Using Complex Surveys to Estimate theL₁‐Median of a Functional Variable: Application to Electricity Load Curves

Abstract

Mean profiles are widely used as indicators of the electricity consumption habits of customers. Currently, inÉlectricité 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.

Clustering functional data using forward search based on functional spatial ranks with medical applications

“…And this functional spatial median has been used as a robust measure of center for a data set of electricity loading curves. 22 The kernelized functional spatial depth (KFSD) has been proposed 23 for the classification of functional data. It is based on the functional spatial depth introduced by Serfling and Wijesuriya.…”

Conditional Bias Robust Estimation of the Total of Curve Data by Sampling in a Finite Population: An Illustration on Electricity Load Curves

“…For the robust estimatort (R4) Y given in (26) (section 5.1) computed by using functional truncation methods based on depth, a variance estimator may be computed by using (29) withẐ iα (t) = π i ψ α (B HT 1i (t)) −B HT 1i (t) . Using linearization techniques, we can write for the robust estimatort (R2) Y given in (23): Chaouch and Goga (2012)) with Γ given in (14). We also have…”

“…. , K. A natural estimator of the geometric median m N is given by the solution m of the following non linear estimating equation (see Chaouch and Goga (2012)),…”

“…The total consumption curve or the load curve of each population of interest is then estimated by using the curves of the customers belonging to the sample. The estimation with survey sampling techniques of parameters of interest such as the total or the mean, the median or the principal components when the data are curves has been developed over the last years: Cardot et al (2010), Cardot and Josserand (2011), Cardot et al (2013a) and Chaouch and Goga (2012). Several sampling designs and estimators have been compared by means of simulation on real electricity data set in Cardot et al (2013b) and some asymptotic properties have been established in Cardot et al (2013c) and Cardot et al (2014).…”

Recursive estimation of the conditional geometric median in Hilbert spaces

“…Nevertheless, these computational techniques may not be able to handle very large samples of high-dimensional data since they require to store all the data. An alternative approach, developed by Chaouch and Goga (2012) and which can cope with this issue, consists in considering unequal probability sampling techniques in order to select, in a effective way, subsamples with sizes much smaller than the initial sample size.…”