Automatic Generation of Membership Functions and Rules in a Fuzzy Logic System

Transcription

1 Proceedigs of the Fifth Iteratioal Coferece o Iformatics ad Applicatios, Takamatsu, Japa, 2016 Automatic Geeratio of Membership Fuctios ad Rules i a Fuzzy Logic System Mohammed A. A. Refaey Iformatio Techology Departmet Faculty of Computers ad Iformatio Cairo Uiversity, Egypt m.refaey@fci-cu.edu.eg ABSTRACT Fuzzy logic is playig a sigificat role i may cotrol ad classificatio systems. This arises from its simplicity, atural laguage based costructio, dealig with ambiguity, ad its ability to model liear ad o-liear complex systems. But, with larger umber of iput ad output variables, the buildig process of fuzzy system maually becomes dautig ad errorproe process. I this work we suggest ew methods for automatically creatio ad geeratio of fuzzy membership fuctios ad fuzzy rule base respectively. The membership fuctios are created adaptively from the traiig data set usig histogram of each feature idividually. Ad i the begiig, a rule is geerated for each membership fuctio, the accordig to the weight assiged to each rule ad membership fuctio, the membership fuctios are merged accordig to successiveess of their domai or support. Also, the rules are started to be co-operated ad merged to ehace the classificatio process. The resulted system is flexible, ad is able to receive more rules ad/or membership fuctios if eeded. KEYWORDS Fuzzy Logic ; Automatic membership fuctios Creatio ; Automatic Rule-Base Geeratio ; Usupervised Classificatio ; Dyamic Fuzzy Logic System ; Adaptive Fuzzy Logic System. 1- INTRODUCTION Fuzzy Logic is itroduced by Lotfi Zadeh i 1960s [1]. From that date, it becomes a pivotal player i the cotrollig systems ad classificatio processes. This is comes as a result for a may advatages that it has. Fuzzy logic ca be cosidered as a superset of covetioal biary logic, where it exteds the cocept of true or false to the cocept of partially-true or partially false, which arises i may faces i our life where a thig or a evet may be partially true ad partially false. This methodology gives the scietists a lot of flexibility i may situatios. The fuzzy logic has may advatages: it is developed usig atural laguage, the words of every oe life. Hece, it is coceptually easy to maipulate ad easy to be uderstood. Also, this makes it flexible, as there are o restrictios ad o costraits. Ad oe of its powerful stregths is its ability to hadle ambiguity. It takes imprecise data as iput ad gives a precise output, which makes it tolerat for udetermied situatios. Aother useful advatage is its ability to model liear ad liear complex systems, which broade the scope of its applicatios. I additio, as it is built usig atural laguage, it ca be built upo the top of experiece of experts. Fially, it ca be bleded with other covetioal systems. The fuzzy logic hadles the partial truth, but it is ot a probability. As there are three clear differeces betwee fuzzy logic ad ISBN: SDIWC 117

2 Proceedigs of the Fifth Iteratioal Coferece o Iformatics ad Applicatios, Takamatsu, Japa, 2016 probability. Firstly, the probabilities of somethig is add to oe, while i fuzzy logic it ca ay defiitio ad it is ot a coditio to adds all to oe. Secodly, the probability talks about chaces or likelihood, while fuzzy logic talks about degree of membership. Thirdly, i probability the evet is completely occurred, while i fuzzy logic the evet may be partially happeed. The buildig of fuzzy logic system goes through a umber of steps till it becomes a stadaloe system: first, you should to defie the iputs ad outputs of the system, ad they called iput variables ad output variables. Secod, for each variable you should to defie the domai or the Uiverse of Discourse, which are all possible values that the variable ca take. Third, for each variable you should to defie a set of adectives. For example if the iput is huma legth, the the uiverse of discourse could be from 25 cm to 300 cm, ad the adectives could be Short, Normal, ad Tall. Forth, you should to defie the domai of each adective. Fifth, for each adective ad its domai of values, you should to defie the Membership Fuctio, which assigs a degree of membership to ay value of the domai. For example, 160 cm belogs to Normal by 0.8 ad to short by 0.4. Fially, you should to defie the set of IF-The rules, which is called Rule Base that maps the iputs to the outputs. To do this, you should to defie how OR ad Ad operators to be executed, ad what is the Implicatio method, which determie how the coditio or the premise of each rule affects its cosequet. After that, the Aggregatio Method should to be defied that defie how all rules are added together. The last thig to be defied is the Defuzzificatio Method, which is used to iterpret the fuzzy output ad covert it to a umerical clear value. I this work, a suggested method for automatically defie the umber of adectives of each variable, ad how automatically assig a membership fuctio for each term. I additio, a method for how automatically ad adaptively to geerate the set of rules of the fuzzy logic system is preseted. The rest of the paper is orgaized as follow: a skim review of some of related ad previous research work is preseted i sectio 2. The automatic membership creatio method is explaied i sectio 3. Sectio 4 presets the method of adaptively geeratig the fuzzy rules. Ad paper is cocluded with the discussio of the proposed methods i sectio 5 ad the paper is trailed by a list of refereces. 2 RELATED WORK The work preseted i [1] suggested a iductive reasoig-based membership fuctios ad rules geeratio for a fuzzy system. The authors formulate the problem as etropy miimizatio problem to make each rule simpler ad more reliable. The uiverse of discourse of each fuzzy variable is partitioed usig threshold(s) that miimize the etropy o its both sides. They suggest that each fuzzy variable has 7-parts of its rage of possible values. The triagle shape is adopted as the membership fuctio of each sub rage. The rules are assumed to differetiate betwee two classes 0 ad 1. Oe rule is geerated for each fuzzy set. Where, each sample traiig data is give a seve-digit biary umber. Ad the rule is give a weight calculated as the mea probability derived from wrog classificatios. The research explaied i [2] discusses how to create fuzzy rules usig dyamic fuzzy eural etwork. The authors assume that the iput uiverse of discourse parameters could be estimated without partitioig. As, the ellipsoidal basis fuctios are adopted, ad the parameters are set o-lie ot radomly iitialized. They basically try to geeralize the dyamic eural etworks. Where, the width of membership fuctios is adaptively set accordig to its role i overall system performace. The proposed mechaism structures the eural etwork as four layers: the first layer for iput variables, ode for each iput variable. The secod layer is set for membership fuctios, as ay iput variable has a umber of membership fuctios. The third layer is for rules, as each rule ca use a umber of membership fuctios. Fially, the ISBN: SDIWC 118

3 Proceedigs of the Fifth Iteratioal Coferece o Iformatics ad Applicatios, Takamatsu, Japa, 2016 forth layer is used for output variables, ot for each output variable. Authors of [3] suggest a model for electroic agets i electroic market as a bargaiig game. This model adopts a fuzzy logic cotroller for deadlie calculatio, which is the best time to make a profitable deal. The umber of combiatios betwee iput ad output parameters is predefied. The subtractive clusterig is used to get the ceter of each class as the umber of classes is give i advace, ad Euclidia distace is used to measure how close the poits to oe specific poit, ad poit with the highest umber of close poits is take to be the first ceter of class. The it is excluded for ext iteratio for ext ceter, ad so o. If the distace betwee ceter ad predefied ceters is less tha a threshold it is eglected. The fuzzy C-mea is used to determie for which cluster is a poit belogig to. The work preseted i [4] talks about usig swarm particle optimizatio ad fuzzy clusterig to automatically geerate fuzzy rules to cotrol a water supplyig system. But the authors assume a fixed umber of iput variables ad a three membership fuctios for each iput variable. The fuzzy clusterig is used to build the parameters of each membership fuctio as allows a data poit to belog to more tha oe membership fuctio. I [5], a algorithm usig adaptive geetic algorithm is suggested to geerate the fuzzy rules. A suggested automatic fuzzy rule base geeratig method is preseted i [6], to be used i a electroic egotiatig system. Where, the data are first classified ito clusters usig machie learig methods like K-Meas ad Fuzzy Clusterig. The, a rule for every cluster is created. If the the ew created rule does ot add a valuable cotributio it is ot added to the rule base. The research itroduced i [7] ad [8] suggests a method to automatically geeratig the fuzzy sets ad fuzzy rules. The iput data is clustered usig a semi-supervised techiques, the it have bee passed to semi-supervised Fuzzy C-Mea to geerate the fuzzy membership fuctios for each label. Ad the rules base is geerated usig geetic algorithms. I [9], the authors suggest a clusterig techique that is based o specificity (how a compact of a cluster) ad cardiality (how a cluster cotais more samples) measures for usupervised clusterig, which is used directly to create the fuzzy rules usig euro-fuzzy traiig. However, there is a threshold of maximum graulatio to be predetermied. The geetic algorithms are also used to geerate the fuzzy rules i [10]. 3 MEMBERSHIP CREATION The first step i the creatio process of a fuzzy system is to determie the iput variables ad output variables. Each variable has a domai of possible values; this domai is called support or uiverse of discourse. Ad every variable is described by a set of terms or sometimes called adectives. For example if we desig a system to cotrol the process of air coditioer, the iput variables could be the temperature s degree i curret room ad amout or percetage of humidity, while the output variable may be the speed of coolig or heatig. The uiverse of discourse of the temperature ca be defied as 0 o C -to-100 o C, ad the uiverse of discourse for the humidity 0%-to-100%, while for coolig or heatig output variables the support is ot equally sigificat to its adectives, so oe may defie their support to be 0-to-10. The temperature adectives may be very low, low, fair, high, ad hot. The same could be used for humidity. The adectives for the speed of coolig or heatig output variables could set as stop, low, medium, ad rapid. After settig all of this, the problem of assigig a membership fuctio for each adective is arise. These membership fuctios determie degree of which each poit of the uiverse of discourse is belogig to a specific adective or term. For example, temperature degree of 19 o C is belogig to low by 0.7 degree of membership, ad is belogig to fair by 0.4 degree of membership. The process, explaied i the above example of settig the terms or adectives of each iput or output variables, is a dautig ad may be complex process with the icreased umber of iput/output variables or with a large domai of support of each variable. The matter that makes ISBN: SDIWC 119

4 Proceedigs of the Fifth Iteratioal Coferece o Iformatics ad Applicatios, Takamatsu, Japa, 2016 the maual maipulatio is difficult ad errorproe. So, i the followig steps we suggest a method for automatically describe each uiverse of discourse by a set of terms, ad subsequetly assig a membership fuctio for each term or adective. The suggest method is goig as follow: 1- I the begiig, there is a traiig data set cosists of M samples, each sample is associated with it output label y k, ad each sample is cosists of N features, i.e. it has a feature vector xi { f1, f 2, f3,, f N }, where i 1,2,,M. Ad K is the umber of output classes, k 1,2,, K. So, we defie the iput fuzzy variables as F ad 1,2,3,, N. 2- A feature histogram is built for each feature, or fuzzy variable, F { xi ( )} is the vector of the feature umber from all traiig samples x i, give that its uiverse of discourse is U, U mi U max, the the histogram is calculated as: H 1 0 M i1 ; x i l ; otherwise (1) (2) ceters. These could be defied as the peaks of the histogram. So, we get the idex of each peak i the histogram ad assig it as the ceter of certai sub-class of the uderlyig feature. The peak P is the idex of the feature value that has H ( P ) H ( P 1) ad H ( P ) H ( P 1). Ad if it is foud it is assiged to the ceter C, the we ca use equatio (3): P l : H ( l 1) H C P 1 H ( l 1) (3) This is repeated till there are o more peaks. Where, the sub-class couter is iitially set to zero. Ad fial value of is set as the curret total umber of subclasses, i.e. Nc. 4- But, however, the feature values may have a lot of variatios, the fact with which the umber of sub-classes or ceters will be ridicules. This meas that may be a variatio i the feature values eve if for same sub-class. So, i order to make the process more logical, we eed the feature values to mootoic either icreases or decreases. This could be doe by low-pass filterig each feature values, usig a ruig averagig filter, or eve a media filter. For the average filter, each H (l) should be replaced by: Where l U U 1, U 2, mi U mi max,, mi 3- As the data set is divided ito classes, the the features by ature should exist i groups. Buildig upo this, we ca skim the histogram of each feature seekig for these groupigs ad its H ( l ) s k( ls) H 2s 1 ( k) (4) Where ( 2s 1) is the average legth ad it is the oly maually set (experimetally) parameter i the ISBN: SDIWC 120

5 Proceedigs of the Fifth Iteratioal Coferece o Iformatics ad Applicatios, Takamatsu, Japa, 2016 suggested method. This lowers the umber of possible sub-classes by removig the fake peaks that does ot belogig to a real sub-class, ad makes the feature values mootoically icreased the mootoically decreased aroud each peak or equivaletly aroud the ceters of the resulted sub-classes. 5- As we get the peaks, we calculate the positio of valleys betwee peaks. We ca calculate it as follow: 7- Report the computed ' s, C ' s, ad MF ' s Figure 1 shows a example with v1= 5, v2 = 25, v3 = 45, c1 = 15, c2 = 35, sig1 = 10, ad sig2 = l : H U mi ( l 1) H ad H N c ( l 1) U max (5) This assures that every peak P lies betwee 1 ad. So, we get the umber of sub-classes, the idex of ceters of them which are the idex of peaks, ad the idex of valleys which could be used as the boudary betwee of sub-classes. 6- The we assig a two-dimesioal Gaussia membership fuctio for each sub-class cetered o C of that subclass. Where, the tail of each membership fuctio is set proportioally to the differece betwee idex betwee adacet valleys ad peaks: MF l 2 l C 2 2 e (6) Figure 1: Example of membership fuctio creatio 4 RULES GENERATION The sub-classes resulted from previous method are the terms or the adectives of the fuzzy variables. After fiishig classificatio of each feature or equivaletly fuzzy variable, we create a fuzzy rule for each sub-class. The rules creatio could be doe usig the resulted adectives as its atecedet ad we try i loop every output aloe as the cosequet of the rule. The cosequet that has the maximum classificatio rate with the specified atecedet is the fial cosequet of the rule. Ad the total true classificatio percetage of the rule is assiged to it as its weight. C 1 C l Where, 2 2 ad ad 1 1 C c 0 U ; C mi N 1 U max CONCLUSIONS AND DISCUSSION I this research work we itroduced a method to automatically create a fuzzy membership fuctios ad fuzzy rules from a set of represetative data set. The proposed method does ot the umber of adectives for ay fuzzy variable. The oly maually set parameter is the averagig size that could be ehaced i the future to deped o the feature values itself. ISBN: SDIWC 121

6 Proceedigs of the Fifth Iteratioal Coferece o Iformatics ad Applicatios, Takamatsu, Japa, 2016 REFERENCES [1] C. J. Kim ad B. D. Russell, "Automatic geeratio of membership fuctio ad fuzzy rule usig iductive reasoig," Idustrial Fuzzy Cotrol ad Itelliget Systems, 1993., IFIS '93., Third Iteratioal Coferece o, Housto, TX, 1993, pp [6] R. Arapoglou, K. Kolomvatsos ad S. Hadiefthymiades, "Buyer aget decisio process based o automatic fuzzy rules geeratio methods," Fuzzy Systems (FUZZ), 2010 IEEE Iteratioal Coferece o, Barceloa, 2010, pp [2] Shiqia Wu, Meg Joo Er ad Yag Gao, "A fast approach for automatic geeratio of fuzzy rules by geeralized dyamic fuzzy eural etworks," i IEEE Trasactios o Fuzzy Systems, vol. 9, o. 4, pp , Aug [7] P. d. A. Lopes ad H. d. A. Camargo, "Automatic labelig by meas of semi-supervised fuzzy clusterig as a boostig mechaism i the geeratio of fuzzy rules," Iformatio Reuse ad Itegratio (IRI), 2012 IEEE 13th Iteratioal Coferece o, Las egas, N, 2012, pp [3] K. Kolomvatsos ad S. Hadiefthymiades, "Automatic Fuzzy rules geeratio for the deadlie calculatio of a seller aget," 2009 Iteratioal Symposium o Autoomous Decetralized Systems, Athes, 2009, pp [4] R. Mohaa, M. Juea, D. Kumar ad M. S. Bedi, "Automatic geeratio of fuzzy rules from data by fuzzy clusterig ad particle swarm optimizatio," Itelliget Aget & Multi-Aget Systems, IAMA Iteratioal Coferece o, Cheai, 2009, pp [5] H. x. Zhag, B. Zhag ad F. Wag, "Automatic Fuzzy Rules Geeratio Usig Fuzzy Geetic Algorithm," Fuzzy Systems ad Kowledge Discovery, FSKD '09. Sixth Iteratioal Coferece o, Tiai, 2009, pp [8] Imtiaz, S. ; P. A. Lopes ad H. A. Camargo. Semisupervised clusterigi fuzzy rule geeratio. I III Ecotro Nacioal de Iteligiecia Artificial - ENIA2011, volume 1, pages , [9] M. Al-Shammaa ad M. F. Abbod, "Automatic geeratio of fuzzy classificatio rules usig graulatio-based adaptive clusterig," Systems Coferece (SysCo), th Aual IEEE Iteratioal, acouver, BC, 2015, pp [10] Hartmut Surma ad Alexader Seleschtschikow, Automatic geeratio of fuzzy logic rule bases: Examples I, Procceedig Of the NF2002: First Iteratioal ICSC Coferece o Neuro-Fuzzy Techologies, PP 75, CUBA JAN ISBN: SDIWC 122