Exploring chaotic time series and phase spaces

Abstract

Technology advances have allowed and inspired the study of data produced along time from applications such as health treatment, biology, sentiment analysis, and entertainment. Those types of data, typically referred to as time series or data streams, have motivated several studies mainly in the area of Machine Learning and Statistics to infer models for performing prediction and classification. However, several studies either employ batchdriven strategies to address temporal data or do not consider chaotic observations, thus missing recurrent patterns and other temporal dependencies especially in real-world data. In that scenario, we consider Dynamical Systems and Chaos Theory tools to improve datastream modeling and forecasting by investigating time-series phase spaces, reconstructed according to Takens' embedding theorem. This theorem relies on two essential embedding parameters, known as embedding dimension and time delay , which are complex to be estimated for real-world scenarios. Such difficulty derives from inconsistencies related to phase space partitioning, computation of probabilities, the curse of dimensionality, and noise. Moreover, an optimal phase space may be represented by attractors with different structures for different systems, which also aggregates to the problem.