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.

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.

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.

Environmental data analysis based on fuzzy clustering method

2020 ◽  
pp. 002072092092854
Author(s):  
Yongyi Li ◽  
Zhongqiang Yang ◽  
Kaixu Han
Keyword(s):  
Membership Function ◽  
Fuzzy Clustering ◽  
Clustering Algorithm ◽  
Fuzzy Theory ◽  
Living Environment ◽  
Environmental Data ◽  
Fuzzy Evaluation ◽  
Clustering Method ◽  
Fuzzy Clustering Method ◽  
The Membership Function

With the growing complexity of the human living environment, the environment-related industries have also been flourishing. Clustering of environmental data is a key task of environment research. The environmental data are characterized by diversification, boundary fuzzification, incompleteness, etc. This paper conducts a cluster analysis of environmental data based on fuzzy theory. To start with, the basic principle of fuzzy theory is analyzed, and the focus is on studying the membership function and the [Formula: see text]-cut-set knowledge. Following that, the fuzzy clustering method and its process are studied. Finally, fuzzy evaluation is used to build the membership function after experiment on the MATLAB platform to evaluate the environmental quality. The fuzzy C-means clustering algorithm is used to realize the target identification of environmental data. In the process of fuzzy clustering, fuzzy evaluation of the seawater quality is realized, and the redundant data of the monitoring station are removed. Through experiment and analysis, experimental results are in line with the practical situations and show a high consistency with the data characteristics. Compared with the traditional clustering algorithm, fuzzy clustering is more suitable for environmental data processing in environmental data research and analysis.

A Hard C-Means Clustering Algorithm Incorporating Membership KL Divergence and Local Data Information for Noisy Image Segmentation

2017 ◽  
Vol 32 (04) ◽  
pp. 1850012 ◽  
Author(s):  
R. R. Gharieb ◽  
G. Gendy ◽  
H. Selim
Keyword(s):  
Image Segmentation ◽  
Membership Function ◽  
Clustering Algorithm ◽  
Clustering Algorithms ◽  
Cluster Center ◽  
Local Data ◽  
Cluster Membership ◽  
Kl Divergence ◽  
Clustering Approach ◽  
Center Distance

In this paper, the standard hard C-means (HCM) clustering approach to image segmentation is modified by incorporating weighted membership Kullback–Leibler (KL) divergence and local data information into the HCM objective function. The membership KL divergence, used for fuzzification, measures the proximity between each cluster membership function of a pixel and the locally-smoothed value of the membership in the pixel vicinity. The fuzzification weight is a function of the pixel to cluster-centers distances. The used pixel to a cluster-center distance is composed of the original pixel data distance plus a fraction of the distance generated from the locally-smoothed pixel data. It is shown that the obtained membership function of a pixel is proportional to the locally-smoothed membership function of this pixel multiplied by an exponentially distributed function of the minus pixel distance relative to the minimum distance provided by the nearest cluster-center to the pixel. Therefore, since incorporating the locally-smoothed membership and data information in addition to the relative distance, which is more tolerant to additive noise than the absolute distance, the proposed algorithm has a threefold noise-handling process. The presented algorithm, named local data and membership KL divergence based fuzzy C-means (LDMKLFCM), is tested by synthetic and real-world noisy images and its results are compared with those of several FCM-based clustering algorithms.

scholarly journals Improved Fuzzy C-Means Clustering for Transformer Fault Diagnosis Using Dissolved Gas Analysis Data

Energies ◽  
2018 ◽  
Vol 11 (9) ◽  
pp. 2344 ◽  
Author(s):  
Enwen Li ◽  
Linong Wang ◽  
Bin Song ◽  
Siliang Jian
Keyword(s):  
Fault Diagnosis ◽  
Membership Function ◽  
Data Clustering ◽  
Clustering Algorithm ◽  
Gas Analysis ◽  
Dissolved Gas ◽  
Fuzzy C Means ◽  
Dissolved Gas Analysis ◽  
Fcm Clustering ◽  
Transformer Fault

Dissolved gas analysis (DGA) of the oil allows transformer fault diagnosis and status monitoring. Fuzzy c-means (FCM) clustering is an effective pattern recognition method, but exhibits poor clustering accuracy for dissolved gas data and usually fails to subsequently correctly classify transformer faults. The existing feasible approach involves combination of the FCM clustering algorithm with other intelligent algorithms, such as neural networks and support vector machines. This method enables good classification; however, the algorithm complexity is greatly increased. In this paper, the FCM clustering algorithm itself is improved and clustering analysis of DGA data is realized. First, the non-monotonicity of the traditional clustering membership function with respect to the sample distance and its several local extrema are discussed, which mainly explain the poor classification accuracy of DGA data clustering. Then, an exponential form of the membership function is proposed to obtain monotony with respect to distance, thereby improving the dissolved gas data clustering. Likewise, a similarity function to determine the degree of membership is derived. Test results for large datasets show that the improved clustering algorithm can be successfully applied for DGA-data-based transformer fault detection.

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.

Identification of Circulating Fluidized Bed Boiler Bed Temperature Based on Hyper-Plane-Shaped Fuzzy C-Regression Model

2020 ◽  
Vol 19 (04) ◽  
pp. 2050029
Author(s):  
Jianzhong Shi
Keyword(s):  
Regression Model ◽  
Fluidized Bed ◽  
Membership Function ◽  
Clustering Algorithm ◽  
Circulating Fluidized Bed ◽  
Fuzzy Model ◽  
Identification Algorithm ◽  
Identification Process ◽  
Bed Temperature ◽  
Hyper Plane

Bed temperature in dense-phase zone is the key parameter of circulating fluidized bed (CFB) boiler for stable combustion and economic operation. It is difficult to establish an accurate bed temperature model as the complexity of circulating fluidized bed combustion system. T-S fuzzy model was widely applied in the system identification for it can approximate complex nonlinear system with high accuracy. Fuzzy c-regression model (FCRM) clustering based on hyper-plane-shaped distance has the advantages in describing T-S fuzzy model, and Gaussian function was adapted in antecedent membership function of T-S fuzzy model. However, Gaussian fuzzy membership function was more suitable for clustering algorithm using point to point distance, such as fuzzy c-means (FCM). In this paper, a hyper-plane-shaped FCRM clustering algorithm for T-S fuzzy model identification algorithm is proposed. The antecedent membership function of proposed identification algorithm is defined by a hyper-plane-shaped membership function and an improved fuzzy partition method is applied. To illustrate the efficiency of the proposed identification algorithm, the algorithm is applied in four nonlinear systems which shows higher identification accuracy and simplified identification process. At last, the algorithm is used in a circulating fluidized bed boiler bed temperature identification process, and gets better identification result.

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