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Multivariate Entropy Measures and Their Applications

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A special issue of Entropy (ISSN 1099-4300).

Deadline for manuscript submissions: closed (31 October 2016) | Viewed by 69912

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Special Issue Editor

Dr. Anne Humeau-Heurtier

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Univ Angers, LARIS, SFR MATHSTIC, F-49000 Angers, France
Interests: entropy; multiscale entropy; nonlinear analysis; empirical mode decomposition; biomedical data
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Special Issue Information

Dear Colleagues,

Standard entropy measures have found applications in a large variety of fields. However, they are adapted to single-channel data and fail to account for dynamical relationships existing between variables. Real systems are very often multivariate in nature: They include several different kinds of variable. Multivariate extensions of entropy measures therefore become increasingly interesting to study real data.

The aim of this Special Issue is to encourage researchers to present original and recent developments on multivariate entropy. Papers presenting concepts or applications are welcome. Applications can include (but are not limited to) biomedical engineering, chemical engineering, hydrology, pharmaceutical sciences, financial analyses, neurosciences, industrial engineering, geosciences, information sciences, etc.

Prof. Anne Humeau-Heurtier
Guest Editor

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Published Papers (9 papers)

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2925 KiB

Open AccessArticle

Tunable-Q Wavelet Transform Based Multivariate Sub-Band Fuzzy Entropy with Application to Focal EEG Signal Analysis

by Abhijit Bhattacharyya, Ram Bilas Pachori and U. Rajendra Acharya

Entropy 2017, 19(3), 99; https://doi.org/10.3390/e19030099 - 3 Mar 2017

Cited by 90 | Viewed by 9185

Abstract

This paper analyses the complexity of multivariate electroencephalogram (EEG) signals in different frequency scales for the analysis and classification of focal and non-focal EEG signals. The proposed multivariate sub-band entropy measure has been built based on tunable-Q wavelet transform (TQWT). In the field of multivariate entropy analysis, recent studies have performed analysis of biomedical signals with a multi-level filtering approach. This approach has become a useful tool for measuring inherent complexity of the biomedical signals. However, these methods may not be well suited for quantifying the complexity of the individual multivariate sub-bands of the analysed signal. In this present study, we have tried to resolve this difficulty by employing TQWT for analysing the sub-band signals of the analysed multivariate signal. It should be noted that higher value of Q factor is suitable for analysing signals with oscillatory nature, whereas the lower value of Q factor is suitable for analysing signals with non-oscillatory transients in nature. Moreover, with an increased number of sub-bands and a higher value of Q-factor, a reasonably good resolution can be achieved simultaneously in high and low frequency regions of the considered signals. Finally, we have employed multivariate fuzzy entropy (mvFE) to the multivariate sub-band signals obtained from the analysed signal. The proposed Q-based multivariate sub-band entropy has been studied on the publicly available bivariate Bern Barcelona focal and non-focal EEG signals database to investigate the statistical significance of the proposed features in different time segmented signals. Finally, the features are fed to random forest and least squares support vector machine (LS-SVM) classifiers to select the best classifier. Our method has achieved the highest classification accuracy of 84.67% in classifying focal and non-focal EEG signals with LS-SVM classifier. The proposed multivariate sub-band fuzzy entropy can also be applied to measure complexity of other multivariate biomedical signals. Full article

(This article belongs to the Special Issue Multivariate Entropy Measures and Their Applications)

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1126 KiB

Open AccessArticle

A Comparison of Postural Stability during Upright Standing between Normal and Flatfooted Individuals, Based on COP-Based Measures

by Tsui-Chiao Chao and Bernard C. Jiang

Entropy 2017, 19(2), 76; https://doi.org/10.3390/e19020076 - 16 Feb 2017

Cited by 7 | Viewed by 6353

Abstract

Aging causes foot arches to collapse, possibly leading to foot deformities and falls. This paper proposes a set of measures involving an entropy-based method used for two groups of young adults with dissimilar foot arches to explore and quantize postural stability on a force plate in an upright position. Fifty-four healthy young adults aged 18–30 years participated in this study. These were categorized into two groups: normal (37 participants) and flatfooted (17 participants). We collected the center of pressure (COP) displacement trajectories of participants during upright standing, on a force plate, in a static position, with eyes open (EO), or eyes closed (EC). These nonstationary time-series signals were quantized using entropy-based measures and traditional measures used to assess postural stability, and the results obtained from these measures were compared. The appropriate combinations of entropy-based measures revealed that, with respect to postural stability, the two groups differed significantly (p < 0.05) under both EO and EC conditions. The traditional commonly-used COP-based measures only revealed differences under EO conditions. Entropy-based measures are thus suitable for examining differences in postural stability for flatfooted people, and may be used by clinicians after further refinement. Full article

(This article belongs to the Special Issue Multivariate Entropy Measures and Their Applications)

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2566 KiB

Open AccessArticle

Univariate and Multivariate Generalized Multiscale Entropy to Characterise EEG Signals in Alzheimer’s Disease

by Hamed Azami, Daniel Abásolo, Samantha Simons and Javier Escudero

Entropy 2017, 19(1), 31; https://doi.org/10.3390/e19010031 - 12 Jan 2017

Cited by 50 | Viewed by 7742

Abstract

Alzheimer’s disease (AD) is a degenerative brain disorder leading to memory loss and changes in other cognitive abilities. The complexity of electroencephalogram (EEG) signals may help to characterise AD. To this end, we propose an extension of multiscale entropy based on variance (MSE_σ²) to multichannel signals, termed multivariate MSE_σ² (mvMSE_σ²), to take into account both the spatial and time domains of time series. Then, we investigate the mvMSE_σ² of EEGs at different frequency bands, including the broadband signals filtered between 1 and 40 Hz, θ, α, and β bands, and compare it with the previously-proposed multiscale entropy based on mean (MSE_µ), multivariate MSE_µ (mvMSE_µ), and MSE_σ², to distinguish different kinds of dynamical properties of the spread and the mean in the signals. Results from 11 AD patients and 11 age-matched controls suggest that the presence of broadband activity of EEGs is required for a proper evaluation of complexity. MSE_σ² and mvMSE_σ² results, showing a loss of complexity in AD signals, led to smaller p-values in comparison with MSE_µ and mvMSE_µ ones, suggesting that the variance-based MSE and mvMSE can characterise changes in EEGs as a result of AD in a more detailed way. The p-values for the slope values of the mvMSE curves were smaller than for MSE at large scale factors, also showing the possible usefulness of multivariate techniques. Full article

(This article belongs to the Special Issue Multivariate Entropy Measures and Their Applications)

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2957 KiB

Open AccessArticle

Information Decomposition in Multivariate Systems: Definitions, Implementation and Application to Cardiovascular Networks

by Luca Faes, Alberto Porta, Giandomenico Nollo and Michal Javorka

Entropy 2017, 19(1), 5; https://doi.org/10.3390/e19010005 - 24 Dec 2016

Cited by 68 | Viewed by 9456

Abstract

The continuously growing framework of information dynamics encompasses a set of tools, rooted in information theory and statistical physics, which allow to quantify different aspects of the statistical structure of multivariate processes reflecting the temporal dynamics of complex networks. Building on the most recent developments in this field, this work designs a complete approach to dissect the information carried by the target of a network of multiple interacting systems into the new information produced by the system, the information stored in the system, and the information transferred to it from the other systems; information storage and transfer are then further decomposed into amounts eliciting the specific contribution of assigned source systems to the target dynamics, and amounts reflecting information modification through the balance between redundant and synergetic interaction between systems. These decompositions are formulated quantifying information either as the variance or as the entropy of the investigated processes, and their exact computation for the case of linear Gaussian processes is presented. The theoretical properties of the resulting measures are first investigated in simulations of vector autoregressive processes. Then, the measures are applied to assess information dynamics in cardiovascular networks from the variability series of heart period, systolic arterial pressure and respiratory activity measured in healthy subjects during supine rest, orthostatic stress, and mental stress. Our results document the importance of combining the assessment of information storage, transfer and modification to investigate common and complementary aspects of network dynamics; suggest the higher specificity to alterations in the network properties of the measures derived from the decompositions; and indicate that measures of information transfer and information modification are better assessed, respectively, through entropy-based and variance-based implementations of the framework. Full article

(This article belongs to the Special Issue Multivariate Entropy Measures and Their Applications)

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913 KiB

Open AccessArticle

A Multivariate Multiscale Fuzzy Entropy Algorithm with Application to Uterine EMG Complexity Analysis

by Mosabber U. Ahmed, Theerasak Chanwimalueang, Sudhin Thayyil and Danilo P. Mandic

Entropy 2017, 19(1), 2; https://doi.org/10.3390/e19010002 - 22 Dec 2016

Cited by 68 | Viewed by 8509

Abstract

The recently introduced multivariate multiscale entropy (MMSE) has been successfully used to quantify structural complexity in terms of nonlinear within- and cross-channel correlations as well as to reveal complex dynamical couplings and various degrees of synchronization over multiple scales in real-world multichannel data. However, the applicability of MMSE is limited by the coarse-graining process which defines scales, as it successively reduces the data length for each scale and thus yields inaccurate and undefined entropy estimates at higher scales and for short length data. To that cause, we propose the multivariate multiscale fuzzy entropy (MMFE) algorithm and demonstrate its superiority over the MMSE on both synthetic as well as real-world uterine electromyography (EMG) short duration signals. Based on MMFE features, an improvement in the classification accuracy of term-preterm deliveries was achieved, with a maximum area under the curve (AUC) value of 0.99. Full article

(This article belongs to the Special Issue Multivariate Entropy Measures and Their Applications)

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3369 KiB

Open AccessArticle

Multivariate Surprisal Analysis of Gene Expression Levels

by Francoise Remacle, Andrew S. Goldstein and Raphael D. Levine

Entropy 2016, 18(12), 445; https://doi.org/10.3390/e18120445 - 11 Dec 2016

Cited by 5 | Viewed by 7282

Abstract

We consider here multivariate data which we understand as the problem where each data point i is measured for two or more distinct variables. In a typical situation there are many data points i while the range of the different variables is more limited. If there is only one variable then the data can be arranged as a rectangular matrix where i is the index of the rows while the values of the variable label the columns. We begin here with this case, but then proceed to the more general case with special emphasis on two variables when the data can be organized as a tensor. An analysis of such multivariate data by a maximal entropy approach is discussed and illustrated for gene expressions in four different cell types of six different patients. The different genes are indexed by i, and there are 24 (4 by 6) entries for each i. We used an unbiased thermodynamic maximal-entropy based approach (surprisal analysis) to analyze the multivariate transcriptional profiles. The measured microarray experimental data is organized as a tensor array where the two minor orthogonal directions are the different patients and the different cell types. The entries are the transcription levels on a logarithmic scale. We identify a disease signature of prostate cancer and determine the degree of variability between individual patients. Surprisal analysis determined a baseline expression level common for all cells and patients. We identify the transcripts in the baseline as the “housekeeping” genes that insure the cell stability. The baseline and two surprisal patterns satisfactorily recover (99.8%) the multivariate data. The two patterns characterize the individuality of the patients and, to a lesser extent, the commonality of the disease. The immune response was identified as the most significant pathway contributing to the cancer disease pattern. Delineating patient variability is a central issue in personalized diagnostics and it remains to be seen if additional data will confirm the power of multivariate analysis to address this key point. The collapsed limits where the data is compacted into two dimensional arrays are contained within the proposed formalism. Full article

(This article belongs to the Special Issue Multivariate Entropy Measures and Their Applications)

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995 KiB

Open AccessArticle

Multivariable Fuzzy Measure Entropy Analysis for Heart Rate Variability and Heart Sound Amplitude Variability

by Lina Zhao, Shoushui Wei, Hong Tang and Chengyu Liu

Entropy 2016, 18(12), 430; https://doi.org/10.3390/e18120430 - 3 Dec 2016

Cited by 13 | Viewed by 6332

Abstract

Simultaneously analyzing multivariate time series provides an insight into underlying interaction mechanisms of cardiovascular system and has recently become an increasing focus of interest. In this study, we proposed a new multivariate entropy measure, named multivariate fuzzy measure entropy (mvFME), for the analysis of multivariate cardiovascular time series. The performances of mvFME, and its two sub-components: the local multivariate fuzzy entropy (mvFEL) and global multivariate fuzzy entropy (mvFEG), as well as the commonly used multivariate sample entropy (mvSE), were tested on both simulation and cardiovascular multivariate time series. Simulation results on multivariate coupled Gaussian signals showed that the statistical stability of mvFME is better than mvSE, but its computation time is higher than mvSE. Then, mvSE and mvFME were applied to the multivariate cardiovascular signal analysis of R wave peak (RR) interval, and first (S1) and second (S2) heart sound amplitude series from three positions of heart sound signal collections, under two different physiological states: rest state and after stair climbing state. The results showed that, compared with rest state, for univariate time series analysis, after stair climbing state has significantly lower mvSE and mvFME values for both RR interval and S1 amplitude series, whereas not for S2 amplitude series. For bivariate time series analysis, all mvSE and mvFME report significantly lower values for after stair climbing. For trivariate time series analysis, only mvFME has the discrimination ability for the two physiological states, whereas mvSE does not. In summary, the new proposed mvFME method shows better statistical stability and better discrimination ability for multivariate time series analysis than the traditional mvSE method. Full article

(This article belongs to the Special Issue Multivariate Entropy Measures and Their Applications)

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979 KiB

Open AccessArticle

Multivariate Generalized Multiscale Entropy Analysis

by Anne Humeau-Heurtier

Entropy 2016, 18(11), 411; https://doi.org/10.3390/e18110411 - 17 Nov 2016

Cited by 14 | Viewed by 7438

Abstract

Multiscale entropy (MSE) was introduced in the 2000s to quantify systems’ complexity. MSE relies on (i) a coarse-graining procedure to derive a set of time series representing the system dynamics on different time scales; (ii) the computation of the sample entropy for each coarse-grained time series. A refined composite MSE (rcMSE)—based on the same steps as MSE—also exists. Compared to MSE, rcMSE increases the accuracy of entropy estimation and reduces the probability of inducing undefined entropy for short time series. The multivariate versions of MSE (MMSE) and rcMSE (MrcMSE) have also been introduced. In the coarse-graining step used in MSE, rcMSE, MMSE, and MrcMSE, the mean value is used to derive representations of the original data at different resolutions. A generalization of MSE was recently published, using the computation of different moments in the coarse-graining procedure. However, so far, this generalization only exists for univariate signals. We therefore herein propose an extension of this generalized MSE to multivariate data. The multivariate generalized algorithms of MMSE and MrcMSE presented herein (MGMSE and MGrcMSE, respectively) are first analyzed through the processing of synthetic signals. We reveal that MGrcMSE shows better performance than MGMSE for short multivariate data. We then study the performance of MGrcMSE on two sets of short multivariate electroencephalograms (EEG) available in the public domain. We report that MGrcMSE may show better performance than MrcMSE in distinguishing different types of multivariate EEG data. MGrcMSE could therefore supplement MMSE or MrcMSE in the processing of multivariate datasets. Full article

(This article belongs to the Special Issue Multivariate Entropy Measures and Their Applications)

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Review

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Open AccessReview

Possibility of Using Entropy Method to Evaluate the Distracting Effect of Mobile Phones on Pedestrians

by Nurul Retno Nurwulan and Bernard C. Jiang

Entropy 2016, 18(11), 390; https://doi.org/10.3390/e18110390 - 4 Nov 2016

Cited by 15 | Viewed by 5619

Abstract

The number of mobile phone users keeps increasing every year and mobile phones have become a primary need for most people. Ordinarily, people are not aware of the risk from a common dual-task study, such as using a mobile phone while walking or simply standing. This study reviewed the methodology in evaluating the distracting effect of mobile phones on pedestrians. A comprehensive review of literature revealed that the most common method in quantifying pedestrian performance is to evaluate postural task performance. Since using a mobile phone while crossing the road is a type of dual-task study, it would give more clarity to investigate it using entropy methods that have been proven more sensitive than the traditional center of pressure (COP) in discriminating the changes in human balance. The descriptions of commonly used entropy methods were also given in order to give a broad overview of the possibility in applying the methods to further clarify the distracting effect of mobile phones. Full article

(This article belongs to the Special Issue Multivariate Entropy Measures and Their Applications)

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