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Multiscale Entropy Approaches and Their Applications II

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

Deadline for manuscript submissions: closed (17 May 2021) | Viewed by 26679

Special Issue Editor

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.

Furthermore, recently entropy measures for multidimensional data (images, volumes) have been proposed.

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 and multidimensional data are welcome. Applications can include biomedical engineering, chemical engineering, hydrology, pharmaceutical sciences, financial analyses, neurosciences, industrial engineering, geosciences, information sciences, etc.

Dr. Anne Humeau-Heurtier
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (12 papers)

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Research

15 pages, 5108 KiB  
Article
Multiscale Entropy Analysis of Short Signals: The Robustness of Fuzzy Entropy-Based Variants Compared to Full-Length Long Signals
by Airton Monte Serrat Borin, Jr., Anne Humeau-Heurtier, Luiz Eduardo Virgílio Silva and Luiz Otávio Murta, Jr.
Entropy 2021, 23(12), 1620; https://doi.org/10.3390/e23121620 - 1 Dec 2021
Cited by 8 | Viewed by 1547
Abstract
Multiscale entropy (MSE) analysis is a fundamental approach to access the complexity of a time series by estimating its information creation over a range of temporal scales. However, MSE may not be accurate or valid for short time series. This is why previous [...] Read more.
Multiscale entropy (MSE) analysis is a fundamental approach to access the complexity of a time series by estimating its information creation over a range of temporal scales. However, MSE may not be accurate or valid for short time series. This is why previous studies applied different kinds of algorithm derivations to short-term time series. However, no study has systematically analyzed and compared their reliabilities. This study compares the MSE algorithm variations adapted to short time series on both human and rat heart rate variability (HRV) time series using long-term MSE as reference. The most used variations of MSE are studied: composite MSE (CMSE), refined composite MSE (RCMSE), modified MSE (MMSE), and their fuzzy versions. We also analyze the errors in MSE estimations for a range of incorporated fuzzy exponents. The results show that fuzzy MSE versions—as a function of time series length—present minimal errors compared to the non-fuzzy algorithms. The traditional multiscale entropy algorithm with fuzzy counting (MFE) has similar accuracy to alternative algorithms with better computing performance. For the best accuracy, the findings suggest different fuzzy exponents according to the time series length. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications II)
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14 pages, 691 KiB  
Article
The Impact of Linear Filter Preprocessing in the Interpretation of Permutation Entropy
by Antonio Dávalos, Meryem Jabloun, Philippe Ravier and Olivier Buttelli
Entropy 2021, 23(7), 787; https://doi.org/10.3390/e23070787 - 22 Jun 2021
Cited by 6 | Viewed by 2297
Abstract
Permutation Entropy (PE) is a powerful tool for measuring the amount of information contained within a time series. However, this technique is rarely applied directly on raw signals. Instead, a preprocessing step, such as linear filtering, is applied in order to remove noise [...] Read more.
Permutation Entropy (PE) is a powerful tool for measuring the amount of information contained within a time series. However, this technique is rarely applied directly on raw signals. Instead, a preprocessing step, such as linear filtering, is applied in order to remove noise or to isolate specific frequency bands. In the current work, we aimed at outlining the effect of linear filter preprocessing in the final PE values. By means of the Wiener–Khinchin theorem, we theoretically characterize the linear filter’s intrinsic PE and separated its contribution from the signal’s ordinal information. We tested these results by means of simulated signals, subject to a variety of linear filters such as the moving average, Butterworth, and Chebyshev type I. The PE results from simulations closely resembled our predicted results for all tested filters, which validated our theoretical propositions. More importantly, when we applied linear filters to signals with inner correlations, we were able to theoretically decouple the signal-specific contribution from that induced by the linear filter. Therefore, by providing a proper framework of PE linear filter characterization, we improved the PE interpretation by identifying possible artifact information introduced by the preprocessing steps. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications II)
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14 pages, 1989 KiB  
Article
Influence of Ectopic Beats on Heart Rate Variability Analysis
by Lina Zhao, Peng Li, Jianqing Li and Chengyu Liu
Entropy 2021, 23(6), 648; https://doi.org/10.3390/e23060648 - 22 May 2021
Cited by 7 | Viewed by 2835
Abstract
The analysis of heart rate variability (HRV) plays a dominant role in the study of physiological signal variability. HRV reflects the information of the adjustment of sympathetic and parasympathetic nerves on the cardiovascular system and, thus, is widely used to evaluate the functional [...] Read more.
The analysis of heart rate variability (HRV) plays a dominant role in the study of physiological signal variability. HRV reflects the information of the adjustment of sympathetic and parasympathetic nerves on the cardiovascular system and, thus, is widely used to evaluate the functional status of the cardiovascular system. Ectopic beats may affect the analysis of HRV. However, the quantitative relationship between the burden of ectopic beats and HRV indices, including entropy measures, has not yet been investigated in depth. In this work, we analyzed the effects of different numbers of ectopic beats on several widely accepted HRV parameters in time-domain (SDNN), frequency-domain (LF/HF), as well as non-linear features (SampEn and Pt-SampEn (physical threshold-based SampEn)). The results showed that all four indices were influenced by ectopic beats, and the degree of influence was roughly increased with the increase of the number of ectopic beats. Ectopic beats had the greatest impact on the frequency domain index LF/HF, whereas the Pt-SampEn was minimally accepted by ectopic beats. These results also indicated that, compared with the other three indices, Pt-SampEn had better robustness for ectopic beats. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications II)
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10 pages, 2090 KiB  
Article
Optimal Frequency and Amplitude of Vertical Viewpoint Oscillation for Improving Vection Strength and Reducing Neural Constrains on Gait
by Wei Wang, Kaiming Yang and Yu Zhu
Entropy 2021, 23(5), 541; https://doi.org/10.3390/e23050541 - 28 Apr 2021
Viewed by 1715
Abstract
Inducing self-motion illusions referred as vection are critical for improving the sensation of walking in virtual environments (VE). Adding viewpoint oscillations to a constant forward velocity in VE is effective for improving vection strength under static conditions. However, the effects of oscillation frequency [...] Read more.
Inducing self-motion illusions referred as vection are critical for improving the sensation of walking in virtual environments (VE). Adding viewpoint oscillations to a constant forward velocity in VE is effective for improving vection strength under static conditions. However, the effects of oscillation frequency and amplitude on vection strength under treadmill walking conditions are still unclear. Besides, due to the visuomotor entrainment mechanism, these visual oscillations would affect gait patterns and be detrimental for achieving natural walking if not properly designed. This study was aimed at determining the optimal frequency and amplitude of vertical viewpoint oscillations for improving vection strength and reducing gait constraints. Seven subjects walked on a treadmill while watching a visual scene. The visual scene presented a constant forward velocity equal to the treadmill velocity with different vertical viewpoint oscillations added. Five oscillation patterns with different combinations of frequency and amplitude were tested. Subjects gave verbal ratings of vection strength. The mediolateral (M-L) center of pressure (CoP) complexity was calculated to indicate gait constraints. After the experiment, subjects were asked to give the best and the worst oscillation pattern based on their walking experience. The oscillation frequency and amplitude had strong positive correlations with vection strength. The M-L CoP complexity was reduced under oscillations with low frequency. The medium oscillation amplitude had greater M-L CoP complexity than the small and large amplitude. Besides, subjects preferred those oscillation patterns with large gait complexity. We suggested that the oscillation amplitude with largest M-L CoP complexity should first be chosen to reduce gait constraints. Then, increasing the oscillation frequency to improve vection strength until individual preference or the boundary of motion sickness. These findings provide important guidelines to promote the sensation of natural walking in VE. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications II)
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16 pages, 7568 KiB  
Article
Refined Composite Multi-Scale Reverse Weighted Permutation Entropy and Its Applications in Ship-Radiated Noise
by Yuxing Li, Bo Geng and Shangbin Jiao
Entropy 2021, 23(4), 476; https://doi.org/10.3390/e23040476 - 17 Apr 2021
Cited by 10 | Viewed by 2233
Abstract
Ship-radiated noise is one of the important signal types under the complex ocean background, which can well reflect physical properties of ships. As one of the valid measures to characterize the complexity of ship-radiated noise, permutation entropy (PE) has the advantages of high [...] Read more.
Ship-radiated noise is one of the important signal types under the complex ocean background, which can well reflect physical properties of ships. As one of the valid measures to characterize the complexity of ship-radiated noise, permutation entropy (PE) has the advantages of high efficiency and simple calculation. However, PE has the problems of missing amplitude information and single scale. To address the two drawbacks, refined composite multi-scale reverse weighted PE (RCMRWPE), as a novel measurement technology of describing the signal complexity, is put forward based on refined composite multi-scale processing (RCMP) and reverse weighted PE (RWPE). RCMP is an improved method of coarse-graining, which not only solves the problem of single scale, but also improves the stability of traditional coarse-graining; RWPE has been proposed more recently, and has better inter-class separability and robustness performance to noise than PE, weighted PE (WPE), and reverse PE (RPE). Additionally, a feature extraction scheme of ship-radiated noise is proposed based on RCMRWPE, furthermore, RCMRWPE is combined with discriminant analysis classifier (DAC) to form a new classification method. After that, a large number of comparative experiments of feature extraction schemes and classification methods with two artificial random signals and six ship-radiated noise are carried out, which show that the proposed feature extraction scheme has better performance in distinguishing ability and stability than the other three similar feature extraction schemes based on multi-scale PE (MPE), multi-scale WPE (MWPE), and multi-scale RPE (MRPE), and the proposed classification method also has the highest recognition rate. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications II)
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11 pages, 13077 KiB  
Article
Using Bidimensional Multiscale Entropy Analysis of Ultrasound Images to Assess the Effect of Various Walking Intensities on Plantar Soft Tissues
by Ben-Yi Liau, Fu-Lien Wu, Keying Zhang, Chi-Wen Lung, Chunmei Cao and Yih-Kuen Jan
Entropy 2021, 23(3), 264; https://doi.org/10.3390/e23030264 - 24 Feb 2021
Cited by 4 | Viewed by 2115
Abstract
Walking performance is usually assessed by linear analysis of walking outcome measures. However, human movements consist of both linear and nonlinear complexity components. The purpose of this study was to use bidimensional multiscale entropy analysis of ultrasound images to evaluate the effects of [...] Read more.
Walking performance is usually assessed by linear analysis of walking outcome measures. However, human movements consist of both linear and nonlinear complexity components. The purpose of this study was to use bidimensional multiscale entropy analysis of ultrasound images to evaluate the effects of various walking intensities on plantar soft tissues. Twelve participants were recruited to perform six walking protocols, consisting of three speeds (slow at 1.8 mph, moderate at 3.6 mph, and fast at 5.4 mph) for two durations (10 and 20 min). A B-mode ultrasound was used to assess plantar soft tissues before and after six walking protocols. Bidimensional multiscale entropy (MSE2D) and the Complexity Index (CI) were used to quantify the changes in irregularity of the ultrasound images of the plantar soft tissues. The results showed that the CI of ultrasound images after 20 min walking increased when compared to before walking (CI4: 0.39 vs. 0.35; CI5: 0.48 vs. 0.43, p < 0.05). When comparing 20 and 10 min walking protocols at 3.6 mph, the CI was higher after 20 min walking than after 10 min walking (CI4: 0.39 vs. 0.36, p < 0.05; and CI5: 0.48 vs. 0.44, p < 0.05). This is the first study to use bidimensional multiscale entropy analysis of ultrasound images to assess plantar soft tissues after various walking intensities. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications II)
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9 pages, 479 KiB  
Article
Multiscale Sample Entropy of Two-Dimensional Decaying Turbulence
by Ildoo Kim
Entropy 2021, 23(2), 245; https://doi.org/10.3390/e23020245 - 20 Feb 2021
Cited by 7 | Viewed by 1776
Abstract
Multiscale sample entropy analysis has been developed to quantify the complexity and the predictability of a time series, originally developed for physiological time series. In this study, the analysis was applied to the turbulence data. We measured time series data for the velocity [...] Read more.
Multiscale sample entropy analysis has been developed to quantify the complexity and the predictability of a time series, originally developed for physiological time series. In this study, the analysis was applied to the turbulence data. We measured time series data for the velocity fluctuation, in either the longitudinal or transverse direction, of turbulent soap film flows at various locations. The research was to assess the feasibility of using the entropy analysis to qualitatively characterize turbulence, without using any conventional energetic analysis of turbulence. The study showed that the application of the entropy analysis to the turbulence data is promising. From the analysis, we successfully captured two important features of the turbulent soap films. It is indicated that the turbulence is anisotropic from the directional disparity. In addition, we observed that the most unpredictable time scale increases with the downstream distance, which is an indication of the decaying turbulence. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications II)
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17 pages, 6561 KiB  
Article
Composite Multiscale Cross-Sample Entropy Analysis for Long-Term Structural Health Monitoring of Residential Buildings
by Tzu-Kang Lin and Dong-You Lee
Entropy 2021, 23(1), 60; https://doi.org/10.3390/e23010060 - 31 Dec 2020
Cited by 2 | Viewed by 2234
Abstract
This study proposesd a novel, entropy-based structural health monitoring (SHM) system for measuring microvibration signals generated by actual buildings. A structural health diagnosis interface was established for demonstration purposes. To enhance the reliability and accuracy of entropy evaluation at various scales, composite multiscale [...] Read more.
This study proposesd a novel, entropy-based structural health monitoring (SHM) system for measuring microvibration signals generated by actual buildings. A structural health diagnosis interface was established for demonstration purposes. To enhance the reliability and accuracy of entropy evaluation at various scales, composite multiscale cross-sample entropy (CMSCE) was adopted to increase the number of coarse-grained time series. The degree of similarity and asynchrony between ambient vibration signals measured on adjacent floors was used as an in-dicator for structural health assessment. A residential building that has been monitored since 1994 was selected for long-term monitoring. The accumulated database, including both the earthquake and ambient vibrations in each seismic event, provided the possibility to evaluate the practicability of the CMSCE-based method. Entropy curves obtained for each of the years, as well as the stable trend of the corresponding damage index (DI) graphs, demonstrated the relia-bility of the proposed SHM system. Moreover, two large earthquake events that occurred near the monitoring site were analyzed. The results revealed that the entropy values may have been slightly increased after the earthquakes. Positive DI values were obtained for higher floors, which could provide an early warning of structural instability. The proposed SHM system is highly stable and practical. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications II)
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19 pages, 1166 KiB  
Article
Improvement of Statistical Performance of Ordinal Multiscale Entropy Techniques Using Refined Composite Downsampling Permutation Entropy
by Antonio Dávalos, Meryem Jabloun, Philippe Ravier and Olivier Buttelli
Entropy 2021, 23(1), 30; https://doi.org/10.3390/e23010030 - 28 Dec 2020
Cited by 4 | Viewed by 1913
Abstract
Multiscale Permutation Entropy (MPE) analysis is a powerful ordinal tool in the measurement of information content of time series. MPE refinements, such as Composite MPE (cMPE) and Refined Composite MPE (rcMPE), greatly increase the precision of the entropy estimation by modifying the original [...] Read more.
Multiscale Permutation Entropy (MPE) analysis is a powerful ordinal tool in the measurement of information content of time series. MPE refinements, such as Composite MPE (cMPE) and Refined Composite MPE (rcMPE), greatly increase the precision of the entropy estimation by modifying the original method. Nonetheless, these techniques have only been proposed as algorithms, and are yet to be described from the theoretical perspective. Therefore, the purpose of this article is two-fold. First, we develop the statistical theory behind cMPE and rcMPE. Second, we propose an alternative method, Refined Composite Downsampling Permutation Entropy (rcDPE) to further increase the entropy estimation’s precision. Although cMPE and rcMPE outperform MPE when applied on uncorrelated noise, the results are higher than our predictions due to inherent redundancies found in the composite algorithms. The rcDPE method, on the other hand, not only conforms to our theoretical predictions, but also greatly improves over the other methods, showing the smallest bias and variance. By using MPE, rcMPE and rcDPE to classify faults in bearing vibration signals, rcDPE outperforms the multiscaling methods, enhancing the difference between faulty and non-faulty signals, provided we apply a proper anti-aliasing low-pass filter at each time scale. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications II)
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12 pages, 2707 KiB  
Article
Effect of Different Local Vibration Frequencies on the Multiscale Regularity of Plantar Skin Blood Flow
by Fuyuan Liao, Keying Zhang, Lingling Zhou, Yanni Chen, Jeannette Elliott and Yih-Kuen Jan
Entropy 2020, 22(11), 1288; https://doi.org/10.3390/e22111288 - 13 Nov 2020
Cited by 6 | Viewed by 2210
Abstract
Local vibration has shown promise in improving skin blood flow (SBF). However, there is no consensus on the selection of the best vibration frequency. An important reason may be that previous studies utilized time- and frequency-domain parameters to characterize vibration-induced SBF responses. These [...] Read more.
Local vibration has shown promise in improving skin blood flow (SBF). However, there is no consensus on the selection of the best vibration frequency. An important reason may be that previous studies utilized time- and frequency-domain parameters to characterize vibration-induced SBF responses. These parameters are unable to characterize the structural features of the SBF response to local vibrations, thus contributing to the inconsistent findings seen in vibration research. The objective of this study was to provide evidence that nonlinear dynamics of SBF responses would be an important aspect for assessing the effect of local vibration on SBF. Local vibrations at 100 Hz, 35 Hz, and 0 Hz (sham vibration) with an amplitude of 1 mm were randomly applied to the right first metatarsal head of 12 healthy participants for 10 min. SBF at the same site was measured for 10 min before and after local vibration. The degree of regularity of SBF was quantified using a multiscale sample entropy algorithm. The results showed that 100 Hz vibration significantly increased multiscale regularity of SBF but 35 Hz and 0 Hz (sham vibration) did not. The significant increase of regularity of SBF after 100 Hz vibration was mainly attributed to increased regularity of SBF oscillations within the frequency interval at 0.0095–0.15 Hz. These findings support the use of multiscale regularity to assess effectiveness of local vibration on improving skin blood flow. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications II)
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19 pages, 766 KiB  
Article
Construction and Application of Functional Brain Network Based on Entropy
by Lingyun Zhang, Taorong Qiu, Zhiqiang Lin, Shuli Zou and Xiaoming Bai
Entropy 2020, 22(11), 1234; https://doi.org/10.3390/e22111234 - 30 Oct 2020
Cited by 6 | Viewed by 2325
Abstract
Functional brain network (FBN) is an intuitive expression of the dynamic neural activity interaction between different neurons, neuron clusters, or cerebral cortex regions. It can characterize the brain network topology and dynamic properties. The method of building an FBN to characterize the features [...] Read more.
Functional brain network (FBN) is an intuitive expression of the dynamic neural activity interaction between different neurons, neuron clusters, or cerebral cortex regions. It can characterize the brain network topology and dynamic properties. The method of building an FBN to characterize the features of the brain network accurately and effectively is a challenging subject. Entropy can effectively describe the complexity, non-linearity, and uncertainty of electroencephalogram (EEG) signals. As a relatively new research direction, the research of the FBN construction method based on EEG data of fatigue driving has broad prospects. Therefore, it is of great significance to study the entropy-based FBN construction. We focus on selecting appropriate entropy features to characterize EEG signals and construct an FBN. On the real data set of fatigue driving, FBN models based on different entropies are constructed to identify the state of fatigue driving. Through analyzing network measurement indicators, the experiment shows that the FBN model based on fuzzy entropy can achieve excellent classification recognition rate and good classification stability. In addition, when compared with the other model based on the same data set, our model could obtain a higher accuracy and more stable classification results even if the length of the intercepted EEG signal is different. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications II)
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15 pages, 1417 KiB  
Article
Revised Stability Scales of the Postural Stability Index for Human Daily Activities
by Yu Ping Chang, Bernard C. Jiang and Nurul Retno Nurwulan
Entropy 2020, 22(10), 1188; https://doi.org/10.3390/e22101188 - 21 Oct 2020
Cited by 2 | Viewed by 2131
Abstract
Evaluation of human postural stability is important to prevent falls. Recent studies have been carried out to develop postural stability evaluation in an attempt to fall prevention. The postural stability index (PSI) was proposed as a measure to evaluate the stability of human [...] Read more.
Evaluation of human postural stability is important to prevent falls. Recent studies have been carried out to develop postural stability evaluation in an attempt to fall prevention. The postural stability index (PSI) was proposed as a measure to evaluate the stability of human postures in performing daily activities. The objective of this study was to use the PSI in developing the stability scales for human daily activities. The current study used two open datasets collected from mobile devices. In addition, we also conducted three experiments to evaluate the effect of age, velocity, step counts, and devices on PSI values. The collected datasets were preprocessed using the ensemble empirical mode decomposition (EEMD), then the complexity index from each intrinsic mode function (IMF) was calculated using the multiscale entropy (MSE). From the evaluation, it can be concluded that the PSI can be applied to do daily monitoring of postural stability for both young and older adults, and the PSI is not affected by age. The revised stability scales developed in this current study can give better suggestions to users than the original one. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications II)
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