A Short Survey of Topological Data Analysis in Time Series and Systems Analysis

Abstract

Topological Data Analysis (TDA) is the collection of mathematical tools that capture the structure of shapes in data. Despite computational topology and computational geometry, the utilization of TDA in time series and signal processing is relatively new. In some recent contributions, TDA has been utilized as an alternative to the conventional signal processing methods. Specifically, TDA is been considered to deal with noisy signals and time series. In these applications, TDA is used to find the shapes in data as the main properties, while the other properties are assumed much less informative. In this paper, we will review recent developments and contributions where topological data analysis especially persistent homology has been applied to time series analysis, dynamical systems and signal processing. We will cover problem statements such as stability determination, risk analysis, systems behaviour, and predicting critical transitions in financial markets.

References in corpus (9)

• Statistical topological data analysis using persistence landscapes
• Topological strata of weighted complex networks
• Topological Data Analysis of Biological Aggregation Models
• Persistent homology of time-dependent functional networks constructed from coupled time series
• Correlation and Network Topologies in Global and Local Stock Indices
• Principal Component Analysis of Persistent Homology Rank Functions with case studies of Spatial Point Patterns, Sphere Packing and Colloids
• On Characterizing the Capacity of Neural Networks using Algebraic Topology
• Evasion Paths in Mobile Sensor Networks
• Exploring the Topology of Dynamical Reconstructions

Cited by in corpus (7)

• Time Series Featurization via Topological Data Analysis
• Topological Persistence Guided Knowledge Distillation for Wearable Sensor Data
• Topological Data Analysis in Text Classification: Extracting Features with Additive Information
• A Novel Method of Extracting Topological Features from Word Embeddings
• Topological Data Analysis for Directed Dependence Networks of Multivariate Time Series Data
• Modeling and Simulating Dependence in Networks Using Topological Data Analysis
• Role of Mixup in Topological Persistence Based Knowledge Distillation for Wearable Sensor Data