On Quantization Errors in Approximate and Sample Entropy

Approximate and sample entropies are acclaimed tools for quantifying the regularity and unpredictability of time series. This paper analyses the causes of their inconsistencies. It is shown that the major problem is a coarse quantization of matching probabilities, causing a large error between their estimated and true values. Error distribution is symmetric, so in sample entropy, where matching probabilities are directly summed, errors cancel each other. In approximate entropy, errors are accumulating, as sums involve logarithms of matching probabilities. Increasing the time series length increases the number of quantization levels, and errors in entropy disappear both in approximate and in sample entropies. The distribution of time series also affects the errors. If it is asymmetric, the matching probabilities are asymmetric as well, so the matching probability errors cease to be mutually canceled and cause a persistent entropy error. Despite the accepted opinion, the influence of self-matching is marginal as it just shifts the error distribution along the error axis by the matching probability quant. Artificial lengthening the time series by interpolation, on the other hand, induces large error as interpolated samples are statistically dependent and destroy the level of unpredictability that is inherent to the original signal.

The Effect of Cognitive Task on Postural Control Dynamic Regularity of Athletes With Chronic Ankle Instability When Standing on an Unstable Surface

Background and Objectives: Postural control disorder is a common complication in patients with Chronic Ankle Instability (CAI). The present study aimed to investigate the effect of dual cognitive task on postural control behavior with regard to the Center of Pressure (CoP) signal regularity while standing on an unstable surface in athletes with CAI. Methods: In the present study, 58 men participated in two groups of healthy and patients with CAI. The CoP signal was examined in 4 different unstable states on the wobble board located at the center of the force plate. The regularity of the signals recorded from the force plate was investigated using sample entropy in two directions: anterior-posterior and medial-lateral. Results: In both groups, there was a significant difference in CoP’s sample entropy signal when performing a cognitive task with a postural task (P<0.001). There was a significant difference between the two groups in the cognitive task and the single task in the anteroposterior direction while standing on two legs. Conclusion: During dual tasks, the patients with CAI have a more dynamic regularity in the CoP signal than their normal counterparts. In the dual-task condition, more irregularities are observed in the CoP signal of healthy individuals. In unstable conditions, patients with CAI decrease the adaptability of postural control behavior with increasing CoP signal regularity.

Short-Term Load Probabilistic Forecasting Based on Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise Reconstruction and Salp Swarm Algorithm

Short-term load forecasting is an important part of load forecasting, which is of great significance to the optimal power flow and power supply guarantee of the power system. In this paper, we proposed the load series reconstruction method combined improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) with sample entropy (SE). The load series is decomposed by ICEEMDAN and is reconstructed into a trend component, periodic component, and random component by comparing with the sample entropy of the original series. Extreme learning machine optimized by salp swarm algorithm (SSA-ELM) is used to predict respectively, and the final prediction value is obtained by superposition of the prediction results of the three components. Then, the prediction error of the training set is divided into four load intervals according to the predicted value, and the kernel probability density is estimated to obtain the error distribution of the training set. Combining the predicted value of the prediction set with the error distribution of the corresponding load interval, the prediction load interval can be obtained. The prediction method is verified by taking the hourly load data of a region in Denmark in 2019 as an example. The final experimental results show that the proposed method has a high prediction accuracy for short-term load forecasting.

Nonlinear signal processing, spectral, and fractal based stridor auscultation: A machine learning approach

The work reported in the paper analyses the adventitious stridor breath sound (ST) and the normal bronchial breath sound (BR) using spectral, fractal, and nonlinear signal processing methods. The sixty breath sound signals are subjected to power spectral density (PSD) and wavelet analyses to understand the temporal evolution of the frequency components. The energy envelope of the PSD plot of ST shows three peaks labelled as A (256 Hz), B (369 Hz), and C (540 Hz), of which A alone is present in BR at 265 Hz. The appearance of B and C in the PSD plot of ST is due to the obstructions in the trachea and upper airways caused by lesions. The phase portrait analysis of the time series data of ST and BR gives information about the randomness and the sample entropy of the dynamical system. The study reveals that the fractal dimension and sample entropy values are higher for BR, which may be due to the musical ordered behaviour of ST. The machine learning techniques based on the features extracted from the PSD data and phase portrait parameters offer good predictability, besides the classification of BR and ST, and thereby revealing its potential in pulmonary auscultation.

Variational Embedding Multiscale Sample Entropy: A Tool for Complexity Analysis of Multichannel Systems

Entropy-based methods have received considerable attention in the quantification of structural complexity of real-world systems. Among numerous empirical entropy algorithms, conditional entropy-based methods such as sample entropy, which are associated with amplitude distance calculation, are quite intuitive to interpret but require excessive data lengths for meaningful evaluation at large scales. To address this issue, we propose the variational embedding multiscale sample entropy (veMSE) method and conclusively demonstrate its ability to operate robustly, even with several times shorter data than the existing conditional entropy-based methods. The analysis reveals that veMSE also exhibits other desirable properties, such as the robustness to the variation in embedding dimension and noise resilience. For rigor, unlike the existing multivariate methods, the proposed veMSE assigns a different embedding dimension to every data channel, which makes its operation independent of channel permutation. The veMSE is tested on both stimulated and real world signals, and its performance is evaluated against the existing multivariate multiscale sample entropy methods. The proposed veMSE is also shown to exhibit computational advantages over the existing amplitude distance-based entropy methods.

A Chaotic Quadratic Bistable Hyperjerk System with Hidden Attractors and a Wide Range of Sample Entropy: Impulsive Stabilization

Hidden attractors generated by the interactions of dynamical variables may have no equilibrium point in their basin of attraction. They have grabbed the attention of mathematicians who investigate strange attractors. Besides, quadratic hyperjerk systems are under the magnifying glass of these mathematicians because of their elegant structures. In this paper, a quadratic hyperjerk system is introduced that can generate chaotic attractors. The dynamical behaviors of the oscillator are investigated by plotting their Lyapunov exponents and bifurcation diagrams. The multistability of the hyperjerk system is investigated using the basin of attraction. It is revealed that the system is bistable when one of its attractors is hidden. Besides, the complexity of the systems’ attractors is investigated using sample entropy as the complexity feature. It is revealed how changing the parameters can affect the complexity of the systems’ time series. In addition, one of the hyperjerk system equilibrium points is stabilized using impulsive control. All real initial conditions become the equilibrium points of the basin of attraction using the stabilizing method.

Role of Heart Rate Variability in Association Between Glomerular Hyperfiltration and All‐Cause Mortality

Background Glomerular hyperfiltration (GHF) is paradoxically associated with increased cardiovascular events in healthy individuals, but the pathogenesis remains unclear. We aim to investigate whether GHF is associated with mortality and whether decreased heart rate variability (HRV) is associated with GHF. Methods and Results We retrospectively analyzed 1615 participants (aged 66.1±17.3 years, 61.9% men) without prior cardiovascular events. The glomerular filtration rate was estimated using the Chronic Kidney Disease Epidemiology Collaboration equation. GHF was defined as glomerular filtration rate >the 95th percentile after stratification for age and sex, whereas normal filtration was defined as the 25th to 75th percentiles. HRV indexes, including time domain, frequency domain, and sample entropy, were measured using 24‐hour ambulatory electrocardiography. Clinical outcomes were defined as all‐cause mortality at 2 years. During a mean follow‐up of 16.5±8.2 months, there were 117 deaths (7.2%). GHF was associated with a higher risk of death (hazard ratio and 95% CIs, 1.97 [1.15–3.37]). Reduced HRV indexes, including time domain, frequency domain, and sample entropy (odds ratio and 95% CIs, 0.79 [0.70–0.89]) were all independently associated with the presence of GHF after accounting for age, sex, mean heart rate, morbidities, and medications. In subgroup analysis, reduced HRV was more predictive of GHF in the young than the elderly. Mediation analysis revealed a significant mediation effect between HRV and GHF in addition to their respective detrimental effects on survival. Conclusions Reduced HRV was independently associated with the presence of GHF. Autonomic dysfunction may be involved in the pathogenesis of adverse outcomes of GHF in individuals without prior cardiovascular events.

A Chaotification model based on sine and cosecant functions for enhancing chaos

Chaotic maps with higher chaotic complexity are urgently needed in many application scenarios. This paper proposes a chaotification model based on sine and cosecant functions (CMSC) to improve the dynamic properties of existing chaotic maps. CMSC can generate a new map with higher chaotic complexity by using the existing one-dimensional (1D) chaotic map as a seed map. To discuss the performance of CMSC, the chaos properties of CMSC are analyzed based on the mathematical definition of the Lyapunov exponent (LE). Then, three new maps are generated by applying three classical 1D chaotic maps to CMSC respectively, and the dynamic behaviors of the new maps are analyzed in terms of fixed point, bifurcation diagram, sample entropy (SE), etc. The results of the analysis demonstrate that the new maps have a larger chaotic region and excellent chaotic characteristics.