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Données fonctionnelles multivariées : extraction de caractéristiques géométriques et apprentissage parcimonieux de la dynamique

Clément Lejeune 1
1 IRIT-SIG - Systèmes d’Informations Généralisées
IRIT - Institut de recherche en informatique de Toulouse
Abstract : A multivariate time series is a time-indexed sequence of multidimensional samples. Such a kind of data appears in many fields since they are the observation of dynamic systems (eg mechanics, biology), which often involve multiple time-dependent variables. The constituting variables of a multivariate time series are often related to each other. This multidimensionality renders the analysis of the phenomenon underlying the data more complex than with univariate time series. In this thesis, a dataset comprises multiple multivariate time series. We interest in the detection of abnormal phenomena, which is commonly referred as emph{outlier} or emph{anomaly} detection. Furthermore, we aim to discover the model of the dynamics underlying a given phenomenon. Such a model can provide indepth knowledge on an abnormal phenomenon. To address these two points, we have made two contributions, wherein both of them we represent a time series as a function over time. Our first contribution cite{lejeuneEDBT,lejeuneKOS} deals with the detection of outliers in a functional data perspective. We observed that, due to atypical relationships between the variables of a multivariate time series, the outlyingness can result in its curve shape. To highlight the shape outlyingness, we proposed to aggregate the variables in several geometric manners and used the output functional representation as input of existing outlier detection algorithms. We have empirically showed that our approach outperforms state-of-the-art. Our second contribution cite{lejeune2021} tackles the data-driven discovery of a deterministic model underlying the dynamics between the variables of a multivariate time series. We focus on the case where this unknown model is a system of ordinary differential equations whose solution is the function representing an observed time series. To discover such a model in closed form, we proposed a penalized multi-task learning algorithm where each task aims at learning a single equation. Since the equations are coupled to each other, our regularizer enforces both sparsity within tasks and similarity between tasks. Contrary to state-of-the-art multi-task regularizers, which are convex, ours is non-convex and thus enables to learn the model with unbiasedness. We have empirically showed, on datasets simulated from known systems of differential equations, that learning in a multi-task way with nonconvex sparsity outperforms state-of-the-art approaches.
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Submitted on : Wednesday, October 13, 2021 - 7:38:37 PM
Last modification on : Tuesday, October 19, 2021 - 2:23:38 PM

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Clément Lejeune. Données fonctionnelles multivariées : extraction de caractéristiques géométriques et apprentissage parcimonieux de la dynamique. Artificial Intelligence [cs.AI]. Université Toulouse 1 Capitole, 2021. English. ⟨tel-03376966⟩

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