Fuzzy model tuning based on a training set with fuzzy model output values
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FUZZY MODEL TUNING BASED ON A TRAINING SET WITH FUZZY MODEL OUTPUT VALUES
UDC 658.012
S. D. Shtovba
The paper considers the training of a fuzzy model with the help of a training set with fuzzy model output values. Two ways are proposed for constructing fuzzy rule-based multifactor models that produce fuzzy numbers at their outputs. The problem of tuning such fuzzy models on the basis of a fuzzy training set is formulated, methods of its solution are considered, and relevant examples are presented. Computational experiments showed that training based on fuzzy data improves the modeling accuracy for both crisp and fuzzy test sets. Keywords: tuning, fuzzy model, fuzzy inference, fuzzy training set. INTRODUCTION In this article, models are considered in which the dependence “inputs-output” is described by a knowledgebase consisting of fuzzy IF—THEN rules. The tuning of a fuzzy model is an iterative procedure of changing its parameters with a view to minimizing the deviation of the results of logical inference from experimental data. The tuning of fuzzy models with knowledgebases of different formats was investigated in many works among which we note [1–9]. A distinctive feature of these investigations lies in the use of a training set with crisp values of input and output variables. Many practical identification problems in medicine, biology, economy, political science, sociology, sports, and other fields include training sets with fuzzy data. In [10], a method is proposed for training a fuzzy model with the help of a training set with fuzzy values of inputs. This article extends this method to the case of a training set with fuzzy output values. This article is organized as follows: fuzzy models transforming input numerical data into fuzzy output values are first proposed, then the tuning of a fuzzy model with the help of a fuzzy set is stated as an optimization problem, and, at the end of this article, the results of computer experiments on the tuning of the proposed models with the help of a fuzzy training set are presented. FUZZY MODELS WITH ONE FUZZY OUTPUT In the case being considered, a fuzzy model must realize some fuzzy function, i.e., must map crisp values of inputs X = ( x1 , x 2 , . . . , x n ) into a fuzzy number ~y at its output, ( x1 , x 2 , . . . , x n ) ® ~y. Typical models of fuzzy inference produce crisp output values. We propose two methods of synthesis of fuzzy numbers on the basis of models with fuzzy knowledgebases. The first method consists of the elimination of the defuzzification operation from a typical fuzzy model. Then such a model based on the Mamdani-type knowledgebase produces a fuzzy set of the type ~y =
ò
m ~y ( y ) / y,
(1)
yÎ[ y, y]
where m ~y ( y ) is the membership function of the fuzzy set ~y whose carrier is [ y, y] .
Vinnitsa National Technical University, Vinnitsa, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 26–32, May–June 2007. Original article submitted June 22, 2006.
334
1060-0396/07/4303-0334
©
2007 Springer
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