Three Perspectives on Complexity - Entropy, Compression, Subsymmetry

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'enon 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.