A Geometric Analysis of Time Series Leading to Information Encoding and a New Entropy Measure

TL;DR

This paper introduces a geometric approach to analyze time series, modeling them as particle trajectories to define a new entropy measure called semantic entropy, which captures information encoding based on shape configurations.

Contribution

It presents a novel geometric framework for time series analysis and introduces a new entropy measure, semantic entropy, based on shape configurations and their frequencies.

Findings

Semantic entropy effectively captures information encoding in time series.

The ratio E/P indicates synchronous behavior in signals like epileptic EEG.

Discrete time series can be represented as strings of 13 symbols based on shape configurations.

Abstract

A time series is uniquely represented by its geometric shape, which also carries information. A time series can be modelled as the trajectory of a particle moving in a force field with one degree of freedom. The force acting on the particle shapes the trajectory of its motion, which is made up of elementary shapes of infinitesimal neighborhoods of points in the trajectory. It has been proved that an infinitesimal neighborhood of a point in a continuous time series can have at least 29 different shapes or configurations. So information can be encoded in it in at least 29 different ways. A 3-point neighborhood (the smallest) in a discrete time series can have precisely 13 different shapes or configurations. In other words, a discrete time series can be expressed as a string of 13 symbols. Across diverse real as well as simulated data sets it has been observed that 6 of them occur more frequently and the remaining 7 occur less frequently. Based on frequency distribution of 13 configurations or 13 different ways of information encoding a novel entropy measure, called semantic entropy (E), has been defined. Following notion of power in Newtonian mechanics of the moving particle whose trajectory is the time series, a notion of information power (P) has been introduced for time series. E/P turned out to be an important indicator of synchronous behaviour of time series as observed in epileptic EEG signals.