Fuzzy rule bases with generalized belief structure inputs
Introduction
Fuzzy systems modeling Mendel, 2017, Pedrycz and Gomide, 2007 is clearly the most successful application of Zadeh’s fuzzy set theory (Zadeh, 1965). Using this idea we are able to model complex input/output functions by imprecisely partitioning the input space using fuzzy sets, and then associating with each fuzzy set in this partitioning an appropriate output. One advantage of this approach is that it allows humans to more easily express the component relationships. In fuzzy systems modeling each of these input/output pairs have come to be called a rule whose antecedent is a fuzzy set and the collection of these component rules is called a rule base. The determination of the output of a fuzzy systems model for given value for the input is implemented via a very natural process. The first step is to obtain the relevance of a rule to the given input, this is called the rule firing level. The systems output is then obtained as a kind of weighted average of individual rule outputs where the weight associated with a given rule is based on its firing level for the system input. The original formalism for fuzzy systems modeling was due to Mamdani and Assilian (1975) and Mamdani (1976) however the prevalent approach to fuzzy systems modeling is based on the ideas of Takagi and Sugeno (1985). Most work on fuzzy systems modeling has focused on two types of systems input values; precise values and imprecise values captured using fuzzy sets. When the input to the system is a precise value a rule firing level is simply the membership grade of the input in the rule antecedent fuzzy sets. When the input is a fuzzy set obtaining the rule firing level involves the determination of the satisfaction of one imprecise object, fuzzy set, by another imprecise object, fuzzy set. As we shall see a reasonable value for the firing level in this case is an interval. In this work we go beyond these two situations. First we consider the situation were the system input value is modeled by a Dempster–Shafer belief structure Dempster, 1967, Shafer, 1976, Dempster, 2008, Yager and Liu, 2008. This type of input has probabilistic uncertainty as well imprecision. Next we consider the situation were the input is modeled by generalized belief structure Yager, 2017, Yager, 2018 here we have measure guided uncertainty (Yager, 2016) as well imprecision. In these situations where the systems input manifests uncertainty as well as imprecision the determination of rule firing level as well the systems outcome becomes more complex. We see that the novelty and benefit of this paper is that it provides tools for working with fuzzy systems models for various types of uncertain inputs.
The structure of the paper is as follows we first discuss the idea of fuzzy systems modeling and look at the determination of the firing level of a rule for fuzzy set inputs. We then consider the situation when input is uncertain and modeled by a Dempster–Shafer belief structure with fuzzy focal elements. We next look the case where our input is modeled via a generalized belief structure.
Section snippets
Basics of fuzzy systems modeling
We now describe the basic framework of fuzzy systems models Mendel, 2017, Pedrycz and Gomide, 2007, Ross, 2010. Assume for to are a collection of variables taking their values in the spaces respectively, these are called the input variables. We let be another variable taking its value in the space , this is called the output variable. We let for to t be another collection of variables which can contain some of the . Central to the fuzzy system modeling technique is a rule
Determination of antecedent firing levels
We now consider the determination of the firing levels for the individual antecedent components, the is , given the knowledge Inf() about the value of the variable . In the following, in order to avoid unnecessary notational complexity we shall, when it does not involve a loss of generality, consider one generic rule In this case where we know the exact value of , then , the membership grade of in . Once we introduce some
Fuzzy rule bases with Dempster–Shafer inputs
If is an uncertain variable that can take its value in the space { for to } the Dempster–Shafer belief structure provides a formulation that can be used for modeling various types of uncertain information associated with the variable (Yager and Liu, 2008). Associated with a D-S belief structure is a collection of normal fuzzy subsets of , , called the focal elements, and their corresponding weights [0,1] where . We recall a fuzzy subset is called
Pignistic firing level
We have previously shown that for a rule base with multi-antecedent rules of the form where our knowledge about each antecedent variable is a Dempster–Shafer belief structure is on the domain of we get as our output an interval valued solution for where Even in the case of a rule base with rules of the form if is the is
Generalized D-S belief structures
In Yager (2017) we considered a generalization of the Dempster–Shafer belief structure, , on the space that can provide a model of uncertain information of about a variable which takes its value in the space . Associated with are a collection of normal fuzzy subsets on , F = {, …, } called the focal elements. We can again, for our purposes view the generalized D-S belief structure as corresponding to a collection of statements is for to . Here again there is some
Conclusion
We described the basics of fuzzy systems modeling and noted that central to this is a collection of rules, a rule base, in which the rule antecedents are fuzzy subsets. We first looked at issue of the determination of the firing level of a rule for fuzzy set inputs and the subsequent rule base output. We noted a number of different approaches for obtaining the satisfaction of one fuzzy subset by another, among these were possibility and certainty. We then considered the situation where the
References (25)
The Dempster–Shafer calculus for statisticians
Internat. J. Approx. Reason.
(2008)Advances in the linguistic synthesis of fuzzy controllers
Int. J. Man–Mach. Stud.
(1976)- et al.
An experiment in linguistic synthesis with a fuzzy logic controller
Int. J. Man–Mach. Stud.
(1975) - et al.
The transferable belief model
Artificial Intelligence
(1994) Satisfying uncertain targets using measure generalized Dempster–Shafer belief structures
Knowl. Based Syst.
(2018)Fuzzy sets
Inf. Control
(1965)Probability measures of fuzzy events
J. Math. Anal. Appl.
(1968)- et al.
Aggregation Functions: A Guide for Practitioners
(2007) Upper and lower probabilities induced by a multi-valued mapping
Ann. Math. Statist.
(1967)- et al.
A universal integral as common frame for Choquet and Sugeno
IEEE Trans. Fuzzy Syst.
(2010)
Uncertain Rule-Based Fuzzy Systems: Introduction and New Directions
Fuzzy measures and fuzzy integrals
Cited by (37)
A new complex belief entropy of χ<sup>2</sup> divergence with its application in cardiac interbeat interval time series analysis
2023, Chaos, Solitons and FractalsAn information fusion method based on deep learning and fuzzy discount-weighting for target intention recognition
2022, Engineering Applications of Artificial IntelligenceCitation Excerpt :Therefore, it is necessary to use information fusion method to construct a reliable intention recognition model of target formation, so as to improve the accuracy of the global intention recognition (Liu et al., 2017). Various theories have been proposed to realize information fusion, such as Bayesian theory (Zhang et al., 2016; Jiang et al., 2020a), Dempster–Shafer theory (DST) (Luo and Deng, 2020b; Deng, 2020; Zhang et al., 2018a), fuzzy theory (Yager, 2018; Vidal et al., 2020), Z-number (Liu et al., 2019a), D-number (Deng and Jiang, 2019), and so on Fu et al. (2019b). As a classic method of information theory, Bayesian theory is a mathematical model based on probabilistic reasoning, which can effectively deal with the problems of uncertainty and incompleteness.
Whale optimization algorithm-based Sugeno fuzzy logic controller for fault ride-through improvement of grid-connected variable speed wind generators
2020, Engineering Applications of Artificial IntelligenceIslanding and non-islanding disturbance detection in microgrid using optimized modes decomposition based robust random vector functional link network
2019, Engineering Applications of Artificial IntelligenceCitation Excerpt :In literature there are many machine learning methods that have been used for power signal disturbance classification such as Fuzzy (Biswal et al., 2009; Chilukuri and Dash, 2004; Yager, 2018), SVM (Janik and Lobos, 2006; Erişti and Demir, 2010; Si et al., 2019), ANN (Perunicic et al., 1998), and ELM (Erişti et al., 2014; Ahila et al., 2015; Pulido et al., 2019; Ryman-Tubb et al., 2018), etc.
Uncertain database retrieval with measure-based belief function attribute values
2019, Information SciencesCitation Excerpt :In the case using Yager's rule we have m(A1) = 0.0001 and m(X) = 0.9999. In [25,28,29] we introduced a more general form of the Dempster-Shafer belief structure called a measure-based belief structure, MBBS, it is described in the following. Assume V, our variable of interest, is an uncertain variable taking its value in the space X = {xi for i = 1 to n}.
A generalized Hellinger distance for multisource information fusion and its application in pattern classification
2024, Computational and Applied Mathematics