Detecting nonlinearity in short and noisy time series using the permutation entropy

Some recent papers were concerned with applicability of the Bayesian (statistical) approach to reconstruction of dynamic systems (DS) from experimental data. A significant merit of the approach is its universality. But, being correct in terms of meeting conditions of the underlying theorem, the Bayesian approach to reconstruction of DS is hard to realize in the most interesting case of noisy chaotic time series (TS). In this work we consider a modification of the Bayesian approach that can be used for reconstruction of DS from noisy TS. We demonstrate efficiency of the modified approach for solution of two types of problems: (1) finding values of parameters of a known DS by noisy TS; (2) classification of modes of behavior of such a DS by short TS with pronounced noise.

Heart rate variability (HRV) contains important information about the modulation of the cardiovascular system. Various methods of nonlinear dynamics (e.g., estimating Lyapunov exponents) and complexity measures (e.g., correlation dimension or entropies) have been applied to HRV analysis. Permutation entropy, which was proposed recently, has been widely used in many fields due to its conceptual and computational simplicity. It maps a time series onto a symbolic sequence of permutation ranks. The original permutation entropy assumes the time series under study has a continuous distribution, thus equal values are rare and can be ignored by ranking them according to their order of emergence, or broken by adding small random perturbations to ensure every symbol in a sequence is different. However, when the observed time series is digitized with lower resolution leading to a greater number of equal values, or the equalities represent certain characteristic sequential patterns of the system, it may not be rational to simply ignore or break them. In the present paper, a modified permutation entropy is proposed that, by mapping the equal value onto the same symbol (rank), allows for a more accurate characterization of system states. The application of the modified permutation entropy to the analysis of HRV is investigated using clinically collected data. Results show that modified permutation entropy can greatly improve the ability to distinguish the HRV signals under different physiological and pathological conditions. It can characterize the complexity of HRV more effectively than the original permutation entropy.

This paper deals with the distinction between white noise and deterministic chaos in multivariate noisy time series. Our method is combinatorial in the sense that it is based on the properties of topological permutation entropy, and it becomes especially interesting when the noise is so high that the standard denoising techniques fail, so a detection of determinism is the most one can hope for. It proceeds by i) counting the number of the so-called ordinal patterns in independent samples of length L from the data sequence and ii) performing a χ2 test based on the results of i), the null hypothesis being that the data are white noise. Holds the null hypothesis, so should all possible ordinal patterns of a given length be visible and evenly distributed over sufficiently many samples, contrarily to what happens in the case of noisy deterministic data. We present numerical evidence in two dimensions for the efficiency of this method. A brief comparison with two common tests for independence, namely, the calculation of the autocorrelation function and the BDS algorithm, is also performed.

We present a hybrid symplectic geometry and central tendency measure (CTM) method for detection of determinism in noisy time series. CTM is effective for detecting determinism in short time series and has been applied in many areas of nonlinear analysis. However, its performance significantly degrades in the presence of strong noise. In order to circumvent this difficulty, we propose to use symplectic principal component analysis (SPCA), a new chaotic signal de-noising method, as the first step to recover the system dynamics. CTM is then applied to determine whether the time series arises from a stochastic process or has a deterministic component. Results from numerical experiments, ranging from six benchmark deterministic models to 1/f noise, suggest that the hybrid method can significantly improve detection of determinism in noisy time series by about 20 dB when the data are contaminated by Gaussian noise. Furthermore, we apply our algorithm to study the mechanomyographic (MMG) signals arising from contraction of human skeletal muscle. Results obtained from the hybrid symplectic principal component analysis and central tendency measure demonstrate that the skeletal muscle motor unit dynamics can indeed be deterministic, in agreement with previous studies. However, the conventional CTM method was not able to definitely detect the underlying deterministic dynamics. This result on MMG signal analysis is helpful in understanding neuromuscular control mechanisms and developing MMG-based engineering control applications.

Spontaneous EEG of 21 healthy human subjects obtained by standard procedure of recording is analysed using non-linear prediction methods to check whether the signals were generated by a non-linear dynamics process or by a linear stochastic process. The test for non-linearity is performed by surrogate data method with non-linear prediction error as the test statistic. The null hypothesis that EEG signal (in rest, with eyes closed) is generated by linear stochastic process can be rejected in 17 cases (5%) out of the 336 (21 subjects, 16 channels) studied epochs. However, most of these rejections concern 3 subjects. The 88% of rejections of the null hypothesis concern frontal channels. The null hypothesis is not rejected for epochs recorded with eyes open and during photostimulation.

Particle swarm optimization (PSO) can be interpreted physically as a particular discretization of a stochastic damped mass-spring system. Knowledge of this analogy has been crucial to derive the PSO continuous model and to introduce different PSO family members including the generalized PSO (GPSO) algorithm, which is the generalization of PSO for any time discretization step. In this paper, we present the stochastic analysis of the linear continuous and generalized PSO models for the case of a stochastic center of attraction. Analysis of the GPSO second order trajectories is performed and clarifies the roles of the PSO parameters and that of the cost function through the algorithm execution: while the PSO parameters mainly control the eigenvalues of the dynamical systems involved, the mean trajectory of the center of attraction and its covariance functions with the trajectories and their derivatives (or the trajectories in the near past) act as forcing terms to update first and second order trajectories. The similarity between the oscillation center dynamics observed for different kinds of benchmark functions might explain the PSO success for a broad range of optimization problems. Finally, a comparison between real simulations and the linear continuous PSO and GPSO models is shown. As expected, the GPSO tends to the continuous PSO when time step approaches zero. Both models account fairly well for the dynamics (first and second order moments) observed in real runs. This analysis constitutes so far the most realistic attempt to better understand and approach the real PSO dynamics from a stochastic point of view.

Nonlinear detection of disordered voice productions from short time series based on a Volterra–Wiener–Korenberg model

A distributed optical fiber acoustic sensing system (DAS) based on phase-sensitive optical time domain reflectometry ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Phi $ </tex-math></inline-formula> -OTDR) can detect the vibration information along an optical fiber and has been widely used for various engineering detections; however, the high-coherence detection light source used in a DAS will cause coherent fading in the fiber, leading to system fading noise and affecting the monitoring of the target vibration events. In order to solve the problem of the fading noise in the DAS, this study considered a method of suppressing the fading noise based on the genetic algorithm (GA) and variational mode decomposition (VMD). Furthermore, the parameter selection problem in the VMD method is improved by introducing the GA. By introducing sample entropy as a fitness function and combining it with the GA, this method adaptively searches for the optimal decomposition parameters by iterating the minimum value of the fitness function according to the type of input signal. Then, it decomposes the noisy DAS data into multiple intrinsic mode functions (IMFs), and the permutation entropy and time-frequency domain information of each component were used to extract the effective IMF components for signal reconstruction to obtain the denoised signal. Experimental results show that compared with empirical mode decomposition (EMD), ensemble EMD (EEMD), and complete EEMD (CEEMD) methods, the proposed method can eliminate the mode aliasing problem, effectively suppress the fading noise of the DAS, improve the signal-to-noise ratio, and reduce the root-mean-square error. It is, thus, possible to more accurately monitor vibration events, improving the practical application value of the DAS.

Battery capacity estimation based on a co-learning framework with few-labeled and noisy data

Optical performance monitoring (OPM) becomes an inviting topic in high speed optical communication networks. In this paper, a novel technique of OPM based on a new elaborated computation approach of singular spectrum analysis (SSA) for time series prediction is presented. Indeed, various optical impairments among chromatic dispersion (CD), polarization mode dispersion (PMD) and amplified spontaneous emission (ASE) noise are a major factors limiting quality of transmission data in the systems with data rates lager than 40 Gbit/s. This technique proposed an independent and simultaneous multi-impairments monitoring, where we used SSA of time series analysis and forecasting. It has proven their usefulness in the temporal analysis of short and noisy time series in several fields, that it is based on the singular value decomposition (SVD). Also, advanced optical modulation formats (100 Gbit/s non-return-to zero dual-polarization quadrature phase shift keying (NRZ-DP-QPSK) and 160 Gbit/s DP-16 quadrature amplitude modulation (DP-16QAM)) offering high spectral efficiencies have been successfully employed by analyzing their asynchronously sampled amplitude. The simulated results proved that our method is efficient on CD, first-order PMD, Q-factor and OSNR monitoring, which enabled large monitoring ranges, the CD in the range of 170-1700 ps/nm.Km and 170-1110 ps/nm.Km for 100 Gbit/s NRZ-DP-QPSK and 160 Gbit/s DP-16QAM respectively, and also the DGD up to 20 ps is monitored. We could accurately monitor the OSNR in the range of 10-40 dB with monitoring error remains less than 1 dB in the presence of large accumulated CD.

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Vibration data from rotating machinery working in different conditions display different properties in spatial and temporal scales. As a result, insights into spatial- and temporal-scale structures of vibration data of rotating machinery are fundamental for describing running conditions of rotating machinery. However, common temporal statistics and typical nonlinear measures have difficulties in describing spatial and temporal scales of data. Recently, statistical linguistic analysis (SLA) has been pioneered in analyzing complex vibration data from rotating machinery. Nonetheless, SLA can examine data in spatial scales but not in temporal scales. To improve SLA, this paper develops symbolic-dynamics entropy for quantifying word-frequency series obtained by SLA. By introducing multiscale analysis to SLA, this paper proposes adaptive multiscale symbolic-dynamics entropy (AMSDE). By AMSDE, spatial and temporal properties of data can be characterized by a set of symbolic-dynamics entropy, each of which corresponds to a specific temporal scale. Afterward, AMSDE is employed to deal with vibration data from defective gears and rolling bearings. Moreover, the performance of AMSDE is benchmarked against five common temporal statistics (mean, standard deviation, root mean square, skewness and kurtosis) and three typical nonlinear measures (approximate entropy, sample entropy and permutation entropy). The results suggest that AMSDE performs better than these benchmark methods in characterizing running conditions of rotating machinery.

This study investigates the feasibility of using tear sample analysis, based on protein corona formation on gold nanoparticles combined with electrospray ionization mass spectrometry (ESI-MS) and machine learning techniques, as a non-invasive approach for the detection of choroidal melanoma. The aim is to assess whether protein-nanoparticle interactions can support early and reliable identification of this ocular condition. Tear samples were collected using Schirmer strips from six healthy individuals and six patients diagnosed with choroidal melanoma, with subsequent augmentation to 18 samples per group. Gold nanoparticles (AuNPs, ~ 20 nm) were synthesized via citrate reduction and incubated with tear samples to form protein coronas, which were analyzed using ESI-MS. Eight statistical and entropy-based features (mean, variance, skewness, kurtosis, Shannon entropy, approximate entropy, sample entropy, and permutation entropy) were extracted from spectral data. Additionally, Continuous Wavelet Transform (CWT) with Mexican hat wavelet was applied to convert mass spectrometry data into 128 × 128 RGB images for deep learning analysis. Classification was performed using traditional machine learning models (Random Forest, Support Vector Machine, Decision Tree, Deep Neural Network) and transfer learning with pre-trained CNNs (VGG16, ResNet50, Xception), evaluated through 5-fold cross-validation. Significant differences in spectral intensity parameters were observed between healthy individuals and choroidal melanoma patients (p < 0.001), with notably lower Mean_Intensity values in cancer patients (56.41 ± 46.06 vs. 111.02 ± 10.01, Cohen’s d = 1.64). While m/z parameters showed moderate differences that didn’t reach statistical significance (p = 0.082), entropy-based features demonstrated strong discriminative power. Among traditional machine learning models, Random Forest achieved the highest accuracy (0.959 ± 0.003) and ROC AUC (0.993 ± 0.000) with remarkable computational efficiency (3.90 s per fold). For deep learning approaches using CWT-generated images, VGG16 demonstrated superior performance (Accuracy: 0.976 ± 0.008, ROC AUC: 0.997 ± 0.002) despite requiring significantly higher computational resources (1349.52 s per fold). This study demonstrates that tear sample analysis using protein corona formation on gold nanoparticles with ESI-MS and advanced machine learning techniques offers a promising non-invasive approach for choroidal melanoma detection with performance metrics that compare favorably to existing methods. The significant differences in spectral intensity parameters between groups suggest distinctive proteomic signatures that can be leveraged for diagnostic purposes. While both traditional machine learning and deep learning approaches achieved exceptional performance, each offers distinct advantages in terms of computational efficiency and feature extraction capabilities.

The automated classification of cognitive workload tasks based on the analysis of multi-channel EEG signals is vital for human–computer interface (HCI) applications. In this paper, we propose a computerized approach for categorizing mental-arithmetic-based cognitive workload tasks using multi-channel electroencephalogram (EEG) signals. The approach evaluates various entropy features, such as the approximation entropy, sample entropy, permutation entropy, dispersion entropy, and slope entropy, from each channel of the EEG signal. These features were fed to various recurrent neural network (RNN) models, such as long-short term memory (LSTM), bidirectional LSTM (BLSTM), and gated recurrent unit (GRU), for the automated classification of mental-arithmetic-based cognitive workload tasks. Two cognitive workload classification strategies (bad mental arithmetic calculation (BMAC) vs. good mental arithmetic calculation (GMAC); and before mental arithmetic calculation (BFMAC) vs. during mental arithmetic calculation (DMAC)) are considered in this work. The approach was evaluated using the publicly available mental arithmetic task-based EEG database. The results reveal that our proposed approach obtained classification accuracy values of 99.81%, 99.43%, and 99.81%, using the LSTM, BLSTM, and GRU-based RNN classifiers, respectively for the BMAC vs. GMAC cognitive workload classification strategy using all entropy features and a 10-fold cross-validation (CV) technique. The slope entropy features combined with each RNN-based model obtained higher classification accuracy compared with other entropy features for the classification of the BMAC vs. GMAC task. We obtained the average classification accuracy values of 99.39%, 99.44%, and 99.63% for the classification of the BFMAC vs. DMAC tasks, using the LSTM, BLSTM, and GRU classifiers with all entropy features and a hold-out CV scheme. Our developed automated mental arithmetic task system is ready to be tested with more databases for real-world applications.

This paper presents a new method to forecast short rainfall time-series. The new framework is by means of Bayesian enhanced modified combined approach (BEMCA) using permutation and relative entropy with Bayesian inference. The aim at the proposed filter is focused on short datasets consisting of at least 36 samples. The structure of the artificial neural networks (ANNs) change according to data model selected, such as the Bayesian approach can be combined with the entropic information of the series. Then computational results are assessed on time series competition and rainfall series, afterwards they are compared with ANN nonlinear approaches proposed in recent work and naïve linear technique such us ARMA. To show a better performance of BEMCA filter, results are analyzed in their forecast horizons by SMAPE and RMSE indices. BEMCA filter shows an increase of accuracy in 3-6 prediction horizon analyzing the dynamic behavior of chaotic series for short series predictions.