Persistent homology: functional summaries of persistence diagrams for time series analysis

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2021, Director: Miquel Crusells i Josep Vives i Santa Eulàlia[en] Topological data analysis (TDA) is a recently emerged field of study , a point of confluence between Algebraic Topology, Statistics and Computation Theory, born to develop a new set of tools capable of extracting qualitative and quantitative information from the data’s underlying geometrical and topological structure. In these notes, we first present the theoretical foundations of the flagship tool of TDA, persistent homology. Later, we provide a framework that allows us to understand homological persistence from a statistical perspective. The latter comprises a set of maps called functional summaries, which map persistence diagrams -a common representation of persistence homology- to L-Lipschitz functions, a more convenient representation for data analysis. We present persistence landscapes, silhouettes, and a new functional summary candidate based on persistence entropy under this framework. From this point on, we will focus our work on TDA for time series [31], and more specifically, financial time series [35][32][17]. First, we present the time-delay embedding and the sliding window approach, two methodologies that will allow us to transform time series into sequences of point clouds, which is essential for applying homological persistence tools. Subsequently, mainly motivated by [1], an article that we could consider an extension of [17], we prove that the results reflected on the dependency relationship between persistence landscapes functional norms and variance-covariance for multivariate time series embedded via the sliding window method are valid for time series understanding them as a realization of a weakly stationary stochastic process, and assuming that the point clouds are obtained by means of time delay embedding. Furthermore, we assert that the validity of the results provided in [1] and the validity of our adaptation hold for silhouettes. Moreover, we provide two distinct Python frameworks for classical and topological data analysis, which serves us to validate results. First, we provide a collection of tools to simulate time series from standard probability distributions and time series models such as AR(1), ARCH(1), GARCH(1,1). We present the necessary tools to embed, plot, and compute, topological summaries such persistence diagrams, persistence landscapes (and their corresponding L p norms), silhouettes (and their corresponding L p norms), persistence entropy, ES and NES functions. Analogous work is done regarding statistical and topological data analysis of stock index data