Three perspectives on complexity: entropy, compression, subsymmetry
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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 relatively a new complexity measure. In this paper, we also propose a novel normalized complexity 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 Hénon map, (C) analyzing the complexity of stochastic time series generated from 2-state Markov chains, and (D) 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.
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- 4.N. Nagaraj, K.R. Sahasranand, Neural signal multiplexing via compressed sensing, in IEEE Int. Conf. on Signal Processing Communications (IEEE SPCOM) 2016, IISc, Bengaluru (2016), doi: 10.1109/SPCOM.2016.7746641
- 7.G.T. Toussaint, N.S. Onea, Q.H. Vuong, Measuring the complexity of two-dimensional binary patterns – sub-symmetries versus papentin complexity, in 2015 14th IAPR International Conference on Machine Vision Applications (MVA) (2015), pp. 480–483 Google Scholar
- 11.T.M. Cover, J.A. Thomas, Elements of Information Theory (John Wiley & Sons, 2012) Google Scholar
- 12.N. Nagaraj, K. Balasubramanian, in Handbook of Research on Applied Cybernetics and Systems Science (IGI Global, 2017), pp. 301–334 Google Scholar
- 13.M. Li, P. Vitányi, An Introduction to Kolmogorov Complexity and Its Applications (Springer Science & Business Media, 2009) Google Scholar
- 21.M. Virmani, N. Nagaraj, A compression-complexity measure of integrated information, arXiv:1608.08450v2 (2016)
- 26.K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos (Springer, 1997) Google Scholar
- 30.S.-T. Pan, Y.-H. Wu, Y.-L. Kung, H.-C. Chen, Heartbeat recognition from ECG signals using hidden Markov model with adaptive features, in 14th ACIS International Conference on Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed Computing (SNPD) (2013), pp. 586–591 Google Scholar
- 31.M.S. Waterman, Mathematical methods for DNA sequences (CRC Press Inc., 1989) Google Scholar
- 35.A. Varga, R. Moore, Hidden Markov model decomposition of speech and noise, in International Conference on Acoustics, Speech and Signal Processing (ICASSP) (1990), pp. 845–848 Google Scholar
- 40.G.A. Fink, Markov models for pattern recognition: from theory to applications (Springer Science & Business Media, 2014) Google Scholar
- 44.M. Svoboda, L. Lukas, Application of Markov chain analysis to trend prediction of stock indices, in Proceedings of 30th International Conference Mathematical Methodsin Economics (Silesian University, School of Business Administration, Karviná, 2012), pp. 848–853 Google Scholar