The Multiscale Entropy Algorithm and Its Variants: A Review

Multiscale Entropy Analysis: Application to Cardio-Respiratory Coupling

It is known that in pathological conditions, physiological systems develop changes in the multiscale properties of physiological signals. However, in real life, little is known about how changes in the function of one of the two coupled physiological systems induce changes in function of the other one, especially on their multiscale behavior. Hence, in this work we aimed to examine the complexity of cardio-respiratory coupled systems control using multiscale entropy (MSE) analysis of cardiac intervals MSE (RR), respiratory time series MSE (Resp), and synchrony of these rhythms by cross multiscale entropy (CMSE) analysis, in the heart failure (HF) patients and healthy subjects. We analyzed 20 min of synchronously recorded RR intervals and respiratory signal during relaxation in the supine position in 42 heart failure patients and 14 control healthy subjects. Heart failure group was divided into three subgroups, according to the RR interval time series characteristics (atrial fibrillation (HFAF), sinus rhythm (HFSin), and sinus rhythm with ventricular extrasystoles (HFVES)). Compared with healthy control subjects, alterations in respiratory signal properties were observed in patients from the HFSin and HFVES groups. Further, mean MSE curves of RR intervals and respiratory signal were not statistically different only in the HFSin group (p = 0.43). The level of synchrony between these time series was significantly higher in HFSin and HFVES patients than in control subjects and HFAF patients (p < 0.01). In conclusion, depending on the specific pathologies, primary alterations in the regularity of cardiac rhythm resulted in changes in the regularity of the respiratory rhythm, as well as in the level of their asynchrony.

Multi-scale Entropy Evaluates the Proarrhythmic Condition of Persistent Atrial Fibrillation Patients Predicting Early Failure of Electrical Cardioversion

Atrial fibrillation (AF) is nowadays the most common cardiac arrhythmia, being associated with an increase in cardiovascular mortality and morbidity. When AF lasts for more than seven days, it is classified as persistent AF and external interventions are required for its termination. A well-established alternative for that purpose is electrical cardioversion (ECV). While ECV is able to initially restore sinus rhythm (SR) in more than 90% of patients, rates of AF recurrence as high as 20-30% have been found after only a few weeks of follow-up. Hence, new methods for evaluating the proarrhythmic condition of a patient before the intervention can serve as efficient predictors about the high risk of early failure of ECV, thus facilitating optimal management of AF patients. Among the wide variety of predictors that have been proposed to date, those based on estimating organization of the fibrillatory (f-) waves from the surface electrocardiogram (ECG) have reported very promising results. However, the existing methods are based on traditional entropy measures, which only assess a single time scale and often are unable to fully characterize the dynamics generated by highly complex systems, such as the heart during AF. The present work then explores whether a multi-scale entropy (MSE) analysis of the f-waves may provide early prediction of AF recurrence after ECV. In addition to the common MSE, two improved versions have also been analyzed, composite MSE (CMSE) and refined MSE (RMSE). When analyzing 70 patients under ECV, of which 31 maintained SR and 39 relapsed to AF after a four week follow-up, the three methods provided similar performance. However, RMSE reported a slightly better discriminant ability of 86%, thus improving the other multi-scale-based outcomes by 3-9% and other previously proposed predictors of ECV by 15-30%. This outcome suggests that investigation of dynamics at large time scales yields novel insights about the underlying complex processes generating f-waves, which could provide individual proarrhythmic condition estimation, thus improving preoperative predictions of ECV early failure.

Long-Range Dependence in Financial Markets: A Moving Average Cluster Entropy Approach

A perspective is taken on the intangible complexity of economic and social systems by investigating the dynamical processes producing, storing and transmitting information in financial time series. An extensive analysis based on the moving average cluster entropy approach has evidenced market and horizon dependence in highest-frequency data of real world financial assets. The behavior is scrutinized by applying the moving average cluster entropy approach to long-range correlated stochastic processes as the Autoregressive Fractionally Integrated Moving Average (ARFIMA) and Fractional Brownian motion (FBM). An extensive set of series is generated with a broad range of values of the Hurst exponent H and of the autoregressive, differencing and moving average parameters p,d,q. A systematic relation between moving average cluster entropy and long-range correlation parameters H, d is observed. This study shows that the characteristic behaviour exhibited by the horizon dependence of the cluster entropy is related to long-range positive correlation in financial markets. Specifically, long range positively correlated ARFIMA processes with differencing parameter d≃0.05, d≃0.15 and d≃0.25 are consistent with moving average cluster entropy results obtained in time series of DJIA, S&P500 and NASDAQ. The findings clearly point to a variability of price returns, consistently with a price dynamics involving multiple temporal scales and, thus, short- and long-run volatility components. An important aspect of the proposed approach is the ability to capture detailed horizon dependence over relatively short horizons (one to twelve months) and thus its relevance to define risk analysis indices.

Neurophysiological Hallmarks of Neurodegenerative Cognitive Decline: The Study of Brain Connectivity as A Biomarker of Early Dementia

Neurodegenerative processes of various types of dementia start years before symptoms, but the presence of a "neural reserve", which continuously feeds and supports neuroplastic mechanisms, helps the aging brain to preserve most of its functions within the "normality" frame. Mild cognitive impairment (MCI) is an intermediate stage between dementia and normal brain aging. About 50% of MCI subjects are already in a stage that is prodromal-to-dementia and during the following 3 to 5 years will develop clinically evident symptoms, while the other 50% remains at MCI or returns to normal. If the risk factors favoring degenerative mechanisms are modified during early stages (i.e., in the prodromal), the degenerative process and the loss of abilities in daily living activities will be delayed. It is therefore extremely important to have biomarkers able to identify-in association with neuropsychological tests-prodromal-to-dementia MCI subjects as early as possible. MCI is a large (i.e., several million in EU) and substantially healthy population; therefore, biomarkers should be financially affordable, largely available and non-invasive, but still accurate in their diagnostic prediction. Neurodegeneration initially affects synaptic transmission and brain connectivity; methods exploring them would represent a 1st line screening. Neurophysiological techniques able to evaluate mechanisms of synaptic function and brain connectivity are attracting general interest and are described here. Results are quite encouraging and suggest that by the application of artificial intelligence (i.e., learning-machine), neurophysiological techniques represent valid biomarkers for screening campaigns of the MCI population.

Obtaining Information about Operation of Centrifugal Compressor from Pressure by Combining EEMD and IMFE

Based on entropy characteristics, some complex nonlinear dynamics of the dynamic pressure at the outlet of a centrifugal compressor are analyzed, as the centrifugal compressor operates in a stable and unstable state. First, the 800-kW centrifugal compressor is tested to gather the time sequence of dynamic pressure at the outlet by controlling the opening of the anti-surge valve at the outlet, and both the stable and unstable states are tested. Then, multi-scale fuzzy entropy and an improved method are introduced to analyze the gathered time sequence of dynamic pressure. Furthermore, the decomposed signals of dynamic pressure are obtained using ensemble empirical mode decomposition (EEMD), and are decomposed into six intrinsic mode functions and one residual signal, and the intrinsic mode functions with large correlation coefficients in the frequency domain are used to calculate the improved multi-scale fuzzy entropy (IMFE). Finally, the statistical reliability of the method is studied by modifying the original data. After analysis of the relationships between the dynamic pressure and entropy characteristics, some important intrinsic dynamics are captured. The entropy becomes the largest in the stable state, but decreases rapidly with the deepening of the unstable state, and it becomes the smallest in the surge. Compared with multi-scale fuzzy entropy, the curve of the improved method is smoother and could show the change of entropy exactly under different scale factors. For the decomposed signals, the unstable state is captured clearly for higher order intrinsic mode functions and residual signals, while the unstable state is not apparent for lower order intrinsic mode functions. In conclusion, it can be observed that the proposed method can be used to accurately identify the unstable states of a centrifugal compressor in real-time fault diagnosis.

Entropy in Heart Rate Dynamics Reflects How HRV-Biofeedback Training Improves Neurovisceral Complexity during Stress-Cognition Interactions

Despite considerable appeal, the growing appreciation of biosignals complexity reflects that system complexity needs additional support. A dynamically coordinated network of neurovisceral integration has been described that links prefrontal-subcortical inhibitory circuits to vagally-mediated heart rate variability. Chronic stress is known to alter network interactions by impairing amygdala functional connectivity. HRV-biofeedback training can counteract stress defects. We hypothesized the great value of an entropy-based approach of beat-to-beat biosignals to illustrate how HRVB training restores neurovisceral complexity, which should be reflected in signal complexity. In thirteen moderately-stressed participants, we obtained vagal tone markers and psychological indexes (state anxiety, cognitive workload, and Perceived Stress Scale) before and after five-weeks of daily HRVB training, at rest and during stressful cognitive tasking. Refined Composite Multiscale Entropy (RCMSE) was computed over short time scales as a marker of signal complexity. Heightened vagal tone at rest and during stressful tasking illustrates training benefits in the brain-to-heart circuitry. The entropy index reached the highest significance levels in both variance and ROC curves analyses. Restored vagal activity at rest correlated with gain in entropy. We conclude that HRVB training is efficient in restoring healthy neurovisceral complexity and stress defense, which is reflected in HRV signal complexity. The very mechanisms that are involved in system complexity remain to be elucidated, despite abundant literature existing on the role played by amygdala in brain interconnections.

Nonlinear Methods Most Applied to Heart-Rate Time Series: A Review

The heart-rate dynamics are one of the most analyzed physiological interactions. Many mathematical methods were proposed to evaluate heart-rate variability. These methods have been successfully applied in research to expand knowledge concerning the cardiovascular dynamics in healthy as well as in pathological conditions. Notwithstanding, they are still far from clinical practice. In this paper, we aim to review the nonlinear methods most used to assess heart-rate dynamics. We focused on methods based on concepts of chaos, fractality, and complexity: Poincaré plot, recurrence plot analysis, fractal dimension (and the correlation dimension), detrended fluctuation analysis, Hurst exponent, Lyapunov exponent entropies (Shannon, conditional, approximate, sample entropy, and multiscale entropy), and symbolic dynamics. We present the description of the methods along with their most notable applications.

A Novel Multiscale Cross-Entropy Method Applied to Navigation Data Acquired with a Bike Simulator

We present herein a new approach - the so-called time-shift multiscale cross-distribution entropy (TSMCDE) - to quantify the complexity between two sequences. By analyzing biomedical data, we reveal that TSMCDE over-performs other cross-entropy measures. Thus, TSMCDE, multiscale cross-sample entropy (MCSE), multiscale cross-distribution entropy (MCDE), and time-shift multiscale cross-sample entropy (TSMCSE) were applied to handlebar angle and speed time series recorded from a bike simulator. Twenty-four subjects divided into two groups (12 subjects each) participated to the study. The first group corresponds to young healthy subjects. The second group corresponds to older adults with loss of autonomy. Our results show that a link may exist between complexity and the age and physical state of a population. Moreover, TSMCDE leads to a better differentiation of the two groups than MCSE, MCDE, and TSMCSE. TSMCDE should now be tested on other types of data and on larger datasets to prove its usefulness and its efficiency.

(Multiscale) Cross-Entropy Methods: A Review

Cross-entropy was introduced in 1996 to quantify the degree of asynchronism between two time series. In 2009, a multiscale cross-entropy measure was proposed to analyze the dynamical characteristics of the coupling behavior between two sequences on multiple scales. Since their introductions, many improvements and other methods have been developed. In this review we offer a state-of-the-art on cross-entropy measures and their multiscale approaches.

Classification of Hepatitis Viruses from Sequencing Chromatograms Using Multiscale Permutation Entropy and Support Vector Machines

Classifying nucleic acid trace files is an important issue in molecular biology researches. For the purpose of obtaining better classification performance, the question of which features are used and what classifier is implemented to best represent the properties of nucleic acid trace files plays a vital role. In this study, different feature extraction methods based on statistical and entropy theory are utilized to discriminate deoxyribonucleic acid chromatograms, and distinguishing their signals visually is almost impossible. Extracted features are used as the input feature set for the classifiers of Support Vector Machines (SVM) with different kernel functions. The proposed framework is applied to a total number of 200 hepatitis nucleic acid trace files which consist of Hepatitis B Virus (HBV) and Hepatitis C Virus (HCV). While the use of statistical-based feature extraction methods allows representing the properties of hepatitis nucleic acid trace files with descriptive measures such as mean, median and standard deviation, entropy-based feature extraction methods including permutation entropy and multiscale permutation entropy enable quantifying the complexity of these files. The results indicate that using statistical and entropy-based features produces exceptionally high performances in terms of accuracies (reached at nearly 99%) in classifying HBV and HCV.

Complexity-Based Measures of Postural Sway during Walking at Different Speeds and Durations Using Multiscale Entropy

: Participation in various physical activities requires successful postural control in response to the changes in position of our body. It is important to assess postural control for early detection of falls and foot injuries. Walking at various speeds and for various durations is essential in daily physical activities. The purpose of this study was to evaluate the changes in complexity of the center of pressure (COP) during walking at different speeds and for different durations. In this study, a total of 12 participants were recruited for walking at two speeds (slow at 3 km/h and moderate at 6 km/h) for two durations (10 and 20 minutes). An insole-type plantar pressure measurement system was used to measure and calculate COP as participants walked on a treadmill. Multiscale entropy (MSE) was used to quantify the complexity of COP. Our results showed that the complexity of COP significantly decreased (p < 0.05) after 20 min of walking (complexity index, CI = −3.51) compared to 10 min of walking (CI = −3.20) while walking at 3 km/h, but not at 6 km/h. Our results also showed that the complexity index of COP indicated a significant difference (p < 0.05) between walking at speeds of 3 km/h (CI = −3.2) and 6 km/h (CI = −3.6) at the walking duration of 10 minutes, but not at 20 minutes. This study demonstrated an interaction between walking speeds and walking durations on the complexity of COP.

Short-Term Model-Based Multiscale Complexity Analysis of Cardiac Control Provides Complementary Information to Single-Scale Approaches

The study compares a recently proposed shortterm model-based linear multiscale complexity approach to a single-scale application of the same method and to a model-free nonlinear one based on the computation of conditional entropy with the aim at assessing the complementary information. Comparison was carried out over 24 hours Holter recordings of heart period variability during daytime and nighttime in 12 healthy men (age: 34-55 years). Single-scale methods were able to detect the increased complexity of the cardiac control during nighttime. Multiscale complexity analysis showed that this increase was due to an increase of complexity in the low frequency band (from 0.04 to 0.15 Hz), while complexity in the range of frequencies typical of the respiratory rate was unmodified. Regardless of the method (i.e. linear or nonlinear) single-scale complexity indexes were uncorrelated to the multiscale ones. We conclude that short-term model-based linear multiscale complexity approach provides complementary information to single-scale methods in an application devoted to the analysis of cardiac control from 24 hours Holter recordings.

Related Entropy Theories Application in Condition Monitoring of Rotating Machineries

Rotating machinery plays an important role in various kinds of industrial engineering. How to assess their conditions is a key problem for operating safety and condition-based maintenance. The potential anomaly, fault and failure information can be obtained by analyzing the collected condition monitoring data of the previously deployed sensors in rotating machinery. Among the available methods of analyzing sensors data, entropy and its variants can provide quantitative information contained in these sensing data. For implementing fault detection, diagnosis, and prognostics, this information can be utilized for feature extraction and selecting appropriate training data for machine learning methods. This article aims to review the related entropy theories which have been applied for condition monitoring of rotating machinery. This review consists of typical entropy theories presentation, application, summary, and discussion.

Multiscale Entropy of Cardiac and Postural Control Reflects a Flexible Adaptation to a Cognitive Task

In humans, physiological systems involved in maintaining stable conditions for health and well-being are complex, encompassing multiple interactions within and between system components. This complexity is mirrored in the temporal structure of the variability of output signals. Entropy has been recognized as a good marker of systems complexity, notably when calculated from heart rate and postural dynamics. A degraded entropy is generally associated with frailty, aging, impairments or diseases. In contrast, high entropy has been associated with the elevated capacity to adjust to an ever-changing environment, but the link is unknown between entropy and the capacity to cope with cognitive tasks in a healthy young to middle-aged population. Here, we addressed classic markers (time and frequency domains) and refined composite multiscale entropy (MSE) markers (after pre-processing) of heart rate and postural sway time series in 34 participants during quiet versus cognitive task conditions. Recordings lasted 10 min for heart rate and 51.2 s for upright standing, providing time series lengths of 500–600 and 2048 samples, respectively. The main finding was that entropy increased during cognitive tasks. This highlights the possible links between our entropy measures and the systems complexity that probably facilitates a control remodeling and a flexible adaptability in our healthy participants. We conclude that entropy is a reliable marker of neurophysiological complexity and adaptability in autonomic and somatic systems.

Rotating Machinery Fault Diagnosis Based on Improved Multiscale Amplitude-Aware Permutation Entropy and Multiclass Relevance Vector Machine

The health state of rotating machinery directly affects the overall performance of the mechanical system. The monitoring of the operation condition is very important to reduce the downtime and improve the production efficiency. This paper presents a novel rotating machinery fault diagnosis method based on the improved multiscale amplitude-aware permutation entropy (IMAAPE) and the multiclass relevance vector machine (mRVM) to provide the necessary information for maintenance decisions. Once the fault occurs, the vibration amplitude and frequency of rotating machinery obviously changes and therefore, the vibration signal contains a considerable amount of fault information. In order to effectively extract the fault features from the vibration signals, the intrinsic time-scale decomposition (ITD) was used to highlight the fault characteristics of the vibration signal by extracting the optimum proper rotation (PR) component. Subsequently, the IMAAPE was utilized to realize the fault feature extraction from the PR component. In the IMAAPE algorithm, the coarse-graining procedures in the multi-scale analysis were improved and the stability of fault feature extraction was promoted. The coarse-grained time series of vibration signals at different time scales were firstly obtained, and the sensitivity of the amplitude-aware permutation entropy (AAPE) to signal amplitude and frequency was adopted to realize the fault feature extraction of coarse-grained time series. The multi-classifier based on the mRVM was established by the fault feature set to identify the fault type and analyze the fault severity of rotating machinery. In order to demonstrate the effectiveness and feasibility of the proposed method, the experimental datasets of the rolling bearing and gearbox were used to verify the proposed fault diagnosis method respectively. The experimental results show that the proposed method can be applied to the fault type identification and the fault severity analysis of rotating machinery with high accuracy.

A Study of Brain Neuronal and Functional Complexities Estimated Using Multiscale Entropy in Healthy Young Adults

Brain complexity estimated using sample entropy and multiscale entropy (MSE) has recently gained much attention to compare brain function between diseased or neurologically impaired groups and healthy control groups. Using resting-state functional magnetic resonance imaging (rfMRI) blood oxygen-level dependent (BOLD) signals in a large cohort (n = 967) of healthy young adults, the present study maps neuronal and functional complexities estimated by using MSE of BOLD signals and BOLD phase coherence connectivity, respectively, at various levels of the brain’s organization. The functional complexity explores patterns in a higher dimension than neuronal complexity and may better discern changes in brain functioning. The leave-one-subject-out cross-validation method is used to predict fluid intelligence using neuronal and functional complexity MSE values as features. While a wide range of scales was selected with neuronal complexity, only the first three scales were selected with functional complexity. Fewer scales are advantageous as they preclude the need for long BOLD signals to calculate good estimates of MSE. The presented results corroborate with previous findings and provide a baseline for other studies exploring the use of MSE to examine changes in brain function related to aging, diseases, and clinical disorders.

Decomposition of a Multiscale Entropy Tensor for Sleep Stage Identification in Preterm Infants

Established sleep cycling is one of the main hallmarks of early brain development in preterm infants, therefore, automated classification of the sleep stages in preterm infants can be used to assess the neonate’s cerebral maturation. Tensor algebra is a powerful tool to analyze multidimensional data and has proven successful in many applications. In this paper, a novel unsupervised algorithm to identify neonatal sleep stages based on the decomposition of a multiscale entropy tensor is presented. The method relies on the difference in electroencephalography(EEG) complexity between the neonatal sleep stages and is evaluated on a dataset of 97 EEG recordings. An average sensitivity, specificity, accuracy and area under the receiver operating characteristic curve of 0.80, 0.79, 0.79 and 0.87 was obtained if the rank of the tensor decomposition is selected based on the age of the infant.

Multivariate Multiscale Dispersion Entropy of Biomedical Times Series

Due to the non-linearity of numerous physiological recordings, non-linear analysis of multi-channel signals has been extensively used in biomedical engineering and neuroscience. Multivariate multiscale sample entropy (MSE–mvMSE) is a popular non-linear metric to quantify the irregularity of multi-channel time series. However, mvMSE has two main drawbacks: (1) the entropy values obtained by the original algorithm of mvMSE are either undefined or unreliable for short signals (300 sample points); and (2) the computation of mvMSE for signals with a large number of channels requires the storage of a huge number of elements. To deal with these problems and improve the stability of mvMSE, we introduce multivariate multiscale dispersion entropy (MDE–mvMDE), as an extension of our recently developed MDE, to quantify the complexity of multivariate time series. We assess mvMDE, in comparison with the state-of-the-art and most widespread multivariate approaches, namely, mvMSE and multivariate multiscale fuzzy entropy (mvMFE), on multi-channel noise signals, bivariate autoregressive processes, and three biomedical datasets. The results show that mvMDE takes into account dependencies in patterns across both the time and spatial domains. The mvMDE, mvMSE, and mvMFE methods are consistent in that they lead to similar conclusions about the underlying physiological conditions. However, the proposed mvMDE discriminates various physiological states of the biomedical recordings better than mvMSE and mvMFE. In addition, for both the short and long time series, the mvMDE-based results are noticeably more stable than the mvMSE- and mvMFE-based ones. For short multivariate time series, mvMDE, unlike mvMSE, does not result in undefined values. Furthermore, mvMDE is faster than mvMFE and mvMSE and also needs to store a considerably smaller number of elements. Due to its ability to detect different kinds of dynamics of multivariate signals, mvMDE has great potential to analyse various signals.

Fault Diagnosis of Rolling Bearing Using Multiscale Amplitude-Aware Permutation Entropy and Random Forest

A rolling bearing is an important connecting part between rotating machines. It is susceptible to mechanical stress and wear, which affect the running state of bearings. In order to effectively identify the fault types and analyze the fault severity of rolling bearings, a rolling bearing fault diagnosis method based on multiscale amplitude-aware permutation entropy (MAAPE) and random forest is proposed in this paper. The vibration signals of rolling bearings to be analyzed are decomposed into different coarse-grained time series by using the coarse-graining procedure in multiscale entropy, highlighting the fault dynamic characteristics of vibration signals at different scales. The fault features contained in the coarse-grained time series at different time scales are extracted by using amplitude-aware permutation entropy’s sensitive characteristics to signal amplitude and frequency changes to form fault feature vectors. The fault feature vector set is used to establish the random forest multi-classifier, and the fault type identification and fault severity analysis of rolling bearings is realized through random forest. In order to demonstrate the feasibility and effectiveness of the proposed method, experiments were fully conducted in this paper. The experimental results show that multiscale amplitude-aware permutation entropy can effectively extract fault features of rolling bearings from vibration signals, and the extracted feature vectors have high separability. Compared with other rolling bearing fault diagnosis methods, the proposed method not only has higher fault type identification accuracy, but also can analyze the fault severity of rolling bearings to some extent. The identification accuracy of four fault types is up to 96.0% and the fault recognition accuracy under different fault severity reached 92.8%.

Falk T. Multi-Scale Heart Beat Entropy Measures for Mental Workload Assessment of Ambulant Users

Mental workload assessment is crucial in many real life applications which require constant attention and where imbalance of mental workload resources may cause safety hazards. As such, mental workload and its relationship with heart rate variability (HRV) have been well studied in the literature. However, the majority of the developed models have assumed individuals are not ambulant, thus bypassing the issue of movement-related electrocardiography (ECG) artifacts and changing heart beat dynamics due to physical activity. In this work, multi-scale features for mental workload assessment of ambulatory users is explored. ECG data was sampled from users while they performed different types and levels of physical activity while performing the multi-attribute test battery (MATB-II) task at varying difficulty levels. Proposed features are shown to outperform benchmark ones and further exhibit complementarity when used in combination. Indeed, results show gains over the benchmark HRV measures of 24.41% in accuracy and of 27.97% in F1 score can be achieved even at high activity levels.

Time, frequency and information domain analysis of short-term heart rate variability before and after focal and generalized seizures in epileptic children

OBJECTIVE

In this work we explore the potential of combining standard time and frequency domain indexes with novel information measures, to characterize pre- and post-ictal heart rate variability (HRV) in epileptic children, with the aim of differentiating focal and generalized epilepsy regarding the autonomic control mechanisms.

APPROACH

We analyze short-term HRV in 37 children suffering from generalized or focal epilepsy, monitored 10 s, 300 s, 600 s and 1800 s both before and after seizure episodes. Nine indexes are computed in time (mean, standard deviation of normal-to-normal intervals, root mean square of the successive differences (RMSSD)), frequency (low-to-high frequency power ratio LF/HF, normalized LF and HF power) and information (entropy, conditional entropy and self-entropy) domains. Focal and generalized epilepsy are compared through statistical analysis of the indexes and using linear discriminant analysis (LDA).

MAIN RESULTS

In children with focal epilepsy, early post-ictal phase is characterized by significant tachycardia, depressed HRV, increased LF power and LF/HF, and decreased complexity, progressively recovered across time windows after the episodes. Children with generalized seizures instead show significant tachycardia, lower RMSSD, higher LF power and LF/HF ratio before the seizure. These different behaviors are exploited by LDA analysis to separate focal and generalized epilepsy up to an accuracy of 75%. Results suggest a shift of the sympatho-vagal balance towards sympathetic dominance and vagal withdrawal, noticeable just after the termination of seizure episodes and then reverted in focal epilepsy, and persistent during inter-ictal and pre-ictal periods in generalized epilepsy.

SIGNIFICANCE

Our analysis helps in elucidating the pathophysiology of inter-ictal HRV autonomic control and the differential diagnosis of generalized and focal epilepsy. These findings may have clinical relevance since altered sympatho-vagal control can be related to a higher danger of morbidity and mortality, may reduce thresholds for life-threatening arrhythmias, and could be a biomarker of risk for sudden unexpected death in epilepsy.

Time-Shift Multi-scale Weighted Permutation Entropy and GWO-SVM Based Fault Diagnosis Approach for Rolling Bearing

Multi-scale permutation entropy (MPE) is an effective nonlinear dynamic approach for complexity measurement of time series and it has been widely applied to fault feature representation of rolling bearing. However, the coarse-grained time series in MPE becomes shorter and shorter with the increase of the scale factor, which causes an imprecise estimation of permutation entropy. In addition, the different amplitudes of the same patterns are not considered by the permutation entropy used in MPE. To solve these issues, the time-shift multi-scale weighted permutation entropy (TSMWPE) approach is proposed in this paper. The inadequate process of coarse-grained time series in MPE was optimized by using a time shift time series and the process of probability calculation that cannot fully consider the symbol mode is solved by introducing a weighting operation. The parameter selections of TSMWPE were studied by analyzing two different noise signals. The stability and robustness were also studied by comparing TSMWPE with TSMPE and MPE. Based on the advantages of TSMWPE, an intelligent fault diagnosis method for rolling bearing is proposed by combining it with gray wolf optimized support vector machine for fault classification. The proposed fault diagnostic method was applied to two cases of experimental data analysis of rolling bearing and the results show that it can diagnose the fault category and severity of rolling bearing accurately and the corresponding recognition rate is higher than the rate provided by the existing comparison methods.

Multi-Lag Analysis of Symbolic Entropies on EEG Recordings for Distress Recognition

Distress is a critical problem in developed societies given its long-term negative effects on physical and mental health. The interest in studying this emotion has notably increased during last years, being electroencephalography (EEG) signals preferred over other physiological variables in this research field. In addition, the non-stationary nature of brain dynamics has impulsed the use of non-linear metrics, such as symbolic entropies in brain signal analysis. Thus, the influence of time-lag on brain patterns assessment has not been tested. Hence, in the present study two permutation entropies denominated Delayed Permutation Entropy and Permutation Min-Entropy have been computed for the first time at different time-lags to discern between emotional states of calmness and distress from EEG signals. Moreover, a number of curve-related features were also calculated to assess brain dynamics across different temporal intervals. Complementary information among these variables was studied through sequential forward selection and 10-fold cross-validation approaches. According to the results obtained, the multi-lag entropy analysis has been able to reveal new significant insights so far undiscovered, thus notably improving the process of distress recognition from EEG recordings.

Complexity-Based Measures of Heart Rate Dynamics in Older Adults Following Long- and Short-Term Tai Chi Training: Cross-sectional and Randomized Trial Studies

Measures characterizing the complexity of heart rate (HR) dynamics have been informative in predicting age- and disease-related decline in cardiovascular health, but few studies have evaluated whether mind-body exercise can impact HR complexity. This study evaluated the effects of long-term Tai Chi (TC) practice on the complexity of HR dynamics using an observational comparison of TC experts and age- and gender-matched TC-naïve individuals. Shorter-term effects of TC were assessed by randomly assigning TC-naïve participants to either TC group to receive six months of TC training or to a waitlist control group. 23 TC experts (age = 63.3 ± 8.0 y; 24.6 ± 12.0 y TC experience) and 52 TC-naïve (age = 64.3 ± 7.7 y) were enrolled. In cross-sectional analyses, TC experts had a higher overall complexity index (CI, p = 0.004) and higher entropy at multiple individual time scales (p < 0.05); these findings persisted in models accounting for age, gender, body mass index (BMI), and physical activity levels. Longitudinal changes in complexity index did not differ significantly following random assignment to six months of TC vs. a waitlist control; however, within the TC group, complexity at select time scales showed statistically non-significant trends toward increases. Our study supports that longer-term TC mind-body training may be associated with increased complexity of HR dynamics.

An Improved Refined Composite Multivariate Multiscale Fuzzy Entropy Method for MI-EEG Feature Extraction

Feature extraction of motor imagery electroencephalogram (MI-EEG) has shown good application prospects in the field of medical health. Also, multivariate entropy-based feature extraction methods have been gradually applied to analyze complex multichannel biomedical signals, such as EEG and electromyography. Compared with traditional multivariate entropies, refined composite multivariate multiscale fuzzy entropy (RCmvMFE) overcomes the defect of unstable entropy values caused by the scale factor increase and is beneficial towards obtaining richer feature information. However, the coarse-grained process of RCmvMFE is mean filtered, which weakens Gaussian noise and is powerless against random impulse noise interference. This yields poor quality feature information and low accuracy classification. In this paper, RCmvMFE is improved (IRCmvMFE) by using composite filters in the coarse-grained procedure to enhance filter performance. Median filters are employed to remove the impulse noise interference from multichannel MI-EEG signals, and these filtered MI-EEGs are further smoothed by the mean filters. The multiscale IRCmvMFEs are calculated for all channels of composite filtered MI-EEGs, forming a feature vector, and a support vector machine is used for pattern classification. Based on two public datasets with different motor imagery tasks, the recognition results of 10 × 10-fold cross-validation achieved 99.43% and 99.86%, respectively, and the statistical analysis of experimental results was completed, showing the effectiveness of IRCmvMFE, as well. The proposed IRCmvMFE-based feature extraction method is superior compared to entropy-based and traditional methods.

Multiscale information storage of linear long-range correlated stochastic processes

Information storage, reflecting the capability of a dynamical system to keep predictable information during its evolution over time, is a key element of intrinsic distributed computation, useful for the description of the dynamical complexity of several physical and biological processes. Here we introduce a parametric approach which allows one to compute information storage across multiple timescales in stochastic processes displaying both short-term dynamics and long-range correlations (LRC). Our analysis is performed in the popular framework of multiscale entropy, whereby a time series is first "coarse grained" at the chosen timescale through low-pass filtering and downsampling, and then its complexity is evaluated in terms of conditional entropy. Within this framework, our approach makes use of linear fractionally integrated autoregressive (ARFI) models to derive analytical expressions for the information storage computed at multiple timescales. Specifically, we exploit state space models to provide the representation of lowpass filtered and downsampled ARFI processes, from which information storage is computed at any given timescale relating the process variance to the prediction error variance. This enhances the practical usability of multiscale information storage, as it enables a computationally reliable quantification of a complexity measure which incorporates the effects of LRC together with that of short-term dynamics. The proposed measure is first assessed in simulated ARFI processes reproducing different types of autoregressive dynamics and different degrees of LRC, studying both the theoretical values and the finite sample performance. We find that LRC alter substantially the complexity of ARFI processes even at short timescales, and that reliable estimation of complexity can be achieved at longer timescales only when LRC are properly modeled. Then, we assess multiscale information storage in physiological time series measured in humans during resting state and postural stress, revealing unprecedented responses to stress of the complexity of heart period and systolic arterial pressure variability, which are related to the different role played by LRC in the two conditions.

On Structural Entropy and Spatial Filling Factor Analysis of Colonoscopy Pictures

Colonoscopy is the standard device for diagnosing colorectal cancer, which develops from little lesions on the bowel wall called polyps. The Rényi entropies-based structural entropy and spatial filling factor are two scale- and resolution-independent quantities that characterize the shape of a probability distribution with the help of characteristic curves of the structural entropy–spatial filling factor map. This alternative definition of structural entropy is easy to calculate, independent of the image resolution, and does not require the calculation of neighbor statistics, unlike the other graph-based structural entropies.The distant goal of this study was to help computer aided diagnosis in finding colorectal polyps by making the Rényi entropy based structural entropy more understood. The direct goal was to determine characteristic curves that can differentiate between polyps and other structure on the picture. After analyzing the distribution of colonoscopy picture color channels, the typical structures were modeled with simple geometrical functions and the structural entropy–spatial filling factor characteristic curves were determined for these model structures for various parameter sets. A colonoscopy image analying method, i.e., the line- or column-wise scanning of the picture, was also tested, with satisfactory matching of the characteristic curve and the image.

Multiscale Entropy Analysis of Page Views: A Case Study of Wikipedia

In this study, the Wikipedia page views for four selected topics, namely, education, the economy/finance, medicine, and nature/environment from 2016–2018 are collected and the sample entropies of the three years’ page views are estimated and investigated using a short-time series multiscale entropy (sMSE) algorithm for a comprehensible understanding of the complexity of human website searching activities. The sample entropies of the selected topics are found to exhibit different temporal variations. In the past three years, the temporal characteristics of the sample entropies are vividly revealed, and the sample entropies of the selected topics follow the same tendencies and can be quantitatively ranked. By taking the 95% confidence interval into account, the temporal variations of sample entropies are further validated by statistical analysis (non-parametric), including the Wilcoxon signed-rank test and the Mann-Whitney U-test. The results suggest that the sample entropies estimated by the sMSE algorithm are feasible for analyzing the temporal variations of complexity for certain topics, whereas the regular variations of estimated sample entropies of different selected topics can’t simply be accepted as is. Potential explanations and paths in forthcoming studies are also described and discussed.

Asynchronous Control of P300-Based Brain-Computer Interfaces Using Sample Entropy

Brain–computer interfaces (BCI) have traditionally worked using synchronous paradigms. In recent years, much effort has been put into reaching asynchronous management, providing users with the ability to decide when a command should be selected. However, to the best of our knowledge, entropy metrics have not yet been explored. The present study has a twofold purpose: (i) to characterize both control and non-control states by examining the regularity of electroencephalography (EEG) signals; and (ii) to assess the efficacy of a scaled version of the sample entropy algorithm to provide asynchronous control for BCI systems. Ten healthy subjects participated in the study, who were asked to spell words through a visual oddball-based paradigm, attending (i.e., control) and ignoring (i.e., non-control) the stimuli. An optimization stage was performed for determining a common combination of hyperparameters for all subjects. Afterwards, these values were used to discern between both states using a linear classifier. Results show that control signals are more complex and irregular than non-control ones, reaching an average accuracy of 94.40% in classification. In conclusion, the present study demonstrates that the proposed framework is useful in monitoring the attention of a user, and granting the asynchrony of the BCI system.

Magnetotelluric Signal-Noise Identification and Separation Based on ApEn-MSE and StOMP

Natural magnetotelluric signals are extremely weak and susceptible to various types of noise pollution. To obtain more useful magnetotelluric data for further analysis and research, effective signal-noise identification and separation is critical. To this end, we propose a novel method of magnetotelluric signal-noise identification and separation based on ApEn-MSE and Stagewise orthogonal matching pursuit (StOMP). Parameters with good irregularity metrics are introduced: Approximate entropy (ApEn) and multiscale entropy (MSE), in combination with k-means clustering, can be used to accurately identify the data segments that are disturbed by noise. Stagewise orthogonal matching pursuit (StOMP) is used for noise suppression only in data segments identified as containing strong interference. Finally, we reconstructed the signal. The results show that the proposed method can better preserve the low-frequency slow-change information of the magnetotelluric signal compared with just using StOMP, thus avoiding the loss of useful information due to over-processing, while producing a smoother and more continuous apparent resistivity curve. Moreover, the results more accurately reflect the inherent electrical structure information of the measured site itself.

Using Multiscale Entropy to Assess the Efficacy of Local Cooling on Reactive Hyperemia in People with a Spinal Cord Injury

Pressure ulcers are one of the most common complications of a spinal cord injury (SCI). Prolonged unrelieved pressure is thought to be the primary causative factor resulting in tissue ischemia and eventually pressure ulcers. Previous studies suggested that local cooling reduces skin ischemia of the compressed soft tissues based on smaller hyperemic responses. However, the effect of local cooling on nonlinear properties of skin blood flow (SBF) during hyperemia is unknown. In this study, 10 wheelchair users with SCI and 10 able-bodied (AB) controls underwent three experimental protocols, each of which included a 10-min period as baseline, a 20-min intervention period, and a 20-min period for recovering SBF. SBF was measured using a laser Doppler flowmetry. During the intervention period, a pressure of 60 mmHg was applied to the sacral skin, while three skin temperature settings were tested, including no temperature change, a decrease by 10 °C, and an increase by 10 °C, respectively. A multiscale entropy (MSE) method was employed to quantify the degree of regularity of blood flow oscillations (BFO) associated with the SBF control mechanisms during baseline and reactive hyperemia. The results showed that under pressure with cooling, skin BFO both in people with SCI and AB controls were more regular at multiple time scales during hyperemia compared to baseline, whereas under pressure with no temperature change and particularly pressure with heating, BFO were more irregular during hyperemia compared to baseline. Moreover, the results of surrogate tests indicated that changes in the degree of regularity of BFO from baseline to hyperemia were only partially attributed to changes in relative amplitudes of endothelial, neurogenic, and myogenic components of BFO. These findings support the use of MSE to assess the efficacy of local cooling on reactive hyperemia and assess the degree of skin ischemia in people with SCI.

On the choice of multiscale entropy algorithm for quantification of complexity in gait data

The present study aimed at identifying a suitable multiscale entropy (MSE) algorithm for assessment of complexity in a stride-to-stride time interval time series. Five different algorithms were included (the original MSE, refine composite multiscale entropy (RCMSE), multiscale fuzzy entropy, generalized multiscale entropy and intrinsic mode entropy) and applied to twenty iterations of white noise, pink noise, or a sine wave with added white noise. Based on their ability to differentiate the level of complexity in the three different generated signal types, and their sensitivity and parameter consistency, MSE and RCMSE were deemed most appropriate. These two algorithms were applied to stride-to-stride time interval time series recorded from fourteen healthy subjects during one hour of overground and treadmill walking. In general, acceptable sensitivity and good parameter consistency were observed for both algorithms; however, they were not able to differentiate the complexity of the stride-to-stride time interval time series between the two walking conditions. Thus, the present study recommends the use of either MSE or RCMSE for quantification of complexity in stride-to-stride time interval time series.

Sample Entropy of sEMG Signals at Different Stages of Rectal Cancer Treatment

Information theory provides a spectrum of nonlinear methods capable of grasping an internal structure of a signal together with an insight into its complex nature. In this work, we discuss the usefulness of the selected entropy techniques for a description of the information carried by the surface electromyography signals during colorectal cancer treatment. The electrical activity of the external anal sphincter can serve as a potential source of knowledge of the actual state of the patient who underwent a common surgery for rectal cancer in the form of anterior or lower anterior resection. The calculation of Sample entropy parameters has been extended to multiple time scales in terms of the Multiscale Sample Entropy. The specific values of the entropy measures and their dependence on the time scales were analyzed with regard to the time elapsed since the operation, the type of surgical treatment and also the different depths of the rectum canal. The Mann-Whitney U test and Anova Friedman statistics indicate the statistically significant differences among all of stages of treatment and for all consecutive depths of rectum area for the estimated Sample Entropy. The further analysis at the multiple time scales signify the substantial differences among compared stages of treatment in the group of patients who underwent the lower anterior resection.

Refined Multiscale Fuzzy Entropy to Analyse Post-Exercise Cardiovascular Response in Older Adults With Orthostatic Intolerance

Orthostatic intolerance syndrome occurs when the autonomic nervous system is incapacitated and fails to respond to the demands associated with the upright position. Assessing this syndrome among the elderly population is important in order to prevent falls. However, this problem is still challenging. The goal of this work was to determine the relationship between orthostatic intolerance (OI) and the cardiovascular response to exercise from the analysis of heart rate and blood pressure. More specifically, the behavior of these cardiovascular variables was evaluated in terms of refined composite multiscale fuzzy entropy (RCMFE), measured at different scales. The dataset was composed by 65 older subjects, 44.6% (n = 29) were OI symptomatic and 55.4% (n = 36) were not. Insignificant differences were found in age and gender between symptomatic and asymptomatic OI participants. When heart rate was evaluated, higher differences between groups were observed during the recovery period immediately after exercise. With respect to the blood pressure and other hemodynamic parameters, most significant results were obtained in the post-exercise stage. In any case, the symptomatic OI group exhibited higher irregularity in the measured parameters, as higher RCMFE levels in all time scales were obtained. This information could be very helpful for a better understanding of cardiovascular instability, as well as to recognize risk factors for falls and impairment of functional status.

Application of multiscale Poincaré short-time computation versus multiscale entropy in analyzing fingertip photoplethysmogram amplitudes to differentiate diabetic from non-diabetic subjects

BACKGROUND AND OBJECTIVES

Multiscale Poincaré (MSP) plots have recently been introduced to facilitate the visualization of time series of physiological signals. This study aimed at investigating the feasibility of MSP application in distinguishing subjects with and without diabetes.

METHODS

Using photoplethysmogram (PPG) waveform amplitudes acquired from unilateral fingertip of non-diabetic (n = 34) and diabetic (n = 30) subjects, MSP indices (MSPI) of the two groups were compared using 1000, 500, 250, 100 data points. Data from Poincaré index (short-term variability/long-term variability [i.e. SD1/SD2] ratio, SSR) and multiscale entropy (MSE) were also obtained with the four corresponding data points for comparison.

RESULTS

SSR and MSPI were both negatively related to glycated hemoglobin (HbA1c) and fasting blood sugar levels. Significant negative correlation was also noted between MSPI and pulse pressure. When only 500 and 250 data points were included, significant elevations in the non-diabetic group were only noted in MSPI (both p < 0.01). Furthermore, MSPI was significantly higher in non-diabetic than that in diabetic subjects on all scales (i.e., 1-10) but not using MSE when utilizing 1000 data points.

CONCLUSIONS

The results demonstrated enhanced sensitivity of MSP in differentiating between non-diabetic and diabetic subjects compared to SSR and MSE, highlighting the feasibility of MSP application in biomedical data analysis to reduce computational time and enhance sensitivity.

Multidimensional and fuzzy sample entropy (SampEnMF) for quantifying H&E histological images of colorectal cancer

In this study, we propose to use a method based on the combination of sample entropy with multiscale and multidimensional approaches, along with a fuzzy function. The model was applied to quantify and classify H&E histological images of colorectal cancer. The multiscale approach was defined by analysing windows of different sizes and variations in tolerance for determining pattern similarity. The multidimensional strategy was performed by considering each pixel in the colour image as an n-dimensional vector, which was analysed from the Minkowski distance. The fuzzy strategy was a Gaussian function used to verify the pertinence of the distances between windows. The result was a method capable of computing similarities between pixels contained in windows of various sizes, as well as the information present in the colour channels. The power of quantification was tested in a public colorectal image dataset, which was composed of both benign and malignant classes. The results were given as inputs for classifiers of different categories and analysed by applying the k-fold cross-validation and holdout methods. The derived performances indicate that the proposed association was capable of distinguishing the benign and malignant groups, with values that surpassed those results obtained with important techniques available in the Literature. The best performance was an AUC value of 0.983, an important result, mainly when we consider the difficulties of clinical practice for the diagnosis of the colorectal cancer.

Dynamic Complexity of Spontaneous BOLD Activity in Alzheimer's Disease and Mild Cognitive Impairment Using Multiscale Entropy Analysis

Alzheimer's disease (AD) is characterized by progressive deterioration of brain function among elderly people. Studies revealed aberrant correlations in spontaneous blood oxygen level-dependent (BOLD) signals in resting-state functional magnetic resonance imaging (rs-fMRI) over a wide range of temporal scales. However, the study of the temporal dynamics of BOLD signals in subjects with AD and mild cognitive impairment (MCI) remains largely unexplored. Multiscale entropy (MSE) analysis is a method for estimating the complexity of finite time series over multiple time scales. In this research, we applied MSE analysis to investigate the abnormal complexity of BOLD signals using the rs-fMRI data from the Alzheimer's disease neuroimaging initiative (ADNI) database. There were 30 normal controls (NCs), 33 early MCI (EMCI), 32 late MCI (LMCI), and 29 AD patients. Following preprocessing of the BOLD signals, whole-brain MSE maps across six time scales were generated using the Complexity Toolbox. One-way analysis of variance (ANOVA) analysis on the MSE maps of four groups revealed significant differences in the thalamus, insula, lingual gyrus and inferior occipital gyrus, superior frontal gyrus and olfactory cortex, supramarginal gyrus, superior temporal gyrus, and middle temporal gyrus on multiple time scales. Compared with the NC group, MCI and AD patients had significant reductions in the complexity of BOLD signals and AD patients demonstrated lower complexity than that of the MCI subjects. Additionally, the complexity of BOLD signals from the regions of interest (ROIs) was found to be significantly associated with cognitive decline in patient groups on multiple time scales. Consequently, the complexity or MSE of BOLD signals may provide an imaging biomarker of cognitive impairments in MCI and AD.

Sample Entropy on Multidistance Signal Level Difference for Epileptic EEG Classification

Epilepsy is a disorder of the brain's nerves as a result of excessive brain cell activity. It is generally characterized by the recurrent unprovoked seizures. This neurological abnormality can be detected and evaluated using Electroencephalogram (EEG) signal. Many algorithms have been applied to achieve high performance for the EEG classification of epileptic. However, the complexity and randomness of EEG signals become a challenge to researchers in applying the appropriate algorithms. In this research, sample entropy on Multidistance Signal Level Difference (MSLD) was applied to obtain the characteristic of EEG signals, especially towards the epilepsy patients. The test was performed on three classes of EEG data: EEG signals of epilepsy patient in ictal (seizure), interictal conditions (occurring between seizures) and normal EEG signals from healthy subjects with a closed eye condition. In this study, classification and verification were done using the Support Vector Machine (SVM) method. Through the 5-fold cross-validation, experimental results showed the highest accuracy of 97.7%.

Applications of Variability Analysis Techniques for Continuous Glucose Monitoring Derived Time Series in Diabetic Patients

Methods from non-linear dynamics have enhanced understanding of functional dysregulation in various diseases but received less attention in diabetes. This retrospective cross-sectional study evaluates and compares relationships between indices of non-linear dynamics and traditional glycemic variability, and their potential application in diabetes control. Continuous glucose monitoring provided data for 177 subjects with type 1 (n = 22), type 2 diabetes (n = 143), and 12 non-diabetic subjects. Each time series comprised 576 glucose values. We calculated Poincaré plot measures (SD1, SD2), shape (SFE) and area of the fitting ellipse (AFE), multiscale entropy (MSE) index, and detrended fluctuation exponents (α1, α2). The glycemic variability metrics were the coefficient of variation (%CV) and standard deviation. Time of glucose readings in the target range (TIR) defined the quality of glycemic control. The Poincaré plot indices and α exponents were higher (p < 0.05) in type 1 than in the type 2 diabetes; SD1 (mmol/l): 1.64 ± 0.39 vs. 0.94 ± 0.35, SD2 (mmol/l): 4.06 ± 0.99 vs. 2.12 ± 1.04, AFE (mmol²/l²): 21.71 ± 9.82 vs. 7.25 ± 5.92, and α1: 1.94 ± 0.12 vs. 1.75 ± 0.12, α2: 1.38 ± 0.11 vs. 1.30 ± 0.15. The MSE index decreased consistently from the non-diabetic to the type 1 diabetic group (5.31 ± 1.10 vs. 3.29 ± 0.83, p < 0.001); higher indices correlated with lower %CV values (r = -0.313, p < 0.001). In a subgroup of type 1 diabetes patients, insulin pump therapy significantly decreased SD1 (-0.85 mmol/l), SD2 (-1.90 mmol/l), and AFE (-16.59 mmol²/l²), concomitantly with %CV (-15.60). The MSE index declined from 3.09 ± 0.94 to 1.93 ± 0.40 (p = 0.001), whereas the exponents α1 and α2 did not. On multivariate regression analyses, SD1, SD2, SFE, and AFE emerged as dominant predictors of TIR (β = -0.78, -1.00, -0.29, and -0.58) but %CV as a minor one, though α1 and MSE failed. In the regression models, including SFE, AFE, and α2 (β = -0.32), %CV was not a significant predictor. Poincaré plot descriptors provide additional information to conventional variability metrics and may complement assessment of glycemia, but complexity measures produce mixed results.

Multivariate Multiscale Complexity Analysis of Self-Reproducing Chaotic Systems

Designing a chaotic system with infinitely many attractors is a hot topic. In this paper, multiscale multivariate permutation entropy (MMPE) and multiscale multivariate Lempel–Ziv complexity (MMLZC) are employed to analyze the complexity of those self-reproducing chaotic systems with one-directional and two-directional infinitely many chaotic attractors. The analysis results show that complexity of this class of chaotic systems is determined by the initial conditions. Meanwhile, the values of MMPE are independent of the scale factor, which is different from the algorithm of MMLZC. The analysis proposed here is helpful as a reference for the application of the self-reproducing systems.

Information Theory and Atrial Fibrillation (AF): A Review

Atrial Fibrillation (AF) is the most common cardiac rhythm disorder seen in hospitals and in general practice, accounting for up to a third of arrhythmia related hospitalizations. Unfortunately, AF treatment is in practice complicated by the lack of understanding of the fundamental mechanisms underlying the arrhythmia, which makes detection of effective ablation targets particularly difficult. Various approaches to AF mapping have been explored in the hopes of better pinpointing these effective targets, such as Dominant Frequency (DF) analysis, complex fractionated electrograms (CFAE) and unipolar reconstruction (FIRM), but many of these methods have produced conflicting results or require further investigation. Exploration of AF using information theoretic-based approaches may have the potential to provide new insights into the complex system dynamics of AF, whilst also providing the benefit of being less reliant on empirically derived definitions in comparison to alternate mapping approaches. This work provides an overview of information theory and reviews its applications in AF analysis, with particular focus on AF mapping. The works discussed in this review demonstrate how understanding AF from a signal property perspective can provide new insights into the arrhythmic phenomena, which may have valuable clinical implications for AF mapping and ablation in the future.

Bi-dimensional multiscale entropy: Relation with discrete Fourier transform and biomedical application

The multiscale entropy (MSE_1D) measure is now widely used to quantify the complexity of time series. The development of complexity measures for images is also a long-standing goal. Recently, the bi-dimensional version of MSE_1D has been proposed (MSE_2D) to analyze images. The interpretation of MSE_2D curves and the applications to real data are still emergent. Because the coarse-graining step in the MSE_2D computation changes the frequency content of the image, we hypothesized a possible dependence between MSE_2D and the discrete Fourier transform (DFT). To analyze this dependence, synthetic as well as biomedical images are analyzed. Our results reveal that i) the profile of MSE_2D is sensitive to both the amplitude and phase of the DFT; ii) MSE_2D could find applications in the biomedical field. This work brings valuable information for MSE_2D interpretation and opens possibilities to study images from an entropy point of view through spatial scales.

Health Degradation Monitoring and Early Fault Diagnosis of a Rolling Bearing Based on CEEMDAN and Improved MMSE

Rolling bearings play a crucial role in rotary machinery systems, and their operating state affects the entire mechanical system. In most cases, the fault of a rolling bearing can only be identified when it has developed to a certain degree. At that moment, there is already not much time for maintenance, and could cause serious damage to the entire mechanical system. This paper proposes a novel approach to health degradation monitoring and early fault diagnosis of rolling bearings based on a complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) and improved multivariate multiscale sample entropy (MMSE). The smoothed coarse graining process was proposed to improve the conventional MMSE. Numerical simulation results indicate that CEEMDAN can alleviate the mode mixing problem and enable accurate intrinsic mode functions (IMFs), and improved MMSE can reflect intrinsic dynamic characteristics of the rolling bearing more accurately. During application studies, rolling bearing signals are decomposed by CEEMDAN to obtain IMFs. Then improved MMSE values of effective IMFs are computed to accomplish health degradation monitoring of rolling bearings, aiming at identifying the early weak fault phase. Afterwards, CEEMDAN is performed to extract the fault characteristic frequency during the early weak fault phase. The experimental results indicate the proposed method can obtain a better performance than other techniques in objective analysis, which demonstrates the effectiveness of the proposed method in practical application. The theoretical derivations, numerical simulations, and application studies all confirmed that the proposed health degradation monitoring and early fault diagnosis approach is promising in the field of prognostic and fault diagnosis of rolling bearings.

Complexity Analysis of Carbon Market Using the Modified Multi-Scale Entropy

Carbon markets provide a market-based way to reduce climate pollution. Subject to general market regulations, the major existing emission trading markets present complex characteristics. This paper analyzes the complexity of carbon market by using the multi-scale entropy. Pilot carbon markets in China are taken as the example. Moving average is adopted to extract the scales due to the short length of the data set. Results show a low-level complexity inferring that China's pilot carbon markets are quite immature in lack of market efficiency. However, the complexity varies in different time scales. China's carbon markets (except for the Chongqing pilot) are more complex in the short period than in the long term. Furthermore, complexity level in most pilot markets increases as the markets developed, showing an improvement in market efficiency. All these results demonstrate that an effective carbon market is required for the full function of emission trading.

A practical comparison of algorithms for the measurement of multiscale entropy in neural time series data

There is a broad family of statistical methods for capturing time series regularity, with increasingly widespread adoption by the neuroscientific community. A common feature of these methods is that they permit investigators to quantify the entropy of brain signals - an index of unpredictability/complexity. Despite the proliferation of algorithms for computing entropy from neural time series data there is scant evidence concerning their relative stability and efficiency. Here we evaluated several different algorithmic implementations (sample, fuzzy, dispersion and permutation) of multiscale entropy in terms of their stability across sessions, internal consistency and computational speed, accuracy and precision using a combination of electroencephalogram (EEG) and synthetic 1/ƒ noise signals. Overall, we report fair to excellent internal consistency and longitudinal stability over a one-week period for the majority of entropy estimates, with several caveats. Computational timing estimates suggest distinct advantages for dispersion and permutation entropy over other entropy estimates. Considered alongside the psychometric evidence, we suggest several ways in which researchers can maximize computational resources (without sacrificing reliability), especially when working with high-density M/EEG data or multivoxel BOLD time series signals.