Statistical analysis of the main parameters in the ... - Semantic Scholar
FUZZY
sets and systems ELSEVIER
Fuzzy Sets and Systems 102 (1999) 157 173
Statistical analysis of the main parameters in the fuzzy inference process I. Rojas*, J. Ortega, F.J. Pelayo, A. Prieto Departamento de Electr6nica y Tecnologla de Computadores, Universidad de Granada 18071-Granada, Spain
Received March 1997
Abstract
As there are many possibilities to select the set of basic operators used in the fuzzy inference process, the search for the fuzzy operators that are most suitable for the different steps of a fuzzy system, their characterization and evaluation, can be included among the most important topics in the field of fuzzy logic. A better insight into the performances of the alternative operators would make it easier to develop a fuzzy application. In the present paper, the relevancy and relative importance of the operators involved in the fuzzy inference process are investigated by using a powerful statistical tool, the ANalysis Of the VAriance (ANOVA) [8]. The results obtained show that the defuzzifier and the T-norm operator are the most relevant factors in the fuzzy inference process. Moreover, this statistical analysis is able to establish a classification of the defuzzifiers and T-norms, according to their intrinsic characteristics. The conclusions here obtained justify the present interest, observed in many current papers, in studying both operators [6, 22-24, 33, 64, 67]. Futhermore, our results are confirmed by some experiments dealing with a real control application. @, 1999 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy operators; Fuzzy implication operator; Fuzzy logic controllers; Statistical analysis of fuzzy operators
1. Introduction
Since Zadeh [68] introduced the theory of fuzzy sets, and the Minimum, the M a x i m u m and the Complement as the operators on fuzzy sets, m a n y alternative operators have been proposed in the specialized literature. Thus, providing a set of logical connectives for fuzzy sets constitutes a central issue in theoretical research and in the development of practical applications. There exist m a n y possibil-
*Corresponding author. E-mail: [email protected].
ities to select the set of basic operations in the fuzzy inference process. Thus, there are several alternatives for the defuzzifier, to define the fuzzy implication operators, and to select the conjunction and disjunction operators, the set of membership functions to be used in the definition of the linguistic variables, and so on. For example, in [-47], 215 different configurations with the usual fuzzy primitives are considered. As is shown in papers dealing with real applications, the designer has to select the operator to be used in each phase, and this decision is usually taken in terms of the most c o m m o n operations
0165-0114/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S01 65-01 14(97)001 53-X
158
L Rojas et al. /Fuzzy Sets and Systems 102 (1999) 157 173
performed. Nevertheless, it is very important to know which stages have the greatest influence on the behaviour and performance of the controller. Therefore, the designer should pay more attention to the phase in which the selection of the operator is most statistically significant. In this way, it would be possible to obviate a detailed analysis of different configurations that lead to systems with very similar performance. Consequently, the goal of this paper is to get a better insight into which are the most relevant blocks in a fuzzy system, in order to establish the elemental operations whose alternatives should be carefully studied when a real application is developed. To do this, an appropriate statistical tool has been used: the multifactorial analysis of the variance [13, 51], which consists of a set of statistical techniques that allow the analysis and comparison of experiments, by describing the interactions and interrelations between either the quantitative or qualitative variables (called factors in this context) of the system. In order to perform this analysis, the most relevant variables in the fuzzy inference process have been taken as the main factors. Thus, the membership functions, the type (crisp/fuzzy) of inputs to the system, the T-norm operator, the T-conorm operator, the defuzzifier, and the fuzzy implication function have been considered. First, an analysis was performed in order to determine the most important factors affecting the output variable. Once these factors have been identified, a more detailed study was completed by analysing the behaviour of the fuzzy system for a high number of alternative operators (levels in the terminology of ANOVA) corresponding to those main factors. The paper has been structured into six main sections. After this brief introduction, Section 2 briefly reviews previous papers dealing with the alternatives to defining the blocks of the fuzzy inference process. This section also serves as an introduction to the terminology and notation used in the paper. Section 3 gives some details about ANOVA. This is the tool used in the paper to perform the statistical analysis of the fuzzy inference process, which is given in Section 4. Then, Section 5 shows an application to a real problem and, finally Section 6 gives the conclusions of the paper.
2. Functional blocks in a fuzzy inference process
In this section, the main elements of a fuzzy system and the terminology used are briefly introduced. Previous studies of the alternatives to defining the blocks of a fuzzy system are also considered. Fuzzy controllers implement groups of fuzzy control rules in the form "IF-THEN", such as: IF X is A THEN Y is B, where A and B are fuzzy variables (linguistic variables such as slow, big, high, etc.) described by membership functions in universes of discourse U and V, respectively, where the variables X and Y take their values. A fuzzy rule is represented by means of a fuzzy relation R from set U to set V (or between U and V), that represents the correlation between A and B as follows: R: U × V ~ [0, 1] :(u, v) --, I(fia(U), fiB(V)), V(u, v) e U x V
(1)
where fA and fib are the membership functions of A and B, and I is the implication operator which is defined in terms of the so called T-norm and Tconorm operators. In this way, the elements in the fuzzy inference process are the fuzzy implication operator, I, the form of the membership functions, the T-norm and T-conorm operators, the defuzzification method, and the kind of input signal (crisp/fuzzy) fed to the controller. The motivation of the present statistical study comes from the great variety of alternatives that a designer has to take into account when developing a fuzzy system. Thus, instead of the existing intuitive knowledge, it is necessary to have a more precise understanding of the significance of the different alternatives.
2.1. Selection of the implication function The fuzzy implication operator is a mathematical operator that translates a fuzzy conditional proposition into a fuzzy relation from U to V. Many theoretical and experimental studies have been carried out to get a better understanding of the functionality of the operations involved in the fuzzy inference process. There are many ways in which a fuzzy implication can be defined. Analysing the extense bibliography on this subject, it is clear that
I. Rojas et al. / Fuzzy Sets and Systems 102 (1999) 157 173 the selection of the ideal fuzzy implication operator is a fundamental issue for designing the inference module in a fuzzy controller [2, 9-12, 23, 24, 37, 61]. The variety of implication operators recently proposed and investigated in the specialized literature implies an additional problem. The designer is faced with the selection of the implication operator providing the best performance for a given application. Table 1 lists the implication functions most commonly cited in the bibliography. It also includes a function that, even while not fulfilling the requirements to be an implication function [-58], have been widely employed in this context (Mamdani implication function). Table 2 summarizes the conclusions given by various authors about the performance or quality of different fuzzy implication operators. The " + " sign means the authors consider the fuzzy implication operator to be the most suitable, while the " - " sign indicates the opposite. As can be seen, there is some controversy over which is the best implication function, and there are some implication functions considered to be the best by some authors and the worst by others. As shown in the bibliography [,9-12, 23, 25, 37, 49, 54, 61], to date no fuzzy implication operator
has been found that verifies all the properties desirable for the fuzzy inference process [53]. Some studies attempted to improve the functionality of known operators, using T-norms and T-conorms which are different from the classical minimum and maximum operators. For example, Piskunov [53] concludes that the Mamdani fuzzy implication has best properties when the Mizumoto operator is used as the composition operator. In this way, the search for the ideal implication operator is, for the moment, an open question [-62]. Moreover, the studies carried out on the implication operators in Table 1 do not include an analysis of the influence of the T-norm and T-conorm operators involved in their definition. For example, in [,11], the maximum and minimum operators were used as the conjunction and intersection operators, respectively. Therefore, the generality of the results analysed is not only restricted to the examples under consideration, but also to the specific operators employed in the definition of the implication operator, and in the fuzzy inference. Furthermore, in the research work carried out to date, conclusions have been obtained from the analysis of a particular fuzzy system. In the statistical analysis here presented, several fuzzy systems are analysed, and conclusions are drawn regarding the whole set.
Table 1 Fuzzy implication operators G6del Rg = Lukasiewicz
ffu
v[Fta(u) ~
×
{ 1 if/~a(u)~
sets and systems ELSEVIER
Fuzzy Sets and Systems 102 (1999) 157 173
Statistical analysis of the main parameters in the fuzzy inference process I. Rojas*, J. Ortega, F.J. Pelayo, A. Prieto Departamento de Electr6nica y Tecnologla de Computadores, Universidad de Granada 18071-Granada, Spain
Received March 1997
Abstract
As there are many possibilities to select the set of basic operators used in the fuzzy inference process, the search for the fuzzy operators that are most suitable for the different steps of a fuzzy system, their characterization and evaluation, can be included among the most important topics in the field of fuzzy logic. A better insight into the performances of the alternative operators would make it easier to develop a fuzzy application. In the present paper, the relevancy and relative importance of the operators involved in the fuzzy inference process are investigated by using a powerful statistical tool, the ANalysis Of the VAriance (ANOVA) [8]. The results obtained show that the defuzzifier and the T-norm operator are the most relevant factors in the fuzzy inference process. Moreover, this statistical analysis is able to establish a classification of the defuzzifiers and T-norms, according to their intrinsic characteristics. The conclusions here obtained justify the present interest, observed in many current papers, in studying both operators [6, 22-24, 33, 64, 67]. Futhermore, our results are confirmed by some experiments dealing with a real control application. @, 1999 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy operators; Fuzzy implication operator; Fuzzy logic controllers; Statistical analysis of fuzzy operators
1. Introduction
Since Zadeh [68] introduced the theory of fuzzy sets, and the Minimum, the M a x i m u m and the Complement as the operators on fuzzy sets, m a n y alternative operators have been proposed in the specialized literature. Thus, providing a set of logical connectives for fuzzy sets constitutes a central issue in theoretical research and in the development of practical applications. There exist m a n y possibil-
*Corresponding author. E-mail: [email protected].
ities to select the set of basic operations in the fuzzy inference process. Thus, there are several alternatives for the defuzzifier, to define the fuzzy implication operators, and to select the conjunction and disjunction operators, the set of membership functions to be used in the definition of the linguistic variables, and so on. For example, in [-47], 215 different configurations with the usual fuzzy primitives are considered. As is shown in papers dealing with real applications, the designer has to select the operator to be used in each phase, and this decision is usually taken in terms of the most c o m m o n operations
0165-0114/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S01 65-01 14(97)001 53-X
158
L Rojas et al. /Fuzzy Sets and Systems 102 (1999) 157 173
performed. Nevertheless, it is very important to know which stages have the greatest influence on the behaviour and performance of the controller. Therefore, the designer should pay more attention to the phase in which the selection of the operator is most statistically significant. In this way, it would be possible to obviate a detailed analysis of different configurations that lead to systems with very similar performance. Consequently, the goal of this paper is to get a better insight into which are the most relevant blocks in a fuzzy system, in order to establish the elemental operations whose alternatives should be carefully studied when a real application is developed. To do this, an appropriate statistical tool has been used: the multifactorial analysis of the variance [13, 51], which consists of a set of statistical techniques that allow the analysis and comparison of experiments, by describing the interactions and interrelations between either the quantitative or qualitative variables (called factors in this context) of the system. In order to perform this analysis, the most relevant variables in the fuzzy inference process have been taken as the main factors. Thus, the membership functions, the type (crisp/fuzzy) of inputs to the system, the T-norm operator, the T-conorm operator, the defuzzifier, and the fuzzy implication function have been considered. First, an analysis was performed in order to determine the most important factors affecting the output variable. Once these factors have been identified, a more detailed study was completed by analysing the behaviour of the fuzzy system for a high number of alternative operators (levels in the terminology of ANOVA) corresponding to those main factors. The paper has been structured into six main sections. After this brief introduction, Section 2 briefly reviews previous papers dealing with the alternatives to defining the blocks of the fuzzy inference process. This section also serves as an introduction to the terminology and notation used in the paper. Section 3 gives some details about ANOVA. This is the tool used in the paper to perform the statistical analysis of the fuzzy inference process, which is given in Section 4. Then, Section 5 shows an application to a real problem and, finally Section 6 gives the conclusions of the paper.
2. Functional blocks in a fuzzy inference process
In this section, the main elements of a fuzzy system and the terminology used are briefly introduced. Previous studies of the alternatives to defining the blocks of a fuzzy system are also considered. Fuzzy controllers implement groups of fuzzy control rules in the form "IF-THEN", such as: IF X is A THEN Y is B, where A and B are fuzzy variables (linguistic variables such as slow, big, high, etc.) described by membership functions in universes of discourse U and V, respectively, where the variables X and Y take their values. A fuzzy rule is represented by means of a fuzzy relation R from set U to set V (or between U and V), that represents the correlation between A and B as follows: R: U × V ~ [0, 1] :(u, v) --, I(fia(U), fiB(V)), V(u, v) e U x V
(1)
where fA and fib are the membership functions of A and B, and I is the implication operator which is defined in terms of the so called T-norm and Tconorm operators. In this way, the elements in the fuzzy inference process are the fuzzy implication operator, I, the form of the membership functions, the T-norm and T-conorm operators, the defuzzification method, and the kind of input signal (crisp/fuzzy) fed to the controller. The motivation of the present statistical study comes from the great variety of alternatives that a designer has to take into account when developing a fuzzy system. Thus, instead of the existing intuitive knowledge, it is necessary to have a more precise understanding of the significance of the different alternatives.
2.1. Selection of the implication function The fuzzy implication operator is a mathematical operator that translates a fuzzy conditional proposition into a fuzzy relation from U to V. Many theoretical and experimental studies have been carried out to get a better understanding of the functionality of the operations involved in the fuzzy inference process. There are many ways in which a fuzzy implication can be defined. Analysing the extense bibliography on this subject, it is clear that
I. Rojas et al. / Fuzzy Sets and Systems 102 (1999) 157 173 the selection of the ideal fuzzy implication operator is a fundamental issue for designing the inference module in a fuzzy controller [2, 9-12, 23, 24, 37, 61]. The variety of implication operators recently proposed and investigated in the specialized literature implies an additional problem. The designer is faced with the selection of the implication operator providing the best performance for a given application. Table 1 lists the implication functions most commonly cited in the bibliography. It also includes a function that, even while not fulfilling the requirements to be an implication function [-58], have been widely employed in this context (Mamdani implication function). Table 2 summarizes the conclusions given by various authors about the performance or quality of different fuzzy implication operators. The " + " sign means the authors consider the fuzzy implication operator to be the most suitable, while the " - " sign indicates the opposite. As can be seen, there is some controversy over which is the best implication function, and there are some implication functions considered to be the best by some authors and the worst by others. As shown in the bibliography [,9-12, 23, 25, 37, 49, 54, 61], to date no fuzzy implication operator
has been found that verifies all the properties desirable for the fuzzy inference process [53]. Some studies attempted to improve the functionality of known operators, using T-norms and T-conorms which are different from the classical minimum and maximum operators. For example, Piskunov [53] concludes that the Mamdani fuzzy implication has best properties when the Mizumoto operator is used as the composition operator. In this way, the search for the ideal implication operator is, for the moment, an open question [-62]. Moreover, the studies carried out on the implication operators in Table 1 do not include an analysis of the influence of the T-norm and T-conorm operators involved in their definition. For example, in [,11], the maximum and minimum operators were used as the conjunction and intersection operators, respectively. Therefore, the generality of the results analysed is not only restricted to the examples under consideration, but also to the specific operators employed in the definition of the implication operator, and in the fuzzy inference. Furthermore, in the research work carried out to date, conclusions have been obtained from the analysis of a particular fuzzy system. In the statistical analysis here presented, several fuzzy systems are analysed, and conclusions are drawn regarding the whole set.
Table 1 Fuzzy implication operators G6del Rg = Lukasiewicz
ffu
v[Fta(u) ~
×
{ 1 if/~a(u)~
