Fuzzy Logic-Based Inference Systems

Fuzzy Logic-Based Inference Systems

Copyright: © 2015 |Pages: 31
DOI: 10.4018/978-1-4666-8705-9.ch008
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Abstract

This chapter presents the mathematical formulation of the fuzzy logic-based inference systems, used as means to infer about the response of ill-conditioned systems, based on the field knowledge representation in the fuzzy world. Particular approaches are explored, e.g., Fuzzy Inference System (FIS), Adaptive Networks-based FIS (ANFIS), Intuitionistic FIS (IFIS) and Fuzzy Cognitive Map (FCM), surfacing their potentialities in modeling applications, such as those in the field of learning, examined in the chapters of Part III that follow.
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Introduction

Initiated by the pioneer work of Zadeh (1965) on fuzzy logic (FL), almost fifty years later, FL-based inference systems (FL-ISs) have become one of the most famous fields of FL. The main reason of the latter is the ability of FL to incorporate human’s expert knowledge with its nuances, as well as to express the behavior of the system in an interpretable way for humans.

The historical trajectory of fuzzy systems begins with the effort of Mamdami and Assilian (1975) to use FL to model natural language, using fuzzy rule-based systems. Later, in the mid-eighties, Takagi and Sugeno (1985) proposed an automatic learning from data, breaking new ground towards the FL-based machine learning and data mining (Hullermeier, 2005). Consequently, this approach terminated the age of expert knowledge-based fuzzy systems, shifting the interest to the data driven rule generation methods as the means for the fuzzy system design. This drift has been clearly expressed by Dubois and Prade (1997) that note:

Fuzzy controllers, and fuzzy rule-based modeling which have become the most popular and visible side of applied fuzzy set theory, are only the emerged part of the fuzzy iceberg, and as time passes this technology seems to owe less and less to fuzzy set theory itself, and mainly becomes a tool for approximating functions. (Dubois & Prade, 1997, p. 148)

As a response to the aforementioned, new research pathways were adopted by the FL researchers, considering the fuzzy formalism not sufficient enough to ensure the interpretability of a knowledge base; hence, setting the following conditions that prior have to be fulfilled (Guillaume, 2001):

  • Semantic integrity should be respected within the partition,

  • The number of rules should be small, and

  • In case of complex systems with a large number of input variables, the fuzzy rules must not systematically include all input variables, but only the important ones in the rule context (incomplete rules).

From the above perspective, a possible conflict between the interpretability constrains and the numerical error minimization objective of automatic learning methods is foreseen. To address the latter, under the perspective of approaching a FL-IS as an integrated framework for system modeling, getting the most out of expert knowledge and data, several works have been carried out to propose a trade-off between interpretability and accuracy (Casillas et al., 2003).

Apart from the prediction ability of FL-ISs, since they are structured as fuzzy rule-based systems, they are good candidates to model a specific kind of knowledge, the so-called operational knowledge. In this perspective, Guillaume and Charnomordic (2012) note:

Fuzzy concepts, whose content, value, or boundaries of application can vary according to context, operator and conditions, instead of being fixed once and for all, arise naturally in the operational approach of system modeling, hence the relevance of FIS [Fuzzy Inference Systems] for that matter. (Guillaume & Charnomordic, 2012, p. 8745)

In the vein of the above, this chapter aims at providing an overview of the main characteristics of different FL-ISs, such as Fuzzy Inference System (FIS), Adaptive Networks-based FIS (ANFIS), Intuitionistic FIS (IFIS) and Fuzzy Cognitive Map (FCM), functioning either as predictive or operational analogs to the modeling of real-world (especially ill-defined) systems. This knowledge would contribute to the comprehension of their use in the educational context, examined in the following Part III.

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