The Area of the Convex Hull of Sampled Curves: a Robust Functional Statistical Depth measure

Guillaume Staerman, Pavlo Mozharovskyi, Stéphan Clémen\con
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:570-579, 2020.

Abstract

With the ubiquity of sensors in the IoT era, statistical observations are becoming increasingly available in the form of massive (multivariate) time-series. Formulated as unsupervised anomaly detection tasks, an abundance of applications like aviation safety management, the health monitoring of complex infrastructures or fraud detection can now rely on such functional data, acquired and stored with an ever finer granularity. The concept of \textit{statistical depth}, which reflects centrality of an arbitrary observation w.r.t. a statistical population may play a crucial role in this regard, anomalies corresponding to observations with ’small’ depth. Supported by sound theoretical and computational developments in the recent decades, it has proven to be extremely useful, in particular in functional spaces. However, most approaches documented in the literature consist in evaluating independently the centrality of each point forming the time series and consequently exhibit a certain insensitivity to possible shape changes.In this paper, we propose a novel notion of functional depth based on the area of the convex hull of sampled curves, capturing gradual departures from centrality, even beyond the envelope of the data, in a natural fashion.We discuss practical relevance of commonly imposed axioms on functional depths and investigate which of them are satisfied by the notion of depth we promote here. Estimation and computational issues are also adressed and various numerical experiments provide empirical evidence of the relevance of the approach proposed.

Cite this Paper

BibTeX
@InProceedings{pmlr-v108-staerman20a, title = {The Area of the Convex Hull of Sampled Curves: a Robust Functional Statistical Depth measure}, author = {Staerman, Guillaume and Mozharovskyi, Pavlo and Cl\'emen{\c}on, St\'ephan}, booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics}, pages = {570--579}, year = {2020}, editor = {Chiappa, Silvia and Calandra, Roberto}, volume = {108}, series = {Proceedings of Machine Learning Research}, month = {26--28 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v108/staerman20a/staerman20a.pdf}, url = {https://proceedings.mlr.press/v108/staerman20a.html}, abstract = {With the ubiquity of sensors in the IoT era, statistical observations are becoming increasingly available in the form of massive (multivariate) time-series. Formulated as unsupervised anomaly detection tasks, an abundance of applications like aviation safety management, the health monitoring of complex infrastructures or fraud detection can now rely on such functional data, acquired and stored with an ever finer granularity. The concept of \textit{statistical depth}, which reflects centrality of an arbitrary observation w.r.t. a statistical population may play a crucial role in this regard, anomalies corresponding to observations with ’small’ depth. Supported by sound theoretical and computational developments in the recent decades, it has proven to be extremely useful, in particular in functional spaces. However, most approaches documented in the literature consist in evaluating independently the centrality of each point forming the time series and consequently exhibit a certain insensitivity to possible shape changes.In this paper, we propose a novel notion of functional depth based on the area of the convex hull of sampled curves, capturing gradual departures from centrality, even beyond the envelope of the data, in a natural fashion.We discuss practical relevance of commonly imposed axioms on functional depths and investigate which of them are satisfied by the notion of depth we promote here. Estimation and computational issues are also adressed and various numerical experiments provide empirical evidence of the relevance of the approach proposed.} }
Endnote
%0 Conference Paper %T The Area of the Convex Hull of Sampled Curves: a Robust Functional Statistical Depth measure %A Guillaume Staerman %A Pavlo Mozharovskyi %A Stéphan Clémen\con %B Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2020 %E Silvia Chiappa %E Roberto Calandra %F pmlr-v108-staerman20a %I PMLR %P 570--579 %U https://proceedings.mlr.press/v108/staerman20a.html %V 108 %X With the ubiquity of sensors in the IoT era, statistical observations are becoming increasingly available in the form of massive (multivariate) time-series. Formulated as unsupervised anomaly detection tasks, an abundance of applications like aviation safety management, the health monitoring of complex infrastructures or fraud detection can now rely on such functional data, acquired and stored with an ever finer granularity. The concept of \textit{statistical depth}, which reflects centrality of an arbitrary observation w.r.t. a statistical population may play a crucial role in this regard, anomalies corresponding to observations with ’small’ depth. Supported by sound theoretical and computational developments in the recent decades, it has proven to be extremely useful, in particular in functional spaces. However, most approaches documented in the literature consist in evaluating independently the centrality of each point forming the time series and consequently exhibit a certain insensitivity to possible shape changes.In this paper, we propose a novel notion of functional depth based on the area of the convex hull of sampled curves, capturing gradual departures from centrality, even beyond the envelope of the data, in a natural fashion.We discuss practical relevance of commonly imposed axioms on functional depths and investigate which of them are satisfied by the notion of depth we promote here. Estimation and computational issues are also adressed and various numerical experiments provide empirical evidence of the relevance of the approach proposed.
APA
Staerman, G., Mozharovskyi, P. & Clémen\con, S.. (2020). The Area of the Convex Hull of Sampled Curves: a Robust Functional Statistical Depth measure. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 108:570-579 Available from https://proceedings.mlr.press/v108/staerman20a.html.

Related Material