Learning from Examples Adriano Cruz ©2004 NCE/UFRJ e IM/UFRJ

@2001 Adriano Cruz NCE e IM - UFRJ Reference n L. Wang, J. Mendel, Generating Fuzzy Rules by Learning from Examples, IEEE Transactions on Systems, Man and Cybernetics, vol 22, no.6, november 1992

@2001 Adriano Cruz NCE e IM - UFRJ Information sources n Information used for most real-world control and signal processing problems can be classified into two kinds: –Numerical obtained form sensor, etc –Linguistic information obtained from human experts

@2001 Adriano Cruz NCE e IM - UFRJ Current situation n Solutions are heuristic in nature n Combine standard control processing methods and expert systems n Weakpoints: –Problem dependent –No commom framework

@2001 Adriano Cruz NCE e IM - UFRJ Problems n Linguistic rules may be incomplete due to loss when humans express their knowedge n Input-output data pairs because past experience may not cover all situations

@2001 Adriano Cruz NCE e IM - UFRJ Generating fuzzy rules n Consider a two input-one output problem as an example n So data is available as input-output pairs as n (x 1 (1),x 2 (1) ;y (1) ), (x 1 (2),x 2 (2) ;y (2) ), … n F(x 1,x 2 ) -> y

@2001 Adriano Cruz NCE e IM - UFRJ Step 1 n Divide the input and output space into fuzzy regions n Assign to each region a membership function n Similar to creating fuzzy sets over a universe of discourse

@2001 Adriano Cruz NCE e IM - UFRJ Step1 x1 B1B2CES1S2 x2 CEB1S1S2S3B2B3 X (2) 1 X (1) 1 X (1) 2 X (2) 2

@2001 Adriano Cruz NCE e IM - UFRJ Step1 y B1B2CES1S2 y (1) y (2)

@2001 Adriano Cruz NCE e IM - UFRJ Step2 n Generate fuzzy rules from given data pairs n First determine the degrees of each given x 1 (i),x 2 (i) and y (i) in different regions n Second assign a given x 1 (i),x 2 (i) and y (i) to the regions with maximum degree n Finally, obtain one rule from one pair of input data

@2001 Adriano Cruz NCE e IM - UFRJ Step2 n (x 1 (1), x2 (1); y (1) )= [0.8 in B1, 0.7 in S1; 0.9 in CE] = Rule 1 n If x1 is B1 and x2 is S1 then y is CE n (x 1 (2), x2 (2); y (2) )= [0.6 in B1, 1.0 in CE; 0.7 in B1] = Rule 1 n If x1 is B1 and x2 is CE then y is B1 = Rule 2

@2001 Adriano Cruz NCE e IM - UFRJ Step3 n Assign a degree to each rule n It is high probable that there will be some conflicting rules, i.e., rules that have the same antecedent but different consequents n The degree is calculated as n D(rule)=m A (x 1 )m B (x 2 )m C (z)

@2001 Adriano Cruz NCE e IM - UFRJ Step3 n D(rule)=m A (x 1 )m B (x 2 )m C (z) n Rule 1 = 0.8 * 0.7 * 0.9 = n Rule 2 = 0.6 * 1.0 * 0.7 = 0.42

@2001 Adriano Cruz NCE e IM - UFRJ Step4 n Create a combined fuzzy rule n If there is more than on rule in one box of the fuzzy rule base, use the rule that has maximum degree.

@2001 Adriano Cruz NCE e IM - UFRJ Step5 n Determine a Mapping based on the combined Fuzzy Rule Base. n Combine the antecedents of ith fuzzy rule: m O i = m I1 i (x 1 )m I2 i (x 2 ) (product operator) n m CE 1 =m B1 (x 1 )m S1 (x 2 )

@2001 Adriano Cruz NCE e IM - UFRJ Step5 n Use the following centroid defuzzification method n y i represents the center value of the region O i (the smallest abs value among all values with membership equals to 1) n K is the number of fuzzy rules combined