator satisfies a functional central limit theorem, it is possible to use a fast technique for constructing confidence bands, based on the simulation of Gaussian processes. This approach is compared with bootstrap techniques that have been adapted to take account of the functional nature of the data.

Key Words

Bonferroni; bootstrap; Horvitz-Thompson estimator; covariance function; model-assisted estimator; functional linear model; Hájek formula.

Table of content

1 Introduction

2 Functional data in a finite population

3 Construction of confidence bands

4 Study of the mean electricity consumption curve

5 Conclusion and perspectives for research

 

 

 

 

 

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¹H. Cardot, Université de Bourgogne, Institut de Mathématiques de Bourgogne, 9 av. Alain Savary, 21078 DIJON, FRANCE; A. Dessertaine, LA POSTE - DIRECTION DU COURRIER - DFI – DCPES, 2 Boulevard Newton 77543 MARNE LA VALLEE CEDEX 2 and EDF, R&D, ICAME-SOAD, 1 av. du Général de Gaulle, 92141 CLAMART, France; C.Goga, Université de Bourgogne, Institut de Mathématiques de Bourgogne; E. Josserand, Université de Bourgogne, Institut de Mathématiques de Bourgogne; P. Lardin, Université de Bourgogne, Institut de Mathématiques de Bourgogne and EDF, R&D, ICAME-SOAD