Complexity and Entropy in Physiological Signals (CEPS): Resonance Breathing Rate Assessed Using Measures of Fractal Dimension, Heart Rate Asymmetry and Permutation Entropy

Physiological signal analysis and open science using the Julia language and associated software

In this mini review, we propose the use of the Julia programming language and its software as a strong candidate for reproducible, efficient, and sustainable physiological signal analysis. First, we highlight available software and Julia communities that provide top-of-the-class algorithms for all aspects of physiological signal processing despite the language's relatively young age. Julia can significantly accelerate both research and software development due to its high-level interactive language and high-performance code generation. It is also particularly suited for open and reproducible science. Openness is supported and welcomed because the overwhelming majority of Julia software programs are open source and developed openly on public platforms, primarily through individual contributions. Such an environment increases the likelihood that an individual not (originally) associated with a software program would still be willing to contribute their code, further promoting code sharing and reuse. On the other hand, Julia's exceptionally strong package manager and surrounding ecosystem make it easy to create self-contained, reproducible projects that can be instantly installed and run, irrespective of processor architecture or operating system.

Symmetry in Signals: A New Insight

Symmetry is a fundamental property of many natural systems, which is observable through signals. In most out-of-equilibrium complex dynamic systems, the observed signals are asymmetric. However, for certain operating modes, some systems have demonstrated a resurgence of symmetry in their signals. Research has naturally focused on examining time invariance to quantify this symmetry. Measures based on the statistical and harmonic properties of signals have been proposed, but most of them focused on harmonic distortion without explicitly measuring symmetry. This paper introduces a new mathematical framework based on group theory for analyzing signal symmetry beyond time invariance. It presents new indicators to evaluate different types of symmetry in non-stochastic symmetric signals. Both periodic and non-periodic symmetric signals are analyzed to formalize the problem. The study raises critical questions about the completeness of symmetry in signals and proposes a new classification for periodic and non-periodic signals that goes beyond the traditional classification based on Fourier coefficients. Furthermore, new measures such as "symmetrometry" and "distorsymmetry" are introduced to quantify symmetry. These measures outperform traditional indicators like Total Harmonic Distortion (THD) and provide a more accurate measurement of symmetry in complex signals from applications where duty cycle plays a major role.

Dispersion complexity-entropy curves: An effective method to characterize the structures of nonlinear time series

The complexity-entropy curve (CEC) is a valuable tool for characterizing the structure of time series and finds broad application across various research fields. Despite its widespread usage, the original permutation complexity-entropy curve (PCEC), which is founded on permutation entropy (PE), exhibits a notable limitation: its inability to take the means and amplitudes of time series into considerations. This oversight can lead to inaccuracies in differentiating time series. In this paper, drawing inspiration from dispersion entropy (DE), we propose the dispersion complexity-entropy curve (DCEC) to enhance the capability of CEC in uncovering the concealed structures within nonlinear time series. Our approach initiates with simulated data including the logistic map, color noises, and various chaotic systems. The outcomes of our simulated experiments consistently showcase the effectiveness of DCEC in distinguishing nonlinear time series with diverse characteristics. Furthermore, we extend the application of DCEC to real-world data, thereby asserting its practical utility. A novel approach is proposed, wherein DCEC-based feature extraction is combined with multivariate support vector machine for the diagnosis of various types of bearing faults. This combination achieved a high accuracy rate in our experiments. Additionally, we employ DCEC to assess stock indices from different countries and periods, thereby facilitating an analysis of the complexity inherent in financial markets. Our findings reveal significant insights into the dynamic regularities and distinct structures of these indices, offering a novel perspective for analyzing financial time series. Collectively, these applications underscore the potential of DCEC as an effective tool for the nonlinear time series analysis.

Estimating fractal dimensions: A comparative review and open source implementations

The fractal dimension is a central quantity in nonlinear dynamics and can be estimated via several different numerical techniques. In this review paper, we present a self-contained and comprehensive introduction to the fractal dimension. We collect and present various numerical estimators and focus on the three most promising ones: generalized entropy, correlation sum, and extreme value theory. We then perform an extensive quantitative evaluation of these estimators, comparing their performance and precision using different datasets and comparing the impact of features like length, noise, embedding dimension, and falsify-ability, among many others. Our analysis shows that for synthetic noiseless data, the correlation sum is the best estimator with extreme value theory following closely. For real experimental data, we found the correlation sum to be more strongly affected by noise vs the entropy and extreme value theory. The recent extreme value theory estimator seems powerful as it has some of the advantages of both alternative methods. However, using four different ways for checking for significance, we found that the method yielded "significant" low-dimensional results for inappropriate data like stock market timeseries. This fact, combined with some ambiguities we found in the literature of the method applications, has implications for both previous and future real-world applications using the extreme value theory approach, as, for example, the argument for small effective dimensionality in the data cannot come from the method itself. All algorithms discussed are implemented as performant and easy to use open source code via the DynamicalSystems.jl library.