Showing papers on "Membership function published in 2015"

Picture fuzzy sets

Abstract: In this paper, we introduce the concept of picture fuzzy sets (PFS), which are direct extensions of the fuzzy sets and the intuitonistic fuzzy sets. Then some operations on PFS with some properties are considered. The following sections are devoted to the Zadeh Extension Principle, picture fuzzy relations and picture fuzzy soft sets. Here, the basic preliminaries of PFS theory are presented.

Correlation coefficients for picture fuzzy sets

Abstract: Picture fuzzy sets are extension of Atanassov's intuitionistic fuzzy sets. Picture fuzzy set based models may be adequate in situations when we face human opinions involving more answers of types: yes, abstain, no, refusal. It can be considered as a powerful tool represent an uncertain information in the process of cluster analysis. In this paper, we present a geometrical interpretation of picture fuzzy sets. We propose correlation coefficients for picture fuzzy sets which considers the degree of positive membership, degree of neutral membership, degree of negative membership and the degree of refusal membership. Effectiveness of the proposed correlation coefficient has been established in a bidirectional approximate reasoning systems. We apply the correlation coefficient to clustering analysis under picture fuzzy environments. Advantages of proposed correlation coefficients and drawbacks of existing correlation coefficients have been discussed.

A Novel Risk Decision Making Based on Decision-Theoretic Rough Sets Under Hesitant Fuzzy Information

Abstract: Decision-theoretic rough sets (DTRSs) play a crucial role in risk decision-making problems. With respect to the minimum expected risk, DTRSs deduce the rules of three-way decisions. Considering the new expression of evaluation information with hesitant fuzzy sets (HFSs), we introduce HFSs into DTRSs and explore their decision mechanisms. More specifically, we take into account the losses of DTRSs with hesitant fuzzy elements and propose a new model of hesitant fuzzy decision-theoretic rough sets (HFDTRSs). Some properties of the expected losses and their corresponding scores are carefully investigated under the hesitant fuzzy information. Three-way decisions and the associated cost of each object are further derived. With the above analysis, a novel risk decision-making method with the aid of HFDTRSs is developed. Besides the three-way decisions with DTRSs, the method investigates the ranking and resource allocation by utilizing the associated costs of alternatives and multiobjective 0–1 integer programming. Our study also offers a solution in the aspect of determining losses of DTRS and extends the range of applications.

Conditional spatial fuzzy C-means clustering algorithm for segmentation of MRI images

Abstract: A conditional spatial fuzzy C-means (csFCM) clustering algorithm to improve the robustness of the conventional FCM algorithm is presented.The method incorporates conditional affects and spatial information into the membership functions.The algorithm resolves the problem of sensitivity to noise and intensity inhomogeneity in magnetic resonance imaging (MRI) data.The experimental results on four volumes of simulated and one volume of real-patient MRI brain images, each one having 51 images, support efficiency of the csFCM algorithm.The csFCM algorithm has superior performance in terms of qualitative and quantitative studies on the image segmentation results than the k-means, FCM and some other recently proposed FCM-based algorithms. The fuzzy C-means (FCM) algorithm has got significant importance due to its unsupervised form of learning and more tolerant to variations and noise as compared to other methods in medical image segmentation. In this paper, we propose a conditional spatial fuzzy C-means (csFCM) clustering algorithm to improve the robustness of the conventional FCM algorithm. This is achieved through the incorporation of conditioning effects imposed by an auxiliary (conditional) variable corresponding to each pixel, which describes a level of involvement of the pixel in the constructed clusters, and spatial information into the membership functions. The problem of sensitivity to noise and intensity inhomogeneity in magnetic resonance imaging (MRI) data is effectively reduced by incorporating local and global spatial information into a weighted membership function. The experimental results on four volumes of simulated and one volume of real-patient MRI brain images, each one having 51 images, show that the csFCM algorithm has superior performance in terms of qualitative and quantitative studies such as, cluster validity functions, segmentation accuracy, tissue segmentation accuracy and receiver operating characteristic (ROC) curve on the image segmentation results than the k-means, FCM and some other recently proposed FCM-based algorithms.

Comments on “Interval Type-2 Fuzzy Sets are Generalization of Interval-Valued Fuzzy Sets: Towards a Wide View on Their Relationship”

Abstract: This letter makes some observations about “Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: Towards a wide view on their relationship,” IEEE Trans. Fuzzy Systems that further support the distinction between an interval type-2 fuzzy set (IT2 FS) and an interval-valued fuzzy set (IV FS), points out that all operations, methods, and systems that have been developed and published about IT2 FSs are, so far, only valid in the special case when IT2 FS = IVFS, and suggests some research opportunities.

A new hybrid algorithm based on optimal fuzzy controller in multimachine power system

Abstract: In this article, a new methodology based on fuzzy proportional-integral-derivative PID controller is proposed to damp low frequency oscillation in multimachine power system where the parameters of proposed controller are optimized offline automatically by hybrid genetic algorithm GA and particle swarm optimization PSO techniques. This newly proposed method is more efficient because it cope with oscillations and different operating points. In this strategy, the controller is tuned online from the knowledge base and fuzzy interference. In the proposed method, for achieving the desired level of robust performance exact tuning of rule base and membership functions MF are very important. The motivation for using the GA and PSO as a hybrid method are to reduce fuzzy effort and take large parametric uncertainties in to account. This newly developed control strategy mixed the advantage of GA and PSO techniques to optimally tune the rule base and MF parameters of fuzzy controller that leads to a flexible controller with simple structure while is easy to implement. The proposed method is tested on three machine nine buses and 16 machine power systems with different operating conditions in present of disturbance and nonlinearity. The effectiveness of proposed controller is compared with robust PSS that tune using PSO and the fuzzy controller which is optimized rule base by GA through figure of demerit and integral of the time multiplied absolute value of the error performance indices. The results evaluation shows that the proposed method achieves good robust performance for a wide range of load change in the presents of disturbance and system nonlinearities and is superior to the other controllers. © 2014 Wiley Periodicals, Inc. Complexity 21: 78-93, 2015

Fuzzy Forecasting Based on Two-Factors Second-Order Fuzzy-Trend Logical Relationship Groups and the Probabilities of Trends of Fuzzy Logical Relationships

Abstract: In this paper, we present a new method for fuzzy forecasting based on two-factors second-order fuzzy-trend logical relationship groups and the probabilities of trends of fuzzy-trend logical relationships. Firstly, the proposed method fuzzifies the historical training data of the main factor and the secondary factor into fuzzy sets, respectively, to form two-factors second-order fuzzy logical relationships. Then, it groups the obtained two-factors second-order fuzzy logical relationships into two-factors second-order fuzzy-trend logical relationship groups. Then, it calculates the probability of the “down-trend,” the probability of the “equal-trend” and the probability of the “up-trend” of the two-factors second-order fuzzy-trend logical relationships in each two-factors second-order fuzzy-trend logical relationship group, respectively. Finally, it performs the forecasting based on the probabilities of the down-trend, the equal-trend, and the up-trend of the two-factors second-order fuzzy-trend logical relationships in each two-factors second-order fuzzy-trend logical relationship group. We also apply the proposed method to forecast the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) and the NTD/USD exchange rates. The experimental results show that the proposed method outperforms the existing methods.

Hesitant Fuzzy Power Bonferroni Means and Their Application to Multiple Attribute Decision Making

Abstract: As a useful generalization of fuzzy sets, the hesitant fuzzy set is designed for situations in which it is difficult to determine the membership of an element to a set because of ambiguity between a few different values. In this paper, we define the i th-order polymerization degree function and propose a new ranking method to further compare different hesitant fuzzy sets. In order to obtain much more information in the process of group decision making, we combine the power average operator with the Bonferroni mean in hesitant fuzzy environments and develop the hesitant fuzzy power Bonferroni mean and the hesitant fuzzy power geometric Bonferroni mean. We investigate the desirable properties of these new hesitant fuzzy aggregation operators and discuss some special cases. The new aggregation operators are utilized to present techniques for hesitant fuzzy multiple attribute group decision making. Finally, a numerical example is provided to illustrate the effectiveness of the developed techniques.

Comparison between the Effects of Different Types of Membership Functions on Fuzzy Logic Controller Performance

Abstract: Membership functions (MFs) are the building blocks of fuzzy set theory, i.e., fuzziness in a fuzzy set is determined by its MF. Accordingly, the shapes of MFs are important for a particular problem since they effect on a fuzzy inference system. They may have different shapes like triangular, trapezoidal, Gaussian, etc. The only condition a MF must really satisfy is that it must vary between 0 and 1. In this paper, a straightforward approach for designing a fuzzy logic based controller is presented to evaluate the effect of membership function in fuzzy logic controller, and presents the performance comparison of fuzzy logic controller with three different types of membership function. An attempt has been made to develop a fuzzy based control system for Antenna Azimuth Position control. This was done using Matlab/Simulink module

Multiattribute Decision Making Based on Interval-Valued Intuitionistic Fuzzy Sets, PSO Techniques, and Evidential Reasoning Methodology

Abstract: In this paper, we propose a new multiattribute decision making method based on interval-valued intuitionistic fuzzy sets, particle swarm optimization (PSO) techniques, and the evidential reasoning methodology. The proposed method uses the evidential reasoning methodology to construct objective functions of the programming models and uses PSO techniques to get optimal weights of the attributes to get the aggregated interval-valued intuitionistic fuzzy value of each alternative. Then, it calculates the transformed value of the obtained interval-valued intuitionistic fuzzy value of each alternative. The larger the transformed value, the better the preference order of the alternative. The proposed method can overcome the drawbacks of the existing methods for multiattribute decision making based on interval-valued intuitionistic fuzzy sets. It provides us with a useful way for multiattribute decision making based on interval-valued intuitionistic fuzzy sets, PSO techniques, and the evidential reasoning methodology.

Reactive Power Control of Three-Phase Grid-Connected PV System During Grid Faults Using Takagi–Sugeno–Kang Probabilistic Fuzzy Neural Network Control

Abstract: An intelligent controller based on the Takagi–Sugeno–Kang-type probabilistic fuzzy neural network with an asymmetric membership function (TSKPFNN-AMF) is developed in this paper for the reactive and active power control of a three-phase grid-connected photovoltaic (PV) system during grid faults. The inverter of the three-phase grid-connected PV system should provide a proper ratio of reactive power to meet the low-voltage ride through (LVRT) regulations and control the output current without exceeding the maximum current limit simultaneously during grid faults. Therefore, the proposed intelligent controller regulates the value of reactive power to a new reference value, which complies with the regulations of LVRT under grid faults. Moreover, a dual-mode operation control method of the converter and inverter of the three-phase grid-connected PV system is designed to eliminate the fluctuation of dc-link bus voltage under grid faults. Furthermore, the network structure, the online learning algorithm, and the convergence analysis of the TSKPFNN-AMF are described in detail. Finally, some experimental results are illustrated to show the effectiveness of the proposed control for the three-phase grid-connected PV system.

Intuitionistic fuzzy interaction bonferroni means and its application to multiple attribute decision making.

Abstract: The Bonferroni mean (BM) was originally presented by Bonferroni and had been generalized by many researchers for its capacity to capture the interrelationship between input arguments. Nevertheless, the existing intuitionistic fuzzy BMs only consider the effects of membership function or nonmembership function of different intuitionistic fuzzy sets (IFSs). As complements to the existing generalizations of BM under intuitionistic fuzzy environment, this paper also considers the interactions between the membership function and nonmembership function of different IFSs and develops the intuitionistic fuzzy interaction BM and the weighted intuitionistic fuzzy interaction BM. We investigate the properties of these new extensions of BM and discuss their special cases. Furthermore, the detailed steps of multiple attribute decision making with the presented operators under intuitionistic fuzzy environment are investigated and an example is illustrated to show the validity and feasibility of the new approach.

Interpretability of Fuzzy Systems: Current Research Trends and Prospects

Abstract: Fuzzy systems are universally acknowledged as valuable tools to model complex phenomena while preserving a readable form of knowledge representation. The resort to natural language for expressing the terms involved in fuzzy rules, in fact, is a key factor to conjugate mathematical formalism and logical inference with human-centered interpretability . That makes fuzzy systems specifically suitable in every real-world context where people are in charge of crucial decisions. This is because the self-explanatory nature of fuzzy rules profitably supports expert assessments. Additionally, as far as interpretability is investigated, it appears that (a) the simple adoption of fuzzy sets in modeling is not enough to ensure interpretability; (b) fuzzy knowledge representation must confront the problem of preserving the overall system accuracy, thus yielding a trade-off which is frequently debated. Such issues have attracted a growing interest in the research community and became to assume a central role in the current literature panorama of computational intelligence. This chapter gives an overview of the topics related to fuzzy system interpretability, facing the ambitious goal of proposing some answers to a number of open challenging questions: What is interpretability? Why interpretability is worth considering? How to ensure interpretability, and how to assess (quantify) it? Finally, how to design interpretable fuzzy models?

The fuzzy cross-entropy for intuitionistic hesitant fuzzy sets and their application in multi-criteria decision-making

Abstract: In this paper, the cross-entropy of intuitionistic hesitant fuzzy sets IHFSs is developed by integrating the cross-entropy of intuitionistic fuzzy sets IFSs and hesitant fuzzy sets HFSs. First, several measurement formulae are discussed and their properties are studied. Then, two approaches, which are based on the developed intuitionistic hesitant fuzzy cross-entropy, are proposed for solving multi-criteria decision-making MCDM problems within an intuitionistic hesitant fuzzy environment. For both methods, an optimisation model is established in order to determine the weight vector for MCDM problems with incomplete information on criteria weights. Finally, an example is provided in order to illustrate the practicality and effectiveness of the proposed approaches.

A novel similarity measure on intuitionistic fuzzy sets with its applications

Abstract: The intuitionistic fuzzy set, as a generation of Zadeh' fuzzy set, can express and process uncertainty much better, by introducing hesitation degree. Similarity measures between intuitionistic fuzzy sets (IFSs) are used to indicate the similarity degree between the information carried by IFSs. Although several similarity measures for intuitionistic fuzzy sets have been proposed in previous studies, some of those cannot satisfy the axioms of similarity, or provide counter-intuitive cases. In this paper, we first review several widely used similarity measures and then propose new similarity measures. As the consistency of two IFSs, the proposed similarity measure is defined by the direct operation on the membership function, non-membership function, hesitation function and the upper bound of membership function of two IFS, rather than based on the distance measure or the relationship of membership and non-membership functions. It proves that the proposed similarity measures satisfy the properties of the axiomatic definition for similarity measures. Comparison between the previous similarity measures and the proposed similarity measure indicates that the proposed similarity measure does not provide any counter-intuitive cases. Moreover, it is demonstrated that the proposed similarity measure is capable of discriminating the difference between patterns.

Stability Analysis of T–S Fuzzy Control Systems by Using Set Theory

Abstract: This paper is concerned with the stability analysis for Takagi–Sugeno (T–S) fuzzy control systems. By exploiting the property of the structure of a fuzzy inference engine, an equivalence relation on the index set of the product of fuzzy rule weights is defined. Furthermore, a new stability criterion is proposed by using the equivalence relation and formulated into progressively less-conservative sets of linear matrix inequalities. By using an extension of Polya's theorem, the new criterion is proved to be with no conservatism for quadratic stability analysis of T–S fuzzy control systems with a product inference engine and any possible fuzzy membership functions. A numerical example is given to illustrate the effectiveness of the proposed method.

Event-triggered H ∞ filtering for networked Takagi–Sugeno fuzzy systems with asynchronous constraints

Abstract: This paper addresses event-triggered H ∞ filtering for networked Takagi-Sugeno fuzzy systems with the asynchronous constraints of the membership functions. First, an event-triggered communication (ETC) scheme is proposed to trigger the transmission only when the variation of the sampled vector exceeds a prescribed threshold condition. Second, a model of fuzzy filtering error system is built under consideration of the ETC scheme, asynchronous premise and communication delay in a unified framework. Then, less conservative criteria on H ∞ filtering analysis and design are derived by making use of the re-constructed asynchronous premises. Finally, two examples are used to show the effectiveness of the proposed method.

Wavelet Fuzzy Neural Network With Asymmetric Membership Function Controller for Electric Power Steering System via Improved Differential Evolution

Abstract: A wavelet fuzzy neural network using asymmetric membership function (WFNN-AMF) with improved differential evolution (IDE) algorithm is proposed in this study to control a six-phase permanent magnet synchronous motor (PMSM) for an electric power steering (EPS) system. First, the dynamics of a steer-by-wire EPS system and a six-phase PMSM drive system are described in detail. Moreover, the WFNN-AMF controller, which combines the advantages of wavelet decomposition, fuzzy logic system, and asymmetric membership function (AMF), is developed to achieve the required control performance of the EPS system for the improvement of stability of the vehicle and the comfort of the driver. Furthermore, the online learning algorithm of WFNN-AMF is derived using back-propagation method. However, degenerated or diverged responses will be resulted due to the inappropriate selection of small or large learning rates of the WFNN-AMF. Therefore, an IDE algorithm is proposed to online adapt the learning rates of WFNN-AMF. In addition, a 32-bit floating-point digital signal processor, TMS320F28335, is adopted for the implementation of the proposed intelligent controlled EPS system. Finally, the feasibility of the proposed WFNN-AMF controller with IDE for the EPS system is verified through experimental results.

Horizontal Membership Function and Examples of its Applications

Abstract: The paper introduces horizontal membership functions (HMFs) which define a fuzzy set not in form of commonly used vertical membership functions of type μ = f 1(x) but in the horizontal form x = f 2(μ). Until now, constructing HMFs had seemed impossible because of horizontal ambiguity of this function. Now, however, it became possible thanks to the multidimensional, RDM-interval arithmetic based on relative-distance-measure variables. HMFs enable direct introducing uncertain, interval or fuzzy variable-values in usual mathematical formulas of type y = f(x 1 ,…,x 2) together with crisp values, without using Zadeh’s extension principle. Thus, a relatively easy aggregation of crisp and uncertain knowledge became possible. The paper shows application of HMFs, first on example of a classical mathematical function y = f(x 1 ,x 2) and next, on example of a computing with words challenge problem.

Multi-attribute decision making based on neutral averaging operators for intuitionistic fuzzy information

Abstract: The neutral operation and scalar neutral operation are proposed.The geometric meaning of the PS function is interpreted.The proportional distribution rules of membership function and non-membership function of IFSs are explained.We develop the IFWNA operator and the IFOWNA operator and investigate the properties.The new operators are applied to decision making and given the comparisons. In this paper, we construct the probability sum (PS) function and the proportional distribution rules of membership function and non-membership function of intuitionistic fuzzy sets (IFSs), and give their corresponding geometric interpretations. Based on which, we present the neutrality operation and the scalar neutrality operation on intuitionistic fuzzy numbers (IFNs). We propose the intuitionistic fuzzy weighted neutral averaging (IFWNA) operator and the intuitionistic fuzzy ordered weighted neutral averaging (IFOWNA) operator. The properties of the IFWNA operator and the IFOWNA operator are investigated. The principal advantages of the proposed operators are that both the attitude of the decision makers and the interactions between different intuitionistic fuzzy numbers (IFNs) are considered. Furthermore, approaches to multi-criteria decision making based on the proposed IFWNA and IFOWNA operator are given. Finally, an example is illustrated to show the feasibility and validity of the new approaches to the application of decision making.

Portfolio optimization using a credibility mean-absolute semi-deviation model

Abstract: We present a cardinality constrained credibility mean-absolute semi-deviation model.We prove relationships for possibility and credibility moments for LR-fuzzy variables.The return on a given portfolio is modeled by means of LR-type fuzzy variables.We solve the portfolio selection problem using an evolutionary procedure with a DSS.We select best portfolio from Pareto-front with a ranking strategy based on Fuzzy VaR. We introduce a cardinality constrained multi-objective optimization problem for generating efficient portfolios within a fuzzy mean-absolute deviation framework. We assume that the return on a given portfolio is modeled by means of LR-type fuzzy variables, whose credibility distributions collect the contemporary relationships among the returns on individual assets. To consider credibility measures of risk and return on a given portfolio enables us to work with its Fuzzy Value-at-Risk. The relationship between credibility expected values for LR-type fuzzy variables and possibilistic moments for LR-fuzzy numbers having the same membership function are analyzed. We apply a heuristic approach to approximate the cardinality constrained efficient frontier of the portfolio selection problem considering the below-mean absolute semi-deviation as a measure of risk. We also explore the impact of adding a Fuzzy Value-at-Risk measure that supports the investor's choices. A computational study of our multi-objective evolutionary approach and the performance of the credibility model are presented with a data set collected from the Spanish stock market.