SENSIBILITY ANALYSIS FOR TYPE-1 AND TYPE-2 TSK FUZZY MODELS
Qun RENLuc BARON
Marek BALAZINSKI
Mechanical engineering departmentÉcole Polytechnique de Montréal
IASTED MS2007 2
Outline
IntroductionSubtractive clustering based type-1 Takagi-Sugeno-Kang (TSK) fuzzy modelType-2 TSK model based on subtractiveclusteringSensibility analysis for type-1 and type-2 TSK fuzzy modelsConclusions
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Outline
IntroductionSubtractive clustering based type-1 TSK fuzzy modelType-2 TSK model based on subtractive clusteringSensibility analysis for type-1 and type-2 TSK modelConclusions
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Classification of TSK FLSs
Antecedent MF Type
TSK Model TypeType-1 fuzzy sets
(A1)Type-2 fuzzy sets
(A2)
Crisp numbers (C0)
Type-1(A1-C0)
Type-2 Model II(A2-C0)
Type-1 fuzzysets (C1)
Type-2 Model III(A1-C1) Type-2 Model I
(A2-C1)
Consequent Parameter
Type
* MF: membership function
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Differences between Type-1 and Type-2 TSK FLSs
* there are M rules and each rule has p antecedents
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Outline
IntroductionSubtractive clustering based type-1 TSK fuzzy modelType-2 TSK model based on subtractive clusteringSensibility analysis for type-1 and type-2 TSK modelConclusions
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Subtractive Clustering Based Type-1 TSK FLS
Type-1 TSK model
Subtractiveclustering
Least-squares estimation
Subtractive clustering operates by finding the optimal data point to define a cluster center based on the density of surrounding data points.
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Subtractive Clustering Method [Chiu 1994]
arCluster radius
Squash factor
Reject ratio
Accept ratio
η
−ε−
ε
?
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Initialization of Parameters for Subtractive Clustering
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Outline
IntroductionSubtractive clustering based type-1 TSK fuzzy modelType-2 TSK model based on subtractive clusteringSensibility analysis for type-1 and type-2 TSK modelConclusions
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Type-2 TSK Model Identification Algorithm [Ren 2006]
Original system Type-1 TSK model
LSE (least square error) is small enough?
Subtractive clustering
Type-2 TSK model
End
EXPANDINGCluster center
Consequent parametersDeviation of MFs
YesNo
∑=
−=n
iimis WWLSE
1
2)(
Is this the best model?
Yes
No
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Expanding a Type-1 TSK Model to a Type-2 TSK Model(1)
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−=
∗ 2
21exp
σjkj
jk
xxQ
Antecedent :is the spread percentage of cluster center
.
⎥⎦
⎤⎢⎣
⎡=
−
−jkjkjk μμμ ,
~
jkμ
* k
ja
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Expanding a Type-1 TSK Model to a Type-2 TSK Model(2)
Consequent:
⎥⎦
⎤⎢⎣
⎡+−=
k
j
k
j
k
j
k
j
k
j
k
j
k
j bccbccp *,*~
k
j
k
j cp =
is the spread percentage of fuzzy numberk
jbk
jp~
.
*
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Expanding a Type-1 TSK Model to a Type-2 TSK Model(3)
kth rule:
.
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Uncertain Parameters in Type-1 and Type-2 Fuzzy Modeling algorithm
Type-1 fuzzy modelingCluster radiusSquash factorReject ratioAccept ratio
Type-2 fuzzy modelingSpread percentage of Cluster centerstandard deviation Spread percentage of consequent parameters
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Outline
IntroductionSubtractive clustering based type-1 TSK fuzzy modelType-2 TSK model based on subtractive clusteringSensibility analysis for type-1 and type-2 TSK fuzzy modelsConclusions
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Sensibility Analysis
To ascertain how a type-1 TSK model output (Least Square errors LSE)depends upon the pre-initialized parameters
To determine how a type-2 TSK model output (Root Mean Square Errors RMSE) depends upon spread percentage of cluster centers and consequent parameters
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System Data
( ) [ ]4,015.2 3 ∈++−= xwherexxzSystem
0 0.5 1 1.5 2 2.5 3 3.5 4-15
-10
-5
0
5
10
Input X
Out
put Z
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Sensibility Analysis for Type-1 TSK Fuzzy Model (1)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
8
9
10
LSE
of t
ype-
1 FL
S
cluster radius
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Sensibility Analysis for Type-1 TSK Fuzzy Model (2)
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Sensibility Analysis for Type-1 TSK Fuzzy Model (3)
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Sensibility Analysis for Type-1 TSK Fuzzy Model (4)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
2
2.5
3
LSE
of t
ype-
1 FL
S
squash factor
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Six Rule Type-1 TSK FLS
−
ε
ar
−ε
= 0.25
= 0.15
= 0.5
= 1.25η
RMSE is 0.093037
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Six-Rules Type-1 Premise Membership Functions
0 0.5 1 1.5 2 2.5 3 3.5 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 Premise
Input
mem
bers
hip
func
tion
degr
ee
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Sensibility Analysis for Type-2 TSK Fuzzy Model (1)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
RM
SE
Spread percentage of consequent parameters
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Sensibility Analysis for Type-2 TSK Fuzzy Model (2)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1R
MS
E
Spread percentage of consequent parameters
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Sensibility Analysis for Type-2 TSK Fuzzy Model (3)
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.650
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
RM
SE
standard deviation of Gaussian MF
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Influence of Uncertainties on a Type-2 TSK Model
k
jak
jbk
jσInfluence
RMSE Yes No Yes
Model output Yes Yes, Significant Yes
Gaussian MFs Yes, Significant No Yes
∑=
−=n
jjmjs WW
nRMSE
1
2)(1* are uncertainties in cluster centers, consequent parameters and standard deviation of Gaussian MF
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Six-Rules Type-2 TSK Model
RMSE is 0.081501
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Six-Rules Type-2 Premise Membership Functions
0 0.5 1 1.5 2 2.5 3 3.5 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Input
mem
bers
hip
func
tion
degr
ee
upperlowerprimary (type-1)
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Outline
IntroductionSubtractive Clustering based type-1 Takagi-Sugeno-Kang (TSK) Fuzzy ModelType-2 TSK Model Based on SubtractiveClusteringSensibility analysis for type-1 and type-2 TSK fuzzy modelsConclusions
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Conclusions(1)
It is recommended an enumerative search for parameters to get the optimal model.
arar
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Conclusions(3)
Type-2 TSK model
RMSE is very sensitive to spread percentage of cluster center;
Interval set of output is very sensitive to spread percentage of consequent parameters;
Smaller step sizes need to select for spread percentage of cluster center and consequent parameters.
arar
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Acknowledgment
Financial supporter:
Natural Sciences and Engineering Research Council of Canada
RGPIN-203618
RGPIN-105518
STPGP-269579
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Thank You!
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