Mohamed Chaouch

Properties of design-based functional principal components analysis

Abstract This work aims at performing functional principal components analysis (FPCA) with Horvitz–Thompson estimators when the observations are curves collected with survey sampling techniques. One important motivation for this study is that FPCA is a dimension reduction tool which is the first step to develop model-assisted approaches that can take auxiliary information into account. FPCA relies on the estimation of the eigenelements of the covariance operator which can be seen as nonlinear functionals. Adapting to our functional context the linearization technique based on the influence function developed by Deville [1999. Variance estimation for complex statistics and estimators: linearization and residual techniques. Survey Methodology 25, 193–203], we prove that these estimators are asymptotically design unbiased and consistent. Under mild assumptions, asymptotic variances are derived for the FPCA’ estimators and consistent estimators of them are proposed. Our approach is illustrated with a simulation study and we check the good properties of the proposed estimators of the eigenelements as well as their variance estimators obtained with the linearization approach.

Design-based estimation for geometric quantiles with application to outlier detection

Geometric quantiles are investigated using data collected from a complex survey. Geometric quantiles are an extension of univariate quantiles in a multivariate set-up that uses the geometry of multivariate data clouds. A very important application of geometric quantiles is the detection of outliers in multivariate data by means of quantile contours. A design-based estimator of geometric quantiles is constructed and used to compute quantile contours in order to detect outliers in both multivariate data and survey sampling set-ups. An algorithm for computing geometric quantile estimates is also developed. Under broad assumptions, the asymptotic variance of the quantile estimator is derived and a consistent variance estimator is proposed. Theoretical results are illustrated with simulated and real data.

Using complex surveys to estimate the L1-median of a functional variable: Application to electricity load curves

Mean proles are widely used as indicators of the electricity consumption habits of customers. Currently, Electricit e De France (EDF), estimates class load proles by 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 prole: the L1-median prole 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 prole and avoid storing all of the 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 stratication 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.

Stochastic Approximation for Multivariate and Functional Median

We propose a very simple algorithm in order to estimate the geometric median, also called spatial median, of multivariate (Small (1990)) or functional data (Gervini (2008)) when the sample size is large. A simple and fast iterative approach based on the Robbins-Monro algorithm (Duflo (1997)) as well as its averaged version (Polyak and Juditsky (1992)) are shown to be effective for large samples of high dimension data. They are very fast and only require O(Nd) elementary operations, where N is the sample size and d is the dimension of data. The averaged approach is shown to be more effective and less sensitive to the tuning parameter. The ability of this new estimator to estimate accurately and rapidly (about thirty times faster than the classical estimator) the geometric median is illustrated on a large sample of 18902 electricity consumption curves measured every half an hour during one week.

Functional Principal Components Analysis with Survey Data

This work aims at performing Functional Principal Components Analysis (FPCA) with Horvitz-Thompson estimators when the observations are curves collected with survey sampling techniques. FPCA relies on estimations of the eigenelements of the covariance operator which can be seen as nonlinear functionals. Adapting to our functional context the linearization technique based on the influence function developed by Deville (1999), we prove that these estimators are asymptotically design unbiased and convergent. Under mild assumptions, asymptotic variances are derived for the FPCA’ estimators and convergent estimators of them are proposed. Our approach is illustrated with a simulation study and we check the good properties of the proposed estimators of the eigenelements as well as their variance estimators obtained with the linearization approach.