Improvements to the COR Methodology by means of Weighted Fuzzy Rules - PowerPoint PPT Presentation

7th Online World Conference on Soft Computing In Industrial Applications Improvements to the COR Methodology by means of Weighted Fuzzy Rules R. Alcalá, J. Casillas, O. Cordón, F. Herrera { alcala,casillas,ocordon,herrera} @decsai.ugr.es Department of Computer Science and Artificial Intelligence University of Granada, Spain WSC7, September 23 th , 2002

Summary 1. Introduction: Fuzzy Modeling 2. COR (COoperative Rules) 2.1. Characteristics 2.2. Algorithm 2.3. A simple example 3. Weighted Linguistic Fuzzy Rules 4. WCOR (Weighted COR) 2.1. Description 2.2. A specific genetic algorithm (GA) 5. Experiments 5.1. Learning methods 5.2. Problem description 5.3. Results 5.4. Fuzzy models obtained 6. Concluding Remarks 1 WSC7, September 23 th , 2002 Improvements to the COR Methodology by means of Weighted Fuzzy Rules

1. Introduction: Fuzzy Modeling ! Fuzzy modeling (FM): system modeling with fuzzy rule- CONTENTS based systems ! Two opposite requirements: 1. I ntroduction ! Interpretability 2. COR ! Accuracy 3. Weighted ! Two approaches: linguistic fuzzy rules ! Linguistic FM: interpretability as main objective ! Precise FM: accuracy as main objective 4. WCOR ! Interpretable models have no sense if they are not 5. Experiments accurate enough 6. Concluding remarks ! A good trade-off between them is needed to perform a useful fuzzy modeling 2 WSC7, September 23 th , 2002 Improvements to the COR Methodology by means of Weighted Fuzzy Rules

1. Introduction: Fuzzy Modeling ! Two possibilities to find the desired balance: improve the CONTENTS accuracy in linguistic FM or improve the interpretability in precise FM 1. I ntroduction Linguistic Fuzzy Modeling (interpretability as main objective) 2. COR 1 Accuracy 3. Weighted improvement linguistic fuzzy Good trade-off rules 2 Interpretability improvement 4. WCOR Precise Fuzzy Modeling 5. Experiments (accuracy as main objective) 6. Concluding ! This paper is focused on the first approach proposing a remarks mechanism to improve the accuracy preserving good interpretability 3 WSC7, September 23 th , 2002 Improvements to the COR Methodology by means of Weighted Fuzzy Rules

2. COR: Characteristics COR is a fuzzy system learning methodology that ! CONTENTS exclusively designs the linguistic fuzzy rule set COR obtains the highest interpretability thanks to keeping ! 1. Introduction the membership functions and the model structure unaltered, as well as making fuzzy rule set reduction 2. COR The accuracy is achieved by developing a smart search 3. Weighted ! linguistic fuzzy space reduction and by inducing cooperation among the rules fuzzy rules 4. WCOR COR consists of two stages: ! 5. Experiments 1. Search space construction: A set of fuzzy input subspaces 6. Concluding and a set of candidate rules for each are defined remarks 2. Selection of the most cooperative fuzzy rule set: A combinatorial search is performed to select a fuzzy rule for each subspace (from the candidate rule sets) 4 WSC7, September 23 th , 2002 Improvements to the COR Methodology by means of Weighted Fuzzy Rules

2. COR: Algorithm 1. Search space construction: CONTENTS a) Define the positive example set for each fuzzy input subspace 1. Introduction 2. COR b) Select only those subspaces containing positive examples 3. Weighted linguistic fuzzy rules c) Define the candidate consequent set for each subspace 4. WCOR d) Define the candidate rule set for each subspace 5. Experiments 6. Concluding e) Add the “don’t care” symbol to each candidate rule set to remarks allow fuzzy rule set reduction 5 WSC7, September 23 th , 2002 Improvements to the COR Methodology by means of Weighted Fuzzy Rules

2. COR: Algorithm 2. Selection of the most cooperative fuzzy rule set: CONTENTS A combinatorial search algorithm is used to look for the combination: 1. Introduction 2. COR with the best accuracy 3. Weighted linguistic fuzzy rules The mean square error (that measure the global 4. WCOR cooperation of the rules) is considered to evaluate the 5. Experiments quality of each solution: 6. Concluding remarks 6 WSC7, September 23 th , 2002 Improvements to the COR Methodology by means of Weighted Fuzzy Rules

2. COR: A simple example (b) The examples are located in four (c) Candidate consequent sets for (d) Combinatorial search in the (f) RB generated from the third (a) Data set and DB previously (e) Decision table of the four X 1 (b) P M G P M G S 1 S 2 S 3 S 4 X 2 X 1 linguistc rules obtained different subspaces solution space the four rules combination defined X 1 S 1 Data Set Data Base CONTENTS B 1 B 1 B 2 B 3 P M G X 2 Combinatorial l B 1 l B 1 l B 3 B 3 Y P M G e = ( X 2 x , x , y ) There are There are S 1 P X 1 l 1 2 Inputs B 1 B not examples not examples B 1 B 2 B 2 B 3 P 2 B 1 Search e = (0,2 , 1,0 , 0,3) e 3 P = B Data Set Data Base 1 1 0 2 0 B 1 B 2 B 3 B 3 l l l P B P M G S 2 S 3 e = ( x , x , y ) (-0,35 , 0 , 0,65) e = (0,4 , 0,8 , 1,5) X 1 S 2 S 3 l 1 2 B 1 B 3 B 2 B 3 P M G 1 (d) e 5 2 1. Introduction (e) e 2 X 2 e = (0,2 , 1,0 , 0,3) M M = B 2 P = B B 2 B 3 B 3 B 3 There are (a) 1 e = (0,7 , 0,0 , 0,4) B 2 1 0 2 M B 2 e 1 (c) 3 (-0,35 , 0 , 0,65) B 1 B 2 B B 2 B not examples B 2 B 1 B 2 B 3 e = (0,4 , 0,8 , 1,5) ( 0,35 , 1 , 1,65) P M G M B 2 3 3 2 0 2 X 2 e = (1,0 , 1,2 , 1,6) M = B 2 (a) B 2 B 1 B 3 B 3 S 4 e = (0,7 , 0,0 , 0,4) 4 G = B 3 3 e 4 ( 0,35 , 1 , 1,65) B 1 B 2 B 3 B 2 B 2 B 2 B 3 S 4 0 2 G e = (1,0 , 1,2 , 1,6) 2. COR e = (1,2 , 0,6 , 1,1) Y B 3 4 ( 1,35 , 2 , 2,65) G = B 3 5 B 1 B 2 B 3 B 2 B 2 B 3 B 3 G B e = (1,2 , 0,6 , 1,1) Y ( 1,35 , 2 , 2,65) 5 There are There are e 6 3 e = (1,8 , 1,8 , 2,0) B 2 G B 3 B 2 B 3 2 e = (1,8 , 1,8 , 2,0) 6 not examples not examples 0 2 B 6 0 2 B 2 B 3 B 3 B 3 3 3. Weighted Step 1: Candidate consequents generation X 1 linguistic fuzzy P M G P M G (b) X 2 S 1 X 1 Y X 2 There are There are rules P not examples B 1 B not examples 2 e 3 0 P B S 2 S 3 1 e 5 e 2 No hay M (c) e 1 B 1 B 2 B B 2 B not exampes M B 3 3 2 e 4 S 4 4. WCOR G B There are There are G e 6 3 2 not examples Not exampesn B X 1 3 (b) P M G P M G X 2 S 1 S 2 X 1 S 3 S 4 P S 1 M G Step 2: Combinatorial search inducing cooperation X 1 X 2 B 1 B 1 B 2 B 3 Rule Base 5. Experiments S 1 Combinatorial Y X 2 B 1 B 1 B 3 B 3 There are X 1 There are P P M G X 2 P B 1 B not examples not examples S 1 S 1 S 2 S 3 S 4 B 1 R 1 = IF X 1 is and X 2 is THEN Y is P M P B B 1 B 2 B 2 B 3 2 B 1 e 3 1 B 1 B 1 B 2 B 3 Search S 2 S 3 B 1 B 1 B 3 B 3 Combinatorial 0 S 2 S 3 M (e) P B 1 S 2 B 2 B 3 S 3 B 3 B B 2 R 2 = IF X 1 is and X 2 is THEN Y is B 2 P M B B 1 B 2 B 2 B 3 6. Concluding 1 2 Search (e) M e 5 S 4 B 1 B 3 B 2 B 3 (d) B 1 B 2 B 3 B 3 B 2 (f) G B 2 e 2 B 3 There are B 1 B 3 B 2 B 3 (d) R 3 = IF X 1 is and X 2 is THEN Y is M M B B 2 M B 3 B 3 B 3 e 1 (c) 2 B 2 B 3 B 3 B 3 remarks B 1 B 2 B B 2 B not examples B 2 B 1 B 2 B 3 M S 4 B B 2 B 1 B 2 B 3 3 3 2 B 2 B 1 B 3 B 3 R 4 = IF X 1 is and X 2 is THEN Y is G G B 3 B 2 B 2 B 2 B 3 B 2 B 1 G B 3 B 3 B 3 B 2 B 2 B 3 B 3 e 4 S 4 Rule base B 2 B 3 B 2 B 3 B 2 B 2 B 2 B 3 B 2 B 3 B 3 B 3 G B 2 B 2 B 3 B 3 B R 1 = IF X 1 is M y X 2 is P THEN Y is B 1 There are There are e 6 3 G R 2 = IF X 1 is y X 2 is THEN Y is B 2 B 3 B 2 B 3 P M B not examples not examples B 2 2 (f) 3 R 3 = IF X 1 is M y X 2 is M THEN Y is B B 2 B 3 B 3 B 3 2 R 4 = IF X 1 is y X 2 is THEN Y is G G B 3 7 WSC7, September 23 th , 2002 Improvements to the COR Methodology by means of Weighted Fuzzy Rules

3. Weighted Linguistic Fuzzy Rules ! A way to improve the fuzzy model accuracy involves CONTENTS the use of a weight for each fuzzy rule ! A weight is a parameter that indicates the importance 1. Introduction degree of its associated rule in the inference process 2. COR ! These weights modulate the firing strength of each 3. Weighted linguistic fuzzy rule rules ! They can describe how a rule interacts with its 4. WCOR neighbor ones 5. Experiments ! Therefore, weighted linguistic fuzzy rules represent a 6. Concluding good framework to improve the accuracy preserving remarks interpretability 8 WSC7, September 23 th , 2002 Improvements to the COR Methodology by means of Weighted Fuzzy Rules

3. Weighted Linguistic Fuzzy Rules CONTENTS ! Weighted linguistic fuzzy rule structure: ∈ K IF X is A and and X is A THEN Y is B with [ w ], w [ 0 , 1 ] 1. Introduction 1 1 n n 2. COR 3. Weighted ! With this structure, the fuzzy reasoning must be extended: linguistic fuzzy rules ∑ ⋅ ⋅ m w P Center of gravity weighted 4. WCOR = i i i i y 0 by the matching degree ∑ ⋅ m w and the rule weight i i 5. Experiments i 6. Concluding remarks 9 WSC7, September 23 th , 2002 Improvements to the COR Methodology by means of Weighted Fuzzy Rules