Measuring dynamical uncertainty with Revealed Dynamics Markov Models

Abstract

Concepts and measures of time series uncertainty and complexity have been applied across domains for behavior classification, risk assessments, and event detection/prediction. This paper contributes three new measures based on an encoding of the series' phase space into a descriptive Markov model. Here we describe constructing this kind of “Revealed Dynamics Markov Model” (RDMM) and using it to calculate the three uncertainty measures: entropy, uniformity, and effective edge density. We compare our approach to existing methods such as approximate entropy (ApEn) and permutation entropy using simulated and empirical time series with known uncertainty features. While previous measures capture local noise or the regularity of short patterns, our measures track holistic features of time series dynamics that also satisfy criteria as being approximate measures of information generation (Kolmogorov entropy). As such, we show that they can distinguish dynamical patterns inaccessible to previous measures and more accurately reflect their relative complexity. We also discuss the benefits and limitations of the Markov model encoding as well as requirements on the sample size.