Comprehensive fuzzy concept-oriented three-way decision and its application
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Fuzzy Model for Fuzzy Time Series Data and Its Application in Finance By Agus Maman Abadi 06/09-I/2021/PS Fuzzy time series is a dynamical process of linguistic variables whose fuzzy sets as its linguistic values. The uniqueness of fuzzy time series model are that the model can formulate problems based on expert knowledge only or hybrid of expert knowledge and empirical data In the modeling fuzzy time series data, previous researchers define fuzzy set using discrete membership function. Then Mamdani composition is applied to construct fuzzy relations based on training data. Optimization of the model is done by clustering of fuzzy relations to a group and then defuzzification is applied. This fuzzy relation is used to predict real data or fuzzy sets. In this research, new procedures to modeling fuzzy for fuzzy time series data were established. The procedures consist of constructing fuzzy relations based on composition and individual based inferences using operator s-norm and t-norm, selecting input variables, designing complete fuzzy relations, and constructing optimal fuzzy relations. The fuzzy relations designed by composition based inferences are union of fuzzy relations based on training data. Then this fuzzy relations are used to predict fuzzy set output using operator sup-t. Furthermore, in individual based inferences, for a fuzzy set input, every fuzzy relation resulted from training data determines fuzzy set output using operator sup-t. Then fuzzy set output of fuzzy model is union of fuzzy set output resulted from every fuzzy relation using operator tnorm. The construction of fuzzy relations based on composition and individual based inferences using operator s-norm and t-norm is generalization of construction of fuzzy relations introduced by Song and Chissom. Selection of input variables is done by singular value decomposition method of sensitivity matrix. Columns of this matrix are sensitivity of each input variable. This method is used to determine significant input variables. Position of the significant input variables is equivalent to the position of entry “1” of permutation matrix. Based on the significant input variables, complete fuzzy relations are designed by degree of fuzzy relation method. This method is generalization of Wang’s method. Then singular value decomposition method is applied to firing strength matrix to choose optimal fuzzy relations from complete fuzzy relations. Then the optimal fuzzy relations are used to design fuzzy model. The methods are applied to forecast inflation rate and interest rate of Bank Indonesia certificate. Forecasting inflation rate and interest rate of Bank Indonesia certificate using method of degree of fuzzy relation gives better accuracy than that using standard method for conventional time series model. Then forecasting inflation rate and interest rate of Bank Indonesia certificate using singular value decomposition method gives better accuracy than that using standard method for conventional time series model
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About the Author. Preface to the Third Edition. 1 Introduction. The Case for Imprecision. A Historical Perspective. The Utility of Fuzzy Systems. Limitations of Fuzzy Systems. The Illusion: Ignoring Uncertainty and Accuracy. Uncertainty and Information. The Unknown. Fuzzy Sets and Membership. Chance Versus Fuzziness. Sets as Points in Hypercubes. Summary. References. Problems. 2 Classical Sets and Fuzzy Sets. Classical Sets. Operations on Classical Sets. Properties of Classical (Crisp) Sets. Mapping of Classical Sets to Functions. Fuzzy Sets. Fuzzy Set Operations. Properties of Fuzzy Sets. Alternative Fuzzy Set Operations. Summary. References. Problems. 3 Classical Relations and Fuzzy Relations. Cartesian Product. Crisp Relations. Cardinality of Crisp Relations. Operations on Crisp Relations. Properties of Crisp Relations. Composition. Fuzzy Relations. Cardinality of Fuzzy Relations. Operations on Fuzzy Relations. Properties of Fuzzy Relations. Fuzzy Cartesian Product and Composition. Tolerance and Equivalence Relations. Crisp Equivalence Relation. Crisp Tolerance Relation. Fuzzy Tolerance and Equivalence Relations. Value Assignments. Cosine Amplitude. Max Min Method. Other Similarity Methods. Other Forms of the Composition Operation. Summary. References. Problems. 4 Properties of Membership Functions, Fuzzification, and Defuzzification. Features of the Membership Function. Various Forms. Fuzzification. Defuzzification to Crisp Sets. -Cuts for Fuzzy Relations. Defuzzification to Scalars. Summary. References. Problems. 5 Logic and Fuzzy Systems. Part I Logic. Classical Logic. Proof. Fuzzy Logic. Approximate Reasoning. Other Forms of the Implication Operation. Part II Fuzzy Systems. Natural Language. Linguistic Hedges. Fuzzy (Rule-Based) Systems. Graphical Techniques of Inference. Summary. References. Problems. 6 Development of Membership Functions. Membership Value Assignments. Intuition. Inference. Rank Ordering. Neural Networks. Genetic Algorithms. Inductive Reasoning. Summary. References. Problems. 7 Automated Methods for Fuzzy Systems. Definitions. Batch Least Squares Algorithm. Recursive Least Squares Algorithm. Gradient Method. Clustering Method. Learning From Examples. Modified Learning From Examples. Summary. References. Problems. 8 Fuzzy Systems Simulation. Fuzzy Relational Equations. Nonlinear Simulation Using Fuzzy Systems. Fuzzy Associative Memories (FAMS). Summary. References. Problems. 9 Decision Making with Fuzzy Information. Fuzzy Synthetic Evaluation. Fuzzy Ordering. Nontransitive Ranking. Preference and Consensus. Multiobjective Decision Making. Fuzzy Bayesian Decision Method. Decision Making Under Fuzzy States and Fuzzy Actions. Summary. References. Problems. 10 Fuzzy Classification. Classification by Equivalence Relations. Crisp Relations. Fuzzy Relations. Cluster Analysis. Cluster Validity. c-Means Clustering. Hard c-Means (HCM). Fuzzy c-Means (FCM). Fuzzy c-Means Algorithm. Classification Metric. Hardening the Fuzzy c-Partition. Similarity Relations from Clustering. Summary. References. Problems. 11 Fuzzy Pattern Recognition. Feature Analysis. Partitions of the Feature Space. Single-Sample Identification. Multifeature Pattern Recognition. Image Processing. Summary. References. Problems. 12 Fuzzy Arithmetic and the Extension Principle. Extension Principle. Crisp Functions, Mapping, and Relations. Functions of Fuzzy Sets Extension Principle. Fuzzy Transform (Mapping). Practical Considerations. Fuzzy Arithmetic. Interval Analysis in Arithmetic. Approximate Methods of Extension. Vertex Method. DSW Algorithm. Restricted DSW Algorithm. Comparisons. Summary. References. Problems. 13 Fuzzy Control Systems. Control System Design Problem. Control (Decision) Surface. Assumptions in a Fuzzy Control System Design. Simple Fuzzy Logic Controllers. Examples of Fuzzy Control System Design. Aircraft Landing Control Problem. Fuzzy Engineering Process Control. Classical Feedback Control. Fuzzy Control. Fuzzy Statistical Process Control. Measurement Data Traditional SPC. Attribute Data Traditional SPC. Industrial Applications. Summary. References. Problems. 14 Miscellaneous Topics. Fuzzy Optimization. One-Dimensional Optimization. Fuzzy Cognitive Mapping. Concept Variables and Causal Relations. Fuzzy Cognitive Maps. Agent-Based Models. Summary. References. Problems. 15 Monotone Measures: Belief, Plausibility, Probability, and Possibility. Monotone Measures. Belief and Plausibility. Evidence Theory. Probability Measures. Possibility and Necessity Measures. Possibility Distributions as Fuzzy Sets. Possibility Distributions Derived from Empirical Intervals. Deriving Possibility Distributions from Overlapping Intervals. Redistributing Weight from Nonconsonant to Consonant Intervals. Comparison of Possibility Theory and Probability Theory. Summary. References. Problems. Index.
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Defuzzification is a process that converts a fuzzy set or fuzzy number into a crisp value or number. Defuzzification is used in fuzzy modeling and in fuzzy control system to convert the fuzzy outputs from the systems to crisp values. This process is necessary because all fuzzy sets inferred by fuzzy inference in the fuzzy rules must be aggregated to produce one single number as the output of the fuzzy model.There are numerous techniques for defuzzifying a fuzzy set; some of the more popular techniques are included in fuzzy logic system. In the present chapter some recent defuzzification methods used in the literature are discussed with examples.
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Defuzzification is a process that converts a fuzzy set or fuzzy number into a crisp value or number. Defuzzification is used in fuzzy modeling and in fuzzy control system to convert the fuzzy outputs from the systems to crisp values. This process is necessary because all fuzzy sets inferred by fuzzy inference in the fuzzy rules must be aggregated to produce one single number as the output of the fuzzy model.There are numerous techniques for defuzzifying a fuzzy set; some of the more popular techniques are included in fuzzy logic system. In the present chapter some recent defuzzification methods used in the literature are discussed with examples.
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Many authors discussed the interpolation mechanism of fuzzy logic system applied in control. The input and output of fuzzy control system are always restrained in (fuzzy) numbers-associated with fuzzification and defuzzification procedure. In artificial intelligence area we need use common fuzzy sets to represent vague and fuzzy concept. In this paper, we propose a "local relation mapping" theory, which is a corresponding result of interpolation mechanism of fuzzy logical system without fuzzification and defuzzification. This local relation mapping property holds in Mamdani algorithm (Max-Min fuzzy reasoning model). We find that each CRI fuzzy reasoning method based on the multi-valued logic implication operator has an approximate Mamdani algorithm. The fuzzy relation matrix of CRI fuzzy inference method based on the multivalued logic implication operator has an enveloping surface of Mamdani algorithm. So most CRI fuzzy reasoning methods have local relation mapping property approximately.
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We present a new method to optimize the nonlinear objective function with fuzzy coefficients and fuzzy constraints using Genetic Algorithm (GA). We use GA to solve this fuzzy problem with defining the membership for fuzzy numbers. The proposed approach simulates every fuzzy number by distributing it into certain partition points. The final values obtained after the evolutionary process represent the membership grade of the fuzzy number. The computation of fuzzy equations by GA does not require the conventional extension principle or interval arithmetic and α-cuts for solving fuzzy nonlinear programming. The empirical results show that the proposed approach obtains very good solutions within the given bounds of each fuzzy coefficient compared with other fuzzy methods. The fuzzy concept of GA approach is different but gives better results than other traditional fuzzy methods.
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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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We propose the concept of a conditional fuzzy set and prove that a type-2 fuzzy set is equivalent to a conditional fuzzy set. We show that both the conditional fuzzy sets and the type-2 fuzzy sets are fuzzy relations on the product space of the primary and secondary variables, and the difference is that the primary and secondary variables in the conditional fuzzy set framework are usually independent to each other, whereas in the type-2 fuzzy set framework, the secondary variable depends on the primary variable by definition. It is this dependence between the primary and secondary variables that makes the type-2 fuzzy sets a complex subject, while the conditional fuzzy sets do not have this built-in dependence and, thus, are much easier to analyze. With the fuzzy relation formulation, powerful tools in fuzzy set theory such as Zadeh's compositional rule of inference can be used to obtain the marginal fuzzy sets of the type-2 and conditional fuzzy sets, transforming the type-2 problems back to the conventional type-1 domain. With the help of the marginal fuzzy set concept, we show that a type-2 fuzzy logic system can be designed in the same way as designing a type-1 fuzzy logic system.
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