Hanna B. Rakytyanska

Optimal Design of Rule-Based Systems by Solving Fuzzy Relational Equations

In this chapter, an approach to the design of rule-based systems within the framework of fuzzy relational calculus is proposed. The system of fuzzy relations serves as the generator of the rule-based solutions of fuzzy relational equations. Each solution represents a different trade-off between the classification accuracy and the number of fuzzy rules. The accuracy-complexity trade-off is achieved by optimization of the total number of decision classes for relations and rules.

Diagnosis based on fuzzy IF-THEN rules and genetic algorithms

This paper proposes an approach for inverse problem solving based on the description of the interconnection between unobserved and observed parameters of an object (causes and effects) with the help of fuzzy IF-THEN rules. The essence of the approach proposed consists in formulating and solving the optimization problems, which, on the one hand, find the roots of fuzzy logical equations, corresponding to IF-THEN rules, and on the other hand, tune the fuzzy model on the readily available experimental data. The genetic algorithms are proposed for the optimization problems solving.

Adaptive refinement of fuzzy knowledge bases using trend rules and inverse inference

In this paper, an adaptive approach to refinement of fuzzy classification knowledge bases within the framework of fuzzy relational equations is proposed. The fuzzy classification knowledge base can be built using the system of trend fuzzy rules and inverse inference. The essence of the approach is in constructing and training the composite neuro-fuzzy network isomorphic to linguistic solutions of fuzzy relational equations. The composite network allows adaptive refinement of the expert rules while the bounds of decision classes are changing.

Direct Inference Based on Fuzzy Rules

This chapter is devoted to the methodology aspects of identification and decision making on the basis of intellectual technologies. The essence of intellectuality consists of representation of the structure of the object in the form of linguistic IF-THEN rules, reflecting human reasoning on the common sense and practical knowledge level. The linguistic approach to designing complex systems based on linguistically described models was originally initiated by Zadeh [1] and developed further by Tong [2], Gupta [3], Pedrych [4 – 6], Sugeno [7], Yager [8], Zimmermann [9], Kacprzyk [10], Kandel [11]. The main principles of fuzzy modeling were formulated by Yager [8]. The linguistic model is a knowledge-based system. The set of fuzzy IF-THEN rules takes the place of the usual set of equations used to characterize a system [12 – 14]. The fuzzy sets associated with input and output variables are the parameters of the linguistic model [15]; the number of the rules determines its structure. Different interpretations of the knowledge contained in these rules, which are due to different reasoning mechanisms, result in different types of models.

Inverse Inference Based on Fuzzy Rules

The wide class of the problems, arising from engineering, medicine, economics and other domains, belongs to the class of inverse problems [1]. The typical representative of the inverse problem is the problem of medical and technical diagnosis, which amounts to the restoration and the identification of the unknown causes of the disease or the failure through the observed effects, i.e. the symptoms or the external signs of the failure. The diagnosis problem, which is based on a cause and effect analysis and abductive reasoning can be formally described by neural networks [2] or Bayesian networks [3, 4]. In the cases, when domain experts are involved in developing cause-effect connections, the dependency between unobserved and observed parameters can be modelled using the means of fuzzy sets theory [5, 6]: fuzzy relations and fuzzy IF-THEN rules. Fuzzy relational calculus plays the central role as a uniform platform for inverse problem resolution on various fuzzy approximation operators [7, 8]. In the case of a multiple variable linguistic model, the cause-effect dependency is extended to the multidimensional fuzzy relational structure [6], and the problem of inputs restoration and identification amounts to solving a system of multidimensional fuzzy relational equations [9, 10]. Fuzzy IF-THEN rules enable us to consider complex combinations in cause-effect connections as being simpler and more natural, which are difficult to model with fuzzy relations. In rule-based models, an inputs-outputs connection is described by a hierarchical system of simplified fuzzy relational equations with max-min and dual min-max laws of composition [11 – 13].

Fuzzy Rules Extraction from Experimental Data

The necessary condition for nonlinear object identification on the basis of fuzzy logic is the availability of IF-THEN rules interconnecting linguistic estimations of input and output variables. Earlier we assumed that IF-THEN rules are generated by an expert who knows the object very well. What is to be done when there is no expert? In this case the generation of IF-THEN rules becomes of interest because it means the generation of fuzzy knowledge base from accessible experimental data [1].

Fuzzy Rules Tuning for Direct Inference

The identification of an object consists of the construction of its mathematical model, i.e., an operator of connection between input and output variables from experimental data. Modern identification theory [1 – 3], based on modeling dynamical objects by equations (differential, difference, etc.), is poorly suited for the use of information about an object in the form of expert IF-THEN statements. Such statements are concentrated expertise and play an important role in the process of human solution of various cybernetic problems: control of technological processes, pattern recognition, diagnostics, forecast, etc. The formal apparatus for processing expert information in a natural language is fuzzy set theory [4, 5]. According to this theory, a model of an object is given in the form of a fuzzy knowledge base, which is a set of IF-THEN rules that connect linguistic estimates for input and output object variables. The adequacy of the model is determined by the quality of the membership functions, by means of which linguistic estimates are transformed into a quantitative form. Since membership functions are determined by expert methods [5], the adequacy of the fuzzy model depends on the expert qualification.

Fundamentals of Intellectual Technologies

Intellectual technologies which are used to do the tasks of identification and decision making in this book represent a combination of three independent theories:

Fuzzy Relations Extraction from Experimental Data

In this chapter, a problem of fuzzy genetic object identification expressed mathematically in terms of fuzzy relational equations is considered.

Inverse Inference with Fuzzy Relations Tuning

Diagnosis, i.e. determination of the identity of the observed phenomena, is the most important stage of decision making in different domains of human activity: medicine, engineering, economics, military affairs, and others. In the case of the diagnosis of problems where physical mechanisms are not well known due to high complexity and nonlinearity, a fuzzy relational model may be useful. A fuzzy relational model for simulating cause and effect connections in diagnosing problems has been introduced by Sanchez [1, 2]. A model for diagnosis can be built on the basis of Zadeh’s compositional rule of inference [3], in which the fuzzy matrix of “causes-effects” relations serves as the support of the diagnostic information. In this case, the problem of diagnosis amounts to solving fuzzy relational equations.