Z-number Decision Tree and Its Appliction
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Abstract Fuzzy decision tree(FDT) is an expansion and improvement of decision tree, which expands the application scope of decision tree to deal with the uncertainty and extracts classification rules efficiently. However, there are some problems in the process of sample attributes fuzzifying, such as information loss and the influence of strong personal subjectivity on the results. To solve these problems, a fuzzy decision tree algorithm based on Z-number(Z-FDT) is put forward. we propose a method to construct Z-number, first, membership functions is defined to fuzzify the data set and describe the size of the probability measure of the fuzzy set, then the cluster centers of multi-dimensional data are extracted ,using the fuzzy clustering algorithm., the membership function and cluster centers of different categories fuzzy partition the sample data set based on the features of the sample, and obtain the fuzzy sets. Finally, based on the definition of Z-number, fuzzy sets are constructed into Z-numbers. After the multidimensional data set is transformed into Z-numbers, the uncertainty measure of Z-number is used to select appropriate features to partition the data set, until the stop condition is met. Z-FDT can directly classify and predict samples with continuous attribute values, and has good generalization ability and relatively stable results, at the last, this will be confirmed through the comparative experiment with ID3, Cart and Fuzzy ID3.Keywords:
Defuzzification
Fuzzy logic is an approach that reflects human thinking and decision making by handling uncertainty and vagueness using fuzzy membership functions. When a human is engaged in the design of a fuzzy system, symmetric properties are naturally preferred. Fuzzy c-means clustering is a clustering algorithm that can cluster datasets to produce membership matrix and cluster centers, which results in generating type-1 fuzzy membership functions. However, fuzzy c-means algorithm has a limitation of producing only a single membership function type, Gaussian MF. Generation of multiple fuzzy membership functions is of immense importance as it provides more efficient and optimal solutions to a problem. Therefore, an approach to generate multiple type-1 fuzzy membership functions through fuzzy c-means is required for the optimal and improved results of classification datasets. Hence, to overcome the limitation of the fuzzy c-means algorithm, an approach for the generation of type-1 fuzzy triangular and trapezoidal membership function through fuzzy c-means is considered in this study. The approach is used to calculate and enhance the accuracy of classification datasets called iris, banknote authentication, blood transfusion, and Haberman’s survival. The proposed approach of generating MFs using FCM produce asymmetric MFs, whose results are compared with the MFs produced from grid partitioning (GP), which are symmetric MFs. The results show that the proposed approach of generating type-1 fuzzy membership function through fuzzy c-means is effective and can be adopted.
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From the beginning, the assessment of membership function in fuzzy mathematics is a demanding task. The extraction of the membership function is ambience dependent and thus complication exists in the process of evaluation. In this assessment, the main work deals with the derivation of fuzzy membership function where numerical data is available. The numerical interpolation and defuzzification technique are used here. Many types of fuzzy number have been used to find out the membership function. In this paper, we mainly used triangular fuzzy number to construct the membership function. A case study is furnished to emphasise the advantage of adopting the method.
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Many theories are developed based on probability to deal with incomplete information. The fuzzy logic deals with belief rather than likelihood (probability). Zadeh first defined fuzzy set as a single membership function. The two fold fuzzy sets with two membership functions will give more evidence than a single membership one. Therefore there is need of fuzzy logic with two membership functions. In this paper, The fuzzy set is defined with two membership functions "Belief" and "Disbelief". The fuzzy inference and fuzzy reasoning are studied for "a two fold fuzzy set". The fuzzy certainty factor (FCF) is defined as a single membership function by taking difference between " Belief" and " Disbelief ". The quantification of fuzzy truth variables are studied for "a two fold fuzzy set". The medical expert system shell EMYCIN is given as an application of "a two fold fuzzy set".
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The functional paradigm for fuzzy multisenosr-multitarget tracking systems with data fusion consists of fuzzification, fuzzy knowledge-base, fuzzy inference mechanism, and defuzzification. In fuzzy system design, users start with some fuzzy rules, which are chosen heuristically based on their experience, and membership functions, which in many cases are chosen subjectively based on understanding the problem, and they use the developed system to tune these rules and membership functions. In most publications, in the area of track-to-track association in multitarget tracking systems, the fuzzy membership functions are chosen subjectively according to the underlying problem. The most commonly used membership functions are trapezoidal, triangular, piecewise linear, and Gaussian membership functions. They are chosen by the users based on their experiences. Therefore the problem of constructing optimal fuzzy membership functions is not considered in most publications. This paper addresses the critical issue of constructing optimal fuzzy membership functions for given input information in case of track-to-track association in multitarget tracking systems.
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This chapter introduces various popular forms of converting fuzzy sets to crisp sets or to single scalar values. It discusses the various features and forms of a membership function and the idea of fuzzyifying scalar quantities to make them fuzzy sets. The primary focus of the chapter, however, has been to explain the process of converting from fuzzy membership functions to crisp formats - a process called defuzzification. Since all information contained in a fuzzy set is described by its membership function, it is useful to develop a lexicon of terms to describe various special features of this function. There may be situations where the output of a fuzzy process needs to be a single scalar quantity as opposed to a fuzzy set. Defuzzification is the conversion of a fuzzy quantity to a precise quantity, just as fuzzification is the conversion of a precise quantity to a fuzzy quantity. Controlled Vocabulary Terms fuzzy control; fuzzy set theory
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Fuzzy set theory has been proposed as a means for modeling the vagueness in complex systems. Fuzzy systems usually employ type-1 fuzzy sets, representing uncertainty by numbers in the range [0, 1]. Despite commercial success of fuzzy logic, a type-1 fuzzy set (T1FS) does not capture uncertainty in its manifestations when it arises from vagueness in the shape of the membership function. Such uncertainties need to be depicted by fuzzy sets that have blur boundaries. The imprecise boundaries of a type-2 fuzzy set (T2FS) give rise to truth/membership values that are fuzzy sets in [0], [1], instead of a crisp number. Type-2 fuzzy logic systems (T2FLSs) offer opportunity to model levels of uncertainty which traditional fuzzy logic type1 struggles. This extra dimension gives more degrees of freedom for better representation of uncertainty compared to type-1 fuzzy sets. A type-1 fuzzy logic system (T1FLSs) inference produces a T1FS and the result of defuzzification of the T1FS, a crisp number, whereas a T2FLS inference produces a type-2 fuzzy set, its type-reduced fuzzy set which is a T1FS and the defuzzification of the type-1 fuzzy set. The type-reduced fuzzy set output gives decision-making flexibilities. Thus, FLSs using T2FS provide the capability of handling a higher level of uncertainty and provide a number of missing components that have held back successful deployment of fuzzy systems in decision making.
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Fuzzy logic systems have found extensive use in system identification, decision making, and pattern recognition problems from industries to academics. The membership functions play a pivotal role in overall role in fuzzy representation, as these are considered as the building blocks of fuzzy set theory and they decide the degree of truth in fuzzy logic. The extraction of the membership function is ambience dependent and thus complication exists in the process of evaluation. In this assessment the main work deals with the derivation of fuzzy membership function where numerical data is available. The numerical cubic spline and defuzzification technique are used here. In this paper we mainly used triangular fuzzy number to construct the membership function. A case study is furnished to emphasize the advantage of adopting the method.
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