Topological data analysis for time series

In this thesis, we develop new methods for time series analysis based on topological data analysis. This work is motivated by the study of physiological signals. These signals usually have a certain structure that can be studied using tools such as persistent homology in an unsupervised and interpretable way. We review the state of the art on the use of persistent homology for time series, and propose three contributions. The first contribution is a non-parametric method for analyzing gait signals measured by inertial units placed on the feet, which we have applied to the study of healthy subjects and patients with multiple sclerosis. This method can provide information on the severity of the disease and its evolution over time in a patient. The second contribution is a clustering algorithm based on a new filtration that incorporates information about the density of each point and thus makes it possible to exclude isolated points. We show that our filtration has a stability property and that the algorithm can perform well on various types of data. The third contribution is an unsupervised anomaly detection algorithm for univariate time series. This method is built from a model of time series composed of a succession of patterns, which allows us to define the anomaly detection problem and to study our method theoretically. We show that it is competitive with the state-of-the art and study the influence of parameters and noise.