A New Extension of Fuzzy C-Means Algorithm using Non Euclidean Distance and Kernel

Methods BOUZBIDA Mohamed, TROUDI Ahmed HASSINE Lassad, CHAARI Abdelkader Higher School of Sciences and Techniques of Higher School of Sciences and Techniques of Tunis (ESSTT) Tunis (ESSTT)

Research unit (C3S) Research unit (C3S) Tunisia Tunisia

bouzbida_mohamed@hotmail.fr, troudi.ahmed@yahoo.fr houcinelassad@yahoo.fr , nabil.chaari@yahoo.fr

Abstract— most of processes in industry are characterized by nonlinear and time-varying behavior. Nonlinear system identification is becoming an important tool which can be used to improve control performance [9]. Among the different nonlinear identification techniques, the Takagi Sugeno fuzzy model has attracted most attention of several researches. In literature, several fuzzy clustering algorithms have been proposed to identify the parameters involved in the Takagi-Sugeno fuzzy model, as the Fuzzy C-Means algorithm (FCM) and Fuzzy C-Means algorithm using non-Euclidean distance (NFCM). This paper presents a new Clustering algorithm for Takagi-Sugeno fuzzy model identification. The proposed algorithm is an extension of the NFCM algorithm called New Extension of Fuzzy C-Means algorithm based on kernel method (KNFCM) and non-Euclidean distance, where the non-Euclidean distance using the Gaussian kernel function. The proposed algorithm (KNFCM) can solve the nonlinear separable problems found by FCM and NFCM. So the KNFCM algorithm is more robust than FCM and NFCM.

Keywords— nonlinear system, TS fuzzy model, Fuzzy identification, fuzzy clustering, non-Euclidean distance, Kernel methods.

I. INTRODUCTION Modeling and identification are important steps in the design of control system. Typical applications of these models are the simulation, the prediction or the control system design. Generally, modeling process consists to obtain a parametric model with the same dynamic behavior of the real process. However, when the process is nonlinear and complex, it is very difficult to define the different mathematical or physical laws which describe its behavior [3]. For nonlinear systems, the conventional techniques of modeling and identifications are difficult to implement and sometimes impracticable. However, others techniques based on fuzzy logic are more and more used for modeling this kind of process [2]. Among the different fuzzy modeling techniques, the fuzzy model suggested by Takagi and Sugeno (1985) drawn the attention of several research, this is to their effectiveness in the nonlinear system modeling.

In fact, this model consists of if-then rules with fuzzy antecedents and mathematical functions in the consequent part. The antecedent’s fuzzy sets divide the input space into a number of fuzzy regions, while the consequent functions describe the system’s behavior in these regions [7]. The identification problem consists of estimating the parameters of the model. This framework presents several techniques, which have been developed to identify the parameters involved in the Takagi-Sugeno fuzzy model; among them, we can mention the art of fuzzy clustering .The fuzzy clustering technique [12] [1] constitute one of the best approaches used for the representation of such process. Indeed, this technique is to approximate the nonlinear system overall by Takagi-Sugeno local linear models, in this case, each model represents by a fuzzy rule [5]. The number of rules (clusters) is fixed by an expert according to the type of application considered and the performances required by this last. Several clustering algorithms exist in literature allowing the identification of the parameters intervening in the TS fuzzy model. We can quote as an example the Fuzzy C-Mean algorithm (FCM) [4]. However, FCM is sensitive to noises. To resist the noises some fuzzy clustering algorithms have been proposed. A new fuzzy clustering algorithm, called fuzzy c-means algorithm using non-Euclidean distance (NFCM) [13], has been proposed by Wu and Zhou to deal with noisy data. However, NFCM can not separate clusters which are nonlinearly separable in input space and their boundaries between two clusters are linear. To solve the nonlinear separable problem, kernel methods are regarded as the way of dealing with this problem. In this context, we proposed the New Extension of Fuzzy C-Means algorithm based on kernel method (KNFCM) and non-Euclidean distance ; extension of the NFCM algorithm. The effectiveness of this algorithm (KNFCM) compared to the FCM and NFCM algorithms are tested on a noisy nonlinear system described by difference equation and application to an electro-hydraulic system. This paper is organized as follows: In second part of this work, we introduce the Takagi-Sugeno fuzzy model. The third part is devoted to identify the premise parameters of this model where we used the proposed

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KNFCM. The forth part is dedicated to identify the consequent parameters by the least square algorithm. The simulations results and the model validity of FCM, NFCM, KNFCM are presented in part five. Finally, we conclude this paper with a conclusion.

II. TAKAGI-SUGENO FUZZY MODEL Fuzzy identification is an effective tool for the approximation of uncertain nonlinear systems defined by the recurrent equation on the basis of measured data. The T-S fuzzy model is based on a set of rules in which the consequent use of numeric variable rather than linguistic variables such as the Mamdani model. The consequent can be expressed as a constant, a polynomial or differential equation depending on the antecedent variables. The T-S fuzzy model makes it possible to approximate the nonlinear system into several locally linear subsystems. [11] [12] In general, a Takagi-Sugeno fuzzy model is based on If ... Then rules of the form: 1 1 : is , , is then T

i k i nk in i i k iR if x A x A y a x b= +… (1) The ″if ″ rule function defines the premise part, while the ″ then ″ rule function constitutes the consequent part of the TS fuzzy model. Where

[ ] 1 ,i C∈ … iR : represents the ith rule

[ ]1 2, , ,Ti i i ina a a a= … : is the Parameters vector, such as

nia R∈ .

ib : It is a scalar.

[ ]1 2, , , Tk k k nkx x x x= … : Observations vector.

1 2, , ,i i i nA A A… : represents the fuzzy subsets. The estimated output of the nonlinear model is given by the following equation:

1

ˆ ( )C

i ii

y k yβ=

= ∑ (2)

As

1

1 1

( )( )

( )

n

ij jj

i nC

ij ji j

xk

x

μβ

μ

=

= =

=∏

∑∏ (3)

ijμ : Membership functions. The estimated output of the Takagi-Sugeno fuzzy model can be expressed by:

1

( ) c

Ti i iki

ky a x bβ∧

⎡ ⎤⎢ ⎥⎣ ⎦=

= +∑ (4)

III. IDENTIFICATION ALGORITHM FOR PREMISE PARAMETERS

To identify the premise parameters of a Takagi Sugeno fuzzy model described by equation (1), the Fuzzy c-means algorithm (FCM) and she extended which used.

A. Fuzzy C-means Algorithm ( FCM )

The Fuzzy c-means algorithm (FCM) , which uses Euclidean distance, finds the partition of the collection

of N measures,

specified by k-

dimensional vectors , into C fuzzy

subsets by minimizing the following objective function [4]:

2,

1 1

( , ) ( )C N

mFCM ik A ki

i k

J U V Dμ= =

=∑∑ (5)

Where 1 C N≤ ≤ : The number of clusters,

ikμ : The membership of xk in cluster i satisfying

0 1ikμ< < 1 i C≤ ≤ , 1 k N≤ ≤ (6)

11

C

iki

μ=

=∑

1 k N≤ ≤ (7)

1

0N

ikk

Nμ=

< <∑

1 i C≤ ≤ (8)

22

, ( ) ( )TA ik k i k i k iD x v x v A x v= − = − −

(9)

( )1 2, ,..., ,...,i cA A A A A= : The set of matrices define standards

induced positive.

V: the set of cluster centers (vi ∈ Rp ). m represent the degree of weighting, this parameter directly influences the form clusters in data space . To minimize equation (5), we take its partial derivative of variables,

ikμ

and

iv , equal to zero and obtain the following

equations:

121

,

1 ,

mCA ik

ikj A jk

DD

μ

−

−

=

⎡ ⎤⎛ ⎞⎢ ⎥= ⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

∑ (10)

1

1

( ),

( )

m

m

N

ik kk

i N

ikk

xv i

μ

μ

=

=

= ∀∑

∑ (11)

FCM algorithm

Given a set of observations }{ 1, NX x x= , the FCM algorithm is described by the following steps:

Initialize l = 0 Set the number of clusters C, 1 C N≤ ≤ Set the level of weighting m : 2< m <4 Set the stopping criterionε , 0ε > Repeat l = l +1 Step 1 Update the cluster centers by equation (11). Step 2 Update the membership matrix [ ]ikU μ= by equation (10)

( ) ( )NL ky k g x=

}{ 1, NpX x x= ⊂ ℜ

[ ]1 2, , , Ti k k ikx x x x= …

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If 1l lU U ε−− < , return to step 1, if not stop.

B. Fuzzy C-Means Algorithm based on Non- Euclidean Distance (NFCM)

The fuzzy c-means algorithm uses the Euclidean distance to calculate the fuzzy membership µik. However, in real world, the Euclidean distance is not complex enough to deal with more sophisticated problem. The new fuzzy c-means algorithm NFCM uses a non-Euclidean distance to replace the Euclidean distance

,A ik k iD x v= − used in FCM. The new distance is more robust to the noises than the Euclidean distance. It is defined as [13]:

2

, 1 exp( )A ik k iD x vρ= − − − (12) Here ρ is a positive constant. Considering equation (12), the objective function (5) is transformed into as follows

( ) ( )2

1 1( , ) 1 exp( )

C Nm

NFCM ik k ii k

J U V x vμ ρ= =

= − − −∑∑ (13)

The minimization of criterion (13) using the centers of the classes is obtained directly by canceling the gradient JNFCM with respect to different centers:

2

1

2

1

( ) exp( )

( ) exp( )

m

m

N

ik k i kk

i N

ik k ik

x v xv

x v

μ ρ

μ ρ

=

=

− −=

− −

∑

∑ (14)

Where 1 i C≤ ≤ , 1 k N≤ ≤

1112

21

1 exp( )

1 exp( )

mC

k iik

j k j

x v

x v

ρμ

ρ

−

−

=

⎡ ⎤⎛ ⎞⎢ ⎥− − −⎜ ⎟= ⎢ ⎥⎜ ⎟⎢ ⎥− − −⎝ ⎠⎢ ⎥⎣ ⎦

∑ (15)

NFCM algorithm Given a set of observations }{ 1, NX x x= , NFCM algorithm is described by the following steps: Initialize l = 0 Set the number of clusters c, 1 C N< < Set the level of weighting m : 2< m <4 Set the stopping criterionε , 0ε > Repeat l = l +1 Step 1 Update the cluster centers by equation (14). Step 2 Update the membership matrix [ ]ikU μ= by equation (15)

If 1l lU U ε−− < , return to step 1, if not stop.

C. New Extension of Fuzzy C-Means algorithm based on Non-Euclidean distance and Kernel method:

The NFCM can deal with noisy data better than FCM. However, FCM, NFCM have the same drawback that they can not separate clusters that are nonlinearly separable in input space and their boundaries between two clusters are linear. To solve the nonlinear separable problem and get nonlinear boundaries, kernel methods [10] are regarded as the way of dealing with this problem. I n this context, we proposed a new extension of Fuzzy C-Means algorithm based on kernel method and non-Euclidean distance (KNFCM). The present work proposes a way of increasing the accuracy of the FCM and NFCM algorithms by exploiting Gaussian Kernel function in calculating the distance of data point from the cluster centers mapping the data points from the input space to a feature space in which the distance is measured using a kernel function. A kernel function is a generalization of the distance metric that measures the distance between two data points as the data points are mapped into a future space in which the data are more clearly separable [14][8]. Define a nonlinear map as Φ : x → Φ(x)∈ F, where x∈ X , F the transformed feature space with higher or even infinite dimension .X denotes the data space mapped into F [10],[14]. The proposed algorithm (KNFCM) minimizes the following objective function:

( )2

1 1( , ) 1 exp( ( ) ( ) )

ik

C Nm

KNFCM k ii k

J U V x vμ ρ φ φ= =

= − − −∑∑

(16)

Here ρ is a positive constant.

Then x vik − is mapped into space F

( ) ( )x v x vi ik kφ φ− → −

Where

(17)

K(xk , vi ) = Φ(xk)T Φ(vi) is an inner product kernel function. If we adopt the Gaussian function as a kernel function, i.e.,

2

2( , ) exp where 02

x vikx vikK⎛ ⎞−⎜ ⎟= − σ >⎜ ⎟σ⎝ ⎠

(18)

Then K(xk, xk) = K(vi , vi) =1. Thus (17) can be written as:

(19)

Considering equation (19), the objective function (16) is transformed into as follows:

( )1 1

( , ) 1 exp( 2 (1 ( , )))ik

C Nm

KNFCMi k

ikJ U V K x vμ ρ= =

= − − −∑∑ (20)

2( ) ( ) ( ( ) ( )).( ( ) ( ))

( , ) ( , ) 2 ( , )

( ). ( ) ( ). ( ) 2 ( ). ( )

k k i i k i

x v x v x vi i ik k k

K x x K v v K x vi i ik k k

x x v v x v

φ φ φ φ φ φ

φ φ φ φ φ φ

− = − −

= + −

= + −

2( ) ( ) 2(1 ( , ))x v K x vi ik kφ φ− = −

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So,

2

21 1( , ) 1 exp 2 1 exp

2C N

i k

mKNFCM ik

x vikJ U V μ ρσ= =

⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= − − − −∑∑ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠

(21)

The derivation of the objective function (21) according to

and , define the relationship update of cluster centers

and membership coefficients : • Partial derivative of JKNFCM (U,V) with respect to

( , )KNFCM

i

J U Vv

∂=

∂

22

2 2 21

4 exp exp 2 1 exp ( )2

Nk im k i

ikk

x v x v x vikρ μ ρσ σ σ=

⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞− ⎛ ⎞− −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟− − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠

∑

(22) So,

( )( )( ) 21

( , )4 ( , ) exp 2 1 ( , ) ( )

NmKNFCM k i

ikki

i ik kJ UV x v

K x v K x vv

ρ μ ρσ=

∂ −= − −

∂ ∑ (23) Equating (23) to zero leads to:

( , )0KNFCM

i

J U Vv

∂=

∂ (24)

Then,

( )( )( )( )

1

1

( , ) exp 2 1 ( , )

( , ) exp 2 1 ( , )

Ni ik k kk

i Ni ik kk

mik

mik

K x v K x v xv

K x v K x v

μ ρ

μ ρ=

=

− −∑=

− −∑ (25)

• Partial derivative of JKNFCM (U,V) with respect to

Using the Lagrange multiplier, the relationship of updating fuzzy coefficients µik is obtained by minimizing the following criterion:

( )1 11 1

( , ) 1 exp( 2 (1 ( , ))) . 1

N C

KNFCM k ikk i

C Niki k

mikJ UV K x vμ ρ λ μ

= == =

⎛ ⎞⎛ ⎞= − − − − −∑∑ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∑ ∑

(26) As the columns of the partition matrix U are independent, we can reduce the problem of minimization at each observation (column).

1

( , )( 1) 0

CKNFCM

iki

J U V μλ =

∂= − − =

∂ ∑ (27)

( )1( , )( ) 1 exp( 2 (1 ( , ))) 0mKNFCM

ikik

ikJ U V

m K x vμ ρ λμ

−∂= − − − − =

∂

(28) From the expression (28), we can write ikμ this form:

( )

11

1 exp( 2 (1 ( , )))

m

ikikm K x v

λμρ

−⎛ ⎞⎜ ⎟=⎜ ⎟− − −⎝ ⎠

(29)

Substituting expression (29) in expression (27):

( )

11

1 1

( , , )1

1 exp( 2 (1 ( , )))

mC CKNFCM

jkj j ik

J U T Vm K x v

λμλ ρ

−

= =

⎛ ⎞∂ ⎜ ⎟= = =⎜ ⎟∂ − − −⎝ ⎠

∑ ∑

(30)

Is also

( )

11

11

1

1

11 exp( 2 (1 ( , )))

mk

mC

j jk

m

K x v

λ

ρ

−

−

=

⎛ ⎞ =⎜ ⎟⎝ ⎠ ⎛ ⎞

⎜ ⎟⎜ ⎟− − −⎝ ⎠

∑

(31)

The two expressions (29) and (31) give the following expression:

( )( )

11

1

1

1 exp( 2 (1 ( , )))

1 exp( 2 (1 ( , )))

ikmC

j j

ik

k

K x v

K x v

μρ

ρ

−

=

=⎛ ⎞− − −⎜ ⎟⎜ ⎟− − −⎝ ⎠

∑

(32)

Hence the relationship of updating the partition matrix:

( )( )

111

1

1 exp( 2 (1 ( , )))

1 exp( 2 (1 ( , )))

mC

ikj j

ik

k

K x v

K x v

ρμ

ρ

−

−

=

⎡ ⎤⎛ ⎞− − −⎢ ⎥⎜ ⎟= ⎢ ⎥⎜ ⎟− − −⎢ ⎥⎝ ⎠

⎣ ⎦

∑ (33)

KNFCM algorithm Given a set of observations }{ 1, NX x x= , KNFCM algorithm is described by the following steps: Initialize l = 0 Set the number of clusters c, 1 C N< < Set the level of weighting m : 2< m <4 Set the stopping criterionε , 0ε > Repeat l = l +1 Step 1 Update the cluster centers by equation (25). Step 2 Update the membership matrix [ ]ikU μ= by equation (33)

If 1l lU U ε−− < , return to step 1, if not stop.

ikμ

iv

iv

ikμ

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IV. IDENTIFICATION FOR CONSEQUENT PARAMETERS

The identification of consequent parameters ,Ta bi i iθ ⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= of

the Takagi-Sugeno model described by equation (1), we must build the matrix of regression X and the output vector Y following certain measures which are defined as follows:

1 2, , ,

TT T TNX X X X⎡ ⎤= ⎣ ⎦…

[ ]1 2, , , TNY y y y= …

Since the défuzzification method, used in the Takagi-Sugeno model, is linear with the consistent parameters ai and bi (1), we can use the least square algorithm for estimating these parameters. Either

;T T

i i ia bθ ⎡ ⎤= ⎣ ⎦ represents the vectors of parameters of

rule i, and the matrix [ ];1eX X= represents an extension of

X . Using the method of least square solution of equation given by the following equation:

(34)

Where, is a diagonal matrix of dimension (N × N) contains the coefficients of fuzzy memberships..

V. SIMULATION RESULTS AND VALIDATION MODEL

A. Simulation Results Example 1: Consider a nonlinear system described by the following difference equation [6]:

( ) ( )2 2

( 1) ( 2) 2 ( 1) 2.5( ) ( ) ( )

8.5 ( 1) ( 2)y k y k y k

y k u k e ky k y k

− − + − += + +

+ − + −

(35)

Where y(k), u(k) are the output and the input of the system respectively. e(k) is a noise. In this part, we have applied the three algorithms, FCM, NFCM, KNFCM and the least square method to identify the parameters of fuzzy models which approximate the nonlinear model (35). The shape of the excitation signal used for identification is illustrated in Figure1

Figure1. Input-output sequences

For another input-output sequence, the figures represent the simulation results for identification by the three algorithms (FCM, NFCM and KNFCM).

Figure2. Identification results for the FCM Algorithm

Figure3. Identification results for the NFCM algorithm

e iY X erθ= +1T T

i e i e e iX X X Yθ ψ ψ−

⎡ ⎤= ⎣ ⎦

iψikμ

0 50 100 150 200 250 300-4

-2

0

2

4Input signal

Sample

0 50 100 150 200 250 300-10

0

10Output signal

S l

0 50 100 150 200 250 300-5

0

5

10Real output (y) and estimated output (yest)

Sample

0 50 100 150 200 250 300-1

0

1x 10

-5 Error signal

Sample

yyest

0 50 100 150 200 250 300-5

0

5

10Real output (y) and estimated output (yest)

Sample

0 50 100 150 200 250 300-1

0

1x 10

-5 Error signal

Sample

yyest

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Figure 4. Identification results for the KNFCM

Algorithm

Example 2: application to an electro-hydraulic system. The effectiveness of the proposed algorithm in this paper (KNFCM) is tested on an electro-hydraulic system described by the schematic diagram in Figure 5.

Figure 5. Bloc diagram

1 : ultrasonic level sensor 2 : Tank 1 3 : Tank 2 4 : centrifugal pump 5 : DC motor 6 : Variable speed 7 : manual valve v1 8 : manual valve v2 9 : Pipe 1 10 : Pipe 2

qe: felling flow. qs: outgoing flow of tank 1. P1: pressure at the button of tank. P2: pressure at the button of tank 2. Pp: exit pressure of centrifugal pump. h1: water level in tank 1. h2: water level in tank 2.

ha: difference in altitude between the sites of the two tanks. u: supply voltage of the engine. To identify the parameters of this system, we applied a proposed clustering algorithm. The set of observations we have taken is illustrated in Fig. 6.

Figure 6. Sequences of input-output For another sequences of input-output, the simulation result given by the three algorithms are given as follows:

Figure 7. Identification results for the FCM algorithm

Figure 8. Identification results for the NFCM algorithm

0 50 100 150 200 250 300-5

0

5

10Real output (y) and estimated output (yest)

Sample

0 50 100 150 200 250 300-1

0

1x 10

-5 Error signal

Sample

yyest

0 50 100 150 200 250 300 350 4000

50

100input signal

sample

0 50 100 150 200 250 300 350 4000

100

200output signal

sample

0 50 100 150 200 250 300 350 4000

100

200Real output (y) and estimated output (yest)

Sample

0 50 100 150 200 250 300 350 400-5

0

5x 10

-5 Error signal

Sample

yyest

0 50 100 150 200 250 300 350 4000

100

200Real output (y) and estimated output (yest)

Sample

0 50 100 150 200 250 300 350 400-5

0

5x 10

-5 Error signal

Sample

yyest

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Figure 9. Identification results for the KNFCM algorithm

B. Model Validation

After applying the identification algorithm, it is necessary to validate the Takagi-Sugeno fuzzy model. Several validation tests of the model are bestowed. Among them, we use the Root Mean Square Error (RMSE) test and the Variance Accounting For (VAF) test.

• Root Mean Square Error (RMSE)

This test (RMSE) calculate the mean squared error between the estimated output and real output.

( )2

1

1 ( ) ( )N

estk

RMSE y k y kN =

= −∑ (36)

Where y is the real output yest is the estimated output If the model output and real output are combined, the RMSE equal zero.

• Variance Accounting For (VAF) The performance of the fuzzy model is measured by the VAF which calculates the percentage deviation of the variance between the real output and model output. It is defined by:

( )var100% 1

var( )esty y

VAFy

−⎡ ⎤= −⎢ ⎥

⎣ ⎦ (37)

The test (VAF) tends in 100% when the y and yest are combined.

TABLE I. VALID RESULTS (EXAMPLE 1)

ALGORITHM FCM NFCM KNFCM

RMSE (10-6) 4.126 2.542 2.007

VAF (%) 99.9430 99.9697 99.9968

TABLE II. VALID RESULTS(ELECTRO-HYDRAULIC SYSTEM)

ALGORITHM FCM NFCM KNFCM

RMSE (10-5) 1.561 0.722 0.248

VAF (%) 99.8684 99.9957 99.9992

VI. CONCLUSION In this paper, we proposed a new clustering algorithm (KNFCM) for the nonlinear systems identification. This algorithm is an extension of the Fuzzy C-Means algorithm based on non-Euclidean distance (NFCM) where we exploiting the kernel function in calculating the distance of data point from the cluster centers. The simulation results show that the proposed algorithm KNFCM can overcome the nonlinear separable problems found by the FCM and NFCM algorithms. The validation results RMSE and VAF show a better behavior of the KNFCM algorithm compared to the FCM and NFCM algorithms.

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[12] T. Ahmed, H. Lassad and C. Abdelkader. “Nonlinear system identification using clustering algorithm and Particle Swarm optimization, “ Scientific Research and Essays, vol. 7(13), pp. 1415-1431, 9 April 2012.

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[13] Xiao-Hong Wu,and Jian-Jiang Zhou . “Noise Clustering Using a New Distance.” IEEE Transactions on Systems 0-7803-9521, 2006.

[14] Xiao-Hong Wu ‘‘ A Possibilistic C-Means Clustering Algorithm Based on Kernel Methods’’Information Science & TechnologyNanjing University of Aeronautics and Astronautics Nanjing, 210016, China,2006.

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