Fuzzy Membership Function Design Using Information Theory Measures and Genetic Algorithms
Outlines
Introduction Problem Statement Proposed Approach Results Conclusion
Masoud Makrehchi
Outlines
Introduction Problem Statement Proposed Approach Results Conclusion
introduction . fuzzy membership function
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)
Masoud Makrehchi
introduction . fuzzy membership function
introduction . fuzzy membership function
Fuzzification
A gateway to any fuzzy system applications
Masoud Makrehchi
Fuzzification precise world Fuzzy Information Processing Defuzzification fuzzy world
introduction . fuzzy membership function NS NM 1 Z PM PL
introduction . fuzzy membership function NS NM 1 Z PM PL
(NM,0.9)
-1 data/signal from real process
+1
Masoud Makrehchi
-1 data/signal from real process
+1
Normalization
Normalization
introduction . fuzzy membership function NS NM 1 Z PM PL
introduction . fuzzy membership function NS NM 1 Z PM
(NS, 0.2) (NM,0.5)
PL
-1 data/signal from real process
+1
Masoud Makrehchi
-1 data/signal from real process
+1
Normalization
Normalization
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introduction . fuzzy membership function
introduction . fuzzy membership function
Fuzzy membership function (FMF)
a critical issue in fuzzy information processing, fuzzy control, fuzzy pattern recognition,
Different types of fuzzy membership functions
Masoud Makrehchi
introduction . fuzzy membership function Boundary
introduction . fuzzy membership function
trapezoidal
triangular
1 Prototype
1 Core
Masoud Makrehchi
Support
UD
Support Boundary
UD
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introduction . fuzzy membership function Boundary
introduction . fuzzy membership function
sigmoidal
bell-shaped
1 Prototype
1 Core
Masoud Makrehchi
Support
UD
Support Boundary
UD
introduction . fuzzy membership function
introduction . fuzzy membership function
singleton
Three parts of FMF
support: boundary:
Masoud Makrehchi
xi X : xi > 0
xi X : xi > 0
xi X : 0 < xi < 1
X1
x2
x3
UD
core/prototype:
xi X : xi = 1
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introduction . fuzzy membership function
introduction . fuzzy membership function
Support or fuzzy partition
An essential part of any fuzzy membership function
Masoud Makrehchi Support of partition A
M(x)
information domain-UD (area of interest)
introduction . fuzzy membership function
introduction . fuzzy membership function
MB(x) B A E F D C MA(x) 0
Masoud Makrehchi
M(x)
information domain-UD (area of interest)
X0 X1 X2 X3
X (Support)
Xn-1 Xn
UD
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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
Masoud Makrehchi
introduction . design factors
Outlines
Introduction Problem Statement Proposed Approach Results Conclusion
FMF design factors
Number: number of fuzzy partitions assigned to a Linguistic Variable, influencing the size of fuzzy rule base,
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Masoud Makrehchi
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problem statement
problem statement
Masoud Makrehchi
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
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
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problem statement
Masoud Makrehchi
Outlines
Introduction Problem Statement Proposed Approach Results Conclusion
Masoud Makrehchi
Finding an optimum set of fuzzy partitions related to a given linguistic variable Optimum fuzzy partitions Optimization problem
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 2001
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Masoud Makrehchi
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proposed approach . modeling & optimization
proposed approach . modeling & optimization
Optimization
To search UD for the best set of support values Genetic Algorithms (GA)
Masoud Makrehchi
Performance indices
Fitness function in GA optimization procedure
Shannon entropy Mutual information
proposed approach . modeling & optimization
proposed approach . modeling & optimization
How we relate FMF to information measure?
Mapping the FMF on the histogram of given data Probability~statistics Maximizing the entropy of partitioned histogram based on given number of partitions (n)
Masoud Makrehchi
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PDF~histogram
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proposed approach . modeling & optimization
proposed approach . modeling & optimization
overlaps
In a n-fuzzy-partitioned information, allowed overlaps just between two adjacent partitions, we have n-1 overlaps
Masoud Makrehchi
NS
NM
1 Z PM
PL
How to model the overlaps between partitions?
-1
+1
proposed approach . modeling & optimization
proposed approach . modeling & optimization
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-1) partitions NS
Masoud Makrehchi
NM
1 Z PM
PL
H1
H2
H3
H4
H5
H6
H7
H8
H9
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proposed approach . modeling & optimization
proposed approach . modeling & optimization
NS
NM
1 Z PM
PL
Masoud Makrehchi
H1
H2
H3
H4
H5
H6
H7
H8
H9
proposed approach . modeling & optimization
proposed approach . modeling & optimization
Algorithm Do optimization for given number of partitions
2 n -1 i =1
Masoud Makrehchi
Increased and enhanced overlaps A conservative strategy
In fuzzy control applications,
Longer rise time Less overshoot Smooth convergence
Change width of partitions Until maximum H
H =
Hi
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proposed approach . modeling & optimization
proposed approach . modeling & optimization
Second: Overlaps as conjunction of two joint partitions NS NM 1 Z PM PL
Masoud Makrehchi
NS
NM
1 Z PM
PL
H2 I1,2 I2,3 I3,4 H3
H4 I4,5
H2 I1,2 I2,3 I3,4 H3
H4 I4,5
H1
H5
H1
H5
proposed approach . modeling & optimization
proposed approach . modeling & optimization
Algorithm Do optimization for given number of partitions
Masoud Makrehchi
Change width of partitions Until maximum H
H= Hi - I (i ,i +1 )
i= 1 i= 1
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n- 1
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proposed approach . modeling & optimization
proposed approach . design
Decreased overlaps
In fuzzy control applications
Shorter rise time More overshoot
Masoud Makrehchi
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
proposed approach . design
proposed approach . design
Importance of boundary
Defining a range instead of an exact value P2 P1 -1 P3 P4
Masoud Makrehchi
In two partitions A and B, if : XA1 < XB1In< XA3 <A X two partitions andB3 B, if :
x <x <x <x Well - defined Wellboundary, defined boundary, W ; W ; B
XA1 < XB1 < XA3 < XB3
A1 B1 A2 B2
B
WB (XA3 - XB1 )
P5 +1 xA1 A xA3 xB1 xB3 B
WB (XA3 - XB1 )
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proposed approach . design xA2
a
proposed approach . design
b < a
b < a
b
xA1 xA2 a < b xB1 xA3 xB3
Masoud Makrehchi
xA1
xB1
a <
xA3
xB3
WB=XA3-XA2 WB>XA3-XB1
WB=XA3-XA2 WB<XA3-XB1
proposed approach . design xA2
b'
proposed approach . evaluation
Evaluation procedure
a a'
b < a
b' > a'
a < xB3
Masoud Makrehchi
xA1
xB1
xA3
WB=XA3-XA2 WB<XA3-XB1
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
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Outlines
Introduction Problem Statement Proposed Approach Results Conclusion Conditions
results
Masoud Makrehchi
Normalized data Five partitions Algorithm test in both two modes
results
results
GA optimization parameters
Search space: 35,184,372,088,832 Population size: 400 Chromosome/string length: 45 Pcross-Over: 0.3 Pmutation: 0.01 Minimum generation: 200
First data set:
Hourly temperature of city of Toronto during year 2000
Masoud Makrehchi
Max: 36.88 Min: -20.67 Mean: 8.90 STD: 10.30
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results
results
Temperature vs. time year 2000
Masoud Makrehchi
Temperature vs. time year 2000 Normalized temperature
results
Mode 1: Overlaps as independent partitions
results
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Masoud Makrehchi
Temperature vs. time year 2000 Normalized temperature Histogram
Best strings over generations
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results
Mode 1: Overlaps as independent partitions Mode 1: Overlaps as independent partitions
results
Masoud Makrehchi
Best strings over generations Mean of strings during convergence
Best strings over generations Mean of strings during convergence Resulted fuzzy memberships
results
Mode 2: Overlaps as conjunction of two joint partitions
results
Mode 2: Overlaps as conjunction of two joint partitions
Best strings over generations
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Masoud Makrehchi
Best strings over generations Mean of strings during convergence
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results
Mode 2: Overlaps as conjunction of two joint partitions
results
Second data set:
Hourly temperature of city of Toronto during year 2001
Masoud Makrehchi
Best strings over generations Mean of strings during convergence Resulted fuzzy memberships
Max: 32.23 Min: -23.34 Mean: 9.60 STD: 10.20
results
results
Temperature vs. time year 2001
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Masoud Makrehchi
Temperature vs. time year 2001 Normalized temperature
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results
Mode 1: Overlaps as independent partitions
results
Masoud Makrehchi
Temperature vs. time year 2001 Normalized temperature Histogram
Best strings over generations
results
Mode 1: Overlaps as independent partitions Mode 1: Overlaps as independent partitions
results
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Masoud Makrehchi
Best strings over generations Mean of strings during convergence
Best strings over generations Mean of strings during convergence Resulted fuzzy memberships
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results
Mode 2: Overlaps as conjunction of two joint partitions
results
Mode 2: Overlaps as conjunction of two joint partitions
Best strings over generations
Masoud Makrehchi
Best strings over generations Mean of strings during convergence
results
Mode 2: Overlaps as conjunction of two joint partitions
results
Mode 1: Overlaps as independent partitions
Data set 2000
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Masoud Makrehchi
Best strings over generations Mean of strings during convergence Resulted fuzzy memberships
Data set 2001
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results
Mode 2: Overlaps as conjunction of two joint partitions
Outlines
Introduction Problem Statement Proposed Approach Results Conclusion
Data set 2000
Data set 2001
Masoud Makrehchi
conclusion
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Masoud Makrehchi
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
