Permutation Entropy Method Research Articles

Application of cross-channel multiscale permutation entropy in measuring multichannel data complexity.

Entropy is a pivotal concept in nonlinear dynamics, revealing chaos, self-organization, and information transmission in complex systems. Permutation entropy, due to its computational efficiency and lower data length requirements, has found widespread use in various fields. However, in the age of multi-channel data, existing permutation entropy methods are limited in capturing cross-channel information. This paper presents cross-channel multiscale permutation entropy algorithm, and the proposed algorithm can effectively capture the cross-channel information of multi-channel dataset. The major modification lies in the concurrent frequency counting of specific events during the calculation steps. The algorithm improves phase space reconstruction and mapping, enhancing the capability of multi-channel permutation entropy methods to extract cross-channel information. Simulation and real-world multi-channel data analysis demonstrate the superiority of the proposed algorithm in distinguishing different types of data. The improvement is not limited to one specific algorithm and can be applied to various multi-channel permutation entropy variants, making them more effective in uncovering information across different channels.

Permutation entropy analysis of EEG signals for distinguishing eyes-open and eyes-closed brain states: Comparison of different approaches.

Developing reliable methodologies to decode brain state information from electroencephalogram (EEG) signals is an open challenge, crucial to implementing EEG-based brain-computer interfaces (BCIs). For example, signal processing methods that identify brain states could allow motor-impaired patients to communicate via non-invasive, EEG-based BCIs. In this work, we focus on the problem of distinguishing between the states of eyes closed (EC) and eyes open (EO), employing quantities based on permutation entropy (PE). An advantage of PE analysis is that it uses symbols (ordinal patterns) defined by the ordering of the data points (disregarding the actual values), hence providing robustness to noise and outliers due to motion artifacts. However, we show that for the analysis of multichannel EEG recordings, the performance of PE in discriminating the EO and EC states depends on the symbols' definition and how their probabilities are estimated. Here, we study the performance of PE-based features for EC/EO state classification in a dataset of N=107 subjects with one-minute 64-channel EEG recordings in each state. We analyze features obtained from patterns encoding temporal or spatial information, and we compare different approaches to estimate their probabilities (by averaging over time, over channels, or by "pooling"). We find that some PE-based features provide about 75% classification accuracy, comparable to the performance of features extracted with other statistical analysis techniques. Our work highlights the limitations of PE methods in distinguishing the eyes' state, but, at the same time, it points to the possibility that subject-specific training could overcome these limitations.

Variable contribution analysis in multivariate process monitoring using permutation entropy

Multivariate statistical process monitoring is widely used in industrial processes for performance monitoring and fault detection. Once a fault has been detected, fault diagnoses must be performed to identify the variables that are responsible for the fault or the deviation from normal operation. In this paper, we present a new data-driven method for variable contribution analysis to a fault or abnormal behaviour in a process. The method is based on permutation entropy (PE), which measures the order of fluctuations within a time series. The PE method for variable contribution analysis is independent of fault detection statistics, and is used to evaluate which variables experienced the greatest change in their expected behaviour over time, immediately after the fault has been identified. We show how PE can be used to identify and interpret the variables responsible for, or affected by faults in an industrial process. The well-known Tennessee Eastman Process (TEP) is used to illustrate the application of PE for variable contribution analysis. The results show that PE is an efficient analysis tool for specifying variable contributions to faults.

Disease identification method based on graph features between pulse cycles

The pulse signal at the wrist communicates a range of cardiovascular information and reflects a person’s state of health. In recent years pulse signals have been used extensively in the field of disease identification. As a typical quasi-cyclic physiological signal, many researchers have simply used single-cycle pulse signals for disease diagnosis, ignoring the influence of inter-cycle differences on the diagnosis. In this paper, we propose a disease identification method based on graph features between pulse cycles, to focus on the differences in pulse cycles brought about by the disease. The pulse was pre-processing using a Sliding Window, followed by the construction of the correlation coefficient matrix, the setting of a threshold and the construction of the connection graph to obtain the graph features; the Multiscale Permutation Entropy method was used to extract the pulse’s entropy features. The ReliefF algorithm is then used to select a total of 18 features and a classification algorithm is used to distinguish between patients and healthy people. Both the self-constructed dataset and the open dataset from the PLA 211 Hospital had high recognition rates for the experiments. The accuracy of identifying hypertension was 94.16% with a precision of 90.41%; the accuracy of identifying chronic diseases was 97.23% with a precision of 97.81%. The experimental results from the two datasets demonstrate both the validity of the difference between cycles for disease classification and the applicability of the method proposed in this paper.

A heart sound classification method based on complete ensemble empirical modal decomposition with adaptive noise permutation entropy and support vector machine

Heart sound signal is a kind of physiological signal with nonlinear and nonstationary features. In order to improve the accuracy and efficiency of the phonocardiogram (PCG) classification, a new method was proposed by means of support vector machine (SVM) in which the complete ensemble empirical modal decomposition with adaptive noise (CEEMDAN) permutation entropy was as the eigenvector of heart sound signal. Firstly, the PCG was decomposed by CEEMDAN into a number of intrinsic mode functions (IMFs) from high to low frequency. Secondly, the IMFs were sifted according to the correlation coefficient, energy factor and signal-to-noise ratio. Then the instantaneous frequency was extracted by Hilbert transform, and its permutation entropy was constituted into eigenvector. Finally, the accuracy of the method was verified by using a hundred PCG samples selected from the 2016 PhysioNet/CinC Challenge. The results showed that the accuracy rate of the proposed method could reach up to 87%. In comparison with the traditional EMD and EEMD permutation entropy methods, the accuracy rate was increased by 18%-24%, which demonstrates the efficiency of the proposed method.

Multiscale Weighted Permutation Entropy Analysis of Schizophrenia Magnetoencephalograms.

Schizophrenia is a neuropsychiatric disease that affects the nonlinear dynamics of brain activity. The primary objective of this study was to explore the complexity of magnetoencephalograms (MEG) in patients with schizophrenia. We combined a multiscale method and weighted permutation entropy to characterize MEG signals from 19 schizophrenia patients and 16 healthy controls. When the scale was larger than 42, the MEG signals of schizophrenia patients were significantly more complex than those of healthy controls (). The difference in complexity between patients with schizophrenia and the controls was strongest in the frontal and occipital areas (), and there was almost no difference in the central area. In addition, the results showed that the dynamic range of MEG complexity is wider in healthy individuals than in people with schizophrenia. Overall, the multiscale weighted permutation entropy method reliably quantified the complexity of MEG from schizophrenia patients, contributing to the development of potential magnetoencephalographic biomarkers for schizophrenia.

Permutation-Loewner entropy analysis for 2-dimensional Ising system interface

The structure of the 2-dimensional (2D) Ising system interface has been investigated in various theoretical statistical physics studies. Although the relationships between the interface geometry and physical properties of the system have already been indicated, in this study, we consider their alternative expressions using the chordal Loewner evolution and permutation entropy method. The Loewner driving forces corresponding to the Ising interfaces for below and at the critical temperature Tc were numerically calculated and analyzed by examining their pattern frequency, entropy, and distance to white noise in the theoretical scheme of the permutation entropy method. Using this permutation-Loewner entropy (PLE) method, we found an exponential function-type relation between the PLE and mean energy of the Ising system, which provides a connection between the theory of Loewner evolution and 2D statistical mechanical systems.

Variational Mode Decomposition and Permutation Entropy Method for Denoising of Distributed Optical Fiber Vibration Sensing System

Variational Mode Decomposition and Permutation Entropy Method for Denoising of Distributed Optical Fiber Vibration Sensing System

The impact of total sleep deprivation upon supine and head up tilt hemodynamics using non-linear analysis in firefighters

PurposeThe impact of 31 h of total sleep deprivation (TSD) on cardiovascular autonomic modulation was evaluated in 46 volunteers all professional fire brigade units officers (all men, mean age = 32.11 ± 6.4). MethodsHemodynamic examination was obtained at baseline and after 31 h of Total Sleep Deprivation (TSD). Each measurement period included supine and head-up tilt (HUT) test response. Continuous beat-to-beat systolic blood pressure (sBP), diastolic blood pressure (dBP) and mean blood pressure (mBP), heart rate (HR), stroke volume (SV), left ventricular ejection time (LVET) and cardiac output (CO) were calculated in rest supine position and in response to HUT. Nonlinear analyses were conducted using Permutation Entropy (PE), Amplitude Aware Permutation Entropy (AAPE) and Fractal Dimension (FD). ResultsSupine increase of HR and decrease in RRI, LVET and entropy of LVET were observed after 31-h of TSD. Increase in entropy of mBP was indicated by AAPE and PE methods. Response to HUT, values of spectral analysis of blood pressure (normalized units of high frequency of sBP and dBP) was altered. Permutation entropy of mBP in response to HUT were significantly changed after 31-h of TSD. ConclusionsTSD alters autonomic nervous system modulation of cardiac and vascular functioning while supine and in response to orthostatic stress. TSD resulted in changes in cardiac and both spectral and two methods of analysis of entropy of blood pressure when supine. Values of spectral analysis of blood pressure and permutation entropy of mean blood pressure in response to HUT were significantly changed after 31 h of TSD.

Application of the refined multiscale permutation entropy method to fault detection of rolling bearing

To detect the fault status and identify the fault category of rolling bearings in a timely manner, a method based on refined composite multiscale permutation entropy (RCMPE) and a support vector machine is proposed. Multiscale permutation entropy (MPE) is an improved method based on permutation entropy. Although this method can effectively perform dynamic analysis of nonlinear signals and fault monitoring of rolling bearings, the method is likely to lose some key information when shortening the time series during the identification of coarse-grained fault signals, which reduces the reliability of the results. To overcome this obstacle, the RCMPE algorithm is utilized to extract bearing feature information, and it is compared and analyzed with MPE and the multiscale entropy (MSE). Through simulation and experimental verification of the signal, it is found that as the scale factor increases, RCMPE can retain more useful information. The calculation results of fault identification are more accurate and have better robustness and stability. This method has less dependence on data length and is more stable in entropy estimation. This new diagnostic method is verified by actual bearing test data. The results show that the method can successfully distinguish the fault type and damage degree of rolling bearings and verify the effectiveness and superiority of the proposed method.

Performance Degradation Assessment of Rotary Machinery Based on a Multiscale Tsallis Permutation Entropy Method

Methods based on vibration analysis are currently regarded as the most conclusive means for fault diagnosis and health prognostics in rotary machinery. However, changing working conditions mean that the vibration signals originating from rotary machinery exhibit different levels of complexity. This complexity leads to increased difficulty in constructing health indicators (HIs). In this paper, we propose a multiscale Tsallis permutation entropy (MTPE) to construct the HIs of rotary machinery under different working conditions. MTPE values are a function of an entropy index and scale, which have the universality for handling the complexity of a permutated time series. The health condition of the rotary machinery was effectively represented by the MTPEs in conditional monitoring; the initial point of the unhealthy stage was found using the 3 σ interval. This was set as the alarm threshold according to the varying HI trend. Once this was established, dividing the stages into two-stage health stages (HS) was straightforward. Using a rolling bearing, a run-to-failure experiment was conducted and results suggested that the proposed method effectively assessed the status of the rotary machinery. Taken together, this study provided a novel complexity measure based on a methodology for constructing the HIs of rotary machinery and enriches conditional monitoring theory.

A new bearing fault diagnosis approach combining sensitive statistical features with improved multiscale permutation entropy method

Obtaining the sensitive feature vectors from the vibration signal is crucial to indicate the bearing’s actual condition. Most often, weak feature vectors are the consequence of heavy noise in the original signal and the incompatibility of the methods to deal with the nonlinear and non-stationary nature of vibration signals. In this paper, the first issue is addressed by employing a complementary ensemble empirical mode decomposition method. A new method based on improved multiscale permutation entropy method and dominant statistical parameters is proposed to deal with the second issue. The proposed approach’s denoising capability is first verified on an amplitude modulated and frequency modulated simulated signal. On the experimental front, the proposed method is investigated under a wide range of operating conditions to simultaneously recognize bearing fault type and severity. Since the experimental investigation includes identifying dominant statistical parameters and classifying different bearing faults, a recent method named XGBoost is explored comprehensively. The results show that the classification accuracy with features extracted by the proposed method exceeds 3% to 18% compared to features extracted by other state-of-the-art permutation-based feature extraction methods.

Permutation Entropy Thresholding: A Non-Linear Signal Processing Method for Islanding Detection

ABSTRACT A novel passive method based on Permutation Entropy (PE) is proposed for fast, efficient, and differential detection of islanding from other disturbance events in power system. The extremely remarkable attributes of PE such as efficient and fast detection of change in dynamics as well as computational and conceptual simplicity can be appropriately utilized to devise a thresholding system that can differentiate the change in dynamics of islanding event from the non-islanding ones with negligible Non-Detection Zone (NDZ). As NDZ is the most critical issue of existing passive islanding detection methods, which overshadows its beneficial characteristics, this new approach can prove to be a suitable alternative with an added benefit of faster detection. The method is evaluated with grid connected single as well as multi-DG systems and its robustness against effects of load quality factors, active and reactive power mismatches is investigated. Results show that this method can detect the islanding disturbance and differentiate it from other disturbances within half a cycle. The performance of the proposed method is evaluated with photovoltaic (PV) sources and also validated on IEEE 14 bus model. Results indicate the suitability of the PE method for the common DG source of PV for a wide range of power mismatches including the worst cases and also its robustness against false tripping during DG switching. The results verify the credibility and advantage of the proposed PE thresholding method and its suitability for adopting into existing networks for passive islanding detection.

A Novel Hybrid Strategy for Detecting COD in Surface Water

The prediction of chemical oxygen demand (COD) by ultraviolet–visible absorption spectrum is a common method. Many researchers use the absorbance at the characteristic wavelength to establish COD prediction models. However, selecting the characteristic wavelength is a problem. In this paper, the extreme values of absorption spectrum change rate, was proposed as a new characteristic parameter to determine the characteristic wavelengths. On this basis, a novel hybrid strategy for detecting COD in surface water was proposed. We first proposed to combine the first derivative method with the permutation entropy method (FDPE) to determine the characteristic wavelengths. Then we used partial least square (PLS) to establish a COD prediction model. Experimental results demonstrated the linear correlation coefficient (R2) of the FDPE_PLS was above 0.99 without turbidity interference. Secondly, a dual-wavelength method (DWM) was proposed to determine the turbidity values. The DWM used slopes of absorbance values at 400 nm and 600 nm to predict the turbidity values. Compared with the single-wavelength method, the DWM improves the measurement accuracy of turbidity. Finally, a new turbidity compensation method was proposed to compensate for the interference in the first derivative spectrum. After compensation, FDPE_PLS can predict COD concentrations accurately, whose R2 was 0.99.

Using the Information Provided by Forbidden Ordinal Patterns in Permutation Entropy to Reinforce Time Series Discrimination Capabilities.

Despite its widely tested and proven usefulness, there is still room for improvement in the basic permutation entropy (PE) algorithm, as several subsequent studies have demonstrated in recent years. Some of these new methods try to address the well-known PE weaknesses, such as its focus only on ordinal and not on amplitude information, and the possible detrimental impact of equal values found in subsequences. Other new methods address less specific weaknesses, such as the PE results’ dependence on input parameter values, a common problem found in many entropy calculation methods. The lack of discriminating power among classes in some cases is also a generic problem when entropy measures are used for data series classification. This last problem is the one specifically addressed in the present study. Toward that purpose, the classification performance of the standard PE method was first assessed by conducting several time series classification tests over a varied and diverse set of data. Then, this performance was reassessed using a new Shannon Entropy normalisation scheme proposed in this paper: divide the relative frequencies in PE by the number of different ordinal patterns actually found in the time series, instead of by the theoretically expected number. According to the classification accuracy obtained, this last approach exhibited a higher class discriminating power. It was capable of finding significant differences in six out of seven experimental datasets—whereas the standard PE method only did in four—and it also had better classification accuracy. It can be concluded that using the additional information provided by the number of forbidden/found patterns, it is possible to achieve a higher discriminating power than using the classical PE normalisation method. The resulting algorithm is also very similar to that of PE and very easy to implement.

Permutation Entropy and Bubble Entropy: Possible interactions and synergies between order and sorting relations.

Despite its widely demonstrated usefulness, there is still room for improvement in the basic Permutation Entropy (PE) algorithm, as several subsequent studies have proposed in the recent years. For example, some improved PE variants try to address possible PE weaknesses, such as its only focus on ordinal information, and not on amplitude, or the possible detrimental impact of equal values in subsequences due to motif ambiguity. Other evolved PE methods try to reduce the influence of input parameters. A good representative of this last point is the Bubble Entropy (BE) method. BE is based on sorting relations instead of ordinal patterns, and its promising capabilities have not been extensively assessed yet. The objective of the present study was to comparatively assess the classification performance of this new method, and study and exploit the possible synergies between PE and BE. The claimed superior performance of BE over PE was first evaluated by conducting a series of time series classification tests over a varied and diverse experimental set. The results of this assessment apparently suggested that there is a complementary relationship between PE and BE, instead of a superior/inferior relationship. A second set of experiments using PE and BE simultaneously as the input features of a clustering algorithm, demonstrated that with a proper algorithm configuration, classification accuracy and robustness can benefit from both measures.

Reverse Dispersion Entropy: A New Complexity Measure for Sensor Signal.

Permutation entropy (PE), as one of the powerful complexity measures for analyzing time series, has advantages of easy implementation and high efficiency. In order to improve the performance of PE, some improved PE methods have been proposed through introducing amplitude information and distance information in recent years. Weighted-permutation entropy (W-PE) weight each arrangement pattern by using variance information, which has good robustness and stability in the case of high noise level and can extract complexity information from data with spike feature or abrupt amplitude change. Dispersion entropy (DE) introduces amplitude information by using the normal cumulative distribution function (NCDF); it not only can detect the change of simultaneous frequency and amplitude, but also is superior to the PE method in distinguishing different data sets. Reverse permutation entropy (RPE) is defined as the distance to white noise in the opposite trend with PE and W-PE, which has high stability for time series with varying lengths. To further improve the performance of PE, we propose a new complexity measure for analyzing time series, and term it as reverse dispersion entropy (RDE). RDE takes PE as its theoretical basis and combines the advantages of DE and RPE by introducing amplitude information and distance information. Simulation experiments were carried out on simulated and sensor signals, including mutation signal detection under different parameters, noise robustness testing, stability testing under different signal-to-noise ratios (SNRs), and distinguishing real data for different kinds of ships and faults. The experimental results show, compared with PE, W-PE, RPE, and DE, that RDE has better performance in detecting abrupt signal and noise robustness testing, and has better stability for simulated and sensor signal. Moreover, it also shows higher distinguishing ability than the other four kinds of PE for sensor signals.

Modeling of investment attractiveness of countries using entropy analysis of regional stock markets

The current study focuses on the problem of determining investment attrаctiveness of countries by means of monitoring regional stock markets. The method of using the permutation entropy as a model of investment attractiveness estimation is suggested. We have calculated the permutation entropy for the time series of stock markets of countries for the period from 2005 to 2018. The countries with high, middle or low income in Europe, Central Asia, East Asia, the Pacific, and North America were selected for the study. The article presents the results of modeling and analysis of dynamic properties of regional stock markets using the permutation entropy. The behavior of the permutation entropy and stock markets is analyzed and conclusions about the possibility of rapid monitoring of the investment attractiveness of countries by classifying the states of the stock markets of these countries are drawn. Particular attention is paid to crisis periods. It has been shown that the permutation entropy rapidly decreases in a universal way in the pre-crisis period, which can serve as an indicator of the precursor for crisis phenomena. Determining the pre-crisis, actual crisis and post-crisis periods will allow the investor to make the right decision on time. The advantages of using the permutation entropy method as a tool for high-frequency monitoring of stock markets and modeling the investment attractiveness of countries are pointed out. The results of determining investment attractiveness in terms of the permutation entropy and ratings of the world countries, compiled by the world-wide rating agencies and literature, are compared.

Permutation Entropy and Information Recovery in Nonlinear Dynamic Economic Time Series

The focus of this paper is an information theoretic-symbolic logic approach to extract information from complex economic systems and unlock its dynamic content. Permutation Entropy (PE) is used to capture the permutation patterns-ordinal relations among the individual values of a given time series; to obtain a probability distribution of the accessible patterns; and to quantify the degree of complexity of an economic behavior system. Ordinal patterns are used to describe the intrinsic patterns, which are hidden in the dynamics of the economic system. Empirical applications involving the Dow Jones Industrial Average are presented to indicate the information recovery value and the applicability of the PE method. The results demonstrate the ability of the PE method to detect the extent of complexity (irregularity) and to discriminate and classify admissible and forbidden states.

Permutation entropy: Influence of amplitude information on time series classification performance.

Permutation Entropy (PE) is a very popular complexity analysis tool for time series. De-spite its simplicity, it is very robust and yields goods results in applications related to assessing the randomness of a sequence, or as a quantitative feature for signal classification. It is based on com-puting the Shannon entropy of the relative frequency of all the ordinal patterns found in a time series. However, there is a basic consensus on the fact that only analysing sample order and not amplitude might have a detrimental effect on the performance of PE. As a consequence, a number of methods based on PE have been proposed in the last years to include the possible influence of sample ampli-tude. These methods claim to outperform PE but there is no general comparative analysis that confirms such claims independently. Furthermore, other statistics such as Sample Entropy (SampEn) are based solely on amplitude, and it could be argued that other tools like this one are better suited to exploit the amplitude differences than PE. The present study quantifies the performance of the standard PE method and other amplitude-included PE methods using a disparity of time series to find out if there are really significant performance differences. In addition, the study compares statistics based uniquely on ordinal or amplitude patterns. The objective was to ascertain whether the whole was more than the sum of its parts. The results confirmed that highest classification accuracy was achieved using both types of patterns simultaneously, instead of using standard PE (ordinal patterns), or SampEn (ampli-tude patterns) isolatedly.