Katholieke Universiteit Leuven Departement Elektrotechniek
ESAT-SISTA/TR 1998-15
Constructing Fuzzy Models with Linguistic Integrity -AFRELI Algorithm 1 Jairo J. Espinosa and Joos Vandewalle2 January 1998 Submitted for publication: IEEE Transactions on Fuzzy Systems
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This report is available by anonymous ftp from ftp.esat.kuleuven.ac.be in the directory pub/SISTA/espinosa/reports/afreli.ps.Z
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K.U.Leuven, Dept. of Electrical Engineering (ESAT), Research group SISTA, Kardinaal Mercierlaan 94, 3001 Leuven, Belgium, Tel. 32/16/32 18 03, Fax 32/16/32 19 70, WWW: http://www.esat.kuleuven.ac.be/sista. E-mail:
[email protected]. This work is supported by several institutions: Concerted Research Action GOA-MIPS (Model-based Information Processing Systems), the FWO (Fund for Scienti c Research - Flanders) project G.0262.97 ;Learning and Optimization: an Interdisciplinary Approach, The FWO Research Communities: ICCoS (Identi cation and Control of Complex Systems) and Advanced Numerical Methods for Mathematical Modelling, The Belgian State, Prime Minister's Oce -Federal Oce for Scienti c, Technical and Cultural A airs - Interuniversity Poles of Attraction Programme (IUAP P4-02 (1997-2001): Modeling, Identi cation, Simulation and Control of Complex Systems SCIENCE-ERNSI (European Research Network for System Identi cation): SC1-CT92-0779. Keep In Touch (SYSIDENT-KIT124): Nonlinear System Identi cation Using Unconventional Methods
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Constructing Fuzzy Models with Linguistic Integrity -AFRELI Algorithm
Jairo Espinosa, Joos Vandewalle
Jairo Espinosa and Joos Vandewalle are with ESAT-SISTA Katholieke Universiteit Leuven, Belgium. Email:
[email protected] . August 17, 1998
DRAFT
Abstract We present an algorithm to extract rules relating input-output data. The rules are created in the environment of fuzzy systems. The concept of linguistic integrity is discussed and used as a framework to propose an algorithm for rule extraction (AFRELI). The algorithm is complemented with the use of the FuZion algorithm created to merge consecutive membership functions and guaranteed the distinguishability between fuzzy sets on each domain.
Keywords Fuzzy Modeling, function approximation, knowledge extraction, data minning
I. Introduction
Mathematical models are powerful tools to natural phenomena represent in a systematic way . They open the possibility of studying the behavior of a system via simulation avoiding costly experiments. The sources of information to construct such models are typically natural laws, input-output data (causal models) and linguistic information (heuristics). Fuzzy models were conceived as a systematic way to construct models from linguistic information. A problem with linguistic information is that it is very vague and sometimes incomplete. These facts imply a degradation on the capacity of the model to predict future behavior and even worse for the case of dynamic models this error will be fed back causing cumulative errors and making the model useless for long term prediction. The answer to this drawback had been the use of the so called neuro-fuzzy models. These models are able to extract their parameters from input-output data by means of gradient descent techniques for optimization [7] [4]. To apply these models the structure of the fuzzy model should be xed in advance (number of membership functions, number of rules, etc). Many schemes have been proposed to solve this inconvenience, some of them, are based on the accuracy of the approximation or local error [6] [2] and others are based on fuzzy clustering methods [5] [10] [8]. The results are models with good capabilities on the framework of numerical approximation, but very poor in the context of linguistic information. This paper presents the AFRELI algorithm (Autonomous Fuzzy Rule Extractor with Linguistic Integrity), the algorithm is able to t input-output data while maintaining the semantic integrity of the rule base. 2
The paper is structured as follows. Section II presents the structure of the fuzzy model, section III introduces the AFRELI algorithm, section IV presents the FuZion algorithm to preserve the semantic integrity of the domain, section V shows some application examples and nally, section VI gives the conclusions. II. Structure of the fuzzy model
A fuzzy inference system has many degrees of freedom (shape and number of membership functions, T-norms, aggregation methods, etc). This fact gives high exibility to the fuzzy system but also demands systematic criteria to select these parameters. For the present case some parameters will be xed taking into account some important facts. The membership functions will be triangular and normal (1(x); 2(x); : : : ; n(x)) with a speci c overlap of 21 . It means that the height of the intersection of two successive fuzzy sets is hgt(i \ i1) = 21 : (1) Two main reasons motivated the choice: one is their optimal interface design and the other is its semantic integrity [3] [1]: Optimal interface design { Error-free Reconstruction: In a fuzzy system a numerical value is converted into a linguistic value by means of fuzzi cation. A defuzzi cation method should guaranteed that this linguistic value can be reconstructed in the same numerical value
8x 2 [a; b] :
L?1[L(x)] = x
(2)
where [a; b] is the universe of discourse. The triangular membership functions described before satisfy this requirement (see proof: [3]) Semantic integrity { Distinguishability Each linguistic label should have semantic meaning and the fuzzy set should clearly de ne a range in the universe of discourse. So, the membership functions should be clearly di erent. The assumption of the overlap equal to 21 makes sure that the support of each fuzzy set will be di erent. The distance between the modal values of the membership functions is also very important to make sure that the membership functions 3
can be distinguished. The de ned modal value of a membership function is de ned as the -cut with = 1
i( =1) (x); i = 1; : : : ; N
(3)
{ Justi able Number of Elements The number of sets should be compatible with the
number of \quanti ers" that a human being can handle. This number should not exceed the limit of 7 2 distinct terms. The simple choice of the shape of the membership functions does not guarantee this property. To assure that this requirement is ful lled the FuZion algorithm is presented further in this paper. { Coverage Any element from the universe of discourse should belong to at least one of the fuzzy sets. This concept is also mentioned in the literature [4] as completness { Normalization Due to the fact that each linguistic label has semantic meaning, at least one of the values in the universe of discourse should have a membership degree equal to one. In other words all the fuzzy sets should be normal. Further details about these concepts can be found on [3] [1]. The AND operation on fuzzy sets can be performed by using product or also min operation. The OR operation can be performed by using probabilistic sum or max operation. The aggregation method and the defuzzi cation method will be discussed in the next sections. III. The AFRELI algorithm
The AFRELI algorithm does the clustering and the subsequent rule extraction for given input output data. The AFRELI algorithm proceeds as follows: 1. Collect N points from the inputs (U = fu1; : : : ; uN g) and the output (Y = fy1; : : : ; yN g)
2 66 uk = 66 4
u1k ... unk
3 77 77 5
(4)
where uk 2
