Three Perspectives on Complexity $-$ Entropy, Compression, Subsymmetry

TL;DR

This paper compares different complexity measures—entropy, compression, and subsymmetry—in analyzing sequences and time series, introducing a new normalized subsymmetry measure and evaluating their effectiveness across various tasks.

Contribution

It introduces a novel normalized subsymmetry complexity measure and compares multiple complexity perspectives on sequences and time series.

Findings

Each complexity measure has unique advantages and overlaps.

The new subsymmetry measure effectively characterizes sequence complexity.

Different measures perform variably across tasks like sequence analysis and chaos distinction.

Abstract

There is no single universally accepted definition of "Complexity". There are several perspectives on complexity and what constitutes complex behaviour or complex systems, as opposed to regular, predictable behaviour and simple systems. In this paper, we explore the following perspectives on complexity: "effort-to-describe" (Shannon entropy $H$, Lempel-Ziv complexity $LZ$), "effort-to-compress" ($ETC$ complexity) and "degree-of-order" (Subsymmetry or $SubSym$). While Shannon entropy and $LZ$ are very popular and widely used, $ETC$ is a recently proposed measure for time series. In this paper, we also propose a novel normalized measure $SubSym$ based on the existing idea of counting the number of subsymmetries or palindromes within a sequence. We compare the performance of these complexity measures on the following tasks: a) characterizing complexity of short binary sequences of lengths 4 to 16, b) distinguishing periodic and chaotic time series from 1D logistic map and 2D H\'{e}non map, and c) distinguishing between tonic and irregular spiking patterns generated from the "Adaptive exponential integrate-and-fire" neuron model. Our study reveals that each perspective has its own advantages and uniqueness while also having an overlap with each other.