Special Issue "Multiscale Entropy Approaches and Their Applications"

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Multidisciplinary Applications".

Deadline for manuscript submissions: closed (15 October 2019).

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Dr. Anne Humeau-Heurtier
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Guest Editor
Laboratoire Angevin de Recherche en Ingénierie des Systèmes (LARIS), University of Angers, IUT, GEII Department, 4 boulevard Lavoisier, BP 42018, 49016 Angers cedex, France
Interests: entropy; multiscale entropy; nonlinear analysis; empirical mode decomposition; biomedical data
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Special Issue Information

Dear Colleagues,

Multiscale entropy measures have been proposed from the beginning of the 2000s to evaluate the complexity of time series, by taking into account the multiple time scales in physical systems. Since then, these approaches have received a great deal of attention and have been used in a large range of applications. Multivariate approaches have also been developed.

The algorithms for a multiscale entropy approach are composed of two main steps: i) a coarse-graining procedure to represent the system’s dynamics on different scales; ii) the entropy computation for the original signal and for the coarse-grained time series to evaluate the irregularity for each scale. Moreover, different entropy measures have been associated with the coarse-graining approach, each one having its advantages and drawbacks: approximate entropy, sample entropy, permutation entropy, fuzzy entropy, distribution entropy, dispersion entropy, etc.

In this Special Issue, we would like to collect papers focusing on both the theory and applications of multiscale entropy approaches. Any kind of entropy measure is considered (see above).

The main topics of this Special Issue include (but are not limited to):

  • improvement of the coarse-graining concept
  • improvement in the entropy measure itself
  • applications of the multiscale approach on univariate or multivariate time series; one-dimensional, but also bi-dimensional data are welcome. Applications can include biomedical engineering, chemical engineering, hydrology, pharmaceutical sciences, financial analyses, neurosciences, industrial engineering, geosciences, information sciences, etc.

This issue is to continue with the first issue of Multiscale Entropy,

https://www.mdpi.com/journal/entropy/special_issues/multiscale_entropy

Dr. Anne Humeau-Heurtier
Guest Editor

Manuscript Submission Information

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Editorial

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Editorial
Multiscale Entropy Approaches and Their Applications
Entropy 2020, 22(6), 644; https://doi.org/10.3390/e22060644 - 10 Jun 2020
Cited by 3 | Viewed by 922
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)

Research

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Article
Multivariate and Multiscale Complexity of Long-Range Correlated Cardiovascular and Respiratory Variability Series
Entropy 2020, 22(3), 315; https://doi.org/10.3390/e22030315 - 11 Mar 2020
Cited by 4 | Viewed by 1371
Abstract
Assessing the dynamical complexity of biological time series represents an important topic with potential applications ranging from the characterization of physiological states and pathological conditions to the calculation of diagnostic parameters. In particular, cardiovascular time series exhibit a variability produced by different physiological [...] Read more.
Assessing the dynamical complexity of biological time series represents an important topic with potential applications ranging from the characterization of physiological states and pathological conditions to the calculation of diagnostic parameters. In particular, cardiovascular time series exhibit a variability produced by different physiological control mechanisms coupled with each other, which take into account several variables and operate across multiple time scales that result in the coexistence of short term dynamics and long-range correlations. The most widely employed technique to evaluate the dynamical complexity of a time series at different time scales, the so-called multiscale entropy (MSE), has been proven to be unsuitable in the presence of short multivariate time series to be analyzed at long time scales. This work aims at overcoming these issues via the introduction of a new method for the assessment of the multiscale complexity of multivariate time series. The method first exploits vector autoregressive fractionally integrated (VARFI) models to yield a linear parametric representation of vector stochastic processes characterized by short- and long-range correlations. Then, it provides an analytical formulation, within the theory of state-space models, of how the VARFI parameters change when the processes are observed across multiple time scales, which is finally exploited to derive MSE measures relevant to the overall multivariate process or to one constituent scalar process. The proposed approach is applied on cardiovascular and respiratory time series to assess the complexity of the heart period, systolic arterial pressure and respiration variability measured in a group of healthy subjects during conditions of postural and mental stress. Our results document that the proposed methodology can detect physiologically meaningful multiscale patterns of complexity documented previously, but can also capture significant variations in complexity which cannot be observed using standard methods that do not take into account long-range correlations. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Biometric Identification Method for Heart Sound Based on Multimodal Multiscale Dispersion Entropy
Entropy 2020, 22(2), 238; https://doi.org/10.3390/e22020238 - 20 Feb 2020
Cited by 11 | Viewed by 1183
Abstract
In this paper, a new method of biometric characterization of heart sounds based on multimodal multiscale dispersion entropy is proposed. Firstly, the heart sound is periodically segmented, and then each single-cycle heart sound is decomposed into a group of intrinsic mode functions (IMFs) [...] Read more.
In this paper, a new method of biometric characterization of heart sounds based on multimodal multiscale dispersion entropy is proposed. Firstly, the heart sound is periodically segmented, and then each single-cycle heart sound is decomposed into a group of intrinsic mode functions (IMFs) by improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN). These IMFs are then segmented to a series of frames, which is used to calculate the refine composite multiscale dispersion entropy (RCMDE) as the characteristic representation of heart sound. In the simulation experiments I, carried out on the open heart sounds database Michigan, Washington and Littman, the feature representation method was combined with the heart sound segmentation method based on logistic regression (LR) and hidden semi-Markov models (HSMM), and feature selection was performed through the Fisher ratio (FR). Finally, the Euclidean distance (ED) and the close principle are used for matching and identification, and the recognition accuracy rate was 96.08%. To improve the practical application value of this method, the proposed method was applied to 80 heart sounds database constructed by 40 volunteer heart sounds to discuss the effect of single-cycle heart sounds with different starting positions on performance in experiment II. The experimental results show that the single-cycle heart sound with the starting position of the start of the first heart sound (S1) has the highest recognition rate of 97.5%. In summary, the proposed method is effective for heart sound biometric recognition. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Application of Time-Scale Decomposition of Entropy for Eye Movement Analysis
Entropy 2020, 22(2), 168; https://doi.org/10.3390/e22020168 - 01 Feb 2020
Cited by 7 | Viewed by 994
Abstract
The methods for nonlinear time series analysis were used in the presented research to reveal eye movement signal characteristics. Three measures were used: approximate entropy, fuzzy entropy, and the Largest Lyapunov Exponent, for which the multilevel maps (MMs), being their time-scale decomposition, were [...] Read more.
The methods for nonlinear time series analysis were used in the presented research to reveal eye movement signal characteristics. Three measures were used: approximate entropy, fuzzy entropy, and the Largest Lyapunov Exponent, for which the multilevel maps (MMs), being their time-scale decomposition, were defined. To check whether the estimated characteristics might be useful in eye movement events detection, these structures were applied in the classification process conducted with the usage of the kNN method. The elements of three MMs were used to define feature vectors for this process. They consisted of differently combined MM segments, belonging either to one or several selected levels, as well as included values either of one or all the analysed measures. Such a classification produced an improvement in the accuracy for saccadic latency and saccade, when compared with the previously conducted studies using eye movement dynamics. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Complexity-Based Measures of Postural Sway during Walking at Different Speeds and Durations Using Multiscale Entropy
Entropy 2019, 21(11), 1128; https://doi.org/10.3390/e21111128 - 16 Nov 2019
Cited by 11 | Viewed by 1578
Abstract
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 [...] Read more.
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 min). 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 min, but not at 20 min. This study demonstrated an interaction between walking speeds and walking durations on the complexity of COP. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Voronoi Decomposition of Cardiovascular Dependency Structures in Different Ambient Conditions: An Entropy Study
Entropy 2019, 21(11), 1103; https://doi.org/10.3390/e21111103 - 11 Nov 2019
Cited by 4 | Viewed by 1176
Abstract
This paper proposes a method that maps the coupling strength of an arbitrary number of signals D, D ≥ 2, into a single time series. It is motivated by the inability of multiscale entropy to jointly analyze more than two signals. The [...] Read more.
This paper proposes a method that maps the coupling strength of an arbitrary number of signals D, D ≥ 2, into a single time series. It is motivated by the inability of multiscale entropy to jointly analyze more than two signals. The coupling strength is determined using the copula density defined over a [0 1]D copula domain. The copula domain is decomposed into the Voronoi regions, with volumes inversely proportional to the dependency level (coupling strength) of the observed joint signals. A stream of dependency levels, ordered in time, creates a new time series that shows the fluctuation of the signals’ coupling strength along the time axis. The composite multiscale entropy (CMSE) is then applied to three signals, systolic blood pressure (SBP), pulse interval (PI), and body temperature (tB), simultaneously recorded from rats exposed to different ambient temperatures (tA). The obtained results are consistent with the results from the classical studies, and the method itself offers more levels of freedom than the classical analysis. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Evidence for Maintained Post-Encoding Memory Consolidation Across the Adult Lifespan Revealed by Network Complexity
Entropy 2019, 21(11), 1072; https://doi.org/10.3390/e21111072 - 01 Nov 2019
Cited by 2 | Viewed by 1098
Abstract
Memory consolidation is well known to occur during sleep, but might start immediately after encoding new information while awake. While consolidation processes are important across the lifespan, they may be even more important to maintain memory functioning in old age. We tested whether [...] Read more.
Memory consolidation is well known to occur during sleep, but might start immediately after encoding new information while awake. While consolidation processes are important across the lifespan, they may be even more important to maintain memory functioning in old age. We tested whether a novel measure of information processing known as network complexity might be sensitive to post-encoding consolidation mechanisms in a sample of young, middle-aged, and older adults. Network complexity was calculated by assessing the irregularity of brain signals within a network over time using multiscale entropy. To capture post-encoding mechanisms, network complexity was estimated using functional magnetic resonance imaging (fMRI) during rest before and after encoding of picture pairs, and subtracted between the two rest periods. Participants received a five-alternative-choice memory test to assess associative memory performance. Results indicated that aging was associated with an increase in network complexity from pre- to post-encoding in the default mode network (DMN). Increases in network complexity in the DMN also were associated with better subsequent memory across all age groups. These findings suggest that network complexity is sensitive to post-encoding consolidation mechanisms that enhance memory performance. These post-encoding mechanisms may represent a pathway to support memory performance in the face of overall memory declines. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
A Study of Brain Neuronal and Functional Complexities Estimated Using Multiscale Entropy in Healthy Young Adults
Entropy 2019, 21(10), 995; https://doi.org/10.3390/e21100995 - 12 Oct 2019
Cited by 4 | Viewed by 1812
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
An Effective Bearing Fault Diagnosis Technique via Local Robust Principal Component Analysis and Multi-Scale Permutation Entropy
Entropy 2019, 21(10), 959; https://doi.org/10.3390/e21100959 - 30 Sep 2019
Cited by 8 | Viewed by 1257
Abstract
The acquired bearing fault signal usually reveals nonlinear and non-stationary nature. Moreover, in the actual environment, some other interference components and strong background noise are unavoidable, which lead to the fault feature signal being weak. Considering the above issues, an effective bearing fault [...] Read more.
The acquired bearing fault signal usually reveals nonlinear and non-stationary nature. Moreover, in the actual environment, some other interference components and strong background noise are unavoidable, which lead to the fault feature signal being weak. Considering the above issues, an effective bearing fault diagnosis technique via local robust principal component analysis (LRPCA) and multi-scale permutation entropy (MSPE) was introduced in this paper. Robust principal component analysis (RPCA) has proven to be a powerful de-noising method, which can extract a low-dimensional submanifold structure representing signal feature from the signal trajectory matrix. However, RPCA can only handle single-component signal. Therefore, in order to suppress background noise, an improved RPCA method named LRPCA is proposed to decompose the signal into several single-components. Since MSPE can efficiently evaluate the dynamic complexity and randomness of the signals under different scales, the fault-related single-components can be identified according the MPSE characteristic of the signals. Thereafter, these identified components are combined into a one-dimensional signal to represent the fault feature component for further diagnosis. The numerical simulation experimentation and the analysis of bearing outer race fault data both verified the effectiveness of the proposed technique. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Decomposition of a Multiscale Entropy Tensor for Sleep Stage Identification in Preterm Infants
Entropy 2019, 21(10), 936; https://doi.org/10.3390/e21100936 - 25 Sep 2019
Cited by 2 | Viewed by 1102
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Multivariate Multiscale Dispersion Entropy of Biomedical Times Series
Entropy 2019, 21(9), 913; https://doi.org/10.3390/e21090913 - 19 Sep 2019
Cited by 16 | Viewed by 1754
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Multi-Scale Heart Beat Entropy Measures for Mental Workload Assessment of Ambulant Users
Entropy 2019, 21(8), 783; https://doi.org/10.3390/e21080783 - 10 Aug 2019
Cited by 9 | Viewed by 1887
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Investigation of Linear and Nonlinear Properties of a Heartbeat Time Series Using Multiscale Rényi Entropy
Entropy 2019, 21(8), 727; https://doi.org/10.3390/e21080727 - 25 Jul 2019
Cited by 3 | Viewed by 1331
Abstract
The time series of interbeat intervals of the heart reveals much information about disease and disease progression. An area of intense research has been associated with cardiac autonomic neuropathy (CAN). In this work we have investigated the value of additional information derived from [...] Read more.
The time series of interbeat intervals of the heart reveals much information about disease and disease progression. An area of intense research has been associated with cardiac autonomic neuropathy (CAN). In this work we have investigated the value of additional information derived from the magnitude, sign and acceleration of the RR intervals. When quantified using an entropy measure, these time series show statistically significant differences between disease classes of Normal, Early CAN and Definite CAN. In addition, pathophysiological characteristics of heartbeat dynamics provide information not only on the change in the system using the first difference but also the magnitude and direction of the change measured by the second difference (acceleration) with respect to sequence length. These additional measures provide disease categories to be discriminated and could prove useful for non-invasive diagnosis and understanding changes in heart rhythm associated with CAN. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Quantifying the Multiscale Predictability of Financial Time Series by an Information-Theoretic Approach
Entropy 2019, 21(7), 684; https://doi.org/10.3390/e21070684 - 12 Jul 2019
Cited by 9 | Viewed by 1591
Abstract
Making predictions on the dynamics of time series of a system is a very interesting topic. A fundamental prerequisite of this work is to evaluate the predictability of the system over a wide range of time. In this paper, we propose an information-theoretic [...] Read more.
Making predictions on the dynamics of time series of a system is a very interesting topic. A fundamental prerequisite of this work is to evaluate the predictability of the system over a wide range of time. In this paper, we propose an information-theoretic tool, multiscale entropy difference (MED), to evaluate the predictability of nonlinear financial time series on multiple time scales. We discuss the predictability of the isolated system and open systems, respectively. Evidence from the analysis of the logistic map, Hénon map, and the Lorenz system manifests that the MED method is accurate, robust, and has a wide range of applications. We apply the new method to five-minute high-frequency data and the daily data of Chinese stock markets. Results show that the logarithmic change of stock price (logarithmic return) has a lower possibility of being predicted than the volatility. The logarithmic change of trading volume contributes significantly to the prediction of the logarithmic change of stock price on multiple time scales. The daily data are found to have a larger possibility of being predicted than the five-minute high-frequency data. This indicates that the arbitrage opportunity exists in the Chinese stock markets, which thus cannot be approximated by the effective market hypothesis (EMH). Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
On the Robustness of Multiscale Indices for Long-Term Monitoring in Cardiac Signals
Entropy 2019, 21(6), 594; https://doi.org/10.3390/e21060594 - 15 Jun 2019
Cited by 3 | Viewed by 1416
Abstract
The identification of patients with increased risk of Sudden Cardiac Death (SCD) has been widely studied during recent decades, and several quantitative measurements have been proposed from the analysis of the electrocardiogram (ECG) stored in 1-day Holter recordings. Indices based on nonlinear dynamics [...] Read more.
The identification of patients with increased risk of Sudden Cardiac Death (SCD) has been widely studied during recent decades, and several quantitative measurements have been proposed from the analysis of the electrocardiogram (ECG) stored in 1-day Holter recordings. Indices based on nonlinear dynamics of Heart Rate Variability (HRV) have shown to convey predictive information in terms of factors related with the cardiac regulation by the autonomous nervous system, and among them, multiscale methods aim to provide more complete descriptions than single-scale based measures. However, there is limited knowledge on the suitability of nonlinear measurements to characterize the cardiac dynamics in current long-term monitoring scenarios of several days. Here, we scrutinized the long-term robustness properties of three nonlinear methods for HRV characterization, namely, the Multiscale Entropy (MSE), the Multiscale Time Irreversibility (MTI), and the Multifractal Spectrum (MFS). These indices were selected because all of them have been theoretically designed to take into account the multiple time scales inherent in healthy and pathological cardiac dynamics, and they have been analyzed so far when monitoring up to 24 h of ECG signals, corresponding to about 20 time scales. We analyzed them in 7-day Holter recordings from two data sets, namely, patients with Atrial Fibrillation and with Congestive Heart Failure, by reaching up to 100 time scales. In addition, a new comparison procedure is proposed to statistically compare the poblational multiscale representations in different patient or processing conditions, in terms of the non-parametric estimation of confidence intervals for the averaged median differences. Our results show that variance reduction is actually obtained in the multiscale estimators. The MSE (MTI) exhibited the lowest (largest) bias and variance at large scales, whereas all the methods exhibited a consistent description of the large-scale processes in terms of multiscale index robustness. In all the methods, the used algorithms could turn to give some inconsistency in the multiscale profile, which was checked not to be due to the presence of artifacts, but rather with unclear origin. The reduction in standard error for several-day recordings compared to one-day recordings was more evident in MSE, whereas bias was more patently present in MFS. Our results pave the way of these techniques towards their use, with improved algorithmic implementations and nonparametric statistical tests, in long-term cardiac Holter monitoring scenarios. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
On the Statistical Properties of Multiscale Permutation Entropy: Characterization of the Estimator’s Variance
Entropy 2019, 21(5), 450; https://doi.org/10.3390/e21050450 - 30 Apr 2019
Cited by 8 | Viewed by 1562
Abstract
Permutation Entropy (PE) and Multiscale Permutation Entropy (MPE) have been extensively used in the analysis of time series searching for regularities. Although PE has been explored and characterized, there is still a lack of theoretical background regarding MPE. Therefore, we expand the available [...] Read more.
Permutation Entropy (PE) and Multiscale Permutation Entropy (MPE) have been extensively used in the analysis of time series searching for regularities. Although PE has been explored and characterized, there is still a lack of theoretical background regarding MPE. Therefore, we expand the available MPE theory by developing an explicit expression for the estimator’s variance as a function of time scale and ordinal pattern distribution. We derived the MPE Cramér–Rao Lower Bound (CRLB) to test the efficiency of our theoretical result. We also tested our formulation against MPE variance measurements from simulated surrogate signals. We found the MPE variance symmetric around the point of equally probable patterns, showing clear maxima and minima. This implies that the MPE variance is directly linked to the MPE measurement itself, and there is a region where the variance is maximum. This effect arises directly from the pattern distribution, and it is unrelated to the time scale or the signal length. The MPE variance also increases linearly with time scale, except when the MPE measurement is close to its maximum, where the variance presents quadratic growth. The expression approaches the CRLB asymptotically, with fast convergence. The theoretical variance is close to the results from simulations, and appears consistently below the actual measurements. By knowing the MPE variance, it is possible to have a clear precision criterion for statistical comparison in real-life applications. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Development of Postural Stability Index to Distinguish Different Stability States
Entropy 2019, 21(3), 314; https://doi.org/10.3390/e21030314 - 22 Mar 2019
Cited by 8 | Viewed by 1516
Abstract
A key factor for fall prevention involves understanding the pathophysiology of stability. This study proposes the postural stability index (PSI), which is a novel measure to quantify different stability states on healthy subjects. The results of the x-, y-, and z-axes of the [...] Read more.
A key factor for fall prevention involves understanding the pathophysiology of stability. This study proposes the postural stability index (PSI), which is a novel measure to quantify different stability states on healthy subjects. The results of the x-, y-, and z-axes of the acceleration signals were analyzed from 10 healthy young adults and 10 healthy older adults under three conditions as follows: Normal walking, walking with obstacles, and fall-like motions. The ensemble empirical mode decomposition (EEMD) was used to reconstruct the acceleration signal data. Wearable accelerometers were located on the ankles and knees of the subjects. The PSI indicated a decreasing trend of its values from normal walking to the fall-like motions. Free-walking data were used to determine the stability based on the PSI. The segmented free-walking data indicated changes in the stability states that suggested that the PSI is potentially helpful in quantifying gait stability. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
An Improved Method of Handling Missing Values in the Analysis of Sample Entropy for Continuous Monitoring of Physiological Signals
Entropy 2019, 21(3), 274; https://doi.org/10.3390/e21030274 - 12 Mar 2019
Cited by 10 | Viewed by 2103
Abstract
Medical devices generate huge amounts of continuous time series data. However, missing values commonly found in these data can prevent us from directly using analytic methods such as sample entropy to reveal the information contained in these data. To minimize the influence of [...] Read more.
Medical devices generate huge amounts of continuous time series data. However, missing values commonly found in these data can prevent us from directly using analytic methods such as sample entropy to reveal the information contained in these data. To minimize the influence of missing points on the calculation of sample entropy, we propose a new method to handle missing values in continuous time series data. We use both experimental and simulated datasets to compare the performance (in percentage error) of our proposed method with three currently used methods: skipping the missing values, linear interpolation, and bootstrapping. Unlike the methods that involve modifying the input data, our method modifies the calculation process. This keeps the data unchanged which is less intrusive to the structure of the data. The results demonstrate that our method has a consistent lower average percentage error than other three commonly used methods in multiple common physiological signals. For missing values in common physiological signal type, different data size and generating mechanism, our method can more accurately extract the information contained in continuously monitored data than traditional methods. So it may serve as an effective tool for handling missing values and may have broad utility in analyzing sample entropy for common physiological signals. This could help develop new tools for disease diagnosis and evaluation of treatment effects. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Multiscale Entropy Analysis of Page Views: A Case Study of Wikipedia
Entropy 2019, 21(3), 229; https://doi.org/10.3390/e21030229 - 27 Feb 2019
Cited by 3 | Viewed by 1907
Abstract
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) [...] Read more.
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. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Multiscale Entropy Quantifies the Differential Effect of the Medium Embodiment on Older Adults Prefrontal Cortex during the Story Comprehension: A Comparative Analysis
Entropy 2019, 21(2), 199; https://doi.org/10.3390/e21020199 - 19 Feb 2019
Cited by 7 | Viewed by 1884
Abstract
Todays’ communication media virtually impact and transform every aspect of our daily communication and yet the extent of their embodiment on our brain is unexplored. The study of this topic becomes more crucial, considering the rapid advances in such fields as socially assistive [...] Read more.
Todays’ communication media virtually impact and transform every aspect of our daily communication and yet the extent of their embodiment on our brain is unexplored. The study of this topic becomes more crucial, considering the rapid advances in such fields as socially assistive robotics that envision the use of intelligent and interactive media for providing assistance through social means. In this article, we utilize the multiscale entropy (MSE) to investigate the effect of the physical embodiment on the older people’s prefrontal cortex (PFC) activity while listening to stories. We provide evidence that physical embodiment induces a significant increase in MSE of the older people’s PFC activity and that such a shift in the dynamics of their PFC activation significantly reflects their perceived feeling of fatigue. Our results benefit researchers in age-related cognitive function and rehabilitation who seek for the adaptation of these media in robot-assistive cognitive training of the older people. In addition, they offer a complementary information to the field of human-robot interaction via providing evidence that the use of MSE can enable the interactive learning algorithms to utilize the brain’s activation patterns as feedbacks for improving their level of interactivity, thereby forming a stepping stone for rich and usable human mental model. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Partial Discharge Fault Diagnosis Based on Multi-Scale Dispersion Entropy and a Hypersphere Multiclass Support Vector Machine
Entropy 2019, 21(1), 81; https://doi.org/10.3390/e21010081 - 17 Jan 2019
Cited by 12 | Viewed by 1724
Abstract
Partial discharge (PD) fault analysis is an important tool for insulation condition diagnosis of electrical equipment. In order to conquer the limitations of traditional PD fault diagnosis, a novel feature extraction approach based on variational mode decomposition (VMD) and multi-scale dispersion entropy (MDE) [...] Read more.
Partial discharge (PD) fault analysis is an important tool for insulation condition diagnosis of electrical equipment. In order to conquer the limitations of traditional PD fault diagnosis, a novel feature extraction approach based on variational mode decomposition (VMD) and multi-scale dispersion entropy (MDE) is proposed. Besides, a hypersphere multiclass support vector machine (HMSVM) is used for PD pattern recognition with extracted PD features. Firstly, the original PD signal is decomposed with VMD to obtain intrinsic mode functions (IMFs). Secondly proper IMFs are selected according to central frequency observation and MDE values in each IMF are calculated. And then principal component analysis (PCA) is introduced to extract effective principle components in MDE. Finally, the extracted principle factors are used as PD features and sent to HMSVM classifier. Experiment results demonstrate that, PD feature extraction method based on VMD-MDE can extract effective characteristic parameters that representing dominant PD features. Recognition results verify the effectiveness and superiority of the proposed PD fault diagnosis method. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Performance Evaluation of an Entropy-Based Structural Health Monitoring System Utilizing Composite Multiscale Cross-Sample Entropy
Entropy 2019, 21(1), 41; https://doi.org/10.3390/e21010041 - 09 Jan 2019
Cited by 9 | Viewed by 1534
Abstract
The aim of this study was to develop an entropy-based structural health monitoring system for solving the problem of unstable entropy values observed when multiscale cross-sample entropy (MSCE) is employed to assess damage in real structures. Composite MSCE was utilized to enhance the [...] Read more.
The aim of this study was to develop an entropy-based structural health monitoring system for solving the problem of unstable entropy values observed when multiscale cross-sample entropy (MSCE) is employed to assess damage in real structures. Composite MSCE was utilized to enhance the reliability of entropy values on every scale. Additionally, the first mode of a structure was extracted using ensemble empirical mode decomposition to conduct entropy analysis and evaluate the accuracy of damage assessment. A seven-story model was created to validate the efficiency of the proposed method and the damage index. Subsequently, an experiment was conducted on a seven-story steel benchmark structure including 15 damaged cases to compare the numerical and experimental models. A confusion matrix was applied to classify the results and evaluate the performance over three indices: accuracy, precision, and recall. The results revealed the feasibility of the modified structural health monitoring system and demonstrated its potential in the field of long-term monitoring. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Complexity of Frontal Cortex fNIRS Can Support Alzheimer Disease Diagnosis in Memory and Visuo-Spatial Tests
Entropy 2019, 21(1), 26; https://doi.org/10.3390/e21010026 - 01 Jan 2019
Cited by 23 | Viewed by 2847
Abstract
Decline in visuo-spatial skills and memory failures are considered symptoms of Alzheimer’s Disease (AD) and they can be assessed at early stages employing clinical tests. However, performance in a single test is generally not indicative of AD. Functional neuroimaging, such as functional Near [...] Read more.
Decline in visuo-spatial skills and memory failures are considered symptoms of Alzheimer’s Disease (AD) and they can be assessed at early stages employing clinical tests. However, performance in a single test is generally not indicative of AD. Functional neuroimaging, such as functional Near Infrared Spectroscopy (fNIRS), may be employed during these tests in an ecological setting to support diagnosis. Indeed, neuroimaging should not alter clinical practice allowing free doctor-patient interaction. However, block-designed paradigms, necessary for standard functional neuroimaging analysis, require tests adaptation. Novel signal analysis procedures (e.g., signal complexity evaluation) may be useful to establish brain signals differences without altering experimental conditions. In this study, we estimated fNIRS complexity (through Sample Entropy metric) in frontal cortex of early AD and controls during three tests that assess visuo-spatial and short-term-memory abilities (Clock Drawing Test, Digit Span Test, Corsi Block Tapping Test). A channel-based analysis of fNIRS complexity during the tests revealed AD-induced changes. Importantly, a multivariate analysis of fNIRS complexity provided good specificity and sensitivity to AD. This outcome was compared to cognitive tests performances that were predictive of AD in only one test. Our results demonstrated the capabilities of fNIRS and complexity metric to support early AD diagnosis. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Article
Multiscale Distribution Entropy Analysis of Short-Term Heart Rate Variability
Entropy 2018, 20(12), 952; https://doi.org/10.3390/e20120952 - 11 Dec 2018
Cited by 12 | Viewed by 2867
Abstract
Electrocardiogram (ECG) signal has been commonly used to analyze the complexity of heart rate variability (HRV). For this, various entropy methods have been considerably of interest. The multiscale entropy (MSE) method, which makes use of the sample entropy (SampEn) calculation of coarse-grained time [...] Read more.
Electrocardiogram (ECG) signal has been commonly used to analyze the complexity of heart rate variability (HRV). For this, various entropy methods have been considerably of interest. The multiscale entropy (MSE) method, which makes use of the sample entropy (SampEn) calculation of coarse-grained time series, has attracted attention for analysis of HRV. However, the SampEn computation may fail to be defined when the length of a time series is not enough long. Recently, distribution entropy (DistEn) with improved stability for a short-term time series has been proposed. Here, we propose a novel multiscale DistEn (MDE) for analysis of the complexity of short-term HRV by utilizing a moving-averaging multiscale process and the DistEn computation of each moving-averaged time series. Thus, it provides an improved stability of entropy evaluation for short-term HRV extracted from ECG. To verify the performance of MDE, we employ the analysis of synthetic signals and confirm the superiority of MDE over MSE. Then, we evaluate the complexity of short-term HRV extracted from ECG signals of congestive heart failure (CHF) patients and healthy subjects. The experimental results exhibit that MDE is capable of quantifying the decreased complexity of HRV with aging and CHF disease with short-term HRV time series. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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Review

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Review
(Multiscale) Cross-Entropy Methods: A Review
Entropy 2020, 22(1), 45; https://doi.org/10.3390/e22010045 - 29 Dec 2019
Cited by 14 | Viewed by 1994
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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