PhD defense of Alexandre BOIS

Alexandre BOIS

Topological data analysis for time series

Key words

Topological data analysis. Time series. Persistent homology.

Abstract

In this thesis, we present the development of new methods for time series analysis based on topological data analysis (TDA). This work is motivated by the study of physiological signals in behavioral neurology, and in particular by the study of gait. Indeed, these signals usually have a certain structure (periodicity, repeating patterns) that can be analyzed using tools such as persistent homology in an unsupervised and interpretable way.

After introducing the main mathematical concepts related to time series and topological data analysis which are used in the following chapters, we review the state of the art on the use of persistent homology for time series analysis. We then propose three contributions.

The first contribution is a non-parametric method for analyzing gait signals measured by inertial measurement units (IMUs) placed on the feet, which we have applied to the study of healthy subjects and multiple sclerosis patients. This method, based on sublevel sets of functions and on the bottleneck distance between persistence barcodes, makes it possible to visualize differences between each pair of signals and to deduce information, notably on the severity of the disease or its evolution over time for a given patient. More precisely, we calculate the sublevel filtration of each signal and the corresponding persistence barcode from which the largest bars are removed. We then calculate the bottleneck distance matrix between each pair of barcodes, visualize these distances using the UMAP algorithm, and propose several ways of interpreting the result. In this chapter, we also study a theoretical property of persistent homology of sublevel sets of periodic functions, which motivates the method because it makes the link between the number of observed periods and the multiplicity of bars in the persistence barcode.

The second contribution is a clustering algorithm for points in any metric space, based on a new filtration called NNVR (Nearest Neighbor Vietoris-Rips). This filtration is close to the Vietoris-Rips filtration, but integrates information about the proximity of each point's nearest neighbor. By reading the persistence diagram, we can then exclude isolated points and choose a suitable threshold for hierarchical clustering. We show that our filtration has the same stability property as the Vietoris-Rips filtration with respect to the Gromov-Hausdorff distance, describe the algorithm and show that it can perform well on various types of data, even in its non-parametric version. Finally, we illustrate its usefulness in a pattern detection algorithm applied to gait signals.

The third contribution is an unsupervised anomaly detection algorithm for univariate time series. It uses a delay embedding technique (transformation of time series into point clouds), a filtration based on the notion of distance to a measure (DTM), the extraction of cycles representing persistent homology classes in dimension 1, and a reading of the persistence diagram to define normal cycles. An anomaly score is then defined for each point in the point cloud as its distance from normal cycles, and a score is deduced for each point in the time series. This method is based on a time series model composed of a succession of patterns, which we describe formally at the beginning of this chapter. This model enables us to propose a rigorous definition of an anomaly detection problem and to study our method theoretically. We show that our method is competitive with state-of-the-art anomaly detection methods on various datasets and study the influence of parameters and noise. Finally, we illustrate its use on gait signals.

Direction

• Laurent OUDRE
• Brian TERVIL

Jury

• Frederic CHAZAL Directeur de recherche     INRIA Saclay Examinateur
• Mathilde MOUGEOT Professeur des universités ENS Paris Saclay Examinateur
• Julien TIERNY Directeur de recherche Sorbonne Université Rapporteur
• Elizabeth MUNCH Associate Professor Michigan State University Rapporteur