Entropy

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17 pages, 2778 KB

Open AccessFeature PaperArticle

Evaluation of Bubble Entropy Using Heart Rate Variability

by Dimitrios Platakis, Roberto Sassi and George Manis

Entropy 2026, 28(6), 638; https://doi.org/10.3390/e28060638 - 5 Jun 2026

Viewed by 446

Abstract

Bubble entropy has established its own place in the research community, representing a new and promising definition of entropy. Based on the work required to order a vector in an embedding space of dimension m, Bubble entropy gives a physical interpretation of [...] Read more.

Bubble entropy has established its own place in the research community, representing a new and promising definition of entropy. Based on the work required to order a vector in an embedding space of dimension m, Bubble entropy gives a physical interpretation of what the metric actually computes. In this work, Bubble entropy is evaluated based on its ability to classify RR time series, the time series most commonly considered for entropy-based analysis in the field of biomedical engineering. For this purpose, it is compared with three other definitions of entropy: the most widely used Sample entropy and Approximate entropy, the most relative to Bubble entropy, and also the widely used Permutation entropy. Signals from healthy individuals, in sinus rhythm, are compared with signals from cardiac patients, and machine learning methods are applied to calculate the classification accuracy that each method can achieve. The classifiers chosen are k-Nearest Neighbors, Support Vector Machine, Logistic Regression, and Gaussian Naive Bayes. Feature evaluation methods are also employed to serve as additional measures of effectiveness. Bubble entropy generally manages to achieve better results than Sample entropy, Approximate entropy and Permutation entropy, both in terms of classification accuracy and feature ranking. Full article

(This article belongs to the Special Issue Entropy in Biomedical Engineering, 3rd Edition)

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18 pages, 1819 KB

Open AccessFeature PaperArticle

by Brenda Y. Angulo-Ruiz, Vanesa Muñoz, Elena I. Rodríguez-Martínez and Carlos M. Gómez

Entropy 2026, 28(4), 390; https://doi.org/10.3390/e28040390 - 1 Apr 2026

Cited by 1 | Viewed by 1203

Abstract

The complexity of physiological signals provides insight into the maturation and functional organization of the developing brain. This study investigated age-related changes in electroencephalographic (EEG) signal complexity and their association with behavioral variability in 240 participants aged 6 to 29 years. EEG signals [...] Read more.

The complexity of physiological signals provides insight into the maturation and functional organization of the developing brain. This study investigated age-related changes in electroencephalographic (EEG) signal complexity and their association with behavioral variability in 240 participants aged 6 to 29 years. EEG signals were recorded during the resting state, and Multiscale Entropy (MSE) was computed across 34 temporal scales, grouped into fine, medium, and coarse scales. Behavioral variability was assessed using measures from Oddball and Delayed Match-to-Sample tasks. Quadratic regression analyses characterized age-related changes in MSE across scalp regions, and Pearson correlations evaluated associations between age-adjusted residuals of MSE and behavioral variability. The results showed that MSE changed with age across temporal scales in all cortical regions. Developmentally, MSE showed a significant age-related increase at fine scales across the entire scalp, region-specific decreases at medium scales, and a generalized decrease at coarse scales. Behavioral variability decreased with age across both tasks. Notably, fine-scale age residual MSE in central and posterior regions was negatively correlated with the coefficient of variation in the Oddball task, indicating that higher neural complexity supports more stable performance. These findings suggest scale- and region-specific age-related changes in neural complexity and suggest that fine-scale MSE captures aspects of brain maturation related to behavioral stability beyond traditional variability measures. Full article

(This article belongs to the Special Issue Entropy in Biomedical Engineering, 3rd Edition)

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17 pages, 3650 KB

Open AccessArticle

Multi-Entropy Feature Concatenation for Data-Efficient Cross-Subject Classification of Alzheimer’s Disease and Frontotemporal Dementia from Single-Channel EEG

by Jiawen Li, Chen Ling, Weidong Zhang, Jujian Lv, Xianglei Hu, Kaihan Lin, Jun Yuan, Shuang Zhang and Rongjun Chen

Entropy 2026, 28(2), 212; https://doi.org/10.3390/e28020212 - 12 Feb 2026

Cited by 4 | Viewed by 867

Abstract

Alzheimer’s disease (AD) and frontotemporal dementia (FTD) are neurodegenerative disorders where early detection is vital. However, the need for long-term monitoring is incompatible with data-scarce settings, and methods trained on one subject often fail on another due to cross-subject variability. To address these [...] Read more.

Alzheimer’s disease (AD) and frontotemporal dementia (FTD) are neurodegenerative disorders where early detection is vital. However, the need for long-term monitoring is incompatible with data-scarce settings, and methods trained on one subject often fail on another due to cross-subject variability. To address these limitations, this study proposes a cross-subject, single-channel electroencephalography (EEG)-based method that uses Multi-Entropy Feature Concatenation (MEFC) to classify AD and FTD. First, single-channel EEG is processed through the Discrete Wavelet Transform (DWT) to extract five rhythms: delta, theta, alpha, beta, and gamma. Subsequently, Permutation Entropy (PE), Singular Spectrum Entropy (SSE), and Sample Entropy (SE) are calculated for each rhythm and concatenated to form a combined MEFC to characterize the non-linear dynamic properties of EEG. Lastly, Dynamic Time Warping (DTW), Pearson Correlation Coefficient (PCC), Wavelet Coherence (WC), and Hilbert Transform Correlation (HTC) are employed to measure the similarity between unknown rhythmic MEFC and those from AD, FTD, and Healthy Control (HC) groups, performing a data-driven classification via similarity measurement. Experimental results on 88 subjects in the AHEPA dataset demonstrate that the beta-rhythm with PCC yields a three-class accuracy of 76.14% using single-channel FP2. In another dataset, the Florida-Based dataset, involving 48 subjects, theta-rhythm with WC achieves a two-class accuracy of 83.33% using FP2. Furthermore, a MATLAB R2023b-based toolbox is developed using the proposed method. Such outcomes are impressive, given the limited data per individual (data-efficient), reliable performance across new subjects (cross-subject), and compatibility with wearable devices (single-channel), providing a novel entropy-based approach for EEG-based applications in biomedical engineering. Full article

(This article belongs to the Special Issue Entropy in Biomedical Engineering, 3rd Edition)

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30 pages, 4063 KB

Open AccessFeature PaperArticle

Information Entropy Metrics to Address the Complexity of Cooperative Gating of Ion Channels

by Agata Wawrzkiewicz-Jałowiecka, Paulina Trybek, Michał Wojcik and Przemysław Borys

Entropy 2026, 28(2), 197; https://doi.org/10.3390/e28020197 - 10 Feb 2026

Cited by 2 | Viewed by 1206

Abstract

Ion channels in biological membranes can form spatially localized clusters that exhibit cooperative gating behavior. In this mode, the activity of one channel modulates the opening probability of its neighbors. Understanding such inter-channel interactions is key to elucidating the molecular mechanisms underlying electrochemical [...] Read more.

Ion channels in biological membranes can form spatially localized clusters that exhibit cooperative gating behavior. In this mode, the activity of one channel modulates the opening probability of its neighbors. Understanding such inter-channel interactions is key to elucidating the molecular mechanisms underlying electrochemical signaling and advancing channel-targeted pharmacology. In this study, we introduce a simplified stochastic model of multi-channel gating that allows for systematic analysis of cooperative behavior under controlled conditions. Two information-theoretic metrics, i.e., Shannon entropy and Sample Entropy, are applied to simulated multi-channel datasets, including idealized total current traces and dwell-time sequences of cluster states, to quantify inter-channel cooperativity. We show that the entropic measures display a strong dependency on the strength and type of cooperation (non-, positive, or negative cooperation). The proposed entropy-based framework offers a generalizable and quantitative approach for biomedical data analysis, demonstrating effectiveness in interpreting multi-channel recordings and uncovering cooperative mechanisms in ion channel behavior. The underlying mechanisms by which entropy reflects cooperativity are expected to appear in real recordings, where deviations can further aid in characterizing individual channel features in future work. Full article

(This article belongs to the Special Issue Entropy in Biomedical Engineering, 3rd Edition)

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8 pages, 347 KB

Open AccessArticle

Localizing Synergies of Hidden Factors in Complex Systems: Resting Brain Networks and HeLa GeneExpression Profile as Case Studies

by Marlis Ontivero-Ortega, Gorana Mijatovic, Luca Faes, Fernando E. Rosas, Daniele Marinazzo and Sebastiano Stramaglia

Entropy 2025, 27(8), 820; https://doi.org/10.3390/e27080820 - 1 Aug 2025

Cited by 2 | Viewed by 2525

Abstract

Factor analysis is a well-known statistical method to describe the variability of observed variables in terms of a smaller number of unobserved latent variables called factors. Even though latent factors are conceptually independent of each other, their influence on the observed variables is [...] Read more.

Factor analysis is a well-known statistical method to describe the variability of observed variables in terms of a smaller number of unobserved latent variables called factors. Even though latent factors are conceptually independent of each other, their influence on the observed variables is often joint and synergistic. We propose to quantify the synergy of the joint influence of factors on the observed variables using O-information, a recently introduced metric to assess high-order dependencies in complex systems; in the proposed framework, latent factors and observed variables are jointly analyzed in terms of their joint informational character. Two case studies are reported: analyzing resting fMRI data, we find that DMN and FP networks show the highest synergy, consistent with their crucial role in higher cognitive functions; concerning HeLa cells, we find that the most synergistic gene is STK-12 (AURKB), suggesting that this gene is involved in controlling the HeLa cell cycle. We believe that our approach, representing a bridge between factor analysis and the field of high-order interactions, will find wide application across several domains. Full article

(This article belongs to the Special Issue Entropy in Biomedical Engineering, 3rd Edition)

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17 pages, 461 KB

Open AccessArticle

Weibull-Type Incubation Period and Time of Exposure Using γ-Divergence

by Daisuke Yoneoka, Takayuki Kawashima, Yuta Tanoue, Shuhei Nomura and Akifumi Eguchi

Entropy 2025, 27(3), 321; https://doi.org/10.3390/e27030321 - 19 Mar 2025

Viewed by 1374

Abstract

Accurately determining the exposure time to an infectious pathogen, together with the corresponding incubation period, is vital for identifying infection sources and implementing targeted public health interventions. However, real-world outbreak data often include outliers—namely, tertiary or subsequent infection cases not directly linked to [...] Read more.

Accurately determining the exposure time to an infectious pathogen, together with the corresponding incubation period, is vital for identifying infection sources and implementing targeted public health interventions. However, real-world outbreak data often include outliers—namely, tertiary or subsequent infection cases not directly linked to the initial source—that complicate the estimation of exposure time. To address this challenge, we introduce a robust estimation framework based on a three-parameter Weibull distribution in which the location parameter naturally corresponds to the unknown exposure time. Our method employs a

γ

-divergence criterion—a robust generalization of the standard cross-entropy criterion—optimized via a tailored majorization–minimization (MM) algorithm designed to guarantee a monotonic decrease in the objective function despite the non-convexity typically present in robust formulations. Extensive Monte Carlo simulations demonstrate that our approach outperforms conventional estimation methods in terms of bias and mean squared error as well as in estimating the incubation period. Moreover, applications to real-world surveillance data on COVID-19 illustrate the practical advantages of the proposed method. These findings highlight the method’s robustness and efficiency in scenarios where data contamination from secondary or tertiary infections is common, showing its potential value for early outbreak detection and rapid epidemiological response. Full article

(This article belongs to the Special Issue Entropy in Biomedical Engineering, 3rd Edition)

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17 pages, 887 KB

Open AccessArticle

Bidimensional Increment Entropy for Texture Analysis: Theoretical Validation and Application to Colon Cancer Images

by Muqaddas Abid, Muhammad Suzuri Hitam, Rozniza Ali, Hamed Azami and Anne Humeau-Heurtier

Entropy 2025, 27(1), 80; https://doi.org/10.3390/e27010080 - 17 Jan 2025

Cited by 4 | Viewed by 2389

Abstract

Entropy algorithms are widely applied in signal analysis to quantify the irregularity of data. In the realm of two-dimensional data, their two-dimensional forms play a crucial role in analyzing images. Previous works have demonstrated the effectiveness of one-dimensional increment entropy in detecting abrupt [...] Read more.

Entropy algorithms are widely applied in signal analysis to quantify the irregularity of data. In the realm of two-dimensional data, their two-dimensional forms play a crucial role in analyzing images. Previous works have demonstrated the effectiveness of one-dimensional increment entropy in detecting abrupt changes in signals. Leveraging these advantages, we introduce a novel concept, two-dimensional increment entropy (IncrEn2D), tailored for analyzing image textures. In our proposed method, increments are translated into two-letter words, encoding both the size (magnitude) and direction (sign) of the increments calculated from an image. We validate the effectiveness of this new entropy measure by applying it to MIX_2D(p) processes and synthetic textures. Experimental validation spans diverse datasets, including the Kylberg dataset for real textures and medical images featuring colon cancer characteristics. To further validate our results, we employ a support vector machine model, utilizing multiscale entropy values as feature inputs. A comparative analysis with well-known bidimensional sample entropy (SampEn_2D) and bidimensional dispersion entropy (DispEn_2D) reveals that IncrEn_2D achieves an average classification accuracy surpassing that of other methods. In summary, IncrEn_2D emerges as an innovative and potent tool for image analysis and texture characterization, offering superior performance compared to existing bidimensional entropy measures. Full article

(This article belongs to the Special Issue Entropy in Biomedical Engineering, 3rd Edition)

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Review

Jump to: Research

23 pages, 878 KB

Open AccessReview

Review of Recent (2015–2024) Popular Entropy Definitions Applied to Physiological Signals

by Dimitrios Platakis and George Manis

Entropy 2025, 27(9), 983; https://doi.org/10.3390/e27090983 - 20 Sep 2025

Cited by 7 | Viewed by 1851

Abstract

Entropy estimation is widely used in time series analysis, particularly in the field of Biomedical Engineering. It plays a key role in analyzing a wide range of physiological signals and serves as a measure of signal complexity, which reflects the complexity of the [...] Read more.

Entropy estimation is widely used in time series analysis, particularly in the field of Biomedical Engineering. It plays a key role in analyzing a wide range of physiological signals and serves as a measure of signal complexity, which reflects the complexity of the underlying system. The widespread adoption of entropy in research has led to numerous entropy definitions, with Approximate Entropy and Sample Entropy being among the most widely used. Over the past decade, the field has remained highly active, with a significant number of new entropy definitions being proposed, some inspired by Approximate and Sample Entropy, some by Permutation entropy, while others followed their own course of thought. In this paper, we review and compare the most prominent entropy definitions that have appeared in the last decade (2015–2024). We performed the search on 20 December 2024. We adopt the PRISMA methodology for this purpose, a widely accepted standard for conducting systematic literature reviews. With the included articles, we present statistical results on the number of citations for each method and the application domains in which they have been used. We also conducted a thorough review of the selected articles, documenting for each paper which definition has been employed and on which physiological signal it has been applied. Full article

(This article belongs to the Special Issue Entropy in Biomedical Engineering, 3rd Edition)

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