Time series analysis of non-Euclidean data is highly challenging and crucial for many real-world applications. We address the problem of detecting multiple changes in time series within these complex data spaces. Hadamard spaces, which encompass important data spaces like positive semidefinite matrices, certain Wasserstein spaces, and hyperbolic spaces, provide the right general framework to address this complexity. We propose a computationally efficient two-step iterative optimization algorithm called HOP (Hadamard Optimal Partitioning) that detects changes in the sequence of so-called Fr{é}chet means. Under mild conditions, the proposed method consistently estimates the change point locations. HOP is highly versatile, accommodating structural assumptions such as cyclic patterns and epidemic settings, making it unique in the literature. We validate its performance in synthetic and real-world scenarios, including applications in human gait analysis using EMG data with low SNR and behavioral analysis of animal motion.

Finding patterns in time series is crucial to understanding physical or physiological phenomena monitored with sensors. Convolutional sparse coding (CSC) methods, which approximate signals by a sparse combination of short signal templates (also called atoms), are well-suited for this task. Nevertheless, sparsity results in intractable non-convex optimization problems. This paper introduces an algorithm, based on Multi-path Matching Pursuit, which is novel in convolutional settings, to efficiently and accurately estimate atoms’ localizations in time series. We describe a principled way to improve this greedy procedure by returning several candidate solutions instead of one. Our approach yields better localization and signal reconstruction on simulated data and in a real-world use case, which consists in automatically detect damages, such as cracks and broken wires, in overhead power lines.

We propose a new approach for covariance change point detection applied to graph signals. Specifically, our method draws on the notion of graph stationarity to derive a relevant parameterization of the covariance matrix that can be used in a cost function. This parameterization allows prior graph knowledge to be incorporated into the detection process and reduces the number of coefficients to be estimated. We have experimentally validated this method against relevant baselines, on synthetic and real data, and showed the influence of several parameters. These experiments demonstrated very low computational complexity, improved robustness against certain adverse effects and competitive performance in more general contexts.

Motif Discovery involves identifying recurring patterns and locating their occurrences within a time series without prior knowledge about their shape or location. In practice, Motif Discovery faces several data-related challenges, leading to various definitions of the problem and multiple algorithms addressing these challenges to different extents. However, there has been no systematic evaluation and comparison of these diverse approaches. Consequently, this paper presents a comprehensive literature review covering data-related challenges, motif definitions, and algorithms. We also analyze the strengths and limitations of algorithms carefully chosen to represent the literature diversity. The analysis is structured around key research questions identified from our review. Our experimental findings provide practical guidelines for selecting Motif Discovery algorithms suitable for a given task and suggest directions for future research.

Exploring and comparing non-stationary multivariate time series is an important problem in many domains and real-world applications. In recent work, we introduced dsymb, a symbolic representation that transforms multivariate time series into interpretable symbolic sequences that comes along with a compatible and efficient distance measure to compare the obtained symbolic sequences. We have shown how dsymbcan handle the non-stationarity of multivariate physiological signals, how interpretable the symbolization is, and how suitable the distance measure is compared to Dynamic Time Warping (DTW) variants. We have also empirically shown that the computation time when using dsymbon a clustering time is significantly smaller than with DTW variants (typically 100 times faster). In this demonstration, we present the dsymbplayground, an interactive web-based tool to interpret and compare a large multivariate time series dataset quickly. We showcase the relevance of this tool in several scenarios based on real-world datasets.

Analyzing inter-individual variability of physiological functions is particularly appealing in medical and biological contexts to describe or quantify health conditions. Such analysis can be done by comparing individuals to a reference one with time series as biomedical data. This paper introduces an unsupervised representation learning (URL) algorithm for time series tailored to inter-individual studies. The idea is to represent time series as deformations of a reference time series. The deformations are diffeomorphisms parameterized and learned by our method called TS-LDDMM. Once the deformations and the reference time series are learned, the vector representations of individual time series are given by the parametrization of their corresponding deformation. At the crossroads between URL for time series and shape analysis, the proposed algorithm handles irregularly sampled multivariate time series of variable lengths and provides shape-based representations of temporal data. In this work, we establish a representation theorem for the graph of a time series and derive its consequences on the LDDMM framework. We showcase the advantages of our representation compared to existing methods using synthetic data and real-world examples motivated by biomedical applications.

Motif Discovery consists of finding repeated patterns and locating their occurrences in a time series without prior knowledge about their shape or location. Most state-of-the-art algorithms rely on three core parameters: the number of motifs to discover, the length of the motifs, and a similarity threshold between motif occurrences. Setting these parameters is difficult in practice and often results from a trial-and-error strategy. In this paper, we propose a new algorithm that discovers motifs of variable length given a single motif length and without requiring a similarity threshold. At its core, the algorithm maps a time series onto a graph, summarizes it with persistent homology - a tool from topological data analysis - and identifies the most relevant motifs from the graph summary. We propose two versions of the algorithm, one requiring the number of motifs to discover and another, adaptive, that infers the number of motifs from the graph summary. Empirical evaluation on 9 labeled datasets, including 6 real-world datasets, shows that both algorithm versions significantly outperform state-of-the-art algorithms.

In this article, we propose an automatic procedure to calibrate change point detection algorithms. Our approach expands on the ability of an expert to provide very rough segmentation estimates, called partial annotations, for a few signal examples. Our contribution consists in a supervised strategy to learn a kernel Mahalanobis metric, which, once combined with a detection algorithm, can replicate the expert's segmentation strategy on new signals. Contrary to previous works, our approach is non-parametric, supervised and naturally accommodates partial annotations. Experiments on real-world data show that supervision significantly improves detection performance.