scholarly journals Quantitative Analyses and Development of aq-Incrementation Algorithm for FCM with Tsallis Entropy Maximization

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Makoto Yasuda
Keyword(s):  
Membership Function ◽  
Annealing Temperature ◽  
Data Distribution ◽  
Tsallis Entropy ◽  
Deterministic Annealing ◽  
Entropy Maximization ◽  
Quantitative Analyses ◽  
Statistical Mechanical ◽  
Combinatorial Methods ◽  
The Membership Function

Tsallis entropy is aq-parameter extension of Shannon entropy. By extremizing the Tsallis entropy within the framework of fuzzyc-means clustering (FCM), a membership function similar to the statistical mechanical distribution function is obtained. The Tsallis entropy-based DA-FCM algorithm was developed by combining it with the deterministic annealing (DA) method. One of the challenges of this method is to determine an appropriate initial annealing temperature and aqvalue, according to the data distribution. This is complex, because the membership function changes its shape by decreasing the temperature or by increasingq. Quantitative relationships between the temperature andqare examined, and the results show that, in order to changeuikqequally, inverse changes must be made to the temperature andq. Accordingly, in this paper, we propose and investigate two kinds of combinatorial methods forq-incrementation and the reduction of temperature for use in the Tsallis entropy-based FCM. In the proposed methods,qis defined as a function of the temperature. Experiments are performed using Fisher’s iris dataset, and the proposed methods are confirmed to determine an appropriateqvalue in many cases.

Analysis of Temperature and q-Parameter Dependency of FCM with Tsallis Entropy Maximization

2018 ◽  
Vol 22 (5) ◽  
pp. 666-673
Author(s):  
Makoto Yasuda ◽  
Keyword(s):  
Membership Function ◽  
Clustering Algorithm ◽  
Annealing Temperature ◽  
Tsallis Entropy ◽  
Deterministic Annealing ◽  
Entropy Maximization ◽  
Statistical Mechanical ◽  
Proportional Relationship ◽  
Parameter Dependency ◽  
The Membership Function

The Tsallis entropy is a q-parameter extension of the Shannon entropy. By maximizing it within the framework of fuzzy c-means, statistical mechanical membership functions can be derived. We propose a clustering algorithm that includes the membership function and deterministic annealing. One of the major issues for this method is the determination of an appropriate values for q and an initial annealing temperature for a given data distribution. Accordingly, in our previous study, we investigated the relationship between q and the annealing temperature. We quantitatively compared the area of the membership function for various values of q and for various temperatures. The results showed that the effect of q on the area was nearly the inverse of that of the temperature. In this paper, we analytically investigate this relationship by directly integrating the membership function, and the inversely proportional relationship between q and the temperature is approximately confirmed. Based on this relationship, a q-incrementation deterministic annealing fuzzy c-means (FCM) algorithm is developed. Experiments are performed, and it is confirmed that the algorithm works properly. However, it is also confirmed that differences in the shape of the membership function of the annealing method and that of the q-incrementation method are remained.

Multi-qExtension of Tsallis Entropy Based Fuzzyc-Means Clustering

2014 ◽  
Vol 18 (3) ◽  
pp. 289-296 ◽  
Author(s):  
Makoto Yasuda ◽  
◽  
Yasuyuki Orito
Keyword(s):  
Membership Function ◽  
Shannon Entropy ◽  
Clustering Algorithm ◽  
Numerical Data ◽  
Tsallis Entropy ◽  
Entropy Maximization ◽  
Cluster Distribution ◽  
The Membership Function

Tsallis entropy is aq-parameter extension of Shannon entropy. Based on the Tsallis entropy, we have introduced an entropy maximization method to fuzzyc-means clustering (FCM), and developed a new clustering algorithm using a single-qvalue. In this article, we propose a multi-qextension of the conventional single-qmethod. In this method, theqs are assigned individually to each cluster. Eachqvalue is determined so that the membership function fits the corresponding cluster distribution. This is done to improve the accuracy of clustering over that of the conventional single-qmethod. Experiments are performed on randomly generated numerical data and Fisher’s iris dataset, and it is confirmed that the proposed method improves the accuracy of clustering and is superior to the conventional single-qmethod. If the parameters introduced in the proposed method can be optimized, it is expected that the clusters in data distributions that are composed of clusters of various sizes can be determined more accurately.

Approximate Determination ofq-Parameter for FCM with Tsallis Entropy Maximization

2017 ◽  
Vol 21 (7) ◽  
pp. 1152-1160
Author(s):  
Makoto Yasuda ◽  
Keyword(s):  
Clustering Algorithm ◽  
Annealing Temperature ◽  
Expansion Method ◽  
Tsallis Entropy ◽  
Approximate Determination ◽  
Data Set ◽  
Entropy Maximization ◽  
Fcm Clustering ◽  
Statistical Mechanical ◽  
Series Expansion Method

This paper considers a fuzzyc-means (FCM) clustering algorithm in combination with deterministic annealing and the Tsallis entropy maximization. The Tsallis entropy is aq-parameter extension of the Shannon entropy. By maximizing the Tsallis entropy within the framework of FCM, statistical mechanical membership functions can be derived. One of the major considerations when using this method is how to determine appropriate values forqand the highest annealing temperature,Thigh, for a given data set. Accordingly, in this paper, a method for determining these values simultaneously without introducing any additional parameters is presented, where the membership function is approximated using a series expansion method. The results of experiments indicate that the proposed method is effective, and bothqandThighcan be determined automatically and algebraically from a given data set.

scholarly journals Deterministic Annealing Approach to Fuzzy C-Means Clustering Based on Entropy Maximization

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Makoto Yasuda
Keyword(s):  
Shannon Entropy ◽  
Distribution Functions ◽  
Point Of View ◽  
Tsallis Entropy ◽  
Deterministic Annealing ◽  
Membership Functions ◽  
Entropy Maximization ◽  
Fuzzy C Means ◽  
Fuzzy Clustering Method ◽  
Statistical Mechanical

This paper is dealing with the fuzzy clustering method which combines the deterministic annealing (DA) approach with an entropy, especially the Shannon entropy and the Tsallis entropy. By maximizing the Shannon entropy, the fuzzy entropy, or the Tsallis entropy within the framework of the fuzzy c-means (FCM) method, membership functions similar to the statistical mechanical distribution functions are obtained. We examine characteristics of these entropy-based membership functions from the statistical mechanical point of view. After that, both the Shannon- and Tsallis-entropy-based FCMs are formulated as DA clustering using the very fast annealing (VFA) method as a cooling schedule. Experimental results indicate that the Tsallis-entropy-based FCM is stable with very fast deterministic annealing and suitable for this annealing process.

Possibility-Based Design Optimization Method for Design Problems With Both Statistical and Fuzzy Input Data

2005 ◽  
Vol 128 (4) ◽  
pp. 928-935 ◽  
Author(s):  
Liu Du ◽  
K. K. Choi ◽  
Byeng D. Youn ◽  
David Gorsich
Keyword(s):  
Design Optimization ◽  
Optimum Design ◽  
Membership Function ◽  
Input Data ◽  
Performance Measure ◽  
Sufficient Data ◽  
Design Problems ◽  
Fuzzy Variables ◽  
Fuzzy Input ◽  
The Membership Function

The reliability based design optimization (RBDO) method is prevailing in stochastic structural design optimization by assuming the amount of input data is sufficient enough to create accurate input statistical distribution. If the sufficient input data cannot be generated due to limitations in technical and/or facility resources, the possibility-based design optimization (PBDO) method can be used to obtain reliable designs by utilizing membership functions for epistemic uncertainties. For RBDO, the performance measure approach (PMA) is well established and accepted by many investigators. It is found that the same PMA is a very much desirable approach also for the PBDO problems. In many industry design problems, we have to deal with uncertainties with sufficient data and uncertainties with insufficient data simultaneously. For these design problems, it is not desirable to use RBDO since it could lead to an unreliable optimum design. This paper proposes to use PBDO for design optimization for such problems. In order to treat uncertainties as fuzzy variables, several methods for membership function generation are proposed. As less detailed information is available for the input data, the membership function that provides more conservative optimum design should be selected. For uncertainties with sufficient data, the membership function that yields the least conservative optimum design is proposed by using the possibility-probability consistency theory and the least conservative condition. The proposed approach for design problems with mixed type input uncertainties is applied to some example problems to demonstrate feasibility of the approach. It is shown that the proposed approach provides conservative optimum design.

scholarly journals Controllable Effective Threshold Based Fusion Coverage Algorithm in Mobile Sensor Networks

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Yong Lu ◽  
Na Sun
Keyword(s):  
Sensor Networks ◽  
Membership Function ◽  
Network Lifetime ◽  
Physical Parameters ◽  
Network Coverage ◽  
Target Node ◽  
Effective Threshold ◽  
Coverage Quality ◽  
Monitoring Area ◽  
The Membership Function

The coverage quality and network lifetime are two key parameters in the research of sensor networks. The coverage quality shows direct influences on the network lifetime. Meanwhile, it is influenced by many other factors such as physical parameters and environmental parameters. To reveal the connection between the coverage quality and the parameters of target node concerned, a fusion coverage algorithm with controllable effective threshold is proposed based on the sensing probability model. We give the model for the membership function of coverage intensity as well as the prediction model for the fusion operator. The range for the effective threshold is presented according to the membership function model. Meanwhile, the maximum of the effective coverage intensity for the target nodes within the monitoring area is derived. The derivation of the maximal fusion coverage intensity is elaborated utilizing a processing function on the distances from the target node to the ones in the sensor node set. Furthermore, we investigate different network properties within the monitoring area such as network coverage quality, the dynamic change of parameters, and the network lifetime, based on the probability theory and the geometric theory. Finally, we present numerical simulations to verify the performances of our algorithm. It is shown under different settings that, compared with the demand coverage quality, the proposed algorithm could improve the network coverage quality by 15.66% on average. The simulation experiment results show that our proposed algorithm has an average improvement by 10.12% and 13.23% in terms of the performances on network coverage quality and network lifetime, respectively. The research results are enlightening to the edge coverage and nonlinear coverage problems within the monitoring area.

FUZZY CLUSTERING OF FEATURE VECTORS WITH SOME ORDINAL VALUED ATTRIBUTES USING GRADIENT DESCENT FOR LEARNING

2008 ◽  
Vol 16 (02) ◽  
pp. 195-218
Author(s):  
ROELOF K. BROUWER
Keyword(s):  
Objective Function ◽  
Membership Function ◽  
Fuzzy Clustering ◽  
Gradient Descent ◽  
Ratio Scale ◽  
Interval Scale ◽  
Real Numbers ◽  
Fuzzy C Means Clustering ◽  
Feature Values ◽  
The Membership Function

There are well established methods for fuzzy clustering especially for the cases where the feature values are numerical of ratio or interval scale. Not so well established are methods to be applied when the feature values are ordinal or nominal. In that case there is no one best method it seems. This paper discusses a method where unknown numeric variables are assigned to the ordinal values. Part of minimizing an objective function for the clustering is to find numeric values for these variables. Thus real numbers of interval scale and even ratio scale for that matter are assigned to the original ordinal values. The method uses the same objective function as used in fuzzy c-means clustering but both the membership function and the ordinal to real mapping are determined by gradient descent. Since the ordinal to real mapping is not known it cannot be verified for its legitimacy. However the ordinal to real mapping that is found is best in terms of the clustering produced. Simulations show the method to be quite effective.

Using Cloud Model to Improve the Membership Function in Fuzzy Risk Assessment of Reservoir-Induced Seismicity

2013 ◽  
Vol 664 ◽  
pp. 270-275 ◽  
Author(s):  
Ming Zhong ◽  
Qiu Wen Zhang
Keyword(s):  
Risk Assessment ◽  
Membership Function ◽  
Induced Seismicity ◽  
Cloud Model ◽  
Reservoir Induced Seismicity ◽  
Fuzzy Risk Assessment ◽  
The One ◽  
The Relationship ◽  
The Membership Function

Due to the uncertainty and complexity of the causes in reservoir-induced seismicity, the relationship between the environmental factor and the possible earthquake magnitude can be described by membership function. This study aims to propose a fuzzy method to contribute the membership function in which the normal cloud model is applied. Firstly, the cloud model is introduced in detail. Based on normal cloud model, the one-to-many mapping model is presented to deal with the fuzziness and randomness in the membership function. Finally, the case study in Yangtze Three Gorges Reservoir is presented to illustrate the membership cloud function in fuzzy risk assessment of reservoir-induced seismicity. The obtained results show that the proposed method is the viable approaches in solving the problem when the memberships are vague and imprecise.

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