Distinguishing between classes of time series sampled from dynamic systems is a common challenge in systems and control engineering, for example in the context of health monitoring, fault detection, and quality control. The challenge is increased when no underlying model of a system is known, measurement noise is present, and long signals need to be interpreted. In this paper we address these issues with a new non parametric classifier based on topological signatures. Our model learns classes as weighted kernel density estimates (KDEs) over persistent homology diagrams and predicts new trajectory labels using Sinkhorn divergences on the space of diagram KDEs to quantify proximity. We show that this approach accurately discriminates between states of chaotic systems that are close in parameter space, and its performance is robust to noise.
Time series have previously been analysed using TDA, largely via PH of point clouds constructed using delay embeddings @cite_5 @cite_42 @cite_15 @cite_37 . However delay embeddings require heuristic estimation of the embedding dimension and delay size to be computed beforehand, increase the influence of noise, and lead to a rapid increase in the complexity of computing PH of filtrations on the point cloud, requiring subsampling techniques such as the witness complex @cite_33 @cite_14 @cite_6 . One previous study avoids embeddings but relies on a coarse grained statistic derived from PDs to characterise time series, the persistent entropy @cite_3 . Our method is similar to this since we use a filtration on the time series directly, however we use a metric on the space of persistence diagrams directly rather than on the space of persistent entropy histograms.
@cite_37 @cite_14 @cite_33 @cite_42 @cite_6 @cite_3 @cite_5 @cite_15
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This work introduces a new dataset and framework for the exploration of topological data analysis (TDA) techniques applied to time-series data. We examine the end-toend TDA processing pipeline for persistent homology applied to time-delay embeddings of time series – embeddings that capture the underlying system dynamics from which time series data is acquired. In particular, we consider stability with respect to time series length, the approximation accuracy of sparse filtration methods, and the discriminating ability of persistence diagrams as a feature for learning. We explore these properties across a wide range of time-series datasets spanning multiple domains for single source multi-segment signals as well as multi-source single segment signals. Our analysis and dataset captures the entire TDA processing pipeline and includes time-delay embeddings, persistence diagrams, topological distance measures, as well as kernels for similarity learning and classification tasks for a broad set of time-series data sources. We outline the TDA framework and rationale behind the dataset and provide insights into the role of TDA for time-series analysis as well as opportunities for new work. This paper tackles the problem of computing topological invariants of geometric objects in a robust manner, using only point cloud data sampled from the object. It is now widely recognised that this kind of topological analysis can give qualitative information about data sets which is not readily available by other means. In particular, it can be an aid to visualisation of high dimensional data. Standard simplicial complexes for approximating the topological type of the underlying space (such as Cech, Rips, or a-shape) produce simplicial complexes whose vertex set has the same size as the underlying set of point cloud data. Such constructions are sometimes still tractable, but are wasteful (of computing resources) since the homotopy types of the underlying objects are generally realisable on much smaller vertex sets. We obtain smaller complexes by choosing a set of 'landmark' points from our data set, and then constructing a "witness complex" on this set using ideas motivated by the usual Delaunay complex in Euclidean space. The key idea is that the remaining (non-landmark) data points are used as witnesses to the existence of edges or simplices spanned by combinations of landmark points. Our construction generalises the topology-preserving graphs of Martinetz and Schulten [MS94] in two directions. First, it produces a simplicial complex rather than a graph. Secondly it actually produces a nested family of simplicial complexes, which represent the data at different feature scales, suitable for calculating persistent homology [ELZ00, ZC04]. We find that in addition to the complexes being smaller, they also provide (in a precise sense) a better picture of the homology, with less noise, than the full scale constructions using all the data points. We illustrate the use of these complexes in qualitatively analyzing a data set of 3 × 3 pixel patches studied by David [LPM03]. Topology-based analysis of time-series data from dynamical systems is powerful: it potentially allows for computer-based proofs of the existence of various classes of regular and chaotic invariant sets for high-dimensional dynamics. Standard methods are based on a cubical discretization of the dynamics and use the time series to construct an outer approximation of the underlying dynamical system. The resulting multivalued map can be used to compute the Conley index of isolated invariant sets of cubes. In this paper we introduce a discretization that uses instead a simplicial complex constructed from a witness-landmark relationship. The goal is to obtain a natural discretization that is more tightly connected with the invariant density of the time series itself. The time-ordering of the data also directly leads to a map on this simplicial complex that we call the witness map. We obtain conditions under which this witness map gives an outer approximation of the dynamics and thus can be used to compute the C... We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window embeddings, as seen through the lens of persistent homology. In particular, we show that maximum persistence at the point-cloud level can be used to quantify periodicity at the signal level, prove structural and convergence theorems for the resulting persistence diagrams, and derive estimates for their dependency on window size and embedding dimension. We apply this methodology to quantifying periodicity in synthetic data sets and compare the results with those obtained using state-of-the-art methods in gene expression analysis. We call this new method SW1PerS, which stands for Sliding Windows and 1-Dimensional Persistence Scoring. Topological data analysis (TDA), while abstract, allows a characterization of time-series data obtained from nonlinear and complex dynamical systems. Though it is surprising that such an abstract measure of structure—counting pieces and holes—could be useful for real-world data, TDA lets us compare different systems, and even do membership testing or change-point detection. However, TDA is computationally expensive and involves a number of free parameters. This complexity can be obviated by coarse-graining, using a construct called the witness complex. The parametric dependence gives rise to the concept of persistent homology: how shape changes with scale. Its results allow us to distinguish time-series data from different systems—e.g., the same note played on different musical instruments. Abstract This paper describes a new approach for ascertaining the stability of stochastic dynamical systems in their parameter space by examining their time series using topological data analysis (TDA). We illustrate the approach using a nonlinear delayed model that describes the tool oscillations due to self-excited vibrations in turning. Each time series is generated using the Euler-Maruyama method and a corresponding point cloud is obtained using the Takens embedding. The point cloud can then be analyzed using a tool from TDA known as persistent homology. The results of this study show that the described approach can be used for analyzing datasets of delay dynamical systems generated both from numerical simulation and experimental data. The contributions of this paper include presenting for the first time a topological approach for investigating the stability of a class of nonlinear stochastic delay equations, and introducing a new application of TDA to machining processes. We explore computational topology for clustering.We focus on time series and spatial data.Shape features are extracted based on n-dimensional holes.We run experiments with synthetic and real-world data.Results show improvements when compared to traditional clustering. Topology is the branch of mathematics that studies how objects relate to one another for their qualitative structural properties, such as connectivity and shape. In this paper, we present an approach for data clustering based on topological features computed over the persistence diagram, estimated using the theory of persistent homology. The features indicate topological properties such as Betti numbers, i.e., the number of n-dimensional holes in the discretized data space. The main contribution of our approach is enabling the clustering of time series that have similar recurrent behavior characterized by their attractors in phase space and spatial data that have similar scale-invariant spatial distributions, as traditional clustering techniques ignore that information as they rely on point-to-point dissimilarity measures such as Euclidean distance or elastic measures. We present experiments that confirm the usefulness of our approach with time series and spatial data applications in the fields of biology, medicine and ecology.