Fuzzy Evidence in Identification, Forecasting and Diagnosis

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

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.

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.

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].

Application of a fuzzy methodology in system failure engineering encompasses the fault diagnosis problem [1, 2]. According to Cai [1] by fault we mean a system state which deviates from the desired system state. The task of fault diagnosing may include detecting whether a fault has occurred, diagnosing where the fault occurred, determining the type of fault, assessing the fault damage, and reconfiguring the system to accommodate the fault. Fault diagnosis partially answers one of the basic issues in system failure engineering: why does it fail.

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.

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].

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

Fuzzy relational calculus [1, 2] provides a powerful theoretical background for knowledge extraction from data. Some fuzzy rule base is modelled by a fuzzy relational matrix, discovering the structure of the data set [3 - 5]. Fuzzy relational equations, which connect membership functions of input and output variables, are built on the basis of a fuzzy relational matrix and Zadeh’s compositional rule of inference [6, 7]. The identification problem consists of extraction of an unknown relational matrix which can be translated as a set of fuzzy IF-THEN rules. In fuzzy relational calculus this type of problem relates to inverse problem resolution for the composite fuzzy relational equations [2]. Solvability and approximate solvability conditions of the composite fuzzy relational equations are considered in [2, 8, 9]. While the theoretical foundations of fuzzy relational equations are well developed, they call for more efficient use of their potential in system modeling. The non-optimizing approach [10] is widely used for fuzzy relational identification. Such adaptive recursive techniques are of interest for the most of on-line applications [11 - 13]. Under general conditions, an optimization environment is the convenient tool for fuzzy relational identification [14]. An approach for identification of fuzzy relational models by fuzzy neural networks is proposed in [15 - 17].

Data processing not only in physics and engineering, but also in medicine, biology, sociology, economics, sport, art, and military affairs, amounts to the different statements of identification problems. Fuzzy logic is mistakenly perceived by many specialists in mathematical simulation as a mean of only approximate decisions making in medicine, economics, art, sport and other different from physics and engineering humanitarian domains, where the high level of accuracy is not required. Therefore, one of the main goals of the authors is to show that it is possible to reach the accuracy of modeling, which does not yield to strict quantitative correlations, by tuning fuzzy knowledge bases. Only objects with discrete outputs for the direct inference and discrete inputs for the inverse inference were considered in the previous chapters. Such a problem corresponds to the problem of automatic classification arising in particular from medical and technical diagnosis. The main idea which the authors strive to render is that while tuning the fuzzy knowledge base it is possible to identify nonlinear dependencies with the necessary precision.