Information-theoretical analysis of statistical measures for multiscale dynamics

TL;DR

This paper explores the theoretical connections between multiscale entropy and Allan variance, showing they share foundational principles and exhibit similar properties in analyzing complex physical systems across multiple time scales.

Contribution

It reveals the information-theoretical relationship between MSE and Allan variance and demonstrates their similar behavior in real-world data, providing a unified understanding of multiscale analysis.

Findings

MSE and Allan variance show similar tendencies in biological and physical data.

The conditions for their consistency are related to specific conditional probabilities.

Artificial data can break the similarity, highlighting the conditions' importance.

Abstract

Multiscale entropy (MSE) has been widely used to examine nonlinear systems involving multiple time scales, such as biological and economic systems. Conversely, Allan variance has been used to evaluate the stability of oscillators, such as clocks and lasers, ranging from short to long time scales. Although these two statistical measures were developed independently for different purposes in different fields in the literature, their interest is to examine multiscale temporal structures of physical phenomena under study. We show that, from an information-theoretical perspective, they share some foundations and exhibit similar tendencies. We experimentally confirmed that similar properties of the MSE and Allan variance can be observed in low-frequency fluctuations (LFF) in chaotic lasers and physiological heartbeat data. Furthermore, we calculated the condition under which this consistency between the MSE and Allan variance exists, which is related to certain conditional probabilities. Heuristically, physical systems in nature including the aforementioned LFF and heartbeat data mostly satisfy this condition, and hence the MSE and Allan variance demonstrate similar properties. As a counterexample, an artificially constructed random sequence is demonstrated, for which the MSE and Allan variance exhibit different trends.