A locally optimal algorithm for estimating a generating partition from an observed time series

Abstract

Estimation of a generating partition is critical for symbolization of measurements from discrete-time dynamical systems, where a sequence of symbols from a (finite-cardinality) alphabet uniquely specifies the underlying time series. Such symbolization is useful for computing measures (e.g., Kolmogorov-Sinai entropy) to characterize the (possibly unknown) dynamical system. It is also useful for time series classification and anomaly detection. Previous work attemps to minimize a clustering objective function that measures discrepancy between a set of reconstruction values and the points from the time series. Unfortunately, the resulting algorithm is non-convergent, with no guarantee of finding even locally optimal solutions. The problem is a heuristic “nearest neighbor” symbol assignment step. Alternatively, we introduce a new, locally optimal algorithm. We apply iterative “nearest neighbor” symbol assignments with guaranteed discrepancy descent, by which joint, locally optimal symbolization of the time series is achieved. While some approaches use vector quantization to partition the state space, our approach only ensures a partition in the space consisting of the entire time series (effectively, clustering in an infinite-dimensional space). Our approach also amounts to a novel type of sliding block lossy source coding. We demonstrate improvement, with respect to several measures, over a popular method used in the literature.