Outlines. Fuzzy Membership Function Design Using Information Theory Measures and Genetic Algorithms. Outlines

Transcription

1 Fuzzy Membership Function Design Using Information Theory Measures and Genetic Algorithms Outlines Introduction Problem Statement Proposed Approach Results Conclusion 2 Outlines Introduction Problem Statement Proposed Approach Results Conclusion Fuzzy systems Perfect operation with fuzzy data Precise data from measurement and interfaces Need to have fuzzy data from precise data Conversion from precise to fuzzy (fuzzification)

2 Fuzzification A gateway to any fuzzy system applications precise world Fuzzification Fuzzy Information Processing 5 fuzzy world Defuzzification 6 NS NM Z PM PL NS NM Z PM PL (NM,0.9) data/signal from real process Normalization data/signal from real process Normalization

3 NS NM Z PM PL NS NM Z PM PL (NS, 0.2) (NM,0.5) data/signal from real process Normalization data/signal from real process Normalization 9 0 Fuzzy membership function (FMF) a critical issue in fuzzy information processing, fuzzy control, fuzzy pattern recognition, Different types of fuzzy membership functions

4 Boundary trapezoidal Prototype triangular Core Support UD Boundary Support UD 3 4 Boundary sigmoidal Prototype bell-shaped Core Support UD Boundary Support UD

5 singleton Three parts of FMF support: x X i X : > 0 : > 0 xi i xi boundary: x X : 0 < i xi < X x2 x3 UD core/prototype: x i X : xi = 7 8 Support or fuzzy partition An essential part of any fuzzy membership function Support of partition A M(x) information domain-ud (area of interest) 0

6 M B (x) M(x) A F B C M A (x) E D information domain-ud (area of interest) 0 X 0 X X 2 X 3 X (Support) X n- X n UD 2 22 introduction. design factors introduction. design factors FMF design factors Support: the domain in which the FMF is defined - domain of FMF or a partition of desired information and our interest in which fuzzy information is defined FMF design factors Shape: determining the boundaries and core/prototype and fuzzy behavior of FMF

7 introduction. design factors Outlines FMF design factors Number: number of fuzzy partitions assigned to a Linguistic Variable, influencing the size of fuzzy rule base, Introduction Problem Statement Proposed Approach Results Conclusion problem statement problem statement How can IT measures help in designing FMF? Which parameters can be optimized by IT measures? Number Estimating number of fuzzy partitions is a trade off with fuzzy rules, we can not estimate it independently It can be finalized during optimization of fuzzy rules The number of fuzzy rules is the bottleneck, not the number of fuzzy partitions

8 problem statement problem statement Shape Shape of FMF is still a heuristic issue There is no proven relation between information domain and degree of fuzziness in that domain, completely related to intuition, expertise, and expert knowledge Learning from examples can be a solution Support we can just estimate informational parameters of FMF, not fuzzy issues Support is a part of our information in which an uncertainty is happening IT measures is suitable for estimating support of FMF, or fuzzy partitions problem statement Outlines Finding an optimum set of fuzzy partitions related to a given linguistic variable Optimum fuzzy partitions Optimization problem Introduction Problem Statement Proposed Approach Results Conclusion

9 proposed approach. requirements proposed approach. data Solution requirements Set of data (simulation or real) for partitioning Fuzzy partitions modeling and Optimization technique FMF design Evaluation procedure Data Real data preferred U of Toronto-Mississauga Meteorological Station Temperature information for year 2000 and Optimization To search UD for the best set of support values Genetic Algorithms (GA) Performance indices Fitness function in GA optimization procedure Shannon entropy Mutual information

10 How we relate FMF to information measure? Mapping the FMF on the histogram of given data Probability~statistics PDF~histogram Maximizing the entropy of partitioned histogram based on given number of partitions (n) NS NM Z PM PL overlaps In a n-fuzzy-partitioned information, allowed overlaps just between two adjacent partitions, we have n- overlaps How to model the overlaps between partitions?

11 Two strategy: Overlaps as independent partitions: maximize entropy of independent partitions Overlaps as conjunction of two joint partitions: maximize entropy of joint partitions (considering mutual information) First: Overlaps as independent partitions (2n-) partitions NS NM Z PM PL H H 2 H 3 H 4 H 5 H 6 H 7 H 8 H NS NM Z PM PL H H 2 H 3 H 4 H 5 H 6 H 7 H 8 H9

12 Algorithm Do optimization for given number of partitions Change width of partitions Until maximum H H = 2 n - H i i = Increased and enhanced overlaps A conservative strategy In fuzzy control applications, Longer rise time Less overshoot Smooth convergence Second: Overlaps as conjunction of two joint partitions NS NM Z PM PL NS NM Z PM PL H H 2 H 4 I,2 I 2,3 I 3,4 I 4,5 H 3 H 5 H H 2 H 4 I,2 I 2,3 I 3,4 I 4,5 H 3 H 5

13 Algorithm Do optimization for given number of partitions Change width of partitions Until maximum H n n- H= H i- I (, + ) i= i= i i proposed approach. design Decreased overlaps In fuzzy control applications Shorter rise time More overshoot Ready to design FMFs We have partitions We need values of boundaries to have complete define of FMF A criteria to choose right value for boundary is necessary

14 proposed approach. design proposed approach. design Importance of boundary Defining a range instead of an exact value P2 P4 In two partitions A and B, if : Well X A In two partitions A and B, if : XA < XB < XA3 < XB3 x A x B x A x 2 B2 - defined Wellboundary, defined W B ; W B < X B < X (X W (X - X ) B A3 A3 A3 - X < X B B ) B3 W ; B P P3 P5 - + A x A x B x A3 x B3 B proposed approach. design proposed approach. design x A2 a b b < a a b < a b x A x A2 a < b x B WB=XA3-XA2 W B >X A3 -X B x A3 x B3 x A x B a < b W B =X A3 -X A2 W B <X A3 -X B x A3 x B3

15 proposed approach. design proposed approach. evaluation b ' a a ' b x A x A2 x B a < b W B =X A3 -X A2 W B <X A3 -X B x A3 b < a b ' > a ' x B3 Evaluation procedure Testing membership function in a complete fuzzy system reacting to a process, compare the output with heuristicdefined membership function Applying the algorithm on other set of data and study the behavior of membership function Outlines Introduction Problem Statement Proposed Approach Results Conclusion Conditions Normalized data Five partitions Algorithm test in both two modes

16 GA optimization parameters Search space: 35,84,372,088,832 Population size: 400 Chromosome/string length: 45 P cross-over : 0.3 P mutation : 0.0 Minimum generation: 200 First data set: Hourly temperature of city of Toronto during year 2000 Max: Min: Mean: 8.90 STD: Temperature vs. time year 2000 Temperature vs. time year 2000 Normalized temperature

17 Mode : Overlaps as independent partitions Temperature vs. time year 2000 Normalized temperature Histogram Mode : Overlaps as independent partitions Mode : Overlaps as independent partitions Mean of strings during convergence Mean of strings during convergence Resulted fuzzy memberships

18 Mode 2: Overlaps as conjunction of two joint partitions Mode 2: Overlaps as conjunction of two joint partitions Mean of strings during convergence Mode 2: Overlaps as conjunction of two joint partitions Mean of strings during convergence Resulted fuzzy memberships Second data set: Hourly temperature of city of Toronto during year 200 Max: Min: Mean: 9.60 STD: 0.20

19 Temperature vs. time year 200 Temperature vs. time year 200 Normalized temperature Mode : Overlaps as independent partitions Temperature vs. time year 200 Normalized temperature Histogram

20 Mode : Overlaps as independent partitions Mode : Overlaps as independent partitions Mean of strings during convergence Mean of strings during convergence Resulted fuzzy memberships Mode 2: Overlaps as conjunction of two joint partitions Mode 2: Overlaps as conjunction of two joint partitions Mean of strings during convergence

21 Mode 2: Overlaps as conjunction of two joint partitions Mode : Overlaps as independent partitions Data set 2000 Mean of strings during convergence Resulted fuzzy memberships Data set Mode 2: Overlaps as conjunction of two joint partitions Outlines Data set 2000 Data set 200 Introduction Problem Statement Proposed Approach Results Conclusion

22 conclusion A solution for designing fuzzy membership function Besides fuzzy rules generation, a solution for designing fuzzy system by learning from example The idea: having generic membership function for generic data 85