Entropy

14 pages, 7993 KiB

Open AccessArticle

An Improved Toeplitz Approximation Method for Coherent DOA Estimation in Impulsive Noise Environments

by Jiang’an Dai, Tianshuang Qiu, Shengyang Luan, Quan Tian and Jiacheng Zhang

Entropy 2023, 25(6), 960; https://doi.org/10.3390/e25060960 - 20 Jun 2023

Cited by 2 | Viewed by 1643

Direction of arrival (DOA) estimation is an important research topic in array signal processing and widely applied in practical engineering. However, when signal sources are highly correlated or coherent, conventional subspace-based DOA estimation algorithms will perform poorly due to the rank deficiency in [...] Read more.

Direction of arrival (DOA) estimation is an important research topic in array signal processing and widely applied in practical engineering. However, when signal sources are highly correlated or coherent, conventional subspace-based DOA estimation algorithms will perform poorly due to the rank deficiency in the received data covariance matrix. Moreover, conventional DOA estimation algorithms are usually developed under Gaussian-distributed background noise, which will deteriorate significantly in impulsive noise environments. In this paper, a novel method is presented to estimate the DOA of coherent signals in impulsive noise environments. A novel correntropy-based generalized covariance (CEGC) operator is defined and proof of boundedness is given to ensure the effectiveness of the proposed method in impulsive noise environments. Furthermore, an improved Toeplitz approximation method combined CEGC operator is proposed to estimate the DOA of coherent sources. Compared to other existing algorithms, the proposed method can avoid array aperture loss and perform more effectively, even in cases of intense impulsive noise and low snapshot numbers. Finally, comprehensive Monte-Carlo simulations are performed to verify the superiority of the proposed method under various impulsive noise conditions. Full article

(This article belongs to the Special Issue Entropy and Information Theory in Machine Learning: Theoretical Insights and Applications)

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19 pages, 849 KiB

Open AccessArticle

Selection of an Insurance Company in Agriculture through Hybrid Multi-Criteria Decision-Making

by Adis Puška, Marija Lukić, Darko Božanić, Miroslav Nedeljković and Ibrahim M. Hezam

Entropy 2023, 25(6), 959; https://doi.org/10.3390/e25060959 - 20 Jun 2023

Cited by 3 | Viewed by 1912

Abstract

Crop insurance is used to reduce risk in agriculture. This research is focused on selecting an insurance company that provides the best policy conditions for crop insurance. A total of five insurance companies that provide crop insurance services in the Republic of Serbia [...] Read more.

Crop insurance is used to reduce risk in agriculture. This research is focused on selecting an insurance company that provides the best policy conditions for crop insurance. A total of five insurance companies that provide crop insurance services in the Republic of Serbia were selected. To choose the insurance company that provides the best policy conditions for farmers, expert opinions were solicited. In addition, fuzzy methods were used to assess the weights of the various criteria and to evaluate insurance companies. The weight of each criterion was determined using a combined approach based on fuzzy LMAW (the logarithm methodology of additive weights) and entropy methods. Fuzzy LMAW was used to determine the weights subjectively through expert ratings, while fuzzy entropy was used to determine the weights objectively. The results of these methods showed that the price criterion received the highest weight. The selection of the insurance company was made using the fuzzy CRADIS (compromise ranking of alternatives, from distance to ideal solution) method. The results of this method showed that the insurance company DDOR offers the best conditions for crop insurance for farmers. These results were confirmed by a validation of the results and sensitivity analysis. Based on all of this, it was shown that fuzzy methods can be used in the selection of insurance companies. Full article

(This article belongs to the Special Issue Entropy Methods for Multicriteria Decision Making)

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14 pages, 2416 KiB

Open AccessArticle

Identifying Important Nodes in Trip Networks and Investigating Their Determinants

by Ze-Tao Li, Wei-Peng Nie, Shi-Min Cai, Zhi-Dan Zhao and Tao Zhou

Entropy 2023, 25(6), 958; https://doi.org/10.3390/e25060958 - 20 Jun 2023

Cited by 1 | Viewed by 1683

Abstract

Describing travel patterns and identifying significant locations is a crucial area of research in transportation geography and social dynamics. Our study aims to contribute to this field by analyzing taxi trip data from Chengdu and New York City. Specifically, we investigate the probability [...] Read more.

Describing travel patterns and identifying significant locations is a crucial area of research in transportation geography and social dynamics. Our study aims to contribute to this field by analyzing taxi trip data from Chengdu and New York City. Specifically, we investigate the probability density distribution of trip distance in each city, which enables us to construct long- and short-distance trip networks. To identify critical nodes within these networks, we employ the PageRank algorithm and categorize them using centrality and participation indices. Furthermore, we explore the factors that contribute to their influence and observe a clear hierarchical multi-centre structure in Chengdu’s trip networks, while no such phenomenon is evident in New York City’s. Our study provides insight into the impact of trip distance on important nodes within trip networks in both cities and serves as a reference for distinguishing between long and short taxi trips. Our findings also reveal substantial differences in network structures between the two cities, highlighting the nuanced relationship between network structure and socio-economic factors. Ultimately, our research sheds light on the underlying mechanisms shaping transportation networks in urban areas and offers valuable insights into urban planning and policy making. Full article

(This article belongs to the Special Issue Complexity, Entropy and the Physics of Information)

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15 pages, 604 KiB

Open AccessFeature PaperArticle

Finite-Size Relaxational Dynamics of a Spike Random Matrix Spherical Model

by Pedro H. de Freitas Pimenta and Daniel A. Stariolo

Entropy 2023, 25(6), 957; https://doi.org/10.3390/e25060957 - 20 Jun 2023

Viewed by 1288

Abstract

We present a thorough numerical analysis of the relaxational dynamics of the Sherrington–Kirkpatrick spherical model with an additive non-disordered perturbation for large but finite sizes N. In the thermodynamic limit and at low temperatures, the perturbation is responsible for a phase transition [...] Read more.

We present a thorough numerical analysis of the relaxational dynamics of the Sherrington–Kirkpatrick spherical model with an additive non-disordered perturbation for large but finite sizes N. In the thermodynamic limit and at low temperatures, the perturbation is responsible for a phase transition from a spin glass to a ferromagnetic phase. We show that finite-size effects induce the appearance of a distinctive slow regime in the relaxation dynamics, the extension of which depends on the size of the system and also on the strength of the non-disordered perturbation. The long time dynamics are characterized by the two largest eigenvalues of a spike random matrix which defines the model, and particularly by the statistics concerning the gap between them. We characterize the finite-size statistics of the two largest eigenvalues of the spike random matrices in the different regimes, sub-critical, critical, and super-critical, confirming some known results and anticipating others, even in the less studied critical regime. We also numerically characterize the finite-size statistics of the gap, which we hope may encourage analytical work which is lacking. Finally, we compute the finite-size scaling of the long time relaxation of the energy, showing the existence of power laws with exponents that depend on the strength of the non-disordered perturbation in a way that is governed by the finite-size statistics of the gap. Full article

(This article belongs to the Special Issue Non-equilibrium Phase Transitions)

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10 pages, 251 KiB

Open AccessArticle

Improving the Performance of Quantum Cryptography by Using the Encryption of the Error Correction Data

by Valeria A. Pastushenko and Dmitry A. Kronberg

Entropy 2023, 25(6), 956; https://doi.org/10.3390/e25060956 - 20 Jun 2023

Cited by 9 | Viewed by 2120

Abstract

Security of quantum key distribution (QKD) protocols rely solely on quantum physics laws, namely, on the impossibility to distinguish between non-orthogonal quantum states with absolute certainty. Due to this, a potential eavesdropper cannot extract full information from the states stored in their quantum [...] Read more.

Security of quantum key distribution (QKD) protocols rely solely on quantum physics laws, namely, on the impossibility to distinguish between non-orthogonal quantum states with absolute certainty. Due to this, a potential eavesdropper cannot extract full information from the states stored in their quantum memory after an attack despite knowing all the information disclosed during classical post-processing stages of QKD. Here, we introduce the idea of encrypting classical communication related to error-correction in order to decrease the amount of information available to the eavesdropper and hence improve the performance of quantum key distribution protocols. We analyze the applicability of the method in the context of additional assumptions concerning the eavesdropper’s quantum memory coherence time and discuss the similarity of our proposition and the quantum data locking (QDL) technique. Full article

(This article belongs to the Special Issue Quantum Information: From Fundamental Aspects to Practical Applications)

16 pages, 464 KiB

Open AccessFeature PaperArticle

Shannon Entropy and Herfindahl-Hirschman Index as Team’s Performance and Competitive Balance Indicators in Cyclist Multi-Stage Races

by Marcel Ausloos

Entropy 2023, 25(6), 955; https://doi.org/10.3390/e25060955 - 19 Jun 2023

Cited by 3 | Viewed by 2044

Abstract

It seems that one cannot find many papers relating entropy to sport competitions. Thus, in this paper, I use (i) the Shannon intrinsic entropy (S) as an indicator of “teams sporting value” (or “competition performance”) and (ii) the Herfindahl-Hirschman index (HHi) [...] Read more.

It seems that one cannot find many papers relating entropy to sport competitions. Thus, in this paper, I use (i) the Shannon intrinsic entropy (S) as an indicator of “teams sporting value” (or “competition performance”) and (ii) the Herfindahl-Hirschman index (HHi) as a “teams competitive balance” indicator, in the case of (professional) cyclist multi-stage races. The 2022 Tour de France and 2023 Tour of Oman are used for numerical illustrations and discussion. The numerical values are obtained from classical and and new ranking indices which measure the teams “final time”, on one hand, and “final place”, on the other hand, based on the “best three” riders in each stage, but also the corresponding times and places throughout the race, for these finishing riders. The analysis data demonstrate that the constraint, “only the finishing riders count”, makes much sense for obtaining a more objective measure of “team value” and team performance”, at the end of a multi-stage race. A graphical analysis allows us to distinguish various team levels, each exhibiting a Feller-Pareto distribution, thereby indicating self-organized processes. In so doing, one hopefully better relates objective scientific measures to sport team competitions. Moreover, this analysis proposes some paths to elaborate on forecasting through standard probability concepts. Full article

(This article belongs to the Special Issue Selected Featured Papers from Entropy Editorial Board Members)

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27 pages, 381 KiB

Open AccessArticle

Uniform Treatment of Integral Majorization Inequalities with Applications to Hermite-Hadamard-Fejér-Type Inequalities and f-Divergences

by László Horváth

Entropy 2023, 25(6), 954; https://doi.org/10.3390/e25060954 - 19 Jun 2023

Cited by 4 | Viewed by 2033

Abstract

In this paper, we present a general framework that provides a comprehensive and uniform treatment of integral majorization inequalities for convex functions and finite signed measures. Along with new results, we present unified and simple proofs of classical statements. To apply our results, [...] Read more.

In this paper, we present a general framework that provides a comprehensive and uniform treatment of integral majorization inequalities for convex functions and finite signed measures. Along with new results, we present unified and simple proofs of classical statements. To apply our results, we deal with Hermite-Hadamard-Fejér-type inequalities and their refinements. We present a general method to refine both sides of Hermite-Hadamard-Fejér-type inequalities. The results of many papers on the refinement of the Hermite-Hadamard inequality, whose proofs are based on different ideas, can be treated in a uniform way by this method. Finally, we establish a necessary and sufficient condition for when a fundamental inequality of f-divergences can be refined by another f-divergence. Full article

(This article belongs to the Special Issue Shannon Entropy: Mathematical View)

16 pages, 1731 KiB

Open AccessArticle

Improved Recurrence Plots Compression Distance by Learning Parameter for Video Compression Quality

by Tatsumasa Murai and Hisashi Koga

Entropy 2023, 25(6), 953; https://doi.org/10.3390/e25060953 - 19 Jun 2023

Cited by 1 | Viewed by 1531

Abstract

As the Internet-of-Things is deployed widely, many time-series data are generated everyday. Thus, classifying time-series automatically has become important. Compression-based pattern recognition has attracted attention, because it can analyze various data universally with few model parameters. RPCD (Recurrent Plots Compression Distance) is known [...] Read more.

As the Internet-of-Things is deployed widely, many time-series data are generated everyday. Thus, classifying time-series automatically has become important. Compression-based pattern recognition has attracted attention, because it can analyze various data universally with few model parameters. RPCD (Recurrent Plots Compression Distance) is known as a compression-based time-series classification method. First, RPCD transforms time-series data into an image called “Recurrent Plots (RP)”. Then, the distance between two time-series data is determined as the dissimilarity between their RPs. Here, the dissimilarity between two images is computed from the file size, when an MPEG-1 encoder compresses the video, which serializes the two images in order. In this paper, by analyzing the RPCD, we give an important insight that the quality parameter for the MPEG-1 encoding that controls the resolution of compressed videos influences the classification performance very much. We also show that the optimal parameter value depends extremely on the dataset to be classified: Interestingly, the optimal value for one dataset can make the RPCD fall behind a naive random classifier for another dataset. Supported by these insights, we propose an improved version of RPCD named qRPCD, which searches the optimal parameter value by means of cross-validation. Experimentally, qRPCD works superiorly to the original RPCD by about 4% in terms of classification accuracy. Full article

(This article belongs to the Special Issue Information Theory in Image Processing and Pattern Recognition)

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11 pages, 275 KiB

Open AccessFeature PaperArticle

A Special Relativistic Exploitation of the Second Law of Thermodynamics and Its Non-Relativistic Limit

by Christina Papenfuss

Entropy 2023, 25(6), 952; https://doi.org/10.3390/e25060952 - 17 Jun 2023

Viewed by 1088

Abstract

A thermodynamic process is a solution of the balance equations fulfilling the second law of thermodynamics. This implies restrictions on the constitutive relations. The most general way to exploit these restrictions is the method introduced by Liu. This method is applied here, in [...] Read more.

A thermodynamic process is a solution of the balance equations fulfilling the second law of thermodynamics. This implies restrictions on the constitutive relations. The most general way to exploit these restrictions is the method introduced by Liu. This method is applied here, in contrast to most of the literature on relativistic thermodynamic constitutive theory, which goes back to a relativistic extension of the Thermodynamics of Irreversible Processes. In the present work, the balance equations and the entropy inequality are formulated in the special relativistic four-dimensional form for an observer with four-velocity parallel to the particle current. The restrictions on constitutive functions are exploited in the relativistic formulation. The domain of the constitutive functions, the state space, is chosen to include the particle number density, the internal energy density, the space derivatives of these quantities, and the space derivative of the material velocity for a chosen observer. The resulting restrictions on constitutive functions, as well as the resulting entropy production are investigated in the non-relativistic limit, and relativistic correction terms of the lowest order are derived. The restrictions on constitutive functions and the entropy production in the low energy limit are compared to the results of an exploitation of the non-relativistic balance equations and entropy inequality. In the next order of approximation our results are compared to the Thermodynamics of Irreversible Processes. Full article

(This article belongs to the Special Issue Thermodynamic Constitutive Theory and Its Application)

19 pages, 8211 KiB

Open AccessArticle

Multi-Focus Image Fusion for Full-Field Optical Angiography

by Yuchan Jie, Xiaosong Li, Mingyi Wang and Haishu Tan

Entropy 2023, 25(6), 951; https://doi.org/10.3390/e25060951 - 16 Jun 2023

Cited by 5 | Viewed by 1580

Abstract

Full-field optical angiography (FFOA) has considerable potential for clinical applications in the prevention and diagnosis of various diseases. However, owing to the limited depth of focus attainable using optical lenses, only information about blood flow in the plane within the depth of field [...] Read more.

Full-field optical angiography (FFOA) has considerable potential for clinical applications in the prevention and diagnosis of various diseases. However, owing to the limited depth of focus attainable using optical lenses, only information about blood flow in the plane within the depth of field can be acquired using existing FFOA imaging techniques, resulting in partially unclear images. To produce fully focused FFOA images, an FFOA image fusion method based on the nonsubsampled contourlet transform and contrast spatial frequency is proposed. Firstly, an imaging system is constructed, and the FFOA images are acquired by intensity-fluctuation modulation effect. Secondly, we decompose the source images into low-pass and bandpass images by performing nonsubsampled contourlet transform. A sparse representation-based rule is introduced to fuse the lowpass images to effectively retain the useful energy information. Meanwhile, a contrast spatial frequency rule is proposed to fuse bandpass images, which considers the neighborhood correlation and gradient relationships of pixels. Finally, the fully focused image is produced by reconstruction. The proposed method significantly expands the range of focus of optical angiography and can be effectively extended to public multi-focused datasets. Experimental results confirm that the proposed method outperformed some state-of-the-art methods in both qualitative and quantitative evaluations. Full article

(This article belongs to the Special Issue Methods in Artificial Intelligence and Information Processing II)

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F Imaging system. Mouse ear samples were fixed on the OMP in the experiments. Tt (Tt = 1,2, …, χ) represents multiple multi-focus source images. C denotes the container, L denotes the light source, and EZL denotes the electric zoom lens. Full article ">Figure 2
Schematic diagram of the proposed multi-focus image fusion method. Full article ">Figure 3
Parts of four groups of source images. Full article ">Figure 4
Fusion results produced by different methods. (a1,a2) are two randomly selected source images of group A; (b1–h1) show the fused images obtained by MFF-GAN, U2Fusion, SwinFusion, NSCT, CSSA, CPFA, and the proposed method, respectively; and (b2–h2) are the difference images obtained from the corresponding fused images and (a2), respectively. The red box is a magnification of the local details. Full article ">Figure 5
Fusion results produced by different methods. (a1,a2,i1,i2) are randomly selected source images of groups B and C, respectively, whereas (b1–h1,j1–p1) are the fused images obtained using the MFF-GAN, U2Fusion, SwinFusion, NSCT, CSSA, and CPFA models and the proposed method. (b2–h2) are the difference images obtained by subtracting the corresponding fused image from the source image of (a1), respectively, and (j2–p2) are the difference images obtained by subtracting the corresponding fused image from (i2). The green and blue boxes are enlargements of two local details. Full article ">Figure 6
Quantitative evaluations of different methods for four groups of the source images. Full article ">Figure 7
Quantitative evaluations of the intermediate and final fused images obtained by different methods for four groups of source images. Full article ">Figure 8
Quantitative evaluations of 15 randomly selected intermediate fusion results. Full article ">

15 pages, 320 KiB

Open AccessArticle

Dynamics of Fractional Delayed Reaction-Diffusion Equations

by Linfang Liu and Juan J. Nieto

Entropy 2023, 25(6), 950; https://doi.org/10.3390/e25060950 - 16 Jun 2023

Cited by 1 | Viewed by 1327

Abstract

The long-term behavior of the weak solution of a fractional delayed reaction–diffusion equation with a generalized Caputo derivative is investigated. By using the classic Galerkin approximation method and comparison principal, the existence and uniqueness of the solution is proved in the sense of [...] Read more.

The long-term behavior of the weak solution of a fractional delayed reaction–diffusion equation with a generalized Caputo derivative is investigated. By using the classic Galerkin approximation method and comparison principal, the existence and uniqueness of the solution is proved in the sense of weak solution. In addition, the global attracting set of the considered system is obtained, with the help of the Sobolev embedding theorem and Halanay inequality. Full article

(This article belongs to the Special Issue Selected Papers from the 2022 Scholars International Conference on Physics and Quantum Physics)

20 pages, 754 KiB

Open AccessFeature PaperArticle

Hierarchical Wilson–Cowan Models and Connection Matrices

by W. A. Zúñiga-Galindo and B. A. Zambrano-Luna

Entropy 2023, 25(6), 949; https://doi.org/10.3390/e25060949 - 16 Jun 2023

Cited by 2 | Viewed by 1525

Abstract

This work aims to study the interplay between the Wilson–Cowan model and connection matrices. These matrices describe cortical neural wiring, while Wilson–Cowan equations provide a dynamical description of neural interaction. We formulate Wilson–Cowan equations on locally compact Abelian groups. We show that the [...] Read more.

This work aims to study the interplay between the Wilson–Cowan model and connection matrices. These matrices describe cortical neural wiring, while Wilson–Cowan equations provide a dynamical description of neural interaction. We formulate Wilson–Cowan equations on locally compact Abelian groups. We show that the Cauchy problem is well posed. We then select a type of group that allows us to incorporate the experimental information provided by the connection matrices. We argue that the classical Wilson–Cowan model is incompatible with the small-world property. A necessary condition to have this property is that the Wilson–Cowan equations be formulated on a compact group. We propose a p-adic version of the Wilson–Cowan model, a hierarchical version in which the neurons are organized into an infinite rooted tree. We present several numerical simulations showing that the p-adic version matches the predictions of the classical version in relevant experiments. The p-adic version allows the incorporation of the connection matrices into the Wilson–Cowan model. We present several numerical simulations using a neural network model that incorporates a p-adic approximation of the connection matrix of the cat cortex. Full article

(This article belongs to the Special Issue New Trends in Theoretical and Mathematical Physics)

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13 pages, 2290 KiB

Open AccessArticle

A Novel Evidence Combination Method Based on Improved Pignistic Probability

by Xin Shi, Fei Liang, Pengjie Qin, Liang Yu and Gaojie He

Entropy 2023, 25(6), 948; https://doi.org/10.3390/e25060948 - 16 Jun 2023

Cited by 3 | Viewed by 1484

Abstract

Evidence theory is widely used to deal with the fusion of uncertain information, but the fusion of conflicting evidence remains an open question. To solve the problem of conflicting evidence fusion in single target recognition, we proposed a novel evidence combination method based [...] Read more.

Evidence theory is widely used to deal with the fusion of uncertain information, but the fusion of conflicting evidence remains an open question. To solve the problem of conflicting evidence fusion in single target recognition, we proposed a novel evidence combination method based on an improved pignistic probability function. Firstly, the improved pignistic probability function could redistribute the probability of multi-subset proposition according to the weight of single subset propositions in a basic probability assignment (BPA), which reduces the computational complexity and information loss in the conversion process. The combination of the Manhattan distance and evidence angle measurements is proposed to extract evidence certainty and obtain mutual support information between each piece of evidence; then, entropy is used to calculate the uncertainty of the evidence and the weighted average method is used to correct and update the original evidence. Finally, the Dempster combination rule is used to fuse the updated evidence. Verified by the analysis results of single-subset proposition and multi-subset proposition highly conflicting evidence examples, compared to the Jousselme distance method, the Lance distance and reliability entropy combination method, and the Jousselme distance and uncertainty measure combination method, our approach achieved better convergence and the average accuracy was improved by 0.51% and 2.43%. Full article

(This article belongs to the Special Issue Application of Information Theory to Physical Modeling and State Awareness in Complex Systems)

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26 pages, 4874 KiB

Open AccessFeature PaperArticle

Increasing Extractable Work in Small Qubit Landscapes

by Unnati Akhouri, Sarah Shandera and Gaukhar Yesmurzayeva

Entropy 2023, 25(6), 947; https://doi.org/10.3390/e25060947 - 16 Jun 2023

Viewed by 1294

Abstract

An interesting class of physical systems, including those associated with life, demonstrates the ability to hold thermalization at bay and perpetuate states of high free-energy compared to a local environment. In this work we study quantum systems with no external sources or sinks [...] Read more.

An interesting class of physical systems, including those associated with life, demonstrates the ability to hold thermalization at bay and perpetuate states of high free-energy compared to a local environment. In this work we study quantum systems with no external sources or sinks for energy, heat, work, or entropy that allow for high free-energy subsystems to form and persist. We initialize systems of qubits in mixed, uncorrelated states and evolve them subject to a conservation law. We find that four qubits make up the minimal system for which these restricted dynamics and initial conditions allow an increase in extractable work for a subsystem. On landscapes of eight co-evolving qubits, interacting in randomly selected subsystems at each step, we demonstrate that restricted connectivity and an inhomogeneous distribution of initial temperatures both lead to landscapes with longer intervals of increasing extractable work for individual qubits. We demonstrate the role of correlations that develop on the landscape in enabling a positive change in extractable work. Full article

(This article belongs to the Section Non-equilibrium Phenomena)

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20 pages, 5199 KiB

Open AccessArticle

Combined Gaussian Mixture Model and Pathfinder Algorithm for Data Clustering

by Huajuan Huang, Zepeng Liao, Xiuxi Wei and Yongquan Zhou

Entropy 2023, 25(6), 946; https://doi.org/10.3390/e25060946 - 16 Jun 2023

Cited by 1 | Viewed by 2536

Abstract

Data clustering is one of the most influential branches of machine learning and data analysis, and Gaussian Mixture Models (GMMs) are frequently adopted in data clustering due to their ease of implementation. However, there are certain limitations to this approach that need to [...] Read more.

Data clustering is one of the most influential branches of machine learning and data analysis, and Gaussian Mixture Models (GMMs) are frequently adopted in data clustering due to their ease of implementation. However, there are certain limitations to this approach that need to be acknowledged. GMMs need to determine the cluster numbers manually, and they may fail to extract the information within the dataset during initialization. To address these issues, a new clustering algorithm called PFA-GMM has been proposed. PFA-GMM is based on GMMs and the Pathfinder algorithm (PFA), and it aims to overcome the shortcomings of GMMs. The algorithm automatically determines the optimal number of clusters based on the dataset. Subsequently, PFA-GMM considers the clustering problem as a global optimization problem for getting trapped in local convergence during initialization. Finally, we conducted a comparative study of our proposed clustering algorithm against other well-known clustering algorithms using both synthetic and real-world datasets. The results of our experiments indicate that PFA-GMM outperformed the competing approaches. Full article

(This article belongs to the Section Signal and Data Analysis)

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12 pages, 1043 KiB

Open AccessArticle

Influence of Removing Leaf Node Neighbors on Network Controllability

by Chengpei Wu, Siyi Xu, Zhuoran Yu and Junli Li

Entropy 2023, 25(6), 945; https://doi.org/10.3390/e25060945 - 15 Jun 2023

Viewed by 1727

Abstract

From the perspective of network attackers, finding attack sequences that can cause significant damage to network controllability is an important task, which also helps defenders improve robustness during network constructions. Therefore, developing effective attack strategies is a key aspect of research on network [...] Read more.

From the perspective of network attackers, finding attack sequences that can cause significant damage to network controllability is an important task, which also helps defenders improve robustness during network constructions. Therefore, developing effective attack strategies is a key aspect of research on network controllability and its robustness. In this paper, we propose a Leaf Node Neighbor-based Attack (LNNA) strategy that can effectively disrupt the controllability of undirected networks. The LNNA strategy targets the neighbors of leaf nodes, and when there are no leaf nodes in the network, the strategy attacks the neighbors of nodes with a higher degree to produce the leaf nodes. Results from simulations on synthetic and real-world networks demonstrate the effectiveness of the proposed method. In particular, our findings suggest that removing neighbors of low-degree nodes (i.e., nodes with degree 1 or 2) can significantly reduce the controllability robustness of networks. Thus, protecting such low-degree nodes and their neighbors during network construction can lead to networks with improved controllability robustness. Full article

(This article belongs to the Topic Complex Systems and Network Science)

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17 pages, 338 KiB

Open AccessReview

Irreversible Geometrothermodynamics of Open Systems in Modified Gravity

by Miguel A. S. Pinto, Tiberiu Harko and Francisco S. N. Lobo

Entropy 2023, 25(6), 944; https://doi.org/10.3390/e25060944 - 15 Jun 2023

Cited by 3 | Viewed by 1116

Abstract

In this work, we explore the formalism of the irreversible thermodynamics of open systems and the possibility of gravitationally generated particle production in modified gravity. More specifically, we consider the scalar–tensor representation of

f (R, T)

gravity, in which the [...] Read more.

In this work, we explore the formalism of the irreversible thermodynamics of open systems and the possibility of gravitationally generated particle production in modified gravity. More specifically, we consider the scalar–tensor representation of

f (R, T)

gravity, in which the matter energy–momentum tensor is not conserved due to a nonminimal curvature–matter coupling. In the context of the irreversible thermodynamics of open systems, this non-conservation of the energy–momentum tensor can be interpreted as an irreversible flow of energy from the gravitational sector to the matter sector, which, in general, could result in particle creation. We obtain and discuss the expressions for the particle creation rate, the creation pressure, and the entropy and temperature evolutions. Applied together with the modified field equations of scalar–tensor

f (R, T)

gravity, the thermodynamics of open systems lead to a generalization of the

Λ

CDM cosmological paradigm, in which the particle creation rate and pressure are considered effectively as components of the cosmological fluid energy–momentum tensor. Thus, generally, modified theories of gravity in which these two quantities do not vanish provide a macroscopic phenomenological description of particle production in the cosmological fluid filling the Universe and also lead to the possibility of cosmological models that start from empty conditions and gradually build up matter and entropy. Full article

(This article belongs to the Special Issue Geometrothermodynamics and Its Applications)

13 pages, 6404 KiB

Open AccessArticle

Software-Defined Networking Orchestration for Interoperable Key Management of Quantum Key Distribution Networks

by Dong-Hi Sim, Jongyoon Shin and Min Hyung Kim

Entropy 2023, 25(6), 943; https://doi.org/10.3390/e25060943 - 15 Jun 2023

Cited by 3 | Viewed by 2235

Abstract

This paper demonstrates the use of software-defined networking (SDN) orchestration to integrate regionally separated networks in which different network parts use incompatible key management systems (KMSs) managed by different SDN controllers to ensure end-to-end QKD service provisioning to deliver the QKD keys between [...] Read more.

This paper demonstrates the use of software-defined networking (SDN) orchestration to integrate regionally separated networks in which different network parts use incompatible key management systems (KMSs) managed by different SDN controllers to ensure end-to-end QKD service provisioning to deliver the QKD keys between geographically different QKD networks. The study focuses on scenarios in which different parts of the network are managed separately by different SDN controllers, requiring an SDN orchestrator to coordinate and manage these controllers. In practical network deployments, operators often utilize multiple vendors for their network equipment. This practice also enables the expansion of the QKD network’s coverage by interconnecting various QKD networks equipped with devices from different vendors. However, as coordinating different parts of the QKD network is a complex task, this paper proposes the implementation of an SDN orchestrator which acts as a central entity to manage multiple SDN controllers, ensuring end-to-end QKD service provisioning to address this challenge. For instance, when there are multiple border nodes to interconnect different networks, the SDN orchestrator calculates the path in advance for the end-to-end delivery of keys between initiating and target applications belonging to different networks. This path selection requires the SDN orchestrator to gather information from each SDN controller managing the respective parts of the QKD network. This work shows the practical implementation of SDN orchestration for interoperable KMS in commercial QKD networks in South Korea. By employing an SDN orchestrator, it becomes possible to coordinate multiple SDN controllers and ensure the efficient and secure delivery of QKD keys between different QKD networks with varying vendor equipment. Full article

(This article belongs to the Special Issue Quantum Communications Networks & Cryptography: From Devices to Industrial Practice)

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21 pages, 2415 KiB

Open AccessFeature PaperArticle

Statistical Analysis of Plasma Dynamics in Gyrokinetic Simulations of Stellarator Turbulence

by Aristeides D. Papadopoulos, Johan Anderson, Eun-jin Kim, Michail Mavridis and Heinz Isliker

Entropy 2023, 25(6), 942; https://doi.org/10.3390/e25060942 - 15 Jun 2023

Cited by 1 | Viewed by 1436

Abstract

A geometrical method for assessing stochastic processes in plasma turbulence is investigated in this study. The thermodynamic length methodology allows using a Riemannian metric on the phase space; thus, distances between thermodynamic states can be computed. It constitutes a geometric methodology to understand [...] Read more.

A geometrical method for assessing stochastic processes in plasma turbulence is investigated in this study. The thermodynamic length methodology allows using a Riemannian metric on the phase space; thus, distances between thermodynamic states can be computed. It constitutes a geometric methodology to understand stochastic processes involved in, e.g., order–disorder transitions, where a sudden increase in distance is expected. We consider gyrokinetic simulations of ion-temperature-gradient (ITG)-mode-driven turbulence in the core region of the stellarator W7-X with realistic quasi-isodynamic topologies. In gyrokinetic plasma turbulence simulations, avalanches, e.g., of heat and particles, are often found, and in this work, a novel method for detection is investigated. This new method combines the singular spectrum analysis algorithm with a hierarchical clustering method such that the time series is decomposed into two parts: useful physical information and noise. The informative component of the time series is used for the calculation of the Hurst exponent, the information length, and the dynamic time. Based on these measures, the physical properties of the time series are revealed. Full article

(This article belongs to the Special Issue Energy Transfer and Dissipation in Plasma Turbulence)

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23 pages, 5818 KiB

Open AccessArticle

The Structure Entropy-Based Node Importance Ranking Method for Graph Data

by Shihu Liu and Haiyan Gao

Entropy 2023, 25(6), 941; https://doi.org/10.3390/e25060941 - 15 Jun 2023

Cited by 2 | Viewed by 1714

Abstract

Due to its wide application across many disciplines, how to make an efficient ranking for nodes in graph data has become an urgent topic. It is well-known that most classical methods only consider the local structure information of nodes, but ignore the global [...] Read more.

Due to its wide application across many disciplines, how to make an efficient ranking for nodes in graph data has become an urgent topic. It is well-known that most classical methods only consider the local structure information of nodes, but ignore the global structure information of graph data. In order to further explore the influence of structure information on node importance, this paper designs a structure entropy-based node importance ranking method. Firstly, the target node and its associated edges are removed from the initial graph data. Next, the structure entropy of graph data can be constructed by considering the local and global structure information at the same time, in which case all nodes can be ranked. The effectiveness of the proposed method was tested by comparing it with five benchmark methods. The experimental results show that the structure entropy-based node importance ranking method performs well on eight real-world datasets. Full article

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13 pages, 327 KiB

Open AccessArticle

Exploring a New Application of Construct Specification Equations (CSEs) and Entropy: A Pilot Study with Balance Measurements

by Jeanette Melin, Helena Fridberg, Eva Ekvall Hansson, Daniel Smedberg and Leslie Pendrill

Entropy 2023, 25(6), 940; https://doi.org/10.3390/e25060940 - 15 Jun 2023

Cited by 2 | Viewed by 1283

Abstract

Both construct specification equations (CSEs) and entropy can be used to provide a specific, causal, and rigorously mathematical conceptualization of item attributes in order to provide fit-for-purpose measurements of person abilities. This has been previously demonstrated for memory measurements. It can also be [...] Read more.

Both construct specification equations (CSEs) and entropy can be used to provide a specific, causal, and rigorously mathematical conceptualization of item attributes in order to provide fit-for-purpose measurements of person abilities. This has been previously demonstrated for memory measurements. It can also be reasonably expected to be applicable to other kinds of measures of human abilities and task difficulty in health care, but further exploration is needed about how to incorporate qualitative explanatory variables in the CSE formulation. In this paper we report two case studies exploring the possibilities of advancing CSE and entropy to include human functional balance measurements. In case study I, physiotherapists have formulated a CSE for balance task difficulty by principal component regression of empirical balance task difficulty values from Berg’s Balance Scale transformed using the Rasch model. In case study II, four balance tasks of increasing difficulty due to diminishing bases of support and vision were briefly investigated in relation to entropy as a measure of the amount of information and order as well as physical thermodynamics. The pilot study has explored both methodological and conceptual possibilities and concerns to be considered in further work. The results should not be considered as fully comprehensive or absolute, but rather open up for further discussion and investigations to advance measurements of person balance ability in clinical practice, research, and trials. Full article

(This article belongs to the Special Issue Applications of Entropy in Health Care)

9 pages, 285 KiB

Open AccessArticle

Correspondence between the Energy Equipartition Theorem in Classical Mechanics and Its Phase-Space Formulation in Quantum Mechanics

by Esteban Marulanda, Alejandro Restrepo and Johans Restrepo

Entropy 2023, 25(6), 939; https://doi.org/10.3390/e25060939 - 15 Jun 2023

Viewed by 1384

Abstract

In classical physics, there is a well-known theorem in which it is established that the energy per degree of freedom is the same. However, in quantum mechanics, due to the non-commutativity of some pairs of observables and the possibility of having non-Markovian dynamics, [...] Read more.

In classical physics, there is a well-known theorem in which it is established that the energy per degree of freedom is the same. However, in quantum mechanics, due to the non-commutativity of some pairs of observables and the possibility of having non-Markovian dynamics, the energy is not equally distributed. We propose a correspondence between what is known as the classical energy equipartition theorem and its counterpart in the phase-space formulation in quantum mechanics based on the Wigner representation. Further, we show that in the high-temperature regime, the classical result is recovered. Full article

(This article belongs to the Collection Foundations of Statistical Mechanics)

17 pages, 10798 KiB

Open AccessArticle

Attention-Based Spatial–Temporal Convolution Gated Recurrent Unit for Traffic Flow Forecasting

by Qingyong Zhang, Wanfeng Chang, Conghui Yin, Peng Xiao, Kelei Li and Meifang Tan

Entropy 2023, 25(6), 938; https://doi.org/10.3390/e25060938 - 14 Jun 2023

Cited by 3 | Viewed by 1871

Abstract

Accurate traffic flow forecasting is very important for urban planning and traffic management. However, this is a huge challenge due to the complex spatial–temporal relationships. Although the existing methods have researched spatial–temporal relationships, they neglect the long periodic aspects of traffic flow data, [...] Read more.

Accurate traffic flow forecasting is very important for urban planning and traffic management. However, this is a huge challenge due to the complex spatial–temporal relationships. Although the existing methods have researched spatial–temporal relationships, they neglect the long periodic aspects of traffic flow data, and thus cannot attain a satisfactory result. In this paper, we propose a novel model Attention-Based Spatial–Temporal Convolution Gated Recurrent Unit (ASTCG) to solve the traffic flow forecasting problem. ASTCG has two core components: the multi-input module and the STA-ConvGru module. Based on the cyclical nature of traffic flow data, the data input to the multi-input module are divided into three parts, near-neighbor data, daily-periodic data, and weekly-periodic data, thus enabling the model to better capture the time dependence. The STA-ConvGru module, formed by CNN, GRU, and attention mechanism, can capture both temporal and spatial dependencies of traffic flow. We evaluate our proposed model using real-world datasets and experiments show that the ASTCG model outperforms the state-of-the-art model. Full article

(This article belongs to the Special Issue Application of Information Theory to Physical Modeling and State Awareness in Complex Systems)

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12 pages, 2476 KiB

Open AccessArticle

Neural Network-Based Prediction for Secret Key Rate of Underwater Continuous-Variable Quantum Key Distribution through a Seawater Channel

by Yun Mao, Yiwu Zhu, Hui Hu, Gaofeng Luo, Jinguang Wang, Yijun Wang and Ying Guo

Entropy 2023, 25(6), 937; https://doi.org/10.3390/e25060937 - 14 Jun 2023

Viewed by 1300

Abstract

Continuous-variable quantum key distribution (CVQKD) plays an important role in quantum communications, because of its compatible setup for optical implementation with low cost. For this paper, we considered a neural network approach to predicting the secret key rate of CVQKD with discrete modulation [...] Read more.

Continuous-variable quantum key distribution (CVQKD) plays an important role in quantum communications, because of its compatible setup for optical implementation with low cost. For this paper, we considered a neural network approach to predicting the secret key rate of CVQKD with discrete modulation (DM) through an underwater channel. A long-short-term-memory-(LSTM)-based neural network (NN) model was employed, in order to demonstrate performance improvement when taking into account the secret key rate. The numerical simulations showed that the lower bound of the secret key rate could be achieved for a finite-size analysis, where the LSTM-based neural network (NN) was much better than that of the backward-propagation-(BP)-based neural network (NN). This approach helped to realize the fast derivation of the secret key rate of CVQKD through an underwater channel, indicating that it can be used for improving performance in practical quantum communications. Full article

(This article belongs to the Special Issue Quantum Communication and Quantum Key Distribution)

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2 pages, 181 KiB

Open AccessEditorial

Entropy-Based Statistics and Their Applications

by Zhiyi Zhang

Entropy 2023, 25(6), 936; https://doi.org/10.3390/e25060936 - 14 Jun 2023

Cited by 2 | Viewed by 1447

Abstract

During the last few decades, research activity in modeling the properties of random systems via entropies has grown noticeably across a wide spectrum of fields [...] Full article

(This article belongs to the Special Issue Entropy-Based Statistics and Their Applications)

18 pages, 1064 KiB

Open AccessArticle

Topic Discovery and Hotspot Analysis of Sentiment Analysis of Chinese Text Using Information-Theoretic Method

by Changlu Zhang, Haojie Fan, Jian Zhang, Qiong Yang and Liqian Tang

Entropy 2023, 25(6), 935; https://doi.org/10.3390/e25060935 - 13 Jun 2023

Cited by 1 | Viewed by 1789

Abstract

Currently, sentiment analysis is a research hotspot in many fields such as computer science and statistical science. Topic discovery of the literature in the field of text sentiment analysis aims to provide scholars with a quick and effective understanding of its research trends. [...] Read more.

Currently, sentiment analysis is a research hotspot in many fields such as computer science and statistical science. Topic discovery of the literature in the field of text sentiment analysis aims to provide scholars with a quick and effective understanding of its research trends. In this paper, we propose a new model for the topic discovery analysis of literature. Firstly, the FastText model is applied to calculate the word vector of literature keywords, based on which cosine similarity is applied to calculate keyword similarity, to carry out the merging of synonymous keywords. Secondly, the hierarchical clustering method based on the Jaccard coefficient is used to cluster the domain literature and count the literature volume of each topic. Thirdly, the information gain method is applied to extract the high information gain characteristic words of various topics, based on which the connotation of each topic is condensed. Finally, by conducting a time series analysis of the literature, a four-quadrant matrix of topic distribution is constructed to compare the research trends of each topic within different stages. The 1186 articles in the field of text sentiment analysis from 2012 to 2022 can be divided into 12 categories. By comparing and analyzing the topic distribution matrices of the two phases of 2012 to 2016 and 2017 to 2022, it is found that the various categories of topics have obvious research development changes in different phases. The results show that: ① Among the 12 categories, online opinion analysis of social media comments represented by microblogs is one of the current hot topics. ② The integration and application of methods such as sentiment lexicon, traditional machine learning and deep learning should be enhanced. ③ Semantic disambiguation of aspect-level sentiment analysis is one of the current difficult problems this field faces. ④ Research on multimodal sentiment analysis and cross-modal sentiment analysis should be promoted. Full article

(This article belongs to the Special Issue Information-Theoretic Methods in Data Analytics)

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22 pages, 369 KiB

Open AccessArticle

Genetic Algebras Associated with ξ^(a)-Quadratic Stochastic Operators

by Farrukh Mukhamedov, Izzat Qaralleh, Taimun Qaisar and Mahmoud Alhaj Hasan

Entropy 2023, 25(6), 934; https://doi.org/10.3390/e25060934 - 13 Jun 2023

Cited by 4 | Viewed by 1207

Abstract

The present paper deals with a class of

ξ^{(a)}

-quadratic stochastic operators, referred to as QSOs, on a two-dimensional simplex. It investigates the algebraic properties of the genetic algebras associated with

ξ^{(a)}

-QSOs. Namely, the associativity, characters [...] Read more.

The present paper deals with a class of

ξ^{(a)}

-quadratic stochastic operators, referred to as QSOs, on a two-dimensional simplex. It investigates the algebraic properties of the genetic algebras associated with

ξ^{(a)}

-QSOs. Namely, the associativity, characters and derivations of genetic algebras are studied. Moreover, the dynamics of these operators are also explored. Specifically, we focus on a particular partition that results in nine classes, which are further reduced to three nonconjugate classes. Each class gives rise to a genetic algebra denoted as

A_{i}

, and it is shown that these algebras are isomorphic. The investigation then delves into analyzing various algebraic properties within these genetic algebras, such as associativity, characters, and derivations. The conditions for associativity and character behavior are provided. Furthermore, a comprehensive analysis of the dynamic behavior of these operators is conducted. Full article

(This article belongs to the Special Issue New Trends in Theoretical and Mathematical Physics)

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13 pages, 660 KiB

Open AccessArticle

Robustness of Sparsely Distributed Representations to Adversarial Attacks in Deep Neural Networks

by Nida Sardar, Sundas Khan, Arend Hintze and Priyanka Mehra

Entropy 2023, 25(6), 933; https://doi.org/10.3390/e25060933 - 13 Jun 2023

Cited by 2 | Viewed by 1988

Abstract

Deep learning models have achieved an impressive performance in a variety of tasks, but they often suffer from overfitting and are vulnerable to adversarial attacks. Previous research has shown that dropout regularization is an effective technique that can improve model generalization and robustness. [...] Read more.

Deep learning models have achieved an impressive performance in a variety of tasks, but they often suffer from overfitting and are vulnerable to adversarial attacks. Previous research has shown that dropout regularization is an effective technique that can improve model generalization and robustness. In this study, we investigate the impact of dropout regularization on the ability of neural networks to withstand adversarial attacks, as well as the degree of “functional smearing” between individual neurons in the network. Functional smearing in this context describes the phenomenon that a neuron or hidden state is involved in multiple functions at the same time. Our findings confirm that dropout regularization can enhance a network’s resistance to adversarial attacks, and this effect is only observable within a specific range of dropout probabilities. Furthermore, our study reveals that dropout regularization significantly increases the distribution of functional smearing across a wide range of dropout rates. However, it is the fraction of networks with lower levels of functional smearing that exhibit greater resilience against adversarial attacks. This suggests that, even though dropout improves robustness to fooling, one should instead try to decrease functional smearing. Full article

(This article belongs to the Special Issue Signal and Information Processing in Networks)

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18 pages, 1882 KiB

Open AccessArticle

Unsupervised Low-Light Image Enhancement Based on Generative Adversarial Network

by Wenshuo Yu, Liquan Zhao and Tie Zhong

Entropy 2023, 25(6), 932; https://doi.org/10.3390/e25060932 - 13 Jun 2023

Cited by 2 | Viewed by 2633

Abstract

Low-light image enhancement aims to improve the perceptual quality of images captured under low-light conditions. This paper proposes a novel generative adversarial network to enhance low-light image quality. Firstly, it designs a generator consisting of residual modules with hybrid attention modules and parallel [...] Read more.

Low-light image enhancement aims to improve the perceptual quality of images captured under low-light conditions. This paper proposes a novel generative adversarial network to enhance low-light image quality. Firstly, it designs a generator consisting of residual modules with hybrid attention modules and parallel dilated convolution modules. The residual module is designed to prevent gradient explosion during training and to avoid feature information loss. The hybrid attention module is designed to make the network pay more attention to useful features. A parallel dilated convolution module is designed to increase the receptive field and capture multi-scale information. Additionally, a skip connection is utilized to fuse shallow features with deep features to extract more effective features. Secondly, a discriminator is designed to improve the discrimination ability. Finally, an improved loss function is proposed by incorporating pixel loss to effectively recover detailed information. The proposed method demonstrates superior performance in enhancing low-light images compared to seven other methods. Full article

(This article belongs to the Special Issue Deep Learning Models and Applications to Computer Vision)

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15 pages, 1261 KiB

Open AccessArticle

Collective Dynamics, Diversification and Optimal Portfolio Construction for Cryptocurrencies

by Nick James and Max Menzies

Entropy 2023, 25(6), 931; https://doi.org/10.3390/e25060931 - 13 Jun 2023

Cited by 11 | Viewed by 1916

Abstract

Since its conception, the cryptocurrency market has been frequently described as an immature market, characterized by significant swings in volatility and occasionally described as lacking rhyme or reason. There has been great speculation as to what role it plays in a diversified portfolio. [...] Read more.

Since its conception, the cryptocurrency market has been frequently described as an immature market, characterized by significant swings in volatility and occasionally described as lacking rhyme or reason. There has been great speculation as to what role it plays in a diversified portfolio. For instance, is cryptocurrency exposure an inflationary hedge or a speculative investment that follows broad market sentiment with amplified beta? We have recently explored similar questions with a clear focus on the equity market. There, our research revealed several noteworthy dynamics such as an increase in the market’s collective strength and uniformity during crises, greater diversification benefits across equity sectors (rather than within them), and the existence of a “best value” portfolio of equities. In essence, we can now contrast any potential signatures of maturity we identify in the cryptocurrency market and contrast these with the substantially larger, older and better-established equity market. This paper aims to investigate whether the cryptocurrency market has recently exhibited similar mathematical properties as the equity market. Instead of relying on traditional portfolio theory, which is grounded in the financial dynamics of equity securities, we adjust our experimental focus to capture the presumed behavioral purchasing patterns of retail cryptocurrency investors. Our focus is on collective dynamics and portfolio diversification in the cryptocurrency market, and examining whether previously established results in the equity market hold in the cryptocurrency market and to what extent. The results reveal nuanced signatures of maturity related to the equity market, including the fact that correlations collectively spike around exchange collapses, and identify an ideal portfolio size and spread across different groups of cryptocurrencies. Full article

(This article belongs to the Special Issue Signatures of Maturity in Cryptocurrency Market)

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Figure 1
Normalized leading eigenvalue <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>λ</mi> <mo>˜</mo> </mover> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> of the cross-correlation matrix as a function of time, for (a) the entire collection of cryptocurrencies and (b) the ten deciles. Like the equity market, collective correlations spike during market crises, such as COVID-19, and the collapse of exchanges BitMEX and FTX. Full article ">Figure 2
Uniformity <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> of the leading eigenvector <math display="inline"><semantics> <msub> <mi mathvariant="bold">v</mi> <mn>1</mn> </msub> </semantics></math> of the cross-correlation matrix as a function of time, for (a) the entire collection of cryptocurrencies and (b) the ten deciles. The results are dramatically different compared to the equity market, with numerous deciles exhibiting strikingly low uniformity scores over time. Full article ">Figure 3
Results of hierarchical clustering applied to (<a href="#FD6-entropy-25-00931" class="html-disp-formula">6</a>) between ordered pairs <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics></math>. A large majority cluster confirms the finding of <a href="#entropy-25-00931-t002" class="html-table">Table 2</a> that the (4,4) portfolio is closely similar to the full (10,4) portfolio in its diversification benefit over time. Full article ">

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Entropy, Volume 25, Issue 6 (June 2023) – 129 articles

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