Fuzzy rule bases with generalized belief structure inputs

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Abstract

We first describe the basics of fuzzy systems modeling. Fundamental to this is a collection of rules, a rule base, in which the rule antecedents are fuzzy subsets. We first look at issue of the determination of the firing level of a rule for fuzzy set inputs and the subsequent rule base output. We next consider the situation where the system input is uncertain and modeled by a Dempster–Shafer belief structure. Here our input is a collection of fuzzy subsets and the true input fuzzy set is selected based on a probability distribution over these potential input fuzzy sets. We next consider the situation where our input is modeled via a generalized belief structure where the determination of applicable input fuzzy set is modeled via a measure over these potential input fuzzy sets.

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 Vj for j=1 to r are a collection of variables taking their values in the spaces Xj respectively, these are called the input variables. We let U be another variable taking its value in the space Y, this is called the output variable. We let W for k=1 to t be another collection of variables which can contain some of the Vj. 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 Vj is Aij, given the knowledge Inf(j) about the value of the variable Vj. 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 IfV1isA1andV2isA2,andVnisAnthenUisb.In this case where we know the exact value of Vj, Vj=aj then τj=Aj(aj), the membership grade of aj in Aj. Once we introduce some

Fuzzy rule bases with Dempster–Shafer inputs

If V is an uncertain variable that can take its value in the space X= {xi for i=1 to n} the Dempster–Shafer belief structure provides a formulation that can be used for modeling various types of uncertain information associated with the variable V (Yager and Liu, 2008). Associated with a D-S belief structure m is a collection of q normal fuzzy subsets of X, Fk, called the focal elements, and their corresponding weights m(Fk)= αk  [0,1] where j=1qαk=1. We recall a fuzzy subset is called

Pignistic firing level

We have previously shown that for a rule base with n multi-antecedent rules of the form IfV1isAi1andV2isAi2andandVjisAijandandVrisAirthenUisbiwhere our knowledge about each antecedent variable is a Dempster–Shafer belief structure Vj is mj on the domain Xj of Vj we get as our output an interval valued solution b for U where b=[i=1nbi(j=1rBelmj(Aij))i=1n(j=1rPlmj(Aij)),i=1nbi(j=1rPlmj(Aij))i=1n(j=1rBellmj(Aij))].Even in the case of a rule base with rules of the form if V is Ai the U is

Generalized D-S belief structures

In Yager (2017) we considered a generalization of the Dempster–Shafer belief structure, g, on the space X that can provide a model of uncertain information of about a variable V which takes its value in the space X. Associated with g are a collection of q normal fuzzy subsets on X, F = {F1, …, Fq} called the focal elements. We can again, for our purposes view the generalized D-S belief structure g as corresponding to a collection of statements V is Fk for k=1 to q. 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

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