Measuring and Locating Zones of Chaos and Irregularity by Application of High Spectral Chaotic Global Variants

Abstract

The protocol followed was analogous to that by Garner and Ling [2014]. High spectral Entropy and high spectral Detrended Fluctuation Analysis have been established as more sensitive to the chaotic responses in forward problems in subjects with type 1 diabetes mellitus (T1DM), chronic obstructive pulmonary disease (COPD), youth obesity and child malnutrition. From 50 linearly increasing starting points, we applied the objective functions (hsCTF1 to hsCTF7) to attain an optimized value, [Formula: see text] which should increase the chaotic response when substituted back into the appropriate model. The mean percentage increase in Kolmogorov–Sinai Entropy and [Formula: see text]-values when the models were compared statistically by Kruskal–Wallis and One-way analysis of variance (ANOVA1) tests were monitored. Normal distributions were considered by Anderson–Darling, Ryan–Joiner and Lilliefors tests. Effect sizes by Cohen’ [Formula: see text] and Hedges [Formula: see text] were computed as sometimes the ANOVA1 and Kruskal–Wallis tests were not discriminative. The optimal statistical results were Duffing (hsCTF1: 29.4%); Brusselator (hsCTF6: 24.7%); Lorenz (hsCTF6: 28.8%). Multivariate analysis was required to determine which of the seven combinations of chaotic globals were most influential. Following Principal Component Analysis (PCA) hsCTF1 and hsCTF3 were the preferred objective functions delivering the highest increases in mean KS-Entropy with most influential PCA. These methods could be advantageous to increase yields and consequently profits in industrial processes such as yeast glycolysis for alcoholic beverages, Systems Biology, inverse and optimization problems, generally.

Monitoring Blood Pressure Variability via Chaotic Global Metrics using Local Field Potential Oscillations

“…We are assessing its BPV via the somewhat sinusoidally oscillating BP signal. This is analogous to the Duffing (Bonatto et al 2008), Brusselator (Osipov and Ponizovskaya 2000) and Lorenz (Jeppesen et al 2015) signals in Garner and Ling (Garner and Ling 2021).…”

“…We computed the chaotic global metrics (Chaos Forward Parameter, CFP1 to CFP7) that assess the chaotic response and irregularities of datasets, as described by Garner and Ling in 2014 (Garner and Ling 2014). These methods were later advanced to investigate high spectral variants and applied them to mathematical inverse problems in 2021 (Garner and Ling 2021). They had already been applied to forward problems (Garner et al 2020a(Garner et al , 2017.…”

Chaotic Global Metric Analysis of Heart Rate Variability Following Six Power Spectral Manipulations in Malnourished Children

“…In that case, spectral entropy and sDFA were calculated via the Welch power spectrum [14]. Later the high spectral alternatives which performed better were established and used in both forward [25,27,29] and mathematical inverse problems [11]. These are computed by implementing the MTM power spectrum throughout [1,25].…”

“…Then, with MTM, the parameters are set as the following: (i) 1 Hz for sampling frequency (ii) time bandwidth for the discrete prolate spheroidal sequences (DPSS) often referred to as Slepian sequences [42] is set at 3; (iii) FFT is the larger of 256 and the next power of two greater than the length of the segment; (iv) Thomson's 'adaptive' nonlinear combination method to combine individual spectral estimates is applied. DPSS is intentionally set at 3; not 5 as with Garner et al [9,11] as these time-series are much shorter.…”

“…These were founded on the Welch power spectrum [10]. Later, their high spectral variants, hsEntropy and hsDFA were formulated and used in mathematical inverse problems by Garner et al in 2021 [11] based on the multi-taper method (MTM) power spectrum [12]. These variants were demonstrated to be more sensitive to fluctuations in chaotic response.…”