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

IASTED MS2007 3

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

IASTED MS2007 4

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

IASTED MS2007 5

Differences between Type-1 and Type-2 TSK FLSs

* there are M rules and each rule has p antecedents

IASTED MS2007 6

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

IASTED MS2007 7

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.

IASTED MS2007 8

Subtractive Clustering Method [Chiu 1994]

arCluster radius

Squash factor

Reject ratio

Accept ratio

η

−ε−

ε

?

IASTED MS2007 9

Initialization of Parameters for Subtractive Clustering

IASTED MS2007 10

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

IASTED MS2007 11

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

IASTED MS2007 12

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

IASTED MS2007 13

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~

.

*

IASTED MS2007 14

Expanding a Type-1 TSK Model to a Type-2 TSK Model(3)

kth rule:

.

IASTED MS2007 15

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

IASTED MS2007 16

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

IASTED MS2007 17

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

IASTED MS2007 18

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

IASTED MS2007 19

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

IASTED MS2007 20

Sensibility Analysis for Type-1 TSK Fuzzy Model (2)

IASTED MS2007 21

Sensibility Analysis for Type-1 TSK Fuzzy Model (3)

IASTED MS2007 22

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

IASTED MS2007 23

Six Rule Type-1 TSK FLS

−

ε

ar

−ε

= 0.25

= 0.15

= 0.5

= 1.25η

RMSE is 0.093037

IASTED MS2007 24

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

IASTED MS2007 25

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

IASTED MS2007 26

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

IASTED MS2007 27

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

IASTED MS2007 28

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

IASTED MS2007 29

Six-Rules Type-2 TSK Model

RMSE is 0.081501

IASTED MS2007 30

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)

IASTED MS2007 31

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

IASTED MS2007 32

Conclusions(1)

It is recommended an enumerative search for parameters to get the optimal model.

arar

IASTED MS2007 34

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

IASTED MS2007 35

Acknowledgment

Financial supporter:

Natural Sciences and Engineering Research Council of Canada

RGPIN-203618

RGPIN-105518

STPGP-269579

IASTED MS2007 36

Thank You!

?