Shannon Entropy has been extensively used for characterizing complexity of time series arising from chaotic dynamical systems and stochastic processes such as Markov chains. However, for short and noisy time series, Shannon entropy performs poorly. Complexity measures which are based on lossless compression algorithms are a good substitute in such scenarios. We evaluate the performance of two such Compression-Complexity Measures namely Lempel-Ziv complexity (LZ) and Effort-To-Compress (ET C) on short time series from chaotic dynamical systems in the presence of noise. Both LZ and ET C outperform Shannon entropy (H) in accurately characterizing the dynamical complexity of such systems. For very short binary sequences (which arise in neuroscience applications), ET C has higher number of distinct complexity values than LZ and H, thus enabling a finer resolution. For two-state ergodic Markov chains, we empirically show that ET C converges to a steady state value faster than LZ. Compression-Complexity Measures are promising for applications which involve short and noisy time series.
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 (ET C complexity) and degree-of-order (Subsymmetry or SubSym). While Shannon entropy and LZ are very popular and widely used, ET C 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 Hénon 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.1 Most naturally occurring systems are a hybrid since stochastic noise is inevitable.
As we age, our hearts undergo changes that result in a reduction in complexity of physiological interactions between different control mechanisms. This results in a potential risk of cardiovascular diseases which are the number one cause of death globally. Since cardiac signals are nonstationary and nonlinear in nature, complexity measures are better suited to handle such data. In this study, three complexity measures are used, namely Lempel–Ziv complexity (LZ), Sample Entropy (SampEn) and Effort-To-Compress (ETC). We determined the minimum length of RR tachogram required for characterizing complexity of healthy young and healthy old hearts. All the three measures indicated significantly lower complexity values for older subjects than younger ones. However, the minimum length of heart-beat interval data needed differs for the three measures, with LZ and ETC needing as low as 10 samples, whereas SampEn requires at least 80 samples. Our study indicates that complexity measures such as LZ and ETC are good candidates for the analysis of cardiovascular dynamics since they are able to work with very short RR tachograms.
Progressive loss of heart rate variability (HRV) and complexity are associated with increased risk of mortality in patients with cardiovascular disease and are a candidate marker for patients at risk of sudden cardiac death. HRV is influenced by the cardiac autonomic nervous system (ANS), although it is unclear which arm of the ANS (sympathetic or parasympathetic) needs to be perturbed to increase the complexity of HRV. In this case–control study, we have analyzed the relation between modulation of vagus nerve stimulation (VNS) and changes in complexity of HRV as a function of states of vigilance. We hypothesize that VNS – being a preferential activator of the parasympathetic system – will decrease the heart rate (HR) and increase the complexity of HRV maximum during sleep. The electrocardiogram (EKG) obtained from a 37-year-old, right-handed male with known intractable partial epilepsy and left therapeutic VNS was analyzed during wakefulness and sleep with VNS ON and OFF states. Age-matched control EKG was obtained from five participants (three with intractable epilepsy and two without epilepsy) that had no VNS implant. The study demonstrated the following: (1) VNS increased the complexity of HRV during sleep and decreased it during wakefulness. (2) An increase in parasympathetic tone is associated with increased complexity of HRV even in the presence of decreased HR. These results need to be replicated in a larger cohort before developing patterned stimulation using VNS to stabilize cardiac dysautonomia and prevent fatal arrhythmias.
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