scholarly journals Phase Transitions in Fuzzy Clustering Based on Fuzzy Entropy

2003 ◽  
Vol 7 (3) ◽  
pp. 370-376 ◽  
Author(s):  
Makoto Yasuda ◽  
◽  
Takeshi Furuhashi ◽  
Shigeru Okuma ◽  
◽  
...  
Keyword(s):  
Phase Transitions ◽  
Fuzzy Clustering ◽  
Numerical Experiments ◽  
Mechanical Characteristics ◽  
Point Of View ◽  
Fuzzy Entropy ◽  
Number Of Clusters ◽  
Statistical Mechanical ◽  
Dirac Distribution ◽  
Dirac Function

We studied the statistical mechanical characteristics of fuzzy clustering regularized with fuzzy entropy. We obtained Fermi-Dirac distribution as a membership function by regularizing the fuzzy c-means with fuzzy entropy. We then formulated it as direct annealing clustering, and determined the meanings of the Fermi-Dirac function and fuzzy entropy from the statistical mechanical point of view, and showed that this fuzzy clustering is a part of Fermi-Dirac statistics. We also derived the critical temperature at which phase transition occurs in this fuzzy clustering. Then, with a combination of cluster divisions by phase transitions and an adequate division termination condition, we derived fuzzy clustering that automatically determined the number of clusters, as verified by numerical experiments.

scholarly journals A New Validity Index Based on Fuzzy Energy and Fuzzy Entropy Measures in Fuzzy Clustering Problems

2020 ◽  
Author(s):  
Ferdinando Di Martino ◽  
salvatore sessa
Keyword(s):  
Fuzzy Clustering ◽  
Fuzzy Entropy ◽  
Validity Index ◽  
Number Of Clusters ◽  
Machine Learning Classification ◽  
Time Consumption ◽  
Validity Indices ◽  
Clustering Quality ◽  
Fuzzy Energy

Two well-known drawbacks in fuzzy clustering are the requirement of assign in advance the number of clusters and random initialization of cluster centers.; the quality of the final fuzzy clusters depends heavily on the initial choice of the number of clusters and the initialization of the clusters, then it is necessary to apply a validity index to measure the compactness and the separability of the final clusters and run the clustering algorithm several times. We propose a new fuzzy C-means algorithm in which a validity index based on the concepts of maximum fuzzy energy and minimum fuzzy entropy is applied to initialize the cluster centers and to find the optimal number of clusters and initial cluster centers in order to obtain a good clustering quality, without increasing time consumption. We test our algorithm on UCI machine learning classification datasets comparing the results with the ones obtained by using well-known validity indices and variations of FCM using optimization algorithms in the initialization phase. The comparison results show that our algorithm represents an optimal trade-off between the quality of clustering and the time consumption.

scholarly journals A New Validity Index Based on Fuzzy Energy and Fuzzy Entropy Measures in Fuzzy Clustering Problems

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1200
Author(s):  
Ferdinando Di Martino ◽  
Salvatore Sessa
Keyword(s):  
Fuzzy Clustering ◽  
Fuzzy Entropy ◽  
Validity Index ◽  
Number Of Clusters ◽  
Machine Learning Classification ◽  
Time Consumption ◽  
Fuzzy C Means ◽  
Validity Indices ◽  
Fuzzy Energy

Two well-known drawbacks in fuzzy clustering are the requirement of assigning in advance the number of clusters and random initialization of cluster centers. The quality of the final fuzzy clusters depends heavily on the initial choice of the number of clusters and the initialization of the clusters, then, it is necessary to apply a validity index to measure the compactness and the separability of the final clusters and run the clustering algorithm several times. We propose a new fuzzy C-means algorithm in which a validity index based on the concepts of maximum fuzzy energy and minimum fuzzy entropy is applied to initialize the cluster centers and to find the optimal number of clusters and initial cluster centers in order to obtain a good clustering quality, without increasing time consumption. We test our algorithm on UCI (University of California at Irvine) machine learning classification datasets comparing the results with the ones obtained by using well-known validity indices and variations of fuzzy C-means by using optimization algorithms in the initialization phase. The comparison results show that our algorithm represents an optimal trade-off between the quality of clustering and the time consumption.

scholarly journals Virtual Element Method for the Laplace-Beltrami equation on surfaces

2018 ◽  
Vol 52 (3) ◽  
pp. 965-993 ◽  
Author(s):  
Massimo Frittelli ◽  
Ivonne Sgura
Keyword(s):  
Numerical Experiments ◽  
Beltrami Equation ◽  
Convergence Result ◽  
Point Of View ◽  
Polygonal Approximation ◽  
Virtual Element Method ◽  
Element Space ◽  
First Order ◽  
Virtual Element ◽  
Element Method

We present and analyze a Virtual Element Method (VEM) for the Laplace-Beltrami equation on a surface in ℝ3, that we call Surface Virtual Element Method (SVEM). The method combines the Surface Finite Element Method (SFEM) (Dziuk, Eliott, G. Dziuk and C.M. Elliott., Acta Numer. 22 (2013) 289–396.) and the recent VEM (Beirão da Veiga et al., Math. Mod. Methods Appl. Sci. 23 (2013) 199–214.) in order to allow for a general polygonal approximation of the surface. We account for the error arising from the geometry approximation and in the case of polynomial order k = 1 we extend to surfaces the error estimates for the interpolation in the virtual element space. We prove existence, uniqueness and first order H1 convergence of the numerical solution.We highlight the differences between SVEM and VEM from the implementation point of view. Moreover, we show that the capability of SVEM of handling nonconforming and discontinuous meshes can be exploited in the case of surface pasting. We provide some numerical experiments to confirm the convergence result and to show an application of mesh pasting.

scholarly journals Paths of unitary access to exceptional points

2021 ◽  
Vol 2038 (1) ◽  
pp. 012026
Author(s):  
Miloslav Znojil
Keyword(s):  
Hilbert Space ◽  
Phase Transitions ◽  
Quantum Phase Transitions ◽  
Quantum Systems ◽  
Point Of View ◽  
Explicit Description ◽  
Direct Access ◽  
Exceptional Point ◽  
Exceptional Points ◽  
Innovative Idea

Abstract With an innovative idea of acceptability and usefulness of the non-Hermitian representations of Hamiltonians for the description of unitary quantum systems (dating back to the Dyson’s papers), the community of quantum physicists was offered a new and powerful tool for the building of models of quantum phase transitions. In this paper the mechanism of such transitions is discussed from the point of view of mathematics. The emergence of the direct access to the instant of transition (i.e., to the Kato’s exceptional point) is attributed to the underlying split of several roles played by the traditional single Hilbert space of states ℒ into a triplet (viz., in our notation, spaces K and ℋ besides the conventional ℒ ). Although this explains the abrupt, quantum-catastrophic nature of the change of phase (i.e., the loss of observability) caused by an infinitesimal change of parameters, the explicit description of the unitarity-preserving corridors of access to the phenomenologically relevant exceptional points remained unclear. In the paper some of the recent results in this direction are summarized and critically reviewed.

scholarly journals Crystallization of Binary Alloys and Non-Equilibrium Phase Transitions

2019 ◽  
Vol 489 (6) ◽  
pp. 545-551
Author(s):  
E. V. Radkevich ◽  
O. A. Vasil’eva ◽  
M. I. Sidorov
Keyword(s):  
Phase Transition ◽  
Phase Transitions ◽  
Numerical Experiments ◽  
Binary Alloys ◽  
Equilibrium Phase ◽  
Homogeneous State ◽  
Nonequilibrium Phase Transition ◽  
Nonequilibrium Phase ◽  
Initial Stage ◽  
Melt Cooling

A model was constructed for the reconstruction of the initial stage of crystallization of binary alloys as a nonequilibrium phase transition, the mechanism of which is diffusion stratification. Numerical experiments were performed. Self-excitation of a homogeneous state by the edge control melt cooling condition.

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