research article universal approximation of a class of...

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2013 Article ID 136214 16 pageshttpdxdoiorg1011552013136214

Research ArticleUniversal Approximation of a Class of Interval Type-2 FuzzyNeural Networks in Nonlinear Identification

Oscar Castillo1 Juan R Castro2 Patricia Melin1 and Antonio Rodriguez-Diaz2

1 Tijuana Institute of Technology 22379 Tijuana BCN Mexico2 Baja California Autonomous University (UABC) 22379 Tijuana BCN Mexico

Correspondence should be addressed to Oscar Castillo ocastillohafsamxorg

Received 15 January 2013 Revised 20 June 2013 Accepted 20 June 2013

Academic Editor F Herrera

Copyright copy 2013 Oscar Castillo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Neural networks (NNs) type-1 fuzzy logic systems (T1FLSs) and interval type-2 fuzzy logic systems (IT2FLSs) have been shown tobe universal approximators whichmeans that they can approximate any nonlinear continuous function Recent research shows thatembedding an IT2FLS on an NN can be very effective for a wide number of nonlinear complex systems especially when handlingimperfect or incomplete information In this paper we show based on the Stone-Weierstrass theorem that an interval type-2 fuzzyneural network (IT2FNN) is a universal approximator which uses a set of rules and interval type-2membership functions (IT2MFs)for this purpose Simulation results of nonlinear function identification using the IT2FNN for one and three variables and for theMackey-Glass chaotic time series prediction are presented to illustrate the concept of universal approximation

1 Introduction

Several authors have contributed to universal approximationresults An overview can be found in [1ndash8] further referencesto prime contributors in function approximations by neuralnetworks are in [4 9ndash12] and type-2 fuzzy logic modelingin [13ndash23] It has been shown that a three-layer NN canapproximate any real continuous function [24]The same hasbeen shown for a T1FLS [1 25] using the Stone-Weierstrasstheorem [3] A similar analysis was made by Kosko [2 9]using the concept of fuzzy regions In [3 26] Buckley showsthat with a Sugeno model [27] a T1FLS can be built withthe ability to approximate any nonlinear continuous functionAlso combining the neural and fuzzy logic paradigms [2829] an effective tool can be created for approximatingany nonlinear function [4] In this sense an expert canuse a type-1 fuzzy neural network (T1FNN) [10ndash12 30] orIT2FNN systems and find interpretable solutions [15ndash1731ndash34] In general Takagi-Sugeno-Kang (TSK) T1FLSs areable to approximate by the use of polynomial consequentrules [7 27] This paper uses the Levenberg-Marquardtbackpropagation learning algorithm for adapting antecedentand consequent parameters for an adaptive IT2FNN since

its efficiency and soundness characteristics make them fit forthese optimizing problems An Adaptive IT2FNN is used as auniversal approximator of any nonlinear functions A setΨ ofIT2FNNs is universal if and only if (iff) given any processΩthere is a system Φ isin Ψ such that the difference between theoutput from IT2FNN and output from Ω is less than a given120576

In this paper the main contribution is the proposedIT2FNNs architectures which are shown to be universalapproximators and are illustrated with several benchmarkproblems to verify their applicability for real world problems

2 Interval Type-2 Fuzzy Neural Networks

An IT2FNN [15 31 35] combines a TSK interval type-2fuzzy inference system (TSKIT2FIS) [13 14 33 34] withan adaptive NN in order to take advantage of both modelsbest characteristics In general when representing IT2FNNgraphically rectangles are used to represent adaptive nodesand circles to represent nonadaptive nodes Output values ofpair nodes (green color) and odd nodes (blue color) representuncertainty intervals (Figures 1ndash4) In this kind of interval

2 Advances in Fuzzy Systems

x1

x1x2

x2

120583

120583

120583

120583

120583

120583

120583

120583

|x1||x2|

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

f1

f1

f2

f2

f3

f3

f4

f4

y1l

y1r

y2l

y2r

y3l

y3r

y4l

y4r

yr

yl

12

T

T

T

T

T

T

T

T

12

sum y

Layer 0 Layer 1 Layer 2 Layer 3 Layer 4 Layer 5

Figure 1 IT2FNN-1 architecture

type-2 neurofuzzy adaptive networks nodes represent pro-cessing units called neurons which can be classified into crispand fuzzy neurons

The IT2FNN-1 architecture has 5 layers (Figure 1) [35]and consists of adaptive nodes with equivalent functionto lower-upper membership in fuzzification layer (layer 1)Nonadaptive nodes in rules layer (layer 2) interconnect withthe fuzzification layer (layer 1) in order to generate TSKIT2FIS rules antecedents Adaptive nodes in the consequentlayer (layer 3) are connected to the input layer (layer 0)to generate rules consequents Nonadaptive nodes in type-reduction layer (layer 4) evaluate left-right values with theKarnik andMendel (KM) [13 14] algorithmThe nonadaptivenodes in the defuzzification layer (layer 5) average left-rightvalues

The IT2FNN-3 architecture has 8 layers (Figure 2) [35]and uses IT2FN for fuzzifying the inputs (layers 1-2) Non-adaptive nodes in rules layer (layer 3) interconnect withlower-upper linguistic values layer (layer 2) to generate TSKIT2FIS rules antecedents Adaptive nodes in layer 4 adapt left-right firing strength biasing rules lower-upper trigger forceswith synaptic weights between layers 3 and 4 Layer 5rsquos non-adaptive nodes normalize rules lower-upper firing strengthNonadaptive nodes I consequent layer (layer 6) interconnectwith input layer (layer 0) to generate rules consequentsNonadaptive nodes in type-reduction layer (layer 7) evaluate

left-right values adding lower-upper product of lower-uppertriggering forces normalized by rules consequent left-rightvalues Node in defuzzification layer is adaptive and itsoutput 119910 is defined as biased average of left-right values andparameter 120574 Parameter 120574 (05 by default) adjusts uncertaintyinterval defined by left-right values [119910119897 119910119903]

Architectures IT2FNN-0 and IT2FNN-2 which will beshown in Sections 32 and 33 respectively as universalapproximators are described withmore details in Section 21

21 IT2FNN-0 Architecture An IT2FNN-0 is a seven-layerIT2FNN which integrates a first order TSKIT2FIS (intervaltype-2 fuzzy antecedents and real consequents) with anadaptive NN The IT2FNN-0 (Figure 3) layers are describedas follows

Layer 0 Inputs

1199000

119894= 119909119894 119894 = 1 119899 (1)

Layer 1 Adaptive type-1 fuzzy neuron (T1FN)

net1119896= 11990810

119896119894119909119894 + 119887

1

119896forall119894 = 1 119899 119896 = 1 120599 (2)

1199001

119896= 120583(net1

119896) where the transfer function 120583 is a member-

ship function net1119896is the weighted sum of inputs (119909119894) and

Advances in Fuzzy Systems 3

x1

x1

x2

x2

sum

sum

sum

sum

sum

sum

sum

sum

sum

sum

sum

y

|x1||x2|

Layer 0 Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Layer 6 Layer 7 Layer 8

120583

120583

120583

120583

120583

120583

120583

120583

T T

S T

T T

S T

T

S

T T

S T

T

T

w11

w21

w31

w41

w52

w62

w72

w82

1

1

1

1

1

1

1

1

b1

net1

b2

b3

net2

b4

net4

net3

b5

b6

net5

b7

net7

net6

b8

net8

120583(net1)

120583(net2)

120583(net3)

120583(net4)

120583(net5)

120583(net6)

120583(net7)

120583(net8)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

f1

f2

f3

f4

f1

f2

f3

f4

f1l

f1r

f2l

f2r

f3l

f3r

f4l

f4r

w1

w1

w2

w2

w3

w3

w4

w4

y1l

yl

y1r

y2l

y2r

y3l

y3r

y4l

y4r

yr

120574

1minus 120574


the synaptic weights (11990810119896119894) and 119887

1

119896is the threshold for each

neuron

Layer 2 Nonadaptive T1FN This layer contains T-norm andS-norm fuzzy nodes

1199002

2119896minus1= 1199001

2119896minus1sdot 1199001

2119896forall119896 = 1 120599 T-norm fuzzy node

(3)

where 120599 is the number of nodes in layers 1 and 2

1199002

2119896= 1199001

2119896minus1+ 1199001

2119896minus 1199002

2119896minus1 S-norm fuzzy node (4)

120591 = ℓ119896119894 for all 119896 = 1 119872 and 119894 = 1 119899 where ℓ119896119894 is thetable of indices of the antecedents of the rules120587 = sum

119894minus1

119895=1V119895+|120591|

where 120587 is a vector of indices for each node of layer 2

if 120591 gt 0

120583119896

119894= 1199002

119894119897(120587) 120583

119896

119894= 1199002

119894119906(120587)(5)

else120583119896119894

= null 120583119896119894

= null (6)

end

where 120583119896

119894 120583119896

119894are lower and upper membership function val-

ues respectively 119894119897(120587) and 119894119906(120587) are vectors with even and oddindices of the nodes of layer 2

Layer 3 Lower-upper firing strength (119908119896 119908119896) Having non-

adaptive nodes for generating lower-upper firing strength ofTSK IT2FIS rules (7)

1199003

2119896minus1= 119908119896 119900

3

2119896= 119908119896

119908119896=

119899

prod

119894=1

120583119865119896119894

(119909)

119908119896=

119899

prod

119894=1

120583119865119896119894

(119909)

(7)

where 120583119865119896119894(119909)

isin [120583119865119896119894

(119909) 120583119865119896119894

(119909)] is the Gaussian interval

type-2 membership function igaussmtype2 (119909 [120590119896

1198941119898119896

119894

2119898119896

119894]) defined by

1120583119865119896119894

(119909119894 [120590119896

1198941119898119896

119894]) = exp[

[

minus1

2(119909119894 minus1119898119896

119894

120590119896

119894

)

2

]

]

(8)


Layer 0 Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Layer 6 Layer 7

x1

x1x2

x2

sum

sum

sum

sum

sum

sum

sum

sum

sum

sum

sum

y

yl

yr

120574

1minus 120574

120583

120583

120583

120583

120583

120583

120583

120583

w11

w41

w52

w82

w62

w72

w21

w31

1b1

net1

1b2

net2

1b3

net3

1b4

net4

1b5

net5

1b6

net6

1b7

net7

1b8

net8

120583(net1) 120583(x)

120583(net2) 120583(x)

120583(net3) 120583(x)

120583(net4) 120583(x)

120583(net5) 120583(x)

120583(net6) 120583(x)

120583(net7) 120583(x)

120583(net8) 120583(x)

TT

TS

TT

TS

TT

TS

TT

TS

f1

f1

f2

f2

f3

f3

f4

f4

w1

w1

w2

w2

w3

w3

w4

w4

y1l

y1r

y2l

y2r

y3l

y3r

y4l

y4r

w1y1

w1y1

w2y2

w2y2

w3y3

w3y3

w4y4

w4y4


2120583119865119896119894

(119909119894 [120590119896

1198942119898119896

119894]) = exp[

[

minus1

2(119909119894 minus2119898119896

119894

120590119896

119894

)

2

]

]

(9)

120583119865119896119894

(119909) =

2120583119865119896119894

(119909119894 [120590119896

1198942119898119896

119894]) 119909119894 le

1119898119896

119894

1120583119865119896119894

(119909119894 [120590119896

1198941119898119896

119894]) 119909119894 gt

2119898119896

119894

(10)

120583119865119896119894

(119909) =

1120583119865119896119894

(119909119894 [120590119896

1198941119898119896

119894]) 119909119894 lt

1119898119896

119894

11119898119896

119894le 119909119894 le

2119898119896

119894

2120583119865119896119894

(119909119894 [120590119896

1198942119898119896

119894]) 119909119894 gt

2119898119896

119894

(11)

Layer 4 Lower-upper firing strength rule normalization(120601119896 120601119896

) Nodes in this layer are nonadaptive and the outputis defined as the ratio between the 119896th lower-upper firingstrength rule (119908

119896 119908119896) and the sum of lower-upper firing

strength of all rules (13) and (14)

1199004

2119896minus1= 120601119896 119900

4

2119896= 120601119896

(12)

If we view 120601119896 120601119896

as fuzzy basis functions (FBF) (32) and (33)and 119910

119896(119909) as linear function (16) then 119910(119909) can be viewed as

a linear combination of the basis functions (20) and (21)

120601119896=

119908119896

119863119897

119896 = 1 119872 (13)

where119863119897 = sum119872

119896=1119908119896

120601119896

=119908119896

119863119903

119896 = 1 119872 (14)

where119863119903 = sum119872

119896=1119908119896

Layer 5 Rule consequents Each node is adaptive and itsparameters are 119888

119896

119894 119888119896

0 The nodersquos output corresponds to

partial output of 119896th rule 119910119896 (16)

1199005

2119896=1= 119910119896 119900

5

2119896= 119910119896 (15)

119910119896=

119899

sum

119894=1

119888119896

119894119909119894 + 119888119896

0 119896 = 1 119872 (16)


x1

x1x2

x2

|x1||x2|

Layer 0 Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Layer 6

sum

sum

sum

sum

sum

sum

sum

sum

sum

y

120583

120583

120583

120583

120583

120583

120583

120583

T T

S T

KMT T

S T

T T

S T

T

S T

T

KM

y1l

y1r

y2l

y2r

y3l

y3r

y4l

y4r

yl

yr

12

12

1b1

net1

1b2

net2

1b3

net3

1b4

net4

1b5

net5

1b6

net6

1b7

net7

1b8

net8

w11

w41

w52

w82

w62

w72

w21

w31

120583(net1)

120583(net2)

120583(x)

120583(x)

120583(net3)

120583(net4)

120583(x)

120583(x)

120583(net5)

120583(net6)

120583(x)

120583(net7) 120583(x)

120583(x)

120583(net8) 120583(x)

f1

f1

f2

f2

f3

f3

f4

f4


Layer 6 Estimating left-right interval values [119910119897 119910119903] (18)nodes are nonadaptive with outputs 119910119897 119910119903 Layer 6 output isdefined by

1199006

1= 119910119897 (119909) 119900

6

2= 119910119903 (119909) (17)

where

119910119897 (119909) =

119872

sum

119896=1

120601119896119910119896

119910119903 (119909) =

119872

sum

119896=1

120601119896

119910119896

(18)

Layer 7 Defuzzification This layerrsquos node is adaptive wherethe output 119910 (20) and (21) is defined as weighted averageof left-right values and parameter 120574 Parameter 120574 (default

value 05) adjusts the uncertainty interval defined by left-rightvalues [119910119897 119910119903]

1199007

1= 119910 (119909) (19)

where

119910 (119909) = 120574119910119897 (119909) + (1 minus 120574) 119910119903 (119909) (20)

119910 (119909) =

119872

sum

119896=1

[120574120601119896(119909) + (1 minus 120574) 120601

119896

(119909)] 119910119896(119909) (21)

22 IT2FNN-2 Architecture An IT2FNN-2 [31] is a six-layer IT2FNN which integrates a first order TSKIT2FIS(interval type-2 fuzzy antecedents and interval type-1fuzzy consequents) with an adaptive NN The IT2FNN-2(Figure 4) layers are described in a similar way to the previousarchitectures


3 IT2FNN as a Universal Approximator

Based on the description of the interval type-2 fuzzy neuralnetworks it is possible to prove that under certain conditionsthe resulting IT2FIS has unlimited approximation power tomatch any nonlinear functions on a compact set [36 37] usingthe Stone-Weierstrass theorem [5 6 10 30]

31 Stone-Weierstrass Theorem

Theorem 1 (Stone-Weierstrass theorem) Let119885 be a set of realcontinuous functions on a compact set119880 If (1) 119885 is an algebrathat is the set 119885 is closed under addition multiplication andscalar multiplication (2) 119885 separates points on 119880 that is forevery x y isin 119880 x = y there exists 119891 isin 119885 such that 119891(x) = 119891(y)and (3) 119885 vanishes at no point of119880 that is for each119909 isin 119880 thereexists 119891 isin 119885 such that 119891(x) = 0 then the uniform closure of 119885consists of all real continuous functions on 119880 that is (119885 119889infin)

is dense in (119862[119880] 119889infin) [36ndash38]

Theorem 2 (universal approximation theorem) For anygiven real continuous function 119892(119906) on the compact set119880 sub 119877

119899

and arbitrary 120576 gt 0 there exists119891 isin 119884 such that sup119909isin119880

(|119892(119909)minus

119891(119909)|) lt 120576

32 Applying Stone-Weierstrass Theorem to the IT2FNN-0Architecture In the IT2FNN-0 the domain on which weoperate is almost always compact It is a standard result in realanalysis that every closed and bounded set inR119899 is compactNow we shall apply the Stone-Weierstrass theorem to showthe representational power of IT2FNN with simplified fuzzyif-then rules We now consider a subset of the IT2FNN-0on Figure 5 The set of IT2FNN-0 with singleton fuzzifierproduct inference center of sets type reduction andGaussianinterval type-2 membership function consists of all FBFexpansion functions of the form (38) (40) 119891 119880 sub

119877119899

rarr 119877 119909 = (1199091 1199092 119909119899) isin 119880 120583119865119896119894(119909)

isin [120583119865119896119894

(119909)

120583119865119896119894

(119909)] is the Gaussian interval type-2membership functionigaussmtype2 (119909 [120590

119896

1198941119898119896

1198942119898119896

119894]) defined by (27) and (31) If

we view 120601119896(119909) 120601119896

(119909) as fuzzy basis functions (32) and (33)and 119910

119896(119909) are linear functions (34) then 119910(119909) of (38) and

(40) can be viewed as a linear combination of the fuzzy basisfunctions and then the IT2FNN-0 system is equivalent to anFBF expansion Let 119884 be the set of all the FBF expansions(38) and (40) with 120601

119896(119909) 120601119896

(119909) given by (13) and (38) and let119889infin(1198911 1198912) = sup

119909isin119880(|1198911(119909) minus 1198912(119909)|) be the supmetric then

(119884 119889infin) is a metric space [38] We use the following Stone-Weierstrass theorem to prove our result

Suppose we have two IT2FNN-0s 1198911 1198912 isin 119884 the outputof each system can be expressed as

1198911 (119909) = 1205721199101

119897(119909) + (1 minus 120572) 119910

1

119903(119909) (22)

where

1199101

119897(119909) =

119872

sum

119896=1

120601119896

1(119909) 119911119896

1(119909) =

sum1198721

119896=1119908119896

1(119909) 119911119896

1(119909)

1198631

119897

1199101

119903(119909) =

1198721

sum

119896=1

120601119896

1(119909) 119911119896

1(119909) =

sum1198721

119896=1119908119896

1(119909) 119911119896

1(119909)

1198631119903

(23)

where

1198631

119897=

1198721

sum

119896=1

119899

prod

119894=1

1205831119865119896119894

(119909) 1198631

119903=

1198721

sum

1198961=1

119899

prod

119894=1

120583 1119865119896119894

(119909)

119908119896

1=

119899

prod

119894=1

1205831119865119896119894

(119909) 119908119896

1=

119899

prod

119894=1

120583 1119865119896119894

(119909)

120601119896

1(119909) =

119908119896

1(119909)

1198631

119897

120601119896

1(119909) =

119908119896

1

1198631119903

119911119896

1(119909) =

119899

sum

119894=1

1119888119896

119894119909119894 +1119888119896

0 119896 = 1 119872

(24)

1198912 (119909) = 1205741199102

119897(119909) + (1 minus 120574) 119910

2

119903(119909) (25)

where

1199102

119897(119909) =

119872

sum

119896=1

120601119896

2(119909) 119911119896

2(119909) =

sum1198722

119896=1119908119896

2(119909) 119911119896

2(119909)

1198632

119897

(26)

1199102

119903(119909) =

1198722

sum

119896=1

120601119896

2(119909) 119911119896

2(119909) =

sum1198721

119896=1119908119896

2(119909) 119911119896

2(119909)

1198632119903

(27)

where

119908119896

2=

119899

prod

119894=1

1205832119865119896119894

(119909) 119908119896

2=

119899

prod

119894=1

120583 2119865119896119894

(119909)

1198632

119897=

1198722

sum

119896=1

119899

prod

119894=1

1205832119865119896119894

(119909) 1198632

119903=

1198722

sum

119896=1

119899

prod

119894=1

120583 2119865119896119894

(119909)

120601119896

2(119909) =

119908119896

2(119909)

1198632

119897

120601119896

2(119909) =

119908119896

2(119909)

1198632119903

119911119896

2(119909) =

119899

sum

119894=1

2119888119896

119894119909119894 +2119888119896

0 119896 = 1 119872

(28)

Lemma 3 119884 is closed under addition

Proof The proof of this lemma requires our IT2FNN-0 tobe able to approximate sums of functions Suppose we have


12

1

08

06

04

02

0

Small Medium Large

minus10 minus5 0 5 10

Mem

bers

hip

grad

es

X

(a) Antecedent IT2MFs for fuzzy rules

minus10 minus5 0 5 10X

Z(X

)

z1

z3

z2

8

7

6

5

4

3

2

1

0

(b) Overall IO curve for crisp rules

1

08

06

04

02

0

Small Medium Large


120006|U

0 5 0 5 11100

(c) An example of the interval type-2 fuzzy basis functions


8

7

6

5

4

3

2

1

0

f|r(X

)

(d) Overall IO curve for IT2FNN-0

Figure 5 An example of the IT2FNN-0 architecture

two IT2FNN-0s 1198911(119909) and 1198912(119909) with 1198721 and 1198722 rulesrespectively The output of each system can be expressed as

1198911 (119909) + 1198912 (119909)

=

sum1198721

1198961=1sum1198722

1198962=1[1199081198961

11199081198962

2(1205721199111198961

1+ 1205741199111198962

2)]

1198631

1198971198632

119897

+

sum1198721

1198961=1sum1198722

1198962=1[1199081198961

11199081198962

2(1 minus 120572) 119911

1198961

1+ (1 minus 120574) 119911

1198962

2]

11986311199031198632119903

(29)

and that Φ1198961 1198962

= (1199081198961

11199081198962

2)(1198631

1199031198632

119903) and Φ1198961 1198962

= (1199081198961

11199081198962

2)

(1198631

1199031198632

119903) where the FBFs are known to be nonlinear There-

fore an equivalent to IT2FNN-0 can be constructed underthe addition of 1198911(119909) and 1198912(119909) where the consequents forman addition of 1205721199111198961

1+1205741199111198962

2and (1minus120572)119911

1198961

1+(1minus120574)119911

1198962

2multiplied

by a respective FBFs expansion (Theorem 1) and there exists119891 isin 119884 such that sup

119909isin119880(|119892(119909) minus 119891(119909)|) lt 120576 (Theorem 2)

Since 119891(119909) satisfies Lemma 3 and 119884 isin 119891(119909) = 1198911(119909) + 1198912(119909)

then we can conclude that 119884 is closed under addition Notethat 1199111198961

1and 119911

1198962

2can be linear since the FBFs are a nonlinear

basis interval and therefore the resultant function 119891(119909) isnonlinear interval (see Figure 5)

Lemma 4 119884 is closed under multiplication

Proof Similar to Lemma 3 we model the product of1198911(119909)1198912(119909) of two IT2FNN-0s The product 1198911(119909)1198912(119909) canbe expressed as

1198911 (119909) 1198912 (119909)

=1

1198631

1198971198632

119897

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

21205721205741199111198961

11199111198962

2]

]

+1

1198631

1198971198632119903

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2120572 (1 minus 120574) 119911

1198961

11199111198962

2]

]

+1

11986311199031198632

119897

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2(1 minus 120572) 120574119911

1198961

11199111198962

2]

]

+1

11986311199031198632119903

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2(1 minus 120572) (1 minus 120574) 119911

1198961

11199111198962

2]

]

(30)


Therefore an equivalent to IT2FNN-0 can be constructedunder the multiplication of 1198911(119909) and 1198912(119909) where theconsequents form an addition of 1205721205741199111198961

11199111198962

2 120572(1minus120574)119911

1198961

11199111198962

2 (1minus

120572)1205741199111198961

11199111198962

2 and (1 minus 120572)(1 minus 120574)119911

1198961

11199111198962

2multiplied by a respective

FBFs (Φ11989711989711989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119897)Φ11989711990311989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119903)

Φ119903119897

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119897) and Φ

119903119903

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119903))

expansion (Theorem 1) and there exists 119891 isin 119884 such thatsup119909isin119880

(|119892(119909) minus 119891(119909)|) lt 120576 (Theorem 2) Since 119891(119909) satisfiesLemma 3 and 119884 isin 119891(119909) = 1198911(119909)1198912(119909) then we can concludethat 119884 is closed under multiplication Note that 1199111198961

1and 119911

1198962

2

can be linear since the FBFs are a nonlinear basis interval andtherefore the resultant function 119891(119909) is nonlinear intervalAlso even if 1199111198961

1and 119911

1198962

2were linear their product 1199111198961

11199111198962

2is

evidently polynomial (see Figure 5)

Lemma 5 119884 is closed under scalar multiplication

Proof Let an arbitrary IT2FNN-0 be 119891(119909) (20) the scalarmultiplication of 119888119891(119909) can be expressed as

119888119891 (119909) = 120572119888119910119897 (119909) + (1 minus 120572) 119888119910119903 (119909)

=

sum119872

119896=1[120572119863119903119908

119896(119909) + (1 minus 120572)119863119897119908

119896(119909)] 119888119911

119896(119909)

119863119897119863119903

(31)

Therefore we can construct an IT2FNN-0 that computes119888119911119896(119909) in the form of the proposed IT2FNN-0 119884 is closed

under scalar multiplication

Lemma 6 For every x0 y0 isin 119880 and x0 = y0 there exists119891 isin 119884

such that 119891(x0) = 119891(y0) that is 119884 separates points on 119880

Proof Weprove that (119884 119889infin) separates points on119880 We provethis by constructing a required 119891(119909) (20) that is we specify119891 isin 119884 such that 119891(x0) = 119891(y0) for arbitrarily given (x0 y0) isin

119880 with x0 = y0 We choose two fuzzy rules in the form of (8)for the fuzzy rule base (ie 119872 = 2) Let 1199090 = (119909

0

1 1199090

2 119909

0

119899)

and 1199100= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2 and 119910

0

119894= (1199100

119897119894+

1199100

119903119894)2 with 119909

0

119894= 1199100

119894 we define two interval type-2 fuzzy sets

(1198651

119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(32)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(33)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(34)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(35)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100

119894) that is only one interval type-2 fuzzy set is defined

We define two real value sets 1199111 and 119911

2 with 119911119896(119909) =

sum119899

119894=1119888119896

119894119909119894 + 119888

119896

0 where 119896 = 1 2 Now we have specified all the

design parameters except 119911119896 that is we have already obtaineda function 119891 which is in the form of (10) with 119872 = 2 and(1198651

119894 [1205831198651119894

1205831198651119894

]) given by (18) (20) and (21) With this 119891 wehave

119891 (1199090) = 120572 [120601

1(1199090) 1199111(1199090) + 1206012(1199090) 1199112(1199090)] + (1 minus 120572)

times [1206011

(1199090) 1199111(1199090) + (1 minus 120601

1

(1199090)) 1199112(1199090)]

(36)

where

1206011(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206011

(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

(37)

119891 (1199100) = 120572 [120601

1(1199100) 1199111(1199100) + 1206012(1199100) 1199112(1199100)] + (1 minus 120572)

times [(1 minus 1206012

(1199100)) 1199111(1199100) + 1206012

(1199100) 1199112(1199100)]

(38)

where

1206011(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1

(39)

Since x0 = y0 there must be some 119894 such that 1199090119894= 1199100

119894 hence

we have prod119899

119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090

119894) = 1 If we choose


1199111= 0 and 119911

2= 1 then119891(119909

0) = 120572120601

2(1199090)+(1minus120572)[1minus120601

1

(1199090)] =

1205721206012(1199100) + (1 minus 120572)120601

2

(1199100) = 119891(119910

0) Separability is satisfied

whenever an IT2FNN-0 can compute strictly monotonicfunctions of each input variable This can easily be achievedby adjusting the membership functions of the premise partTherefore (119884 119889infin) separates points on 119880

Lemma7 For each119909 isin 119880 there exists119891 isin 119884 such that119891(119909) =

0 that is 119884 vanishes at no point of 119880

Finally we prove that (119884 119889infin) vanishes at no point of 119880By observing (8)ndash(11) (20) and (21) we simply choose all119910119896(119909) gt 0 (119896 = 1 2 119872) that is any 119891 isin 119884 with 119910

119896(119909) gt 0

serves as the required 119891

Proof of Theorem 2 From (20) and (21) it is evident that119884 is a set of real continuous functions on 119880 which areestablished by using complete interval type-2 fuzzy sets inthe IF parts of fuzzy rules Using Lemmas 3 4 and 5 119884is proved to be an algebra By using the Stone-Weierstrasstheorem together with Lemmas 6 and 7 we establish that theproposed IT2FNN-0 possesses the universal approximationcapability

33 Applying the Stone-Weierstrass Theorem to the IT2FNN-2Architecture We now consider a subset of the IT2FNN-2 onFigure 2 The set of IT2FNN-2 with singleton fuzzifier prod-uct inference type-reduction defuzzifier (KM) [13 14] andGaussian interval type-2 membership function consists of allFBF expansion functions 119891 119880 sub 119877

119899rarr 119877 119909 = (1199091 1199092

119909119899) isin 119880 120583119865119896119894(119909)

isin [120583119865119896119894

(119909) 120583119865119896119894

(119909)] is the Gaussian inter-

val type-2 membership function igaussmtype2 (119909 [120590119896

1198941119898119896

119894

1119898119896

119894]) defined by (8)ndash(11) If we view 120601

119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909)

120601119896

119903(119909) as basis functions (44) (46) (49) (50) and 119910

119896

119897 119910119896

119903are

linear functions (41) then 119910(119909) can be viewed as a linearcombination of the basis functions Let 119884 be the set of allthe FBF expansions with 120601

119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909) 120601

119896

119903(119909) and let

119889infin(1198911 1198912) = sup119909isin119880

(|1198911(119909) minus 1198912(119909)|) be the supmetric then(119884 119889infin) is a metric space [38] The following theorem showsthat (119884 119889infin) is dense in (119862[119880] 119889infin) where119862[119880] is the set of allreal continuous functions defined on119880 We use the followingStone-Weierstrass theorem to prove the theorem


1198911 (119909) = 1205741199101

119897(119909) + (1 minus 120574) 119910

1

119903(119909) (40)

where

1199101

119897(119909) =

1198711

sum

1198961=1

11206011198961

119897(119909)11199111198961

119897(119909) +

1198721

sum

1198961=1198711+1

11206011198961

119897(119909)11199111198961

119897(119909)

=

sum1198711

1198961=11199081198961

1(119909)11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909)11199111198961

119897(119909)

1198631

119897

1199101

119903(119909) =

1198771

sum

1198961=1

11206011198961

119903(119909)11199111198961119903

(119909) +

1198721

sum

1198961=1198771+1

11206011198961

119903(119909)11199111198961119903

(119909)

=

sum1198771

1198961=11199081198961

1(119909)11199111198961119903

(119909) + sum1198721

1198961=1198771+11199081198961

1(119909)11199111198961119903

(119909)

1198631119903

(41)

where

1199081198961

1=

119899

prod

119894=1

12058311198651198961

119894

(119909) 1199081198961

1=

119899

prod

119894=1

120583 11198651198961

119894

(119909)

1198631

119897=

1198711

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198711+1

1199081198961

1

1198631

119903=

1198771

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198771+1

1199081198961

1

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1 1198711 (119909)

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1198711 (119909) + 1 1198721

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1 1198771 (119909)

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1198771 (119909) + 1 1198721

11199111198961

119897=

119899

sum

119894=1

11198881198961

119894119909119894 +1119888119896

0minus

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus11199041198961

0

1198961 = 1 1198721

11199111198961119903

=

119899

sum

119894=1

11198881198961

119894119909119894 +11198881198961

0+

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 +11199041198961

0

1198961 = 1 1198721

1198912 (119909) = 1205741199102

119897(119909) + (1 minus 120574) 119910

2

119903(119909)

(42)

where

1199102

119897(119909)

=

1198712

sum

1198962=1

21206011198962

119897(119909)21199111198962

119897(119909) +

1198722

sum

1198962=1198712+1

21206011198962

119897(119909)21199111198962

119897(119909)

=

sum1198712

1198962=11199081198962

2(119909)21199111198962

119897(119909) + sum

1198722

1198962=1198712+11199081198962

2(119909)21199111198962

119897(119909)

1198632

119897

(43)


1199102

119903(119909)

=

1198772

sum

1198962=1

21206011198962

119903(119909)21199111198962119903

(119909) +

1198722

sum

1198962=1198772+1

21206011198962

119903(119909)21199111198962119903

(119909)

=

sum11987721

1198962=11199081198962

2(119909)21199111198962119903

(119909) + sum1198722

1198962=1198772+11199081198962

2(119909)21199111198962119903

(119909)

1198631119903

(44)

where

1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909) 1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909)

1198632

119897=

1198712

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198712+1

1199081198962

2

1198632

119903=

1198772

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198772+1

1199081198962

2

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1 1198712 (119909)

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1198712 (119909) + 1 1198722

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1 1198772 (119909)

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1198772 (119909) + 1 1198722

21199111198962

119897=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0minus

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus21199041198962

0

1198962 = 1 1198722

21199111198962119903

=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0+

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 +21199041198962

0

1198962 = 1 1198722

(45)


Proof The proof of this lemma requires our IT2FNN-2 tobe able to approximate sums of functions Suppose we havetwo IT2FNN-2s 1198911(119909) and 1198912(119909) with rules 1198721 and 1198722respectively The output of each system can be expressed as

1198911 (119909) + 1198912 (119909)

= ((

1198711

sum

1198961=1

1198712

sum

1198962=1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909) + 120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909)+120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198771

sum

1198961=1

1198772

sum

1198962=1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])times (1198631

1199031198632

119903)minus1

)

+ ((

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])

times (1198631

1199031198632

119903)minus1

)

(46)Therefore an equivalent to IT2FNN-2 can be constructedunder the addition of1198911(119909) and1198912(119909) where the consequentsform an addition of 120572 11199111198961

119897+12057421199111198962

119897and (1minus120572)

11199111198961119903+(1minus120574)

21199111198962119903

multiplied by a respective FBFs expansion (Theorem 1) andthere exists 119891 isin 119884 such that sup

119909isin119880(|119892(119909) minus 119891(119909)|) lt 120576

(Theorem 2) Since 119891(119909) satisfies Lemma 3 and 119884 isin 119891(119909) =

1198911(119909) + 1198912(119909) then we can conclude that 119884 is closed underaddition Note that 1199111198961

1and 119911

1198962

2can be linear interval since

the FBFs are a nonlinear basis and therefore the resultantfunction 119891(119909) is nonlinear interval (see Figure 6)


Proof In a similar way to Lemma 8 we model the productof 1198911(119909)1198912(119909) of two IT2FNN-2s which is the last point weneed to demonstrate before we can conclude that the Stone-Weierstrass theoremcan be applied to the proposed reasoningmechanism The product 1198911(119909)1198912(119909) can be expressed as1198911 (119909) 1198912 (119909)

=120572120574

1198631

1198971198632

119897

times [

[

1198711

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198711+1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)


12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X


z1

z3

z2


8

7

6

5

4

3

2

1

0

Z|r(X

)

(b) Overall IO curve for interval rules

1

08

06

04

02

0

Small Medium Large


120006|U



8

7

6

5

4

3

2

1

0

f|r(X

)



+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)]

]

+120572 (1 minus 120574)

1198631

1198971198632119903

times [

[

1198711

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)]

]

+(1 minus 120572) 120574

11986311199031198632

119897

times [

[

1198771

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198712

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)]

]

+(1 minus 120572) (1 minus 120574)

11986311199031198632119903

times [

[

1198771

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)


+

1198721

sum

1198961=1198771+1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)]

]

(47)

Therefore an equivalent to IT2FNN-2 can be constructedunder the multiplication of 1198911(119909) and 1198912(119909) where the con-sequents form an addition of 120572120574 11199111198961

119897

21199111198962

119897 120572(1 minus 120574)

11199111198961

119897

21199111198962119903

(1 minus 120572)12057411199111198961119903

21199111198962

119897 and (1 minus 120572)(1 minus 120574)

11199111198961119903

21199111198962119903

multiplied bya respective FBFs expansion (Theorem 1) and there exists119891 isin 119884 such that sup

119909isin119880(|119892(119909)minus119891(119909)|) lt 120576 (Theorem 2) Since

119891(119909) satisfies Lemma 3 and 119884 isin 119891(119909) = 1198911(119909)1198912(119909) thenwe can conclude that 119884 is closed under multiplication Notethat 1199111198961

1and 119911

1198962

2can be linear intervals since the FBFs are a

nonlinear basis interval and therefore the resultant function119891(119909) is nonlinear interval Also even if 119911

1198961

1and 119911

1198962

2were

linear their product 119911119896111199111198962

2is evidently polynomial interval

(see Figure 10)



1198881198911 (119909)

= 1205721198881199101

119897(119909) + (1 minus 120572) 119888119910

1

119903(119909)

=

sum1198711

1198961=11199081198961

1(119909) 120572119888

11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909) 120572119888

11199111198961

119897(119909)

1198631

119897

+

sum1198771

1198961=11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

+

sum1198721

1198961=1198771+11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

(48)

Therefore we can construct an IT2FNN-2 that computesall FBF expansion combinations with 120572119888

11199111198961

119897(119909) and (1 minus

120572)11988811199111198961119903(119909) in the form of the proposed IT2FNN-2 and 119884 is

closed under scalar multiplication

Lemma 11 For every (x0 y0) isin 119880 and x0 = y0 there exists119891 isin

119884 such that 119891(x0) = 119891(y0) that is 119884 separates points on 119880

We prove this by constructing a required 119891 that is wespecify 119891 isin 119884 such that 119891(x0) = 119891(y0) for arbitrarily givenx0 y0 isin 119880 with x0 = y0 We choose two fuzzy rules in theform of (8) for the fuzzy rule base (ie 119872 = 2) Let 1199090 =

(1199090

1 1199090

2 119909

0

119899) and 119910

0= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2

and 1199100

119894= (1199100

119897119894+ 1199100

119903119894)2 with 119909

0

119894= 1199100

119894 we define two interval

type-2 fuzzy sets (1198651119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(49)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(50)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(51)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(52)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100


We define two interval value real sets 1199111 isin [1199111

119897 1199111

119903] and 119911

2isin

[1199112

119897 1199112

119903] Now we have specified all the design parameters

except [119911119896119897 119911119896

119903] that is we have already obtained a function

119891 which is in the form of (20) (21) with 119872 = 2 and(119865119896

119894 [120583119865119896119894

120583119865119896119894

]) given by (8)ndash(11) With this 119891 we have

119891 (1199090) = 120572 [120601

1

119897(1199090) 1199111

119897(1199090) + (1 minus 120601

1

119897(1199090)) 1199112

119897(1199090)]

+ (1 minus 120572) [1206011

119903(1199090) 1199111

119903(1199090) + 1206012

119903(1199090) 1199112

119903(1199090)]

(53)

where

1206011

119897(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

1206011

119903(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012

119903(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

119891 (1199100) = 120572 [120601

1

119897(1199100) 1199111

119897(1199100) + 1206012

119897(1199100) 1199112

119897(1199100)] + (1 minus 120572)


119903(1199100)) 1199111

119903(1199100) + 1206012

119903(1199100) 1199112

119903(1199100)]

(54)

where

1206012

119903(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1


1206011

119897(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

119897(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

(55)

Since x0 = y0 there must be some i such that 1199090

119894= 1199100

119894

hence we have prod119899

119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090

119894) = 1 If we

choose 1199111

119897119903= 0 and 119911

2

119897119903= 1 then 119891(119909

0) = 120572(1 minus 120601

1

119897(1199090)) +

(1 minus 120572)1206012

119903(1199090) = 120572120601

2

119897(1199100) + (1 minus 120572)120601

2

119903(1199100) = 119891(119910

0) Therefore

(119884 119889infin) separates point on 119880

Lemma 12 For each 119909 isin 119880 there exists 119891 isin 119884 such that119891(119909) = 0 that is 119884 vanishes at no point of 119880

Finally we prove that (119884 119889infin) vanishes at no point of 119880By observing (8)ndash(11) (20) and (21) we just choose all 119911119896 gt 0

(119896 = 1 2 119872) that is any 119891 isin 119884 with 119911119896gt 0 serves as

required 119891

Proof of Theorem 2 From (20) and (21) it is evident that 119884 isa set of real continuous functions on119880 which are establishedby using complete interval type-2 fuzzy sets in the IF partsof fuzzy rules Using Lemmas 8 9 and 10 119884 is proved to bean algebra By using the Stone-Weierstrass theorem togetherwith Lemmas 11 and 12 we establish that the proposedIT2FNN-2 possesses the universal approximation capability

Therefore by choosing appropriate class of interval type-2membership functions we can conclude that the IT2FNN-0and IT2FNN-2 with simplified fuzzy if-then rules satisfy thefive criteria of the Stone-Weierstrass theorem

4 Application Examples

In this section the results from simulations using ANFISIT2FNN-0 IT2FNN-1 [35] IT2FNN-2 and IT2FNN-3 [35]are presented for nonlinear system identification and fore-casting the Mackey-Glass chaotic time series [39] with 120591 =

60 with different signal noise ratio values SNR(dB) = 0 1020 30 free as uncertainty source These examples are usedas benchmark problems to test the proposed ideas in thepaper We have to mention that the IT2FNN-1 and IT2FNN-3 architectures are very similar to I2FNN-0 and IT2FNN-2 respectively [35] and their results are presented for com-parison purposes The proposed IT2FNN architectures arevalidated using 10-fold cross-validation [40 41] consideringsum of square errors (SSE) or rootmean square error (RMSE)in the training or test phase We use cross-validation tomeasure the variability of the RMSE in the training andtesting phases to compare network architectures IT2FNNCross-validation procedure evaluation is done using Matlabrsquoscrossvalind function Noise is added by Matlabrsquos awgn func-tion

In119870-fold cross-validation [40] the original sample is ran-domly partitioned into 119870 subsamples Of the 119870 subsamples

Table 1 RMSE (CHK) values of ANFIS and IT2FNN with 10-foldcross-validation for identifying non-linearity of Experiment 1

SNR (dB) ANFIS IT2FNN-0 IT2FNN-1 IT2FNN-2 IT2FNN-30 06156 03532 02764 02197 0153510 02375 01153 00986 00683 0045320 00806 00435 00234 00127 0008730 00225 00106 00079 00045 00028Free 00015 00009 00004 00002 00001

100

10minus1

10minus2

10minus3

10minus4CV

-RM

SE

SNR (dB)0 10 20 30 40 50 60 70 80

ANFISIT2FNN-0IT2FNN-1

IT2FNN-2IT2FNN-3

Figure 7 RMSE (CHK) values of ANFIS and IT2FNN using 10-foldcross-validation for identifying nonlinearity in Experiment 1

a single subsample is retained as the validation data for testingthe model and the remaining 119870 minus 1 subsamples are used astraining dataThe cross-validation process is then repeated119870

times (the folds) with each of the 119870 subsamples used exactlyonce as the validation data The 119870 results from the foldsthen can be averaged (or otherwise combined) to produce asingle estimationThe advantage of thismethod over repeatedrandom subsampling is that all observations are used forboth training and validation and each observation is used forvalidation exactly once 10-fold cross-validation is commonlyusedThree application examples are used to illustrate proofsof universality as follows

Experiment 1 (identification of a one variable nonlinearfunction) In this experiment we approximate a nonlinearfunction 119891 R rarr R

119891 (119906) = 06 sin (120587119906) + 03 sin (3120587119906) + 01 sin (5120587119906) + 120578

(56)

(where 120578 is a uniform noise component) using a-one inputone-output IT2FNN 50 training data sets with 10-foldcross-validation with uniform noise levels six IT2MFs typeigaussmtype2 6 rules and 50 epochs Once the ANFISand IT2FNN models are identified a comparison was madetaking into account RMSE statistic values with 10-fold cross-validation Table 1 and Figure 7 show the resulting RMSE


Table 2 Resulting RMSE (CHK) values in ANFIS and IT2FNNfor non-linearity identification in Experiment 2 with 10-fold cross-validation


SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3

100

10minus02

10minus04

10minus06

10minus08

CV-R

MSE

Figure 8 Resulting RMSE (CHK) values obtained by ANFIS andIT2FNN for nonlinearity identification in Experiment 2with 10-foldcross-validation

(CHK) values for ANFIS and IT2FNN it can be seen thatIT2FNN architectures [31] perform better than ANFIS

Experiment 2 (identification of a three variable nonlinearfunction) A three-input one-output IT2FNN is used toapproximate nonlinear Sugeno [27] function 119891 R3 rarr R

119891 (1199091 1199092 1199093) = (1 + radic1199091 +1

1199092

+1

radic1199093

3

)

2

+ 120578 (57)

216 training data sets are generated with 10-fold cross-validation and 125 for tests 2 igaussmtype2 IT2MFs for eachinput 8 rules and 50 epochs Once the ANFIS and IT2FNNmodels are identified a comparison is made with RMSEstatistic values and 10-fold cross-validation Table 2 andFigure 8 show the resultant RMSE (CHK) values for ANFISand IT2FNN It can be seen that IT2FNN architectures [31]perform better than ANFIS

Experiment 3 Predicting the Mackey-Glass chaotic timeseries

SNR (dB)0 10 20 30 40 50 60 70 80

IT2FNN-0 TRNIT2FNN-1 TRN


ANFIS TRN

CV-R

MSE

035

03

025

02

015

01

005

0

Mackey Glass chaotic time series 120591 = 60

Figure 9 RMSE (TRN) values determined by ANFIS and IT2FNNmodels with 10-fold cross-validation for Mackey-Glass chaotic timeseries prediction with 120591 = 60

Mackey-Glass chaotic time series is a well-known bench-mark [39] for systems modeling and is described as follows

(119905) =01119909 (119905 minus 120591)

1 + 11990910 (119905 minus 120591)minus 01119909 (119905) (58)

1200 data sets are generated based on initial conditions119909(0) =12 and 120591 = 60 using fourth order Runge-Kutta methodadding different levels of uniform noise For comparing withother methods an input-output vector is chosen for IT2FNNmodel with the following format

[119909 (119905 minus 18) 119909 (119905 minus 12) 119909 (119905 minus 6) 119909 (119905) 119909 (119905 + 6)] (59)

Four-input and one-output IT2FNN model is used forMackey-Glass chaotic time series prediction choosing 500data sets for training and 500 test data data sets with 10-fold cross-validation test 2 IT2MFs for each input withmembership function igaussmtype2 16 rules and 50 epochsANFIS and IT2FNNmodels are identified comparing RMSEstatistical values with 10-fold cross-validation Table 3 andFigures 9 and 10 show the number of 120589 points out ofuncertainty interval (119909) isin [119910119897(119909) 119910119903(119909)] evaluated byIT2FNNmodel RMSE training values (TRN) and test (CHK)obtained for ANFIS and IT2FNN models It can be seen thatIT2FNN model architectures predict better Mackey-Glasschaotic time series

5 Conclusions

In this paper we have shown that an interval type-2 fuzzyneural network (IT2FNN) is a universal approximator Sim-ulation results of nonlinear function identification using


Table 3 RMSE (TRNCHK) and 120589 values determined by ANFIS and IT2FNNmodels with 10 fold cross-validation for Mackey-Glass chaotictime series prediction with 120591 = 60

SNR (dB)120591 = 60

ANFIS IT2FNN-0 IT2FNN-1 IT2FNN-2 IT2FNN-3TRN CHK 120589 TRN CHK 120589 TRN CHK 120589 TRN CHK 120589 TRN CHK 120589

0 03403 03714 NA 02388 02687 37 02027 02135 35 01495 01536 34 01244 01444 3110 01544 01714 NA 01312 01456 33 01069 01119 30 00805 00972 28 00532 00629 2520 00929 01007 NA 00708 00781 28 00566 00594 26 00424 00485 24 00342 00355 2130 00788 00847 NA 00526 00582 19 00411 00468 18 00309 00332 16 00227 00316 14Free 00749 00799 NA 00414 00477 10 00321 00353 9 00251 00319 6 00215 00293 4

SNR (dB)0 10 20 30 40 50 60 70 80

IT2FNN-0 CHKIT2FNN-1 CHK


ANFIS CHK

CV-R

MSE

04

035

03

025

02

015

01

005

0


Figure 10 RMSE (CHK) values determined by ANFIS and IT2FNNmodels with 10-fold cross-validation for Mackey-Glass chaotic timeseries prediction with 120591 = 60

the IT2FNN for one and three variables and for the Mackey-Glass chaotic time series prediction have been presentedto illustrate the theoretical result In these experiments theestimated RMSE values for nonlinear function identificationwith 10-fold cross-validation for the hybrid architectures(IT2FNN-2A2C0 and IT2FNN-2A2C1) illustrate the proofbased on Stone-Weierstrass theorem that they are universalapproximators for efficient identification of nonlinear func-tions complying with |119892(119909) minus 119891(119909)| lt 120576 Also it can beseen that while increasing the Signal Noise Ratio (SNR)IT2FNN architectures handle uncertainty more efficientlyWe have also illustrated the ideas presented in the paper withthe benchmark problem of Mackey-Glass chaotic time seriesprediction

Acknowledgments

The authors would like to thank CONACYT and DGESTfor the financial support given for this research project

The student J R Castro was supported by a scholarship fromMYCDI UABC-CONACYT

References

[1] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992

[2] B Kosko ldquoFuzzy systems as universal approximatorsrdquo IEEETransactions on Computers vol 43 no 11 pp 1329ndash1333 1994

[3] J J Buckley ldquoUniversal fuzzy controllersrdquo Automatica vol 28no 6 pp 1245ndash1248 1992

[4] J J Buckley and Y Hayashi ldquoCan fuzzy neural nets approximatecontinuous fuzzy functionsrdquo Fuzzy Sets and Systems vol 61 no1 pp 43ndash51 1994

[5] L V Kantorovich and G P Akilov Functional Analysis Perga-mon Oxford UK 2nd edition 1982

[6] H L Royden Real Analysis Macmillan New York NY USA2nd edition 1968

[7] V Kreinovich G C Mouzouris and H T Nguyen ldquoFuzzy rulebased modelling as a universal aproximation toolrdquo in FuzzySystems Modeling and Control H T Nguyen and M SugenoEds pp 135ndash195 Kluwer Academic Publisher Boston MassUSA 1998

[8] M G Joo and J S Lee ldquoUniversal approximation by hierarchi-cal fuzzy system with constraints on the fuzzy rulerdquo Fuzzy Setsand Systems vol 130 no 2 pp 175ndash188 2002

[9] B Kosko Neural Networks and Fuzzy Systems Prentice HallEnglewood Cliffs NJ USA 1992

[10] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993

[11] S M Zhou and L D Xu ldquoA new type of recurrent fuzzyneural network for modeling dynamic systemsrdquo Knowledge-Based Systems vol 14 no 5-6 pp 243ndash251 2001

[12] S M Zhou and L D Xu ldquoDynamic recurrent neural networksfor a hybrid intelligent decision support system for themetallur-gical industryrdquo Expert Systems vol 16 no 4 pp 240ndash247 1999

[13] Q Liang and J M Mendel ldquoInterval type-2 fuzzy logic systemstheory and designrdquo IEEE Transactions on Fuzzy Systems vol 8no 5 pp 535ndash550 2000

[14] J M Mendel Uncertain Rule-Based Fuzzy Logic Systems Intro-duction and New Directions Prentice Hall Upper Sadler RiverNJ USA 2001

[15] C-H Lee T-WHu C-T Lee and Y-C Lee ldquoA recurrent inter-val type-2 fuzzy neural network with asymmetric membership


functions for nonlinear system identificationrdquo in Proceedings ofthe IEEE International Conference on Fuzzy Systems (FUZZ rsquo08)pp 1496ndash1502 Hong Kong June 2008

[16] C H Lee J L Hong Y C Lin and W Y Lai ldquoType-2 fuzzyneural network systems and learningrdquo International Journal ofComputational Cognition vol 1 no 4 pp 79ndash90 2003

[17] C-H Wang C-S Cheng and T-T Lee ldquoDynamical optimaltraining for interval type-2 fuzzy neural network (T2FNN)rdquoIEEETransactions on SystemsMan andCybernetics Part B vol34 no 3 pp 1462ndash1477 2004

[18] J T Rickard J Aisbett and G Gibbon ldquoFuzzy subsethood forfuzzy sets of type-2 and generalized type-nrdquo IEEE Transactionson Fuzzy Systems vol 17 no 1 pp 50ndash60 2009

[19] S-M Zhou J M Garibaldi R I John and F ChiclanaldquoOn constructing parsimonious type-2 fuzzy logic systems viainfluential rule selectionrdquo IEEE Transactions on Fuzzy Systemsvol 17 no 3 pp 654ndash667 2009

[20] N S Bajestani and A Zare ldquoApplication of optimized type 2fuzzy time series to forecast Taiwan stock indexrdquo in Proceedingsof the 2nd International Conference on Computer Control andCommunication pp 1ndash6 February 2009

[21] B Karl Y Kocyiethit and M Korurek ldquoDifferentiating typesof muscle movements using a wavelet based fuzzy clusteringneural networkrdquo Expert Systems vol 26 no 1 pp 49ndash59 2009

[22] S M Zhou H X Li and L D Xu ldquoA variational approachto intensity approximation for remote sensing images usingdynamic neural networksrdquo Expert Systems vol 20 no 4 pp163ndash170 2003

[23] S Panahian Fard and Z Zainuddin ldquoInterval type-2 fuzzyneural networks version of the Stone-Weierstrass theoremrdquoNeurocomputing vol 74 no 14-15 pp 2336ndash2343 2011

[24] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[25] L-X Wang and J M Mendel ldquoGenerating fuzzy rules bylearning from examplesrdquo IEEE Transactions on Systems Manand Cybernetics vol 22 no 6 pp 1414ndash1427 1992

[26] J J Buckley ldquoSugeno type controllers are universal controllersrdquoFuzzy Sets and Systems vol 53 no 3 pp 299ndash303 1993

[27] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[28] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[29] K Hirota andW Pedrycz ldquoORANDneuron inmodeling fuzzyset connectivesrdquo IEEE Transactions on Fuzzy Systems vol 2 no2 pp 151ndash161 1994

[30] J-S R Jang C T Sun and E Mizutani Neuro-Fuzzy and SoftComputing Prentice-Hall New York NY USA 1997

[31] J R Castro O Castillo P Melin and A Rodriguez-DiazldquoA hybrid learning algorithm for interval type-2 fuzzy neuralnetworks the case of time series predictionrdquo in Soft ComputingFor Hybrid Intelligent Systems vol 154 pp 363ndash386 SpringerBerlin Germany 2008

[32] J R Castro O Castillo P Melin and A Rodriguez-DiazldquoHybrid learning algorithm for interval type-2 fuzzy neuralnetworksrdquo in Proceedings of Granular Computing pp 157ndash162Silicon Valley Calif USA 2007

[33] F Lin and P Chou ldquoAdaptive control of two-axis motioncontrol system using interval type-2 fuzzy neural networkrdquoIEEE Transactions on Industrial Electronics vol 56 no 1 pp178ndash193 2009

[34] R Ceylan Y Ozbay and B Karlik ldquoClassification of ECGarrhythmias using type-2 fuzzy clustering neural networkrdquo inProceedings of the 14th National Biomedical EngineeringMeeting(BIYOMUT rsquo09) pp 1ndash4 May 2009

[35] J R Castro O Castillo P Melin and A Rodrıguez-Dıaz ldquoAhybrid learning algorithm for a class of interval type-2 fuzzyneural networksrdquo Information Sciences vol 179 no 13 pp 2175ndash2193 2009

[36] C T Scarborough andAH Stone ldquoProducts of nearly compactspacesrdquo Transactions of the AmericanMathematical Society vol124 pp 131ndash147 1966

[37] R E Edwards Functional Analysis Theory and ApplicationsDover 1995

[38] W Rudin Principles of Mathematical Analysis McGraw-HillNew York NY USA 1976

[39] M C Mackey and L Glass ldquoOscillation and chaos in physio-logical control systemsrdquo Science vol 197 no 4300 pp 287ndash2891977

[40] T Hastie R Tibshirani and J Friedman The Elements ofStatistical Learning Springer 2001

[41] S Theodoridis and K Koutroumbas Pattern Recognition Aca-demic Press 1999

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

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Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

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HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



x1

x1x2

x2

120583

120583

120583

120583

120583

120583

120583

120583

|x1||x2|

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

f1

f1

f2

f2

f3

f3

f4

f4

y1l

y1r

y2l

y2r

y3l

y3r

y4l

y4r

yr

yl

12

T

T

T

T

T

T

T

T

12

sum y

Layer 0 Layer 1 Layer 2 Layer 3 Layer 4 Layer 5


type-2 neurofuzzy adaptive networks nodes represent pro-cessing units called neurons which can be classified into crispand fuzzy neurons

The IT2FNN-1 architecture has 5 layers (Figure 1) [35]and consists of adaptive nodes with equivalent functionto lower-upper membership in fuzzification layer (layer 1)Nonadaptive nodes in rules layer (layer 2) interconnect withthe fuzzification layer (layer 1) in order to generate TSKIT2FIS rules antecedents Adaptive nodes in the consequentlayer (layer 3) are connected to the input layer (layer 0)to generate rules consequents Nonadaptive nodes in type-reduction layer (layer 4) evaluate left-right values with theKarnik andMendel (KM) [13 14] algorithmThe nonadaptivenodes in the defuzzification layer (layer 5) average left-rightvalues

The IT2FNN-3 architecture has 8 layers (Figure 2) [35]and uses IT2FN for fuzzifying the inputs (layers 1-2) Non-adaptive nodes in rules layer (layer 3) interconnect withlower-upper linguistic values layer (layer 2) to generate TSKIT2FIS rules antecedents Adaptive nodes in layer 4 adapt left-right firing strength biasing rules lower-upper trigger forceswith synaptic weights between layers 3 and 4 Layer 5rsquos non-adaptive nodes normalize rules lower-upper firing strengthNonadaptive nodes I consequent layer (layer 6) interconnectwith input layer (layer 0) to generate rules consequentsNonadaptive nodes in type-reduction layer (layer 7) evaluate

left-right values adding lower-upper product of lower-uppertriggering forces normalized by rules consequent left-rightvalues Node in defuzzification layer is adaptive and itsoutput 119910 is defined as biased average of left-right values andparameter 120574 Parameter 120574 (05 by default) adjusts uncertaintyinterval defined by left-right values [119910119897 119910119903]

Architectures IT2FNN-0 and IT2FNN-2 which will beshown in Sections 32 and 33 respectively as universalapproximators are described withmore details in Section 21

21 IT2FNN-0 Architecture An IT2FNN-0 is a seven-layerIT2FNN which integrates a first order TSKIT2FIS (intervaltype-2 fuzzy antecedents and real consequents) with anadaptive NN The IT2FNN-0 (Figure 3) layers are describedas follows

Layer 0 Inputs

1199000

119894= 119909119894 119894 = 1 119899 (1)

Layer 1 Adaptive type-1 fuzzy neuron (T1FN)

net1119896= 11990810

119896119894119909119894 + 119887

1

119896forall119894 = 1 119899 119896 = 1 120599 (2)

1199001

119896= 120583(net1

119896) where the transfer function 120583 is a member-

ship function net1119896is the weighted sum of inputs (119909119894) and


x1

x1

x2

x2

sum

sum

sum

sum

sum

sum

sum

sum

sum

sum

sum

y

|x1||x2|


120583

120583

120583

120583

120583

120583

120583

120583

T T

S T

T T

S T

T

S

T T

S T

T

T

w11

w21

w31

w41

w52

w62

w72

w82

1

1

1

1

1

1

1

1

b1

net1

b2

b3

net2

b4

net4

net3

b5

b6

net5

b7

net7

net6

b8

net8

120583(net1)

120583(net2)

120583(net3)

120583(net4)

120583(net5)

120583(net6)

120583(net7)

120583(net8)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

f1

f2

f3

f4

f1

f2

f3

f4

f1l

f1r

f2l

f2r

f3l

f3r

f4l

f4r

w1

w1

w2

w2

w3

w3

w4

w4

y1l

yl

y1r

y2l

y2r

y3l

y3r

y4l

y4r

yr

120574

1minus 120574



1


neuron


1199002

2119896minus1= 1199001

2119896minus1sdot 1199001


(3)


1199002

2119896= 1199001

2119896minus1+ 1199001

2119896minus 1199002



119894minus1

119895=1V119895+|120591|


if 120591 gt 0

120583119896

119894= 1199002

119894119897(120587) 120583

119896

119894= 1199002

119894119906(120587)(5)

else120583119896119894

= null 120583119896119894

= null (6)

end

where 120583119896

119894 120583119896





1199003

2119896minus1= 119908119896 119900

3

2119896= 119908119896

119908119896=

119899

prod

119894=1

120583119865119896119894

(119909)

119908119896=

119899

prod

119894=1

120583119865119896119894

(119909)

(7)

where 120583119865119896119894(119909)

isin [120583119865119896119894

(119909) 120583119865119896119894



1198941119898119896

119894

2119898119896

119894]) defined by

1120583119865119896119894

(119909119894 [120590119896

1198941119898119896

119894]) = exp[

[

minus1

2(119909119894 minus1119898119896

119894

120590119896

119894

)

2

]

]

(8)



x1

x1x2

x2

sum

sum

sum

sum

sum

sum

sum

sum

sum

sum

sum

y

yl

yr

120574

1minus 120574

120583

120583

120583

120583

120583

120583

120583

120583

w11

w41

w52

w82

w62

w72

w21

w31

1b1

net1

1b2

net2

1b3

net3

1b4

net4

1b5

net5

1b6

net6

1b7

net7

1b8

net8

120583(net1) 120583(x)

120583(net2) 120583(x)

120583(net3) 120583(x)

120583(net4) 120583(x)

120583(net5) 120583(x)

120583(net6) 120583(x)

120583(net7) 120583(x)

120583(net8) 120583(x)

TT

TS

TT

TS

TT

TS

TT

TS

f1

f1

f2

f2

f3

f3

f4

f4

w1

w1

w2

w2

w3

w3

w4

w4

y1l

y1r

y2l

y2r

y3l

y3r

y4l

y4r

w1y1

w1y1

w2y2

w2y2

w3y3

w3y3

w4y4

w4y4


2120583119865119896119894

(119909119894 [120590119896

1198942119898119896

119894]) = exp[

[

minus1

2(119909119894 minus2119898119896

119894

120590119896

119894

)

2

]

]

(9)

120583119865119896119894

(119909) =

2120583119865119896119894

(119909119894 [120590119896

1198942119898119896

119894]) 119909119894 le

1119898119896

119894

1120583119865119896119894

(119909119894 [120590119896

1198941119898119896

119894]) 119909119894 gt

2119898119896

119894

(10)

120583119865119896119894

(119909) =

1120583119865119896119894

(119909119894 [120590119896

1198941119898119896

119894]) 119909119894 lt

1119898119896

119894

11119898119896

119894le 119909119894 le

2119898119896

119894

2120583119865119896119894

(119909119894 [120590119896

1198942119898119896

119894]) 119909119894 gt

2119898119896

119894

(11)





1199004

2119896minus1= 120601119896 119900

4

2119896= 120601119896

(12)

If we view 120601119896 120601119896




120601119896=

119908119896

119863119897

119896 = 1 119872 (13)

where119863119897 = sum119872

119896=1119908119896

120601119896

=119908119896

119863119903

119896 = 1 119872 (14)

where119863119903 = sum119872

119896=1119908119896


119896

119894 119888119896



1199005

2119896=1= 119910119896 119900

5

2119896= 119910119896 (15)

119910119896=

119899

sum

119894=1

119888119896

119894119909119894 + 119888119896

0 119896 = 1 119872 (16)


x1

x1x2

x2

|x1||x2|


sum

sum

sum

sum

sum

sum

sum

sum

sum

y

120583

120583

120583

120583

120583

120583

120583

120583

T T

S T

KMT T

S T

T T

S T

T

S T

T

KM

y1l

y1r

y2l

y2r

y3l

y3r

y4l

y4r

yl

yr

12

12

1b1

net1

1b2

net2

1b3

net3

1b4

net4

1b5

net5

1b6

net6

1b7

net7

1b8

net8

w11

w41

w52

w82

w62

w72

w21

w31

120583(net1)

120583(net2)

120583(x)

120583(x)

120583(net3)

120583(net4)

120583(x)

120583(x)

120583(net5)

120583(net6)

120583(x)

120583(net7) 120583(x)

120583(x)

120583(net8) 120583(x)

f1

f1

f2

f2

f3

f3

f4

f4



1199006

1= 119910119897 (119909) 119900

6

2= 119910119903 (119909) (17)

where

119910119897 (119909) =

119872

sum

119896=1

120601119896119910119896

119910119903 (119909) =

119872

sum

119896=1

120601119896

119910119896

(18)



1199007

1= 119910 (119909) (19)

where

119910 (119909) = 120574119910119897 (119909) + (1 minus 120574) 119910119903 (119909) (20)

119910 (119909) =

119872

sum

119896=1

[120574120601119896(119909) + (1 minus 120574) 120601

119896

(119909)] 119910119896(119909) (21)









119899


(|119892(119909)minus

119891(119909)|) lt 120576


119877119899

rarr 119877 119909 = (1199091 1199092 119909119899) isin 119880 120583119865119896119894(119909)

isin [120583119865119896119894

(119909)

120583119865119896119894


119896

1198941119898119896

1198942119898119896

119894]) defined by (27) and (31) If

we view 120601119896(119909) 120601119896




119896(119909) 120601119896





1198911 (119909) = 1205721199101

119897(119909) + (1 minus 120572) 119910

1

119903(119909) (22)

where

1199101

119897(119909) =

119872

sum

119896=1

120601119896

1(119909) 119911119896

1(119909) =

sum1198721

119896=1119908119896

1(119909) 119911119896

1(119909)

1198631

119897

1199101

119903(119909) =

1198721

sum

119896=1

120601119896

1(119909) 119911119896

1(119909) =

sum1198721

119896=1119908119896

1(119909) 119911119896

1(119909)

1198631119903

(23)

where

1198631

119897=

1198721

sum

119896=1

119899

prod

119894=1

1205831119865119896119894

(119909) 1198631

119903=

1198721

sum

1198961=1

119899

prod

119894=1

120583 1119865119896119894

(119909)

119908119896

1=

119899

prod

119894=1

1205831119865119896119894

(119909) 119908119896

1=

119899

prod

119894=1

120583 1119865119896119894

(119909)

120601119896

1(119909) =

119908119896

1(119909)

1198631

119897

120601119896

1(119909) =

119908119896

1

1198631119903

119911119896

1(119909) =

119899

sum

119894=1

1119888119896

119894119909119894 +1119888119896

0 119896 = 1 119872

(24)

1198912 (119909) = 1205741199102

119897(119909) + (1 minus 120574) 119910

2

119903(119909) (25)

where

1199102

119897(119909) =

119872

sum

119896=1

120601119896

2(119909) 119911119896

2(119909) =

sum1198722

119896=1119908119896

2(119909) 119911119896

2(119909)

1198632

119897

(26)

1199102

119903(119909) =

1198722

sum

119896=1

120601119896

2(119909) 119911119896

2(119909) =

sum1198721

119896=1119908119896

2(119909) 119911119896

2(119909)

1198632119903

(27)

where

119908119896

2=

119899

prod

119894=1

1205832119865119896119894

(119909) 119908119896

2=

119899

prod

119894=1

120583 2119865119896119894

(119909)

1198632

119897=

1198722

sum

119896=1

119899

prod

119894=1

1205832119865119896119894

(119909) 1198632

119903=

1198722

sum

119896=1

119899

prod

119894=1

120583 2119865119896119894

(119909)

120601119896

2(119909) =

119908119896

2(119909)

1198632

119897

120601119896

2(119909) =

119908119896

2(119909)

1198632119903

119911119896

2(119909) =

119899

sum

119894=1

2119888119896

119894119909119894 +2119888119896

0 119896 = 1 119872

(28)




12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X



Z(X

)

z1

z3

z2

8

7

6

5

4

3

2

1

0


1

08

06

04

02

0

Small Medium Large


120006|U

0 5 0 5 11100



8

7

6

5

4

3

2

1

0

f|r(X

)




1198911 (119909) + 1198912 (119909)

=

sum1198721

1198961=1sum1198722

1198962=1[1199081198961

11199081198962

2(1205721199111198961

1+ 1205741199111198962

2)]

1198631

1198971198632

119897

+

sum1198721

1198961=1sum1198722

1198962=1[1199081198961

11199081198962

2(1 minus 120572) 119911

1198961

1+ (1 minus 120574) 119911

1198962

2]

11986311199031198632119903

(29)

and that Φ1198961 1198962

= (1199081198961

11199081198962

2)(1198631

1199031198632

119903) and Φ1198961 1198962

= (1199081198961

11199081198962

2)

(1198631

1199031198632



1+1205741199111198962

2and (1minus120572)119911

1198961

1+(1minus120574)119911

1198962

2multiplied





1and 119911

1198962





1198911 (119909) 1198912 (119909)

=1

1198631

1198971198632

119897

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

21205721205741199111198961

11199111198962

2]

]

+1

1198631

1198971198632119903

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2120572 (1 minus 120574) 119911

1198961

11199111198962

2]

]

+1

11986311199031198632

119897

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2(1 minus 120572) 120574119911

1198961

11199111198962

2]

]

+1

11986311199031198632119903

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2(1 minus 120572) (1 minus 120574) 119911

1198961

11199111198962

2]

]

(30)



11199111198962

2 120572(1minus120574)119911

1198961

11199111198962

2 (1minus

120572)1205741199111198961

11199111198962

2 and (1 minus 120572)(1 minus 120574)119911

1198961

11199111198962


FBFs (Φ11989711989711989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119897)Φ11989711990311989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119903)

Φ119903119897

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119897) and Φ

119903119903

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119903))



1and 119911

1198962

2


1and 119911

1198962


11199111198962

2is




119888119891 (119909) = 120572119888119910119897 (119909) + (1 minus 120572) 119888119910119903 (119909)

=

sum119872

119896=1[120572119863119903119908

119896(119909) + (1 minus 120572)119863119897119908

119896(119909)] 119888119911

119896(119909)

119863119897119863119903

(31)







0

1 1199090

2 119909

0

119899)

and 1199100= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2 and 119910

0

119894= (1199100

119897119894+

1199100

119903119894)2 with 119909

0

119894= 1199100


(1198651

119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(32)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(33)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(34)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(35)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



2 with 119911119896(119909) =

sum119899

119894=1119888119896

119894119909119894 + 119888

119896



119894 [1205831198651119894

1205831198651119894


119891 (1199090) = 120572 [120601

1(1199090) 1199111(1199090) + 1206012(1199090) 1199112(1199090)] + (1 minus 120572)

times [1206011

(1199090) 1199111(1199090) + (1 minus 120601

1

(1199090)) 1199112(1199090)]

(36)

where

1206011(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206011

(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

(37)

119891 (1199100) = 120572 [120601

1(1199100) 1199111(1199100) + 1206012(1199100) 1199112(1199100)] + (1 minus 120572)


(1199100)) 1199111(1199100) + 1206012

(1199100) 1199112(1199100)]

(38)

where

1206011(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1

(39)


119894 hence

we have prod119899

119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090



1199111= 0 and 119911

2= 1 then119891(119909

0) = 120572120601

2(1199090)+(1minus120572)[1minus120601

1

(1199090)] =

1205721206012(1199100) + (1 minus 120572)120601

2

(1199100) = 119891(119910






119896(119909) gt 0




119899rarr 119877 119909 = (1199091 1199092

119909119899) isin 119880 120583119865119896119894(119909)

isin [120583119865119896119894

(119909) 120583119865119896119894



1198941119898119896

119894

1119898119896


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909)

120601119896


119896

119897 119910119896

119903are


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909) 120601

119896

119903(119909) and let

119889infin(1198911 1198912) = sup119909isin119880



1198911 (119909) = 1205741199101

119897(119909) + (1 minus 120574) 119910

1

119903(119909) (40)

where

1199101

119897(119909) =

1198711

sum

1198961=1

11206011198961

119897(119909)11199111198961

119897(119909) +

1198721

sum

1198961=1198711+1

11206011198961

119897(119909)11199111198961

119897(119909)

=

sum1198711

1198961=11199081198961

1(119909)11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909)11199111198961

119897(119909)

1198631

119897

1199101

119903(119909) =

1198771

sum

1198961=1

11206011198961

119903(119909)11199111198961119903

(119909) +

1198721

sum

1198961=1198771+1

11206011198961

119903(119909)11199111198961119903

(119909)

=

sum1198771

1198961=11199081198961

1(119909)11199111198961119903

(119909) + sum1198721

1198961=1198771+11199081198961

1(119909)11199111198961119903

(119909)

1198631119903

(41)

where

1199081198961

1=

119899

prod

119894=1

12058311198651198961

119894

(119909) 1199081198961

1=

119899

prod

119894=1

120583 11198651198961

119894

(119909)

1198631

119897=

1198711

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198711+1

1199081198961

1

1198631

119903=

1198771

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198771+1

1199081198961

1

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1 1198711 (119909)

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1198711 (119909) + 1 1198721

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1 1198771 (119909)

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1198771 (119909) + 1 1198721

11199111198961

119897=

119899

sum

119894=1

11198881198961

119894119909119894 +1119888119896

0minus

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus11199041198961

0

1198961 = 1 1198721

11199111198961119903

=

119899

sum

119894=1

11198881198961

119894119909119894 +11198881198961

0+

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 +11199041198961

0

1198961 = 1 1198721

1198912 (119909) = 1205741199102

119897(119909) + (1 minus 120574) 119910

2

119903(119909)

(42)

where

1199102

119897(119909)

=

1198712

sum

1198962=1

21206011198962

119897(119909)21199111198962

119897(119909) +

1198722

sum

1198962=1198712+1

21206011198962

119897(119909)21199111198962

119897(119909)

=

sum1198712

1198962=11199081198962

2(119909)21199111198962

119897(119909) + sum

1198722

1198962=1198712+11199081198962

2(119909)21199111198962

119897(119909)

1198632

119897

(43)


1199102

119903(119909)

=

1198772

sum

1198962=1

21206011198962

119903(119909)21199111198962119903

(119909) +

1198722

sum

1198962=1198772+1

21206011198962

119903(119909)21199111198962119903

(119909)

=

sum11987721

1198962=11199081198962

2(119909)21199111198962119903

(119909) + sum1198722

1198962=1198772+11199081198962

2(119909)21199111198962119903

(119909)

1198631119903

(44)

where

1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909) 1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909)

1198632

119897=

1198712

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198712+1

1199081198962

2

1198632

119903=

1198772

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198772+1

1199081198962

2

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1 1198712 (119909)

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1198712 (119909) + 1 1198722

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1 1198772 (119909)

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1198772 (119909) + 1 1198722

21199111198962

119897=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0minus

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus21199041198962

0

1198962 = 1 1198722

21199111198962119903

=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0+

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 +21199041198962

0

1198962 = 1 1198722

(45)



1198911 (119909) + 1198912 (119909)

= ((

1198711

sum

1198961=1

1198712

sum

1198962=1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909) + 120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909)+120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198771

sum

1198961=1

1198772

sum

1198962=1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])times (1198631

1199031198632

119903)minus1

)

+ ((

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])

times (1198631

1199031198632

119903)minus1

)


119897+12057421199111198962

119897and (1minus120572)

11199111198961119903+(1minus120574)

21199111198962119903


119909isin119880(|119892(119909) minus 119891(119909)|) lt 120576



1and 119911

1198962





=120572120574

1198631

1198971198632

119897

times [

[

1198711

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198711+1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)


12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X


z1

z3

z2


8

7

6

5

4

3

2

1

0

Z|r(X

)


1

08

06

04

02

0

Small Medium Large


120006|U



8

7

6

5

4

3

2

1

0

f|r(X

)



+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)]

]

+120572 (1 minus 120574)

1198631

1198971198632119903

times [

[

1198711

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)]

]

+(1 minus 120572) 120574

11986311199031198632

119897

times [

[

1198771

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198712

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)]

]

+(1 minus 120572) (1 minus 120574)

11986311199031198632119903

times [

[

1198771

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)


+

1198721

sum

1198961=1198771+1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)]

]

(47)


119897

21199111198962

119897 120572(1 minus 120574)

11199111198961

119897

21199111198962119903

(1 minus 120572)12057411199111198961119903

21199111198962

119897 and (1 minus 120572)(1 minus 120574)

11199111198961119903

21199111198962119903




1and 119911

1198962



1198961

1and 119911

1198962

2were



(see Figure 10)



1198881198911 (119909)

= 1205721198881199101

119897(119909) + (1 minus 120572) 119888119910

1

119903(119909)

=

sum1198711

1198961=11199081198961

1(119909) 120572119888

11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909) 120572119888

11199111198961

119897(119909)

1198631

119897

+

sum1198771

1198961=11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

+

sum1198721

1198961=1198771+11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

(48)


11199111198961

119897(119909) and (1 minus






(1199090

1 1199090

2 119909

0

119899) and 119910

0= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2

and 1199100

119894= (1199100

119897119894+ 1199100

119903119894)2 with 119909

0

119894= 1199100


type-2 fuzzy sets (1198651119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(49)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(50)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(51)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(52)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



119897 1199111

119903] and 119911

2isin

[1199112

119897 1199112


except [119911119896119897 119911119896



119894 [120583119865119896119894

120583119865119896119894


119891 (1199090) = 120572 [120601

1

119897(1199090) 1199111

119897(1199090) + (1 minus 120601

1

119897(1199090)) 1199112

119897(1199090)]

+ (1 minus 120572) [1206011

119903(1199090) 1199111

119903(1199090) + 1206012

119903(1199090) 1199112

119903(1199090)]

(53)

where

1206011

119897(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

1206011

119903(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012

119903(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

119891 (1199100) = 120572 [120601

1

119897(1199100) 1199111

119897(1199100) + 1206012

119897(1199100) 1199112

119897(1199100)] + (1 minus 120572)


119903(1199100)) 1199111

119903(1199100) + 1206012

119903(1199100) 1199112

119903(1199100)]

(54)

where

1206012

119903(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1


1206011

119897(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

119897(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

(55)


119894= 1199100

119894


119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090

119894) = 1 If we

choose 1199111

119897119903= 0 and 119911

2

119897119903= 1 then 119891(119909

0) = 120572(1 minus 120601

1

119897(1199090)) +

(1 minus 120572)1206012

119903(1199090) = 120572120601

2

119897(1199100) + (1 minus 120572)120601

2

119903(1199100) = 119891(119910

0) Therefore





required 119891









100

10minus1

10minus2

10minus3

10minus4CV

-RM

SE

SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3





119891 (119906) = 06 sin (120587119906) + 03 sin (3120587119906) + 01 sin (5120587119906) + 120578

(56)





SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3

100

10minus02

10minus04

10minus06

10minus08

CV-R

MSE




119891 (1199091 1199092 1199093) = (1 + radic1199091 +1

1199092

+1

radic1199093

3

)

2

+ 120578 (57)



SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS TRN

CV-R

MSE

035

03

025

02

015

01

005

0




(119905) =01119909 (119905 minus 120591)

1 + 11990910 (119905 minus 120591)minus 01119909 (119905) (58)




5 Conclusions




SNR (dB)120591 = 60


0 03403 03714 NA 02388 02687 37 02027 02135 35 01495 01536 34 01244 01444 3110 01544 01714 NA 01312 01456 33 01069 01119 30 00805 00972 28 00532 00629 2520 00929 01007 NA 00708 00781 28 00566 00594 26 00424 00485 24 00342 00355 2130 00788 00847 NA 00526 00582 19 00411 00468 18 00309 00332 16 00227 00316 14Free 00749 00799 NA 00414 00477 10 00321 00353 9 00251 00319 6 00215 00293 4

SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS CHK

CV-R

MSE

04

035

03

025

02

015

01

005

0




Acknowledgments



References



















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




x1

x1

x2

x2

sum

sum

sum

sum

sum

sum

sum

sum

sum

sum

sum

y

|x1||x2|


120583

120583

120583

120583

120583

120583

120583

120583

T T

S T

T T

S T

T

S

T T

S T

T

T

w11

w21

w31

w41

w52

w62

w72

w82

1

1

1

1

1

1

1

1

b1

net1

b2

b3

net2

b4

net4

net3

b5

b6

net5

b7

net7

net6

b8

net8

120583(net1)

120583(net2)

120583(net3)

120583(net4)

120583(net5)

120583(net6)

120583(net7)

120583(net8)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

120583(x)

f1

f2

f3

f4

f1

f2

f3

f4

f1l

f1r

f2l

f2r

f3l

f3r

f4l

f4r

w1

w1

w2

w2

w3

w3

w4

w4

y1l

yl

y1r

y2l

y2r

y3l

y3r

y4l

y4r

yr

120574

1minus 120574



1


neuron


1199002

2119896minus1= 1199001

2119896minus1sdot 1199001


(3)


1199002

2119896= 1199001

2119896minus1+ 1199001

2119896minus 1199002



119894minus1

119895=1V119895+|120591|


if 120591 gt 0

120583119896

119894= 1199002

119894119897(120587) 120583

119896

119894= 1199002

119894119906(120587)(5)

else120583119896119894

= null 120583119896119894

= null (6)

end

where 120583119896

119894 120583119896





1199003

2119896minus1= 119908119896 119900

3

2119896= 119908119896

119908119896=

119899

prod

119894=1

120583119865119896119894

(119909)

119908119896=

119899

prod

119894=1

120583119865119896119894

(119909)

(7)

where 120583119865119896119894(119909)

isin [120583119865119896119894

(119909) 120583119865119896119894



1198941119898119896

119894

2119898119896

119894]) defined by

1120583119865119896119894

(119909119894 [120590119896

1198941119898119896

119894]) = exp[

[

minus1

2(119909119894 minus1119898119896

119894

120590119896

119894

)

2

]

]

(8)



x1

x1x2

x2

sum

sum

sum

sum

sum

sum

sum

sum

sum

sum

sum

y

yl

yr

120574

1minus 120574

120583

120583

120583

120583

120583

120583

120583

120583

w11

w41

w52

w82

w62

w72

w21

w31

1b1

net1

1b2

net2

1b3

net3

1b4

net4

1b5

net5

1b6

net6

1b7

net7

1b8

net8

120583(net1) 120583(x)

120583(net2) 120583(x)

120583(net3) 120583(x)

120583(net4) 120583(x)

120583(net5) 120583(x)

120583(net6) 120583(x)

120583(net7) 120583(x)

120583(net8) 120583(x)

TT

TS

TT

TS

TT

TS

TT

TS

f1

f1

f2

f2

f3

f3

f4

f4

w1

w1

w2

w2

w3

w3

w4

w4

y1l

y1r

y2l

y2r

y3l

y3r

y4l

y4r

w1y1

w1y1

w2y2

w2y2

w3y3

w3y3

w4y4

w4y4


2120583119865119896119894

(119909119894 [120590119896

1198942119898119896

119894]) = exp[

[

minus1

2(119909119894 minus2119898119896

119894

120590119896

119894

)

2

]

]

(9)

120583119865119896119894

(119909) =

2120583119865119896119894

(119909119894 [120590119896

1198942119898119896

119894]) 119909119894 le

1119898119896

119894

1120583119865119896119894

(119909119894 [120590119896

1198941119898119896

119894]) 119909119894 gt

2119898119896

119894

(10)

120583119865119896119894

(119909) =

1120583119865119896119894

(119909119894 [120590119896

1198941119898119896

119894]) 119909119894 lt

1119898119896

119894

11119898119896

119894le 119909119894 le

2119898119896

119894

2120583119865119896119894

(119909119894 [120590119896

1198942119898119896

119894]) 119909119894 gt

2119898119896

119894

(11)





1199004

2119896minus1= 120601119896 119900

4

2119896= 120601119896

(12)

If we view 120601119896 120601119896




120601119896=

119908119896

119863119897

119896 = 1 119872 (13)

where119863119897 = sum119872

119896=1119908119896

120601119896

=119908119896

119863119903

119896 = 1 119872 (14)

where119863119903 = sum119872

119896=1119908119896


119896

119894 119888119896



1199005

2119896=1= 119910119896 119900

5

2119896= 119910119896 (15)

119910119896=

119899

sum

119894=1

119888119896

119894119909119894 + 119888119896

0 119896 = 1 119872 (16)


x1

x1x2

x2

|x1||x2|


sum

sum

sum

sum

sum

sum

sum

sum

sum

y

120583

120583

120583

120583

120583

120583

120583

120583

T T

S T

KMT T

S T

T T

S T

T

S T

T

KM

y1l

y1r

y2l

y2r

y3l

y3r

y4l

y4r

yl

yr

12

12

1b1

net1

1b2

net2

1b3

net3

1b4

net4

1b5

net5

1b6

net6

1b7

net7

1b8

net8

w11

w41

w52

w82

w62

w72

w21

w31

120583(net1)

120583(net2)

120583(x)

120583(x)

120583(net3)

120583(net4)

120583(x)

120583(x)

120583(net5)

120583(net6)

120583(x)

120583(net7) 120583(x)

120583(x)

120583(net8) 120583(x)

f1

f1

f2

f2

f3

f3

f4

f4



1199006

1= 119910119897 (119909) 119900

6

2= 119910119903 (119909) (17)

where

119910119897 (119909) =

119872

sum

119896=1

120601119896119910119896

119910119903 (119909) =

119872

sum

119896=1

120601119896

119910119896

(18)



1199007

1= 119910 (119909) (19)

where

119910 (119909) = 120574119910119897 (119909) + (1 minus 120574) 119910119903 (119909) (20)

119910 (119909) =

119872

sum

119896=1

[120574120601119896(119909) + (1 minus 120574) 120601

119896

(119909)] 119910119896(119909) (21)









119899


(|119892(119909)minus

119891(119909)|) lt 120576


119877119899

rarr 119877 119909 = (1199091 1199092 119909119899) isin 119880 120583119865119896119894(119909)

isin [120583119865119896119894

(119909)

120583119865119896119894


119896

1198941119898119896

1198942119898119896

119894]) defined by (27) and (31) If

we view 120601119896(119909) 120601119896




119896(119909) 120601119896





1198911 (119909) = 1205721199101

119897(119909) + (1 minus 120572) 119910

1

119903(119909) (22)

where

1199101

119897(119909) =

119872

sum

119896=1

120601119896

1(119909) 119911119896

1(119909) =

sum1198721

119896=1119908119896

1(119909) 119911119896

1(119909)

1198631

119897

1199101

119903(119909) =

1198721

sum

119896=1

120601119896

1(119909) 119911119896

1(119909) =

sum1198721

119896=1119908119896

1(119909) 119911119896

1(119909)

1198631119903

(23)

where

1198631

119897=

1198721

sum

119896=1

119899

prod

119894=1

1205831119865119896119894

(119909) 1198631

119903=

1198721

sum

1198961=1

119899

prod

119894=1

120583 1119865119896119894

(119909)

119908119896

1=

119899

prod

119894=1

1205831119865119896119894

(119909) 119908119896

1=

119899

prod

119894=1

120583 1119865119896119894

(119909)

120601119896

1(119909) =

119908119896

1(119909)

1198631

119897

120601119896

1(119909) =

119908119896

1

1198631119903

119911119896

1(119909) =

119899

sum

119894=1

1119888119896

119894119909119894 +1119888119896

0 119896 = 1 119872

(24)

1198912 (119909) = 1205741199102

119897(119909) + (1 minus 120574) 119910

2

119903(119909) (25)

where

1199102

119897(119909) =

119872

sum

119896=1

120601119896

2(119909) 119911119896

2(119909) =

sum1198722

119896=1119908119896

2(119909) 119911119896

2(119909)

1198632

119897

(26)

1199102

119903(119909) =

1198722

sum

119896=1

120601119896

2(119909) 119911119896

2(119909) =

sum1198721

119896=1119908119896

2(119909) 119911119896

2(119909)

1198632119903

(27)

where

119908119896

2=

119899

prod

119894=1

1205832119865119896119894

(119909) 119908119896

2=

119899

prod

119894=1

120583 2119865119896119894

(119909)

1198632

119897=

1198722

sum

119896=1

119899

prod

119894=1

1205832119865119896119894

(119909) 1198632

119903=

1198722

sum

119896=1

119899

prod

119894=1

120583 2119865119896119894

(119909)

120601119896

2(119909) =

119908119896

2(119909)

1198632

119897

120601119896

2(119909) =

119908119896

2(119909)

1198632119903

119911119896

2(119909) =

119899

sum

119894=1

2119888119896

119894119909119894 +2119888119896

0 119896 = 1 119872

(28)




12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X



Z(X

)

z1

z3

z2

8

7

6

5

4

3

2

1

0


1

08

06

04

02

0

Small Medium Large


120006|U

0 5 0 5 11100



8

7

6

5

4

3

2

1

0

f|r(X

)




1198911 (119909) + 1198912 (119909)

=

sum1198721

1198961=1sum1198722

1198962=1[1199081198961

11199081198962

2(1205721199111198961

1+ 1205741199111198962

2)]

1198631

1198971198632

119897

+

sum1198721

1198961=1sum1198722

1198962=1[1199081198961

11199081198962

2(1 minus 120572) 119911

1198961

1+ (1 minus 120574) 119911

1198962

2]

11986311199031198632119903

(29)

and that Φ1198961 1198962

= (1199081198961

11199081198962

2)(1198631

1199031198632

119903) and Φ1198961 1198962

= (1199081198961

11199081198962

2)

(1198631

1199031198632



1+1205741199111198962

2and (1minus120572)119911

1198961

1+(1minus120574)119911

1198962

2multiplied





1and 119911

1198962





1198911 (119909) 1198912 (119909)

=1

1198631

1198971198632

119897

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

21205721205741199111198961

11199111198962

2]

]

+1

1198631

1198971198632119903

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2120572 (1 minus 120574) 119911

1198961

11199111198962

2]

]

+1

11986311199031198632

119897

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2(1 minus 120572) 120574119911

1198961

11199111198962

2]

]

+1

11986311199031198632119903

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2(1 minus 120572) (1 minus 120574) 119911

1198961

11199111198962

2]

]

(30)



11199111198962

2 120572(1minus120574)119911

1198961

11199111198962

2 (1minus

120572)1205741199111198961

11199111198962

2 and (1 minus 120572)(1 minus 120574)119911

1198961

11199111198962


FBFs (Φ11989711989711989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119897)Φ11989711990311989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119903)

Φ119903119897

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119897) and Φ

119903119903

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119903))



1and 119911

1198962

2


1and 119911

1198962


11199111198962

2is




119888119891 (119909) = 120572119888119910119897 (119909) + (1 minus 120572) 119888119910119903 (119909)

=

sum119872

119896=1[120572119863119903119908

119896(119909) + (1 minus 120572)119863119897119908

119896(119909)] 119888119911

119896(119909)

119863119897119863119903

(31)







0

1 1199090

2 119909

0

119899)

and 1199100= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2 and 119910

0

119894= (1199100

119897119894+

1199100

119903119894)2 with 119909

0

119894= 1199100


(1198651

119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(32)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(33)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(34)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(35)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



2 with 119911119896(119909) =

sum119899

119894=1119888119896

119894119909119894 + 119888

119896



119894 [1205831198651119894

1205831198651119894


119891 (1199090) = 120572 [120601

1(1199090) 1199111(1199090) + 1206012(1199090) 1199112(1199090)] + (1 minus 120572)

times [1206011

(1199090) 1199111(1199090) + (1 minus 120601

1

(1199090)) 1199112(1199090)]

(36)

where

1206011(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206011

(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

(37)

119891 (1199100) = 120572 [120601

1(1199100) 1199111(1199100) + 1206012(1199100) 1199112(1199100)] + (1 minus 120572)


(1199100)) 1199111(1199100) + 1206012

(1199100) 1199112(1199100)]

(38)

where

1206011(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1

(39)


119894 hence

we have prod119899

119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090



1199111= 0 and 119911

2= 1 then119891(119909

0) = 120572120601

2(1199090)+(1minus120572)[1minus120601

1

(1199090)] =

1205721206012(1199100) + (1 minus 120572)120601

2

(1199100) = 119891(119910






119896(119909) gt 0




119899rarr 119877 119909 = (1199091 1199092

119909119899) isin 119880 120583119865119896119894(119909)

isin [120583119865119896119894

(119909) 120583119865119896119894



1198941119898119896

119894

1119898119896


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909)

120601119896


119896

119897 119910119896

119903are


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909) 120601

119896

119903(119909) and let

119889infin(1198911 1198912) = sup119909isin119880



1198911 (119909) = 1205741199101

119897(119909) + (1 minus 120574) 119910

1

119903(119909) (40)

where

1199101

119897(119909) =

1198711

sum

1198961=1

11206011198961

119897(119909)11199111198961

119897(119909) +

1198721

sum

1198961=1198711+1

11206011198961

119897(119909)11199111198961

119897(119909)

=

sum1198711

1198961=11199081198961

1(119909)11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909)11199111198961

119897(119909)

1198631

119897

1199101

119903(119909) =

1198771

sum

1198961=1

11206011198961

119903(119909)11199111198961119903

(119909) +

1198721

sum

1198961=1198771+1

11206011198961

119903(119909)11199111198961119903

(119909)

=

sum1198771

1198961=11199081198961

1(119909)11199111198961119903

(119909) + sum1198721

1198961=1198771+11199081198961

1(119909)11199111198961119903

(119909)

1198631119903

(41)

where

1199081198961

1=

119899

prod

119894=1

12058311198651198961

119894

(119909) 1199081198961

1=

119899

prod

119894=1

120583 11198651198961

119894

(119909)

1198631

119897=

1198711

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198711+1

1199081198961

1

1198631

119903=

1198771

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198771+1

1199081198961

1

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1 1198711 (119909)

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1198711 (119909) + 1 1198721

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1 1198771 (119909)

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1198771 (119909) + 1 1198721

11199111198961

119897=

119899

sum

119894=1

11198881198961

119894119909119894 +1119888119896

0minus

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus11199041198961

0

1198961 = 1 1198721

11199111198961119903

=

119899

sum

119894=1

11198881198961

119894119909119894 +11198881198961

0+

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 +11199041198961

0

1198961 = 1 1198721

1198912 (119909) = 1205741199102

119897(119909) + (1 minus 120574) 119910

2

119903(119909)

(42)

where

1199102

119897(119909)

=

1198712

sum

1198962=1

21206011198962

119897(119909)21199111198962

119897(119909) +

1198722

sum

1198962=1198712+1

21206011198962

119897(119909)21199111198962

119897(119909)

=

sum1198712

1198962=11199081198962

2(119909)21199111198962

119897(119909) + sum

1198722

1198962=1198712+11199081198962

2(119909)21199111198962

119897(119909)

1198632

119897

(43)


1199102

119903(119909)

=

1198772

sum

1198962=1

21206011198962

119903(119909)21199111198962119903

(119909) +

1198722

sum

1198962=1198772+1

21206011198962

119903(119909)21199111198962119903

(119909)

=

sum11987721

1198962=11199081198962

2(119909)21199111198962119903

(119909) + sum1198722

1198962=1198772+11199081198962

2(119909)21199111198962119903

(119909)

1198631119903

(44)

where

1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909) 1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909)

1198632

119897=

1198712

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198712+1

1199081198962

2

1198632

119903=

1198772

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198772+1

1199081198962

2

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1 1198712 (119909)

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1198712 (119909) + 1 1198722

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1 1198772 (119909)

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1198772 (119909) + 1 1198722

21199111198962

119897=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0minus

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus21199041198962

0

1198962 = 1 1198722

21199111198962119903

=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0+

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 +21199041198962

0

1198962 = 1 1198722

(45)



1198911 (119909) + 1198912 (119909)

= ((

1198711

sum

1198961=1

1198712

sum

1198962=1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909) + 120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909)+120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198771

sum

1198961=1

1198772

sum

1198962=1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])times (1198631

1199031198632

119903)minus1

)

+ ((

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])

times (1198631

1199031198632

119903)minus1

)


119897+12057421199111198962

119897and (1minus120572)

11199111198961119903+(1minus120574)

21199111198962119903


119909isin119880(|119892(119909) minus 119891(119909)|) lt 120576



1and 119911

1198962





=120572120574

1198631

1198971198632

119897

times [

[

1198711

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198711+1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)


12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X


z1

z3

z2


8

7

6

5

4

3

2

1

0

Z|r(X

)


1

08

06

04

02

0

Small Medium Large


120006|U



8

7

6

5

4

3

2

1

0

f|r(X

)



+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)]

]

+120572 (1 minus 120574)

1198631

1198971198632119903

times [

[

1198711

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)]

]

+(1 minus 120572) 120574

11986311199031198632

119897

times [

[

1198771

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198712

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)]

]

+(1 minus 120572) (1 minus 120574)

11986311199031198632119903

times [

[

1198771

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)


+

1198721

sum

1198961=1198771+1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)]

]

(47)


119897

21199111198962

119897 120572(1 minus 120574)

11199111198961

119897

21199111198962119903

(1 minus 120572)12057411199111198961119903

21199111198962

119897 and (1 minus 120572)(1 minus 120574)

11199111198961119903

21199111198962119903




1and 119911

1198962



1198961

1and 119911

1198962

2were



(see Figure 10)



1198881198911 (119909)

= 1205721198881199101

119897(119909) + (1 minus 120572) 119888119910

1

119903(119909)

=

sum1198711

1198961=11199081198961

1(119909) 120572119888

11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909) 120572119888

11199111198961

119897(119909)

1198631

119897

+

sum1198771

1198961=11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

+

sum1198721

1198961=1198771+11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

(48)


11199111198961

119897(119909) and (1 minus






(1199090

1 1199090

2 119909

0

119899) and 119910

0= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2

and 1199100

119894= (1199100

119897119894+ 1199100

119903119894)2 with 119909

0

119894= 1199100


type-2 fuzzy sets (1198651119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(49)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(50)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(51)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(52)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



119897 1199111

119903] and 119911

2isin

[1199112

119897 1199112


except [119911119896119897 119911119896



119894 [120583119865119896119894

120583119865119896119894


119891 (1199090) = 120572 [120601

1

119897(1199090) 1199111

119897(1199090) + (1 minus 120601

1

119897(1199090)) 1199112

119897(1199090)]

+ (1 minus 120572) [1206011

119903(1199090) 1199111

119903(1199090) + 1206012

119903(1199090) 1199112

119903(1199090)]

(53)

where

1206011

119897(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

1206011

119903(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012

119903(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

119891 (1199100) = 120572 [120601

1

119897(1199100) 1199111

119897(1199100) + 1206012

119897(1199100) 1199112

119897(1199100)] + (1 minus 120572)


119903(1199100)) 1199111

119903(1199100) + 1206012

119903(1199100) 1199112

119903(1199100)]

(54)

where

1206012

119903(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1


1206011

119897(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

119897(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

(55)


119894= 1199100

119894


119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090

119894) = 1 If we

choose 1199111

119897119903= 0 and 119911

2

119897119903= 1 then 119891(119909

0) = 120572(1 minus 120601

1

119897(1199090)) +

(1 minus 120572)1206012

119903(1199090) = 120572120601

2

119897(1199100) + (1 minus 120572)120601

2

119903(1199100) = 119891(119910

0) Therefore





required 119891









100

10minus1

10minus2

10minus3

10minus4CV

-RM

SE

SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3





119891 (119906) = 06 sin (120587119906) + 03 sin (3120587119906) + 01 sin (5120587119906) + 120578

(56)





SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3

100

10minus02

10minus04

10minus06

10minus08

CV-R

MSE




119891 (1199091 1199092 1199093) = (1 + radic1199091 +1

1199092

+1

radic1199093

3

)

2

+ 120578 (57)



SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS TRN

CV-R

MSE

035

03

025

02

015

01

005

0




(119905) =01119909 (119905 minus 120591)

1 + 11990910 (119905 minus 120591)minus 01119909 (119905) (58)




5 Conclusions




SNR (dB)120591 = 60


0 03403 03714 NA 02388 02687 37 02027 02135 35 01495 01536 34 01244 01444 3110 01544 01714 NA 01312 01456 33 01069 01119 30 00805 00972 28 00532 00629 2520 00929 01007 NA 00708 00781 28 00566 00594 26 00424 00485 24 00342 00355 2130 00788 00847 NA 00526 00582 19 00411 00468 18 00309 00332 16 00227 00316 14Free 00749 00799 NA 00414 00477 10 00321 00353 9 00251 00319 6 00215 00293 4

SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS CHK

CV-R

MSE

04

035

03

025

02

015

01

005

0




Acknowledgments



References



















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





x1

x1x2

x2

sum

sum

sum

sum

sum

sum

sum

sum

sum

sum

sum

y

yl

yr

120574

1minus 120574

120583

120583

120583

120583

120583

120583

120583

120583

w11

w41

w52

w82

w62

w72

w21

w31

1b1

net1

1b2

net2

1b3

net3

1b4

net4

1b5

net5

1b6

net6

1b7

net7

1b8

net8

120583(net1) 120583(x)

120583(net2) 120583(x)

120583(net3) 120583(x)

120583(net4) 120583(x)

120583(net5) 120583(x)

120583(net6) 120583(x)

120583(net7) 120583(x)

120583(net8) 120583(x)

TT

TS

TT

TS

TT

TS

TT

TS

f1

f1

f2

f2

f3

f3

f4

f4

w1

w1

w2

w2

w3

w3

w4

w4

y1l

y1r

y2l

y2r

y3l

y3r

y4l

y4r

w1y1

w1y1

w2y2

w2y2

w3y3

w3y3

w4y4

w4y4


2120583119865119896119894

(119909119894 [120590119896

1198942119898119896

119894]) = exp[

[

minus1

2(119909119894 minus2119898119896

119894

120590119896

119894

)

2

]

]

(9)

120583119865119896119894

(119909) =

2120583119865119896119894

(119909119894 [120590119896

1198942119898119896

119894]) 119909119894 le

1119898119896

119894

1120583119865119896119894

(119909119894 [120590119896

1198941119898119896

119894]) 119909119894 gt

2119898119896

119894

(10)

120583119865119896119894

(119909) =

1120583119865119896119894

(119909119894 [120590119896

1198941119898119896

119894]) 119909119894 lt

1119898119896

119894

11119898119896

119894le 119909119894 le

2119898119896

119894

2120583119865119896119894

(119909119894 [120590119896

1198942119898119896

119894]) 119909119894 gt

2119898119896

119894

(11)





1199004

2119896minus1= 120601119896 119900

4

2119896= 120601119896

(12)

If we view 120601119896 120601119896




120601119896=

119908119896

119863119897

119896 = 1 119872 (13)

where119863119897 = sum119872

119896=1119908119896

120601119896

=119908119896

119863119903

119896 = 1 119872 (14)

where119863119903 = sum119872

119896=1119908119896


119896

119894 119888119896



1199005

2119896=1= 119910119896 119900

5

2119896= 119910119896 (15)

119910119896=

119899

sum

119894=1

119888119896

119894119909119894 + 119888119896

0 119896 = 1 119872 (16)


x1

x1x2

x2

|x1||x2|


sum

sum

sum

sum

sum

sum

sum

sum

sum

y

120583

120583

120583

120583

120583

120583

120583

120583

T T

S T

KMT T

S T

T T

S T

T

S T

T

KM

y1l

y1r

y2l

y2r

y3l

y3r

y4l

y4r

yl

yr

12

12

1b1

net1

1b2

net2

1b3

net3

1b4

net4

1b5

net5

1b6

net6

1b7

net7

1b8

net8

w11

w41

w52

w82

w62

w72

w21

w31

120583(net1)

120583(net2)

120583(x)

120583(x)

120583(net3)

120583(net4)

120583(x)

120583(x)

120583(net5)

120583(net6)

120583(x)

120583(net7) 120583(x)

120583(x)

120583(net8) 120583(x)

f1

f1

f2

f2

f3

f3

f4

f4



1199006

1= 119910119897 (119909) 119900

6

2= 119910119903 (119909) (17)

where

119910119897 (119909) =

119872

sum

119896=1

120601119896119910119896

119910119903 (119909) =

119872

sum

119896=1

120601119896

119910119896

(18)



1199007

1= 119910 (119909) (19)

where

119910 (119909) = 120574119910119897 (119909) + (1 minus 120574) 119910119903 (119909) (20)

119910 (119909) =

119872

sum

119896=1

[120574120601119896(119909) + (1 minus 120574) 120601

119896

(119909)] 119910119896(119909) (21)









119899


(|119892(119909)minus

119891(119909)|) lt 120576


119877119899

rarr 119877 119909 = (1199091 1199092 119909119899) isin 119880 120583119865119896119894(119909)

isin [120583119865119896119894

(119909)

120583119865119896119894


119896

1198941119898119896

1198942119898119896

119894]) defined by (27) and (31) If

we view 120601119896(119909) 120601119896




119896(119909) 120601119896





1198911 (119909) = 1205721199101

119897(119909) + (1 minus 120572) 119910

1

119903(119909) (22)

where

1199101

119897(119909) =

119872

sum

119896=1

120601119896

1(119909) 119911119896

1(119909) =

sum1198721

119896=1119908119896

1(119909) 119911119896

1(119909)

1198631

119897

1199101

119903(119909) =

1198721

sum

119896=1

120601119896

1(119909) 119911119896

1(119909) =

sum1198721

119896=1119908119896

1(119909) 119911119896

1(119909)

1198631119903

(23)

where

1198631

119897=

1198721

sum

119896=1

119899

prod

119894=1

1205831119865119896119894

(119909) 1198631

119903=

1198721

sum

1198961=1

119899

prod

119894=1

120583 1119865119896119894

(119909)

119908119896

1=

119899

prod

119894=1

1205831119865119896119894

(119909) 119908119896

1=

119899

prod

119894=1

120583 1119865119896119894

(119909)

120601119896

1(119909) =

119908119896

1(119909)

1198631

119897

120601119896

1(119909) =

119908119896

1

1198631119903

119911119896

1(119909) =

119899

sum

119894=1

1119888119896

119894119909119894 +1119888119896

0 119896 = 1 119872

(24)

1198912 (119909) = 1205741199102

119897(119909) + (1 minus 120574) 119910

2

119903(119909) (25)

where

1199102

119897(119909) =

119872

sum

119896=1

120601119896

2(119909) 119911119896

2(119909) =

sum1198722

119896=1119908119896

2(119909) 119911119896

2(119909)

1198632

119897

(26)

1199102

119903(119909) =

1198722

sum

119896=1

120601119896

2(119909) 119911119896

2(119909) =

sum1198721

119896=1119908119896

2(119909) 119911119896

2(119909)

1198632119903

(27)

where

119908119896

2=

119899

prod

119894=1

1205832119865119896119894

(119909) 119908119896

2=

119899

prod

119894=1

120583 2119865119896119894

(119909)

1198632

119897=

1198722

sum

119896=1

119899

prod

119894=1

1205832119865119896119894

(119909) 1198632

119903=

1198722

sum

119896=1

119899

prod

119894=1

120583 2119865119896119894

(119909)

120601119896

2(119909) =

119908119896

2(119909)

1198632

119897

120601119896

2(119909) =

119908119896

2(119909)

1198632119903

119911119896

2(119909) =

119899

sum

119894=1

2119888119896

119894119909119894 +2119888119896

0 119896 = 1 119872

(28)




12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X



Z(X

)

z1

z3

z2

8

7

6

5

4

3

2

1

0


1

08

06

04

02

0

Small Medium Large


120006|U

0 5 0 5 11100



8

7

6

5

4

3

2

1

0

f|r(X

)




1198911 (119909) + 1198912 (119909)

=

sum1198721

1198961=1sum1198722

1198962=1[1199081198961

11199081198962

2(1205721199111198961

1+ 1205741199111198962

2)]

1198631

1198971198632

119897

+

sum1198721

1198961=1sum1198722

1198962=1[1199081198961

11199081198962

2(1 minus 120572) 119911

1198961

1+ (1 minus 120574) 119911

1198962

2]

11986311199031198632119903

(29)

and that Φ1198961 1198962

= (1199081198961

11199081198962

2)(1198631

1199031198632

119903) and Φ1198961 1198962

= (1199081198961

11199081198962

2)

(1198631

1199031198632



1+1205741199111198962

2and (1minus120572)119911

1198961

1+(1minus120574)119911

1198962

2multiplied





1and 119911

1198962





1198911 (119909) 1198912 (119909)

=1

1198631

1198971198632

119897

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

21205721205741199111198961

11199111198962

2]

]

+1

1198631

1198971198632119903

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2120572 (1 minus 120574) 119911

1198961

11199111198962

2]

]

+1

11986311199031198632

119897

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2(1 minus 120572) 120574119911

1198961

11199111198962

2]

]

+1

11986311199031198632119903

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2(1 minus 120572) (1 minus 120574) 119911

1198961

11199111198962

2]

]

(30)



11199111198962

2 120572(1minus120574)119911

1198961

11199111198962

2 (1minus

120572)1205741199111198961

11199111198962

2 and (1 minus 120572)(1 minus 120574)119911

1198961

11199111198962


FBFs (Φ11989711989711989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119897)Φ11989711990311989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119903)

Φ119903119897

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119897) and Φ

119903119903

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119903))



1and 119911

1198962

2


1and 119911

1198962


11199111198962

2is




119888119891 (119909) = 120572119888119910119897 (119909) + (1 minus 120572) 119888119910119903 (119909)

=

sum119872

119896=1[120572119863119903119908

119896(119909) + (1 minus 120572)119863119897119908

119896(119909)] 119888119911

119896(119909)

119863119897119863119903

(31)







0

1 1199090

2 119909

0

119899)

and 1199100= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2 and 119910

0

119894= (1199100

119897119894+

1199100

119903119894)2 with 119909

0

119894= 1199100


(1198651

119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(32)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(33)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(34)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(35)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



2 with 119911119896(119909) =

sum119899

119894=1119888119896

119894119909119894 + 119888

119896



119894 [1205831198651119894

1205831198651119894


119891 (1199090) = 120572 [120601

1(1199090) 1199111(1199090) + 1206012(1199090) 1199112(1199090)] + (1 minus 120572)

times [1206011

(1199090) 1199111(1199090) + (1 minus 120601

1

(1199090)) 1199112(1199090)]

(36)

where

1206011(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206011

(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

(37)

119891 (1199100) = 120572 [120601

1(1199100) 1199111(1199100) + 1206012(1199100) 1199112(1199100)] + (1 minus 120572)


(1199100)) 1199111(1199100) + 1206012

(1199100) 1199112(1199100)]

(38)

where

1206011(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1

(39)


119894 hence

we have prod119899

119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090



1199111= 0 and 119911

2= 1 then119891(119909

0) = 120572120601

2(1199090)+(1minus120572)[1minus120601

1

(1199090)] =

1205721206012(1199100) + (1 minus 120572)120601

2

(1199100) = 119891(119910






119896(119909) gt 0




119899rarr 119877 119909 = (1199091 1199092

119909119899) isin 119880 120583119865119896119894(119909)

isin [120583119865119896119894

(119909) 120583119865119896119894



1198941119898119896

119894

1119898119896


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909)

120601119896


119896

119897 119910119896

119903are


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909) 120601

119896

119903(119909) and let

119889infin(1198911 1198912) = sup119909isin119880



1198911 (119909) = 1205741199101

119897(119909) + (1 minus 120574) 119910

1

119903(119909) (40)

where

1199101

119897(119909) =

1198711

sum

1198961=1

11206011198961

119897(119909)11199111198961

119897(119909) +

1198721

sum

1198961=1198711+1

11206011198961

119897(119909)11199111198961

119897(119909)

=

sum1198711

1198961=11199081198961

1(119909)11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909)11199111198961

119897(119909)

1198631

119897

1199101

119903(119909) =

1198771

sum

1198961=1

11206011198961

119903(119909)11199111198961119903

(119909) +

1198721

sum

1198961=1198771+1

11206011198961

119903(119909)11199111198961119903

(119909)

=

sum1198771

1198961=11199081198961

1(119909)11199111198961119903

(119909) + sum1198721

1198961=1198771+11199081198961

1(119909)11199111198961119903

(119909)

1198631119903

(41)

where

1199081198961

1=

119899

prod

119894=1

12058311198651198961

119894

(119909) 1199081198961

1=

119899

prod

119894=1

120583 11198651198961

119894

(119909)

1198631

119897=

1198711

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198711+1

1199081198961

1

1198631

119903=

1198771

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198771+1

1199081198961

1

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1 1198711 (119909)

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1198711 (119909) + 1 1198721

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1 1198771 (119909)

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1198771 (119909) + 1 1198721

11199111198961

119897=

119899

sum

119894=1

11198881198961

119894119909119894 +1119888119896

0minus

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus11199041198961

0

1198961 = 1 1198721

11199111198961119903

=

119899

sum

119894=1

11198881198961

119894119909119894 +11198881198961

0+

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 +11199041198961

0

1198961 = 1 1198721

1198912 (119909) = 1205741199102

119897(119909) + (1 minus 120574) 119910

2

119903(119909)

(42)

where

1199102

119897(119909)

=

1198712

sum

1198962=1

21206011198962

119897(119909)21199111198962

119897(119909) +

1198722

sum

1198962=1198712+1

21206011198962

119897(119909)21199111198962

119897(119909)

=

sum1198712

1198962=11199081198962

2(119909)21199111198962

119897(119909) + sum

1198722

1198962=1198712+11199081198962

2(119909)21199111198962

119897(119909)

1198632

119897

(43)


1199102

119903(119909)

=

1198772

sum

1198962=1

21206011198962

119903(119909)21199111198962119903

(119909) +

1198722

sum

1198962=1198772+1

21206011198962

119903(119909)21199111198962119903

(119909)

=

sum11987721

1198962=11199081198962

2(119909)21199111198962119903

(119909) + sum1198722

1198962=1198772+11199081198962

2(119909)21199111198962119903

(119909)

1198631119903

(44)

where

1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909) 1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909)

1198632

119897=

1198712

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198712+1

1199081198962

2

1198632

119903=

1198772

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198772+1

1199081198962

2

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1 1198712 (119909)

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1198712 (119909) + 1 1198722

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1 1198772 (119909)

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1198772 (119909) + 1 1198722

21199111198962

119897=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0minus

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus21199041198962

0

1198962 = 1 1198722

21199111198962119903

=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0+

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 +21199041198962

0

1198962 = 1 1198722

(45)



1198911 (119909) + 1198912 (119909)

= ((

1198711

sum

1198961=1

1198712

sum

1198962=1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909) + 120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909)+120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198771

sum

1198961=1

1198772

sum

1198962=1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])times (1198631

1199031198632

119903)minus1

)

+ ((

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])

times (1198631

1199031198632

119903)minus1

)


119897+12057421199111198962

119897and (1minus120572)

11199111198961119903+(1minus120574)

21199111198962119903


119909isin119880(|119892(119909) minus 119891(119909)|) lt 120576



1and 119911

1198962





=120572120574

1198631

1198971198632

119897

times [

[

1198711

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198711+1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)


12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X


z1

z3

z2


8

7

6

5

4

3

2

1

0

Z|r(X

)


1

08

06

04

02

0

Small Medium Large


120006|U



8

7

6

5

4

3

2

1

0

f|r(X

)



+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)]

]

+120572 (1 minus 120574)

1198631

1198971198632119903

times [

[

1198711

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)]

]

+(1 minus 120572) 120574

11986311199031198632

119897

times [

[

1198771

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198712

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)]

]

+(1 minus 120572) (1 minus 120574)

11986311199031198632119903

times [

[

1198771

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)


+

1198721

sum

1198961=1198771+1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)]

]

(47)


119897

21199111198962

119897 120572(1 minus 120574)

11199111198961

119897

21199111198962119903

(1 minus 120572)12057411199111198961119903

21199111198962

119897 and (1 minus 120572)(1 minus 120574)

11199111198961119903

21199111198962119903




1and 119911

1198962



1198961

1and 119911

1198962

2were



(see Figure 10)



1198881198911 (119909)

= 1205721198881199101

119897(119909) + (1 minus 120572) 119888119910

1

119903(119909)

=

sum1198711

1198961=11199081198961

1(119909) 120572119888

11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909) 120572119888

11199111198961

119897(119909)

1198631

119897

+

sum1198771

1198961=11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

+

sum1198721

1198961=1198771+11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

(48)


11199111198961

119897(119909) and (1 minus






(1199090

1 1199090

2 119909

0

119899) and 119910

0= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2

and 1199100

119894= (1199100

119897119894+ 1199100

119903119894)2 with 119909

0

119894= 1199100


type-2 fuzzy sets (1198651119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(49)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(50)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(51)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(52)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



119897 1199111

119903] and 119911

2isin

[1199112

119897 1199112


except [119911119896119897 119911119896



119894 [120583119865119896119894

120583119865119896119894


119891 (1199090) = 120572 [120601

1

119897(1199090) 1199111

119897(1199090) + (1 minus 120601

1

119897(1199090)) 1199112

119897(1199090)]

+ (1 minus 120572) [1206011

119903(1199090) 1199111

119903(1199090) + 1206012

119903(1199090) 1199112

119903(1199090)]

(53)

where

1206011

119897(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

1206011

119903(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012

119903(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

119891 (1199100) = 120572 [120601

1

119897(1199100) 1199111

119897(1199100) + 1206012

119897(1199100) 1199112

119897(1199100)] + (1 minus 120572)


119903(1199100)) 1199111

119903(1199100) + 1206012

119903(1199100) 1199112

119903(1199100)]

(54)

where

1206012

119903(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1


1206011

119897(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

119897(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

(55)


119894= 1199100

119894


119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090

119894) = 1 If we

choose 1199111

119897119903= 0 and 119911

2

119897119903= 1 then 119891(119909

0) = 120572(1 minus 120601

1

119897(1199090)) +

(1 minus 120572)1206012

119903(1199090) = 120572120601

2

119897(1199100) + (1 minus 120572)120601

2

119903(1199100) = 119891(119910

0) Therefore





required 119891









100

10minus1

10minus2

10minus3

10minus4CV

-RM

SE

SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3





119891 (119906) = 06 sin (120587119906) + 03 sin (3120587119906) + 01 sin (5120587119906) + 120578

(56)





SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3

100

10minus02

10minus04

10minus06

10minus08

CV-R

MSE




119891 (1199091 1199092 1199093) = (1 + radic1199091 +1

1199092

+1

radic1199093

3

)

2

+ 120578 (57)



SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS TRN

CV-R

MSE

035

03

025

02

015

01

005

0




(119905) =01119909 (119905 minus 120591)

1 + 11990910 (119905 minus 120591)minus 01119909 (119905) (58)




5 Conclusions




SNR (dB)120591 = 60


0 03403 03714 NA 02388 02687 37 02027 02135 35 01495 01536 34 01244 01444 3110 01544 01714 NA 01312 01456 33 01069 01119 30 00805 00972 28 00532 00629 2520 00929 01007 NA 00708 00781 28 00566 00594 26 00424 00485 24 00342 00355 2130 00788 00847 NA 00526 00582 19 00411 00468 18 00309 00332 16 00227 00316 14Free 00749 00799 NA 00414 00477 10 00321 00353 9 00251 00319 6 00215 00293 4

SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS CHK

CV-R

MSE

04

035

03

025

02

015

01

005

0




Acknowledgments



References



















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




x1

x1x2

x2

|x1||x2|


sum

sum

sum

sum

sum

sum

sum

sum

sum

y

120583

120583

120583

120583

120583

120583

120583

120583

T T

S T

KMT T

S T

T T

S T

T

S T

T

KM

y1l

y1r

y2l

y2r

y3l

y3r

y4l

y4r

yl

yr

12

12

1b1

net1

1b2

net2

1b3

net3

1b4

net4

1b5

net5

1b6

net6

1b7

net7

1b8

net8

w11

w41

w52

w82

w62

w72

w21

w31

120583(net1)

120583(net2)

120583(x)

120583(x)

120583(net3)

120583(net4)

120583(x)

120583(x)

120583(net5)

120583(net6)

120583(x)

120583(net7) 120583(x)

120583(x)

120583(net8) 120583(x)

f1

f1

f2

f2

f3

f3

f4

f4



1199006

1= 119910119897 (119909) 119900

6

2= 119910119903 (119909) (17)

where

119910119897 (119909) =

119872

sum

119896=1

120601119896119910119896

119910119903 (119909) =

119872

sum

119896=1

120601119896

119910119896

(18)



1199007

1= 119910 (119909) (19)

where

119910 (119909) = 120574119910119897 (119909) + (1 minus 120574) 119910119903 (119909) (20)

119910 (119909) =

119872

sum

119896=1

[120574120601119896(119909) + (1 minus 120574) 120601

119896

(119909)] 119910119896(119909) (21)









119899


(|119892(119909)minus

119891(119909)|) lt 120576


119877119899

rarr 119877 119909 = (1199091 1199092 119909119899) isin 119880 120583119865119896119894(119909)

isin [120583119865119896119894

(119909)

120583119865119896119894


119896

1198941119898119896

1198942119898119896

119894]) defined by (27) and (31) If

we view 120601119896(119909) 120601119896




119896(119909) 120601119896





1198911 (119909) = 1205721199101

119897(119909) + (1 minus 120572) 119910

1

119903(119909) (22)

where

1199101

119897(119909) =

119872

sum

119896=1

120601119896

1(119909) 119911119896

1(119909) =

sum1198721

119896=1119908119896

1(119909) 119911119896

1(119909)

1198631

119897

1199101

119903(119909) =

1198721

sum

119896=1

120601119896

1(119909) 119911119896

1(119909) =

sum1198721

119896=1119908119896

1(119909) 119911119896

1(119909)

1198631119903

(23)

where

1198631

119897=

1198721

sum

119896=1

119899

prod

119894=1

1205831119865119896119894

(119909) 1198631

119903=

1198721

sum

1198961=1

119899

prod

119894=1

120583 1119865119896119894

(119909)

119908119896

1=

119899

prod

119894=1

1205831119865119896119894

(119909) 119908119896

1=

119899

prod

119894=1

120583 1119865119896119894

(119909)

120601119896

1(119909) =

119908119896

1(119909)

1198631

119897

120601119896

1(119909) =

119908119896

1

1198631119903

119911119896

1(119909) =

119899

sum

119894=1

1119888119896

119894119909119894 +1119888119896

0 119896 = 1 119872

(24)

1198912 (119909) = 1205741199102

119897(119909) + (1 minus 120574) 119910

2

119903(119909) (25)

where

1199102

119897(119909) =

119872

sum

119896=1

120601119896

2(119909) 119911119896

2(119909) =

sum1198722

119896=1119908119896

2(119909) 119911119896

2(119909)

1198632

119897

(26)

1199102

119903(119909) =

1198722

sum

119896=1

120601119896

2(119909) 119911119896

2(119909) =

sum1198721

119896=1119908119896

2(119909) 119911119896

2(119909)

1198632119903

(27)

where

119908119896

2=

119899

prod

119894=1

1205832119865119896119894

(119909) 119908119896

2=

119899

prod

119894=1

120583 2119865119896119894

(119909)

1198632

119897=

1198722

sum

119896=1

119899

prod

119894=1

1205832119865119896119894

(119909) 1198632

119903=

1198722

sum

119896=1

119899

prod

119894=1

120583 2119865119896119894

(119909)

120601119896

2(119909) =

119908119896

2(119909)

1198632

119897

120601119896

2(119909) =

119908119896

2(119909)

1198632119903

119911119896

2(119909) =

119899

sum

119894=1

2119888119896

119894119909119894 +2119888119896

0 119896 = 1 119872

(28)




12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X



Z(X

)

z1

z3

z2

8

7

6

5

4

3

2

1

0


1

08

06

04

02

0

Small Medium Large


120006|U

0 5 0 5 11100



8

7

6

5

4

3

2

1

0

f|r(X

)




1198911 (119909) + 1198912 (119909)

=

sum1198721

1198961=1sum1198722

1198962=1[1199081198961

11199081198962

2(1205721199111198961

1+ 1205741199111198962

2)]

1198631

1198971198632

119897

+

sum1198721

1198961=1sum1198722

1198962=1[1199081198961

11199081198962

2(1 minus 120572) 119911

1198961

1+ (1 minus 120574) 119911

1198962

2]

11986311199031198632119903

(29)

and that Φ1198961 1198962

= (1199081198961

11199081198962

2)(1198631

1199031198632

119903) and Φ1198961 1198962

= (1199081198961

11199081198962

2)

(1198631

1199031198632



1+1205741199111198962

2and (1minus120572)119911

1198961

1+(1minus120574)119911

1198962

2multiplied





1and 119911

1198962





1198911 (119909) 1198912 (119909)

=1

1198631

1198971198632

119897

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

21205721205741199111198961

11199111198962

2]

]

+1

1198631

1198971198632119903

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2120572 (1 minus 120574) 119911

1198961

11199111198962

2]

]

+1

11986311199031198632

119897

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2(1 minus 120572) 120574119911

1198961

11199111198962

2]

]

+1

11986311199031198632119903

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2(1 minus 120572) (1 minus 120574) 119911

1198961

11199111198962

2]

]

(30)



11199111198962

2 120572(1minus120574)119911

1198961

11199111198962

2 (1minus

120572)1205741199111198961

11199111198962

2 and (1 minus 120572)(1 minus 120574)119911

1198961

11199111198962


FBFs (Φ11989711989711989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119897)Φ11989711990311989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119903)

Φ119903119897

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119897) and Φ

119903119903

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119903))



1and 119911

1198962

2


1and 119911

1198962


11199111198962

2is




119888119891 (119909) = 120572119888119910119897 (119909) + (1 minus 120572) 119888119910119903 (119909)

=

sum119872

119896=1[120572119863119903119908

119896(119909) + (1 minus 120572)119863119897119908

119896(119909)] 119888119911

119896(119909)

119863119897119863119903

(31)







0

1 1199090

2 119909

0

119899)

and 1199100= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2 and 119910

0

119894= (1199100

119897119894+

1199100

119903119894)2 with 119909

0

119894= 1199100


(1198651

119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(32)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(33)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(34)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(35)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



2 with 119911119896(119909) =

sum119899

119894=1119888119896

119894119909119894 + 119888

119896



119894 [1205831198651119894

1205831198651119894


119891 (1199090) = 120572 [120601

1(1199090) 1199111(1199090) + 1206012(1199090) 1199112(1199090)] + (1 minus 120572)

times [1206011

(1199090) 1199111(1199090) + (1 minus 120601

1

(1199090)) 1199112(1199090)]

(36)

where

1206011(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206011

(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

(37)

119891 (1199100) = 120572 [120601

1(1199100) 1199111(1199100) + 1206012(1199100) 1199112(1199100)] + (1 minus 120572)


(1199100)) 1199111(1199100) + 1206012

(1199100) 1199112(1199100)]

(38)

where

1206011(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1

(39)


119894 hence

we have prod119899

119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090



1199111= 0 and 119911

2= 1 then119891(119909

0) = 120572120601

2(1199090)+(1minus120572)[1minus120601

1

(1199090)] =

1205721206012(1199100) + (1 minus 120572)120601

2

(1199100) = 119891(119910






119896(119909) gt 0




119899rarr 119877 119909 = (1199091 1199092

119909119899) isin 119880 120583119865119896119894(119909)

isin [120583119865119896119894

(119909) 120583119865119896119894



1198941119898119896

119894

1119898119896


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909)

120601119896


119896

119897 119910119896

119903are


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909) 120601

119896

119903(119909) and let

119889infin(1198911 1198912) = sup119909isin119880



1198911 (119909) = 1205741199101

119897(119909) + (1 minus 120574) 119910

1

119903(119909) (40)

where

1199101

119897(119909) =

1198711

sum

1198961=1

11206011198961

119897(119909)11199111198961

119897(119909) +

1198721

sum

1198961=1198711+1

11206011198961

119897(119909)11199111198961

119897(119909)

=

sum1198711

1198961=11199081198961

1(119909)11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909)11199111198961

119897(119909)

1198631

119897

1199101

119903(119909) =

1198771

sum

1198961=1

11206011198961

119903(119909)11199111198961119903

(119909) +

1198721

sum

1198961=1198771+1

11206011198961

119903(119909)11199111198961119903

(119909)

=

sum1198771

1198961=11199081198961

1(119909)11199111198961119903

(119909) + sum1198721

1198961=1198771+11199081198961

1(119909)11199111198961119903

(119909)

1198631119903

(41)

where

1199081198961

1=

119899

prod

119894=1

12058311198651198961

119894

(119909) 1199081198961

1=

119899

prod

119894=1

120583 11198651198961

119894

(119909)

1198631

119897=

1198711

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198711+1

1199081198961

1

1198631

119903=

1198771

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198771+1

1199081198961

1

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1 1198711 (119909)

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1198711 (119909) + 1 1198721

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1 1198771 (119909)

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1198771 (119909) + 1 1198721

11199111198961

119897=

119899

sum

119894=1

11198881198961

119894119909119894 +1119888119896

0minus

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus11199041198961

0

1198961 = 1 1198721

11199111198961119903

=

119899

sum

119894=1

11198881198961

119894119909119894 +11198881198961

0+

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 +11199041198961

0

1198961 = 1 1198721

1198912 (119909) = 1205741199102

119897(119909) + (1 minus 120574) 119910

2

119903(119909)

(42)

where

1199102

119897(119909)

=

1198712

sum

1198962=1

21206011198962

119897(119909)21199111198962

119897(119909) +

1198722

sum

1198962=1198712+1

21206011198962

119897(119909)21199111198962

119897(119909)

=

sum1198712

1198962=11199081198962

2(119909)21199111198962

119897(119909) + sum

1198722

1198962=1198712+11199081198962

2(119909)21199111198962

119897(119909)

1198632

119897

(43)


1199102

119903(119909)

=

1198772

sum

1198962=1

21206011198962

119903(119909)21199111198962119903

(119909) +

1198722

sum

1198962=1198772+1

21206011198962

119903(119909)21199111198962119903

(119909)

=

sum11987721

1198962=11199081198962

2(119909)21199111198962119903

(119909) + sum1198722

1198962=1198772+11199081198962

2(119909)21199111198962119903

(119909)

1198631119903

(44)

where

1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909) 1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909)

1198632

119897=

1198712

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198712+1

1199081198962

2

1198632

119903=

1198772

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198772+1

1199081198962

2

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1 1198712 (119909)

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1198712 (119909) + 1 1198722

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1 1198772 (119909)

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1198772 (119909) + 1 1198722

21199111198962

119897=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0minus

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus21199041198962

0

1198962 = 1 1198722

21199111198962119903

=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0+

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 +21199041198962

0

1198962 = 1 1198722

(45)



1198911 (119909) + 1198912 (119909)

= ((

1198711

sum

1198961=1

1198712

sum

1198962=1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909) + 120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909)+120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198771

sum

1198961=1

1198772

sum

1198962=1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])times (1198631

1199031198632

119903)minus1

)

+ ((

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])

times (1198631

1199031198632

119903)minus1

)


119897+12057421199111198962

119897and (1minus120572)

11199111198961119903+(1minus120574)

21199111198962119903


119909isin119880(|119892(119909) minus 119891(119909)|) lt 120576



1and 119911

1198962





=120572120574

1198631

1198971198632

119897

times [

[

1198711

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198711+1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)


12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X


z1

z3

z2


8

7

6

5

4

3

2

1

0

Z|r(X

)


1

08

06

04

02

0

Small Medium Large


120006|U



8

7

6

5

4

3

2

1

0

f|r(X

)



+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)]

]

+120572 (1 minus 120574)

1198631

1198971198632119903

times [

[

1198711

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)]

]

+(1 minus 120572) 120574

11986311199031198632

119897

times [

[

1198771

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198712

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)]

]

+(1 minus 120572) (1 minus 120574)

11986311199031198632119903

times [

[

1198771

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)


+

1198721

sum

1198961=1198771+1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)]

]

(47)


119897

21199111198962

119897 120572(1 minus 120574)

11199111198961

119897

21199111198962119903

(1 minus 120572)12057411199111198961119903

21199111198962

119897 and (1 minus 120572)(1 minus 120574)

11199111198961119903

21199111198962119903




1and 119911

1198962



1198961

1and 119911

1198962

2were



(see Figure 10)



1198881198911 (119909)

= 1205721198881199101

119897(119909) + (1 minus 120572) 119888119910

1

119903(119909)

=

sum1198711

1198961=11199081198961

1(119909) 120572119888

11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909) 120572119888

11199111198961

119897(119909)

1198631

119897

+

sum1198771

1198961=11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

+

sum1198721

1198961=1198771+11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

(48)


11199111198961

119897(119909) and (1 minus






(1199090

1 1199090

2 119909

0

119899) and 119910

0= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2

and 1199100

119894= (1199100

119897119894+ 1199100

119903119894)2 with 119909

0

119894= 1199100


type-2 fuzzy sets (1198651119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(49)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(50)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(51)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(52)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



119897 1199111

119903] and 119911

2isin

[1199112

119897 1199112


except [119911119896119897 119911119896



119894 [120583119865119896119894

120583119865119896119894


119891 (1199090) = 120572 [120601

1

119897(1199090) 1199111

119897(1199090) + (1 minus 120601

1

119897(1199090)) 1199112

119897(1199090)]

+ (1 minus 120572) [1206011

119903(1199090) 1199111

119903(1199090) + 1206012

119903(1199090) 1199112

119903(1199090)]

(53)

where

1206011

119897(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

1206011

119903(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012

119903(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

119891 (1199100) = 120572 [120601

1

119897(1199100) 1199111

119897(1199100) + 1206012

119897(1199100) 1199112

119897(1199100)] + (1 minus 120572)


119903(1199100)) 1199111

119903(1199100) + 1206012

119903(1199100) 1199112

119903(1199100)]

(54)

where

1206012

119903(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1


1206011

119897(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

119897(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

(55)


119894= 1199100

119894


119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090

119894) = 1 If we

choose 1199111

119897119903= 0 and 119911

2

119897119903= 1 then 119891(119909

0) = 120572(1 minus 120601

1

119897(1199090)) +

(1 minus 120572)1206012

119903(1199090) = 120572120601

2

119897(1199100) + (1 minus 120572)120601

2

119903(1199100) = 119891(119910

0) Therefore





required 119891









100

10minus1

10minus2

10minus3

10minus4CV

-RM

SE

SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3





119891 (119906) = 06 sin (120587119906) + 03 sin (3120587119906) + 01 sin (5120587119906) + 120578

(56)





SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3

100

10minus02

10minus04

10minus06

10minus08

CV-R

MSE




119891 (1199091 1199092 1199093) = (1 + radic1199091 +1

1199092

+1

radic1199093

3

)

2

+ 120578 (57)



SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS TRN

CV-R

MSE

035

03

025

02

015

01

005

0




(119905) =01119909 (119905 minus 120591)

1 + 11990910 (119905 minus 120591)minus 01119909 (119905) (58)




5 Conclusions




SNR (dB)120591 = 60


0 03403 03714 NA 02388 02687 37 02027 02135 35 01495 01536 34 01244 01444 3110 01544 01714 NA 01312 01456 33 01069 01119 30 00805 00972 28 00532 00629 2520 00929 01007 NA 00708 00781 28 00566 00594 26 00424 00485 24 00342 00355 2130 00788 00847 NA 00526 00582 19 00411 00468 18 00309 00332 16 00227 00316 14Free 00749 00799 NA 00414 00477 10 00321 00353 9 00251 00319 6 00215 00293 4

SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS CHK

CV-R

MSE

04

035

03

025

02

015

01

005

0




Acknowledgments



References



















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










119899


(|119892(119909)minus

119891(119909)|) lt 120576


119877119899

rarr 119877 119909 = (1199091 1199092 119909119899) isin 119880 120583119865119896119894(119909)

isin [120583119865119896119894

(119909)

120583119865119896119894


119896

1198941119898119896

1198942119898119896

119894]) defined by (27) and (31) If

we view 120601119896(119909) 120601119896




119896(119909) 120601119896





1198911 (119909) = 1205721199101

119897(119909) + (1 minus 120572) 119910

1

119903(119909) (22)

where

1199101

119897(119909) =

119872

sum

119896=1

120601119896

1(119909) 119911119896

1(119909) =

sum1198721

119896=1119908119896

1(119909) 119911119896

1(119909)

1198631

119897

1199101

119903(119909) =

1198721

sum

119896=1

120601119896

1(119909) 119911119896

1(119909) =

sum1198721

119896=1119908119896

1(119909) 119911119896

1(119909)

1198631119903

(23)

where

1198631

119897=

1198721

sum

119896=1

119899

prod

119894=1

1205831119865119896119894

(119909) 1198631

119903=

1198721

sum

1198961=1

119899

prod

119894=1

120583 1119865119896119894

(119909)

119908119896

1=

119899

prod

119894=1

1205831119865119896119894

(119909) 119908119896

1=

119899

prod

119894=1

120583 1119865119896119894

(119909)

120601119896

1(119909) =

119908119896

1(119909)

1198631

119897

120601119896

1(119909) =

119908119896

1

1198631119903

119911119896

1(119909) =

119899

sum

119894=1

1119888119896

119894119909119894 +1119888119896

0 119896 = 1 119872

(24)

1198912 (119909) = 1205741199102

119897(119909) + (1 minus 120574) 119910

2

119903(119909) (25)

where

1199102

119897(119909) =

119872

sum

119896=1

120601119896

2(119909) 119911119896

2(119909) =

sum1198722

119896=1119908119896

2(119909) 119911119896

2(119909)

1198632

119897

(26)

1199102

119903(119909) =

1198722

sum

119896=1

120601119896

2(119909) 119911119896

2(119909) =

sum1198721

119896=1119908119896

2(119909) 119911119896

2(119909)

1198632119903

(27)

where

119908119896

2=

119899

prod

119894=1

1205832119865119896119894

(119909) 119908119896

2=

119899

prod

119894=1

120583 2119865119896119894

(119909)

1198632

119897=

1198722

sum

119896=1

119899

prod

119894=1

1205832119865119896119894

(119909) 1198632

119903=

1198722

sum

119896=1

119899

prod

119894=1

120583 2119865119896119894

(119909)

120601119896

2(119909) =

119908119896

2(119909)

1198632

119897

120601119896

2(119909) =

119908119896

2(119909)

1198632119903

119911119896

2(119909) =

119899

sum

119894=1

2119888119896

119894119909119894 +2119888119896

0 119896 = 1 119872

(28)




12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X



Z(X

)

z1

z3

z2

8

7

6

5

4

3

2

1

0


1

08

06

04

02

0

Small Medium Large


120006|U

0 5 0 5 11100



8

7

6

5

4

3

2

1

0

f|r(X

)




1198911 (119909) + 1198912 (119909)

=

sum1198721

1198961=1sum1198722

1198962=1[1199081198961

11199081198962

2(1205721199111198961

1+ 1205741199111198962

2)]

1198631

1198971198632

119897

+

sum1198721

1198961=1sum1198722

1198962=1[1199081198961

11199081198962

2(1 minus 120572) 119911

1198961

1+ (1 minus 120574) 119911

1198962

2]

11986311199031198632119903

(29)

and that Φ1198961 1198962

= (1199081198961

11199081198962

2)(1198631

1199031198632

119903) and Φ1198961 1198962

= (1199081198961

11199081198962

2)

(1198631

1199031198632



1+1205741199111198962

2and (1minus120572)119911

1198961

1+(1minus120574)119911

1198962

2multiplied





1and 119911

1198962





1198911 (119909) 1198912 (119909)

=1

1198631

1198971198632

119897

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

21205721205741199111198961

11199111198962

2]

]

+1

1198631

1198971198632119903

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2120572 (1 minus 120574) 119911

1198961

11199111198962

2]

]

+1

11986311199031198632

119897

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2(1 minus 120572) 120574119911

1198961

11199111198962

2]

]

+1

11986311199031198632119903

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2(1 minus 120572) (1 minus 120574) 119911

1198961

11199111198962

2]

]

(30)



11199111198962

2 120572(1minus120574)119911

1198961

11199111198962

2 (1minus

120572)1205741199111198961

11199111198962

2 and (1 minus 120572)(1 minus 120574)119911

1198961

11199111198962


FBFs (Φ11989711989711989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119897)Φ11989711990311989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119903)

Φ119903119897

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119897) and Φ

119903119903

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119903))



1and 119911

1198962

2


1and 119911

1198962


11199111198962

2is




119888119891 (119909) = 120572119888119910119897 (119909) + (1 minus 120572) 119888119910119903 (119909)

=

sum119872

119896=1[120572119863119903119908

119896(119909) + (1 minus 120572)119863119897119908

119896(119909)] 119888119911

119896(119909)

119863119897119863119903

(31)







0

1 1199090

2 119909

0

119899)

and 1199100= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2 and 119910

0

119894= (1199100

119897119894+

1199100

119903119894)2 with 119909

0

119894= 1199100


(1198651

119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(32)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(33)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(34)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(35)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



2 with 119911119896(119909) =

sum119899

119894=1119888119896

119894119909119894 + 119888

119896



119894 [1205831198651119894

1205831198651119894


119891 (1199090) = 120572 [120601

1(1199090) 1199111(1199090) + 1206012(1199090) 1199112(1199090)] + (1 minus 120572)

times [1206011

(1199090) 1199111(1199090) + (1 minus 120601

1

(1199090)) 1199112(1199090)]

(36)

where

1206011(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206011

(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

(37)

119891 (1199100) = 120572 [120601

1(1199100) 1199111(1199100) + 1206012(1199100) 1199112(1199100)] + (1 minus 120572)


(1199100)) 1199111(1199100) + 1206012

(1199100) 1199112(1199100)]

(38)

where

1206011(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1

(39)


119894 hence

we have prod119899

119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090



1199111= 0 and 119911

2= 1 then119891(119909

0) = 120572120601

2(1199090)+(1minus120572)[1minus120601

1

(1199090)] =

1205721206012(1199100) + (1 minus 120572)120601

2

(1199100) = 119891(119910






119896(119909) gt 0




119899rarr 119877 119909 = (1199091 1199092

119909119899) isin 119880 120583119865119896119894(119909)

isin [120583119865119896119894

(119909) 120583119865119896119894



1198941119898119896

119894

1119898119896


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909)

120601119896


119896

119897 119910119896

119903are


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909) 120601

119896

119903(119909) and let

119889infin(1198911 1198912) = sup119909isin119880



1198911 (119909) = 1205741199101

119897(119909) + (1 minus 120574) 119910

1

119903(119909) (40)

where

1199101

119897(119909) =

1198711

sum

1198961=1

11206011198961

119897(119909)11199111198961

119897(119909) +

1198721

sum

1198961=1198711+1

11206011198961

119897(119909)11199111198961

119897(119909)

=

sum1198711

1198961=11199081198961

1(119909)11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909)11199111198961

119897(119909)

1198631

119897

1199101

119903(119909) =

1198771

sum

1198961=1

11206011198961

119903(119909)11199111198961119903

(119909) +

1198721

sum

1198961=1198771+1

11206011198961

119903(119909)11199111198961119903

(119909)

=

sum1198771

1198961=11199081198961

1(119909)11199111198961119903

(119909) + sum1198721

1198961=1198771+11199081198961

1(119909)11199111198961119903

(119909)

1198631119903

(41)

where

1199081198961

1=

119899

prod

119894=1

12058311198651198961

119894

(119909) 1199081198961

1=

119899

prod

119894=1

120583 11198651198961

119894

(119909)

1198631

119897=

1198711

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198711+1

1199081198961

1

1198631

119903=

1198771

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198771+1

1199081198961

1

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1 1198711 (119909)

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1198711 (119909) + 1 1198721

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1 1198771 (119909)

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1198771 (119909) + 1 1198721

11199111198961

119897=

119899

sum

119894=1

11198881198961

119894119909119894 +1119888119896

0minus

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus11199041198961

0

1198961 = 1 1198721

11199111198961119903

=

119899

sum

119894=1

11198881198961

119894119909119894 +11198881198961

0+

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 +11199041198961

0

1198961 = 1 1198721

1198912 (119909) = 1205741199102

119897(119909) + (1 minus 120574) 119910

2

119903(119909)

(42)

where

1199102

119897(119909)

=

1198712

sum

1198962=1

21206011198962

119897(119909)21199111198962

119897(119909) +

1198722

sum

1198962=1198712+1

21206011198962

119897(119909)21199111198962

119897(119909)

=

sum1198712

1198962=11199081198962

2(119909)21199111198962

119897(119909) + sum

1198722

1198962=1198712+11199081198962

2(119909)21199111198962

119897(119909)

1198632

119897

(43)


1199102

119903(119909)

=

1198772

sum

1198962=1

21206011198962

119903(119909)21199111198962119903

(119909) +

1198722

sum

1198962=1198772+1

21206011198962

119903(119909)21199111198962119903

(119909)

=

sum11987721

1198962=11199081198962

2(119909)21199111198962119903

(119909) + sum1198722

1198962=1198772+11199081198962

2(119909)21199111198962119903

(119909)

1198631119903

(44)

where

1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909) 1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909)

1198632

119897=

1198712

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198712+1

1199081198962

2

1198632

119903=

1198772

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198772+1

1199081198962

2

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1 1198712 (119909)

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1198712 (119909) + 1 1198722

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1 1198772 (119909)

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1198772 (119909) + 1 1198722

21199111198962

119897=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0minus

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus21199041198962

0

1198962 = 1 1198722

21199111198962119903

=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0+

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 +21199041198962

0

1198962 = 1 1198722

(45)



1198911 (119909) + 1198912 (119909)

= ((

1198711

sum

1198961=1

1198712

sum

1198962=1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909) + 120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909)+120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198771

sum

1198961=1

1198772

sum

1198962=1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])times (1198631

1199031198632

119903)minus1

)

+ ((

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])

times (1198631

1199031198632

119903)minus1

)


119897+12057421199111198962

119897and (1minus120572)

11199111198961119903+(1minus120574)

21199111198962119903


119909isin119880(|119892(119909) minus 119891(119909)|) lt 120576



1and 119911

1198962





=120572120574

1198631

1198971198632

119897

times [

[

1198711

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198711+1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)


12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X


z1

z3

z2


8

7

6

5

4

3

2

1

0

Z|r(X

)


1

08

06

04

02

0

Small Medium Large


120006|U



8

7

6

5

4

3

2

1

0

f|r(X

)



+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)]

]

+120572 (1 minus 120574)

1198631

1198971198632119903

times [

[

1198711

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)]

]

+(1 minus 120572) 120574

11986311199031198632

119897

times [

[

1198771

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198712

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)]

]

+(1 minus 120572) (1 minus 120574)

11986311199031198632119903

times [

[

1198771

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)


+

1198721

sum

1198961=1198771+1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)]

]

(47)


119897

21199111198962

119897 120572(1 minus 120574)

11199111198961

119897

21199111198962119903

(1 minus 120572)12057411199111198961119903

21199111198962

119897 and (1 minus 120572)(1 minus 120574)

11199111198961119903

21199111198962119903




1and 119911

1198962



1198961

1and 119911

1198962

2were



(see Figure 10)



1198881198911 (119909)

= 1205721198881199101

119897(119909) + (1 minus 120572) 119888119910

1

119903(119909)

=

sum1198711

1198961=11199081198961

1(119909) 120572119888

11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909) 120572119888

11199111198961

119897(119909)

1198631

119897

+

sum1198771

1198961=11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

+

sum1198721

1198961=1198771+11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

(48)


11199111198961

119897(119909) and (1 minus






(1199090

1 1199090

2 119909

0

119899) and 119910

0= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2

and 1199100

119894= (1199100

119897119894+ 1199100

119903119894)2 with 119909

0

119894= 1199100


type-2 fuzzy sets (1198651119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(49)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(50)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(51)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(52)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



119897 1199111

119903] and 119911

2isin

[1199112

119897 1199112


except [119911119896119897 119911119896



119894 [120583119865119896119894

120583119865119896119894


119891 (1199090) = 120572 [120601

1

119897(1199090) 1199111

119897(1199090) + (1 minus 120601

1

119897(1199090)) 1199112

119897(1199090)]

+ (1 minus 120572) [1206011

119903(1199090) 1199111

119903(1199090) + 1206012

119903(1199090) 1199112

119903(1199090)]

(53)

where

1206011

119897(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

1206011

119903(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012

119903(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

119891 (1199100) = 120572 [120601

1

119897(1199100) 1199111

119897(1199100) + 1206012

119897(1199100) 1199112

119897(1199100)] + (1 minus 120572)


119903(1199100)) 1199111

119903(1199100) + 1206012

119903(1199100) 1199112

119903(1199100)]

(54)

where

1206012

119903(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1


1206011

119897(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

119897(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

(55)


119894= 1199100

119894


119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090

119894) = 1 If we

choose 1199111

119897119903= 0 and 119911

2

119897119903= 1 then 119891(119909

0) = 120572(1 minus 120601

1

119897(1199090)) +

(1 minus 120572)1206012

119903(1199090) = 120572120601

2

119897(1199100) + (1 minus 120572)120601

2

119903(1199100) = 119891(119910

0) Therefore





required 119891









100

10minus1

10minus2

10minus3

10minus4CV

-RM

SE

SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3





119891 (119906) = 06 sin (120587119906) + 03 sin (3120587119906) + 01 sin (5120587119906) + 120578

(56)





SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3

100

10minus02

10minus04

10minus06

10minus08

CV-R

MSE




119891 (1199091 1199092 1199093) = (1 + radic1199091 +1

1199092

+1

radic1199093

3

)

2

+ 120578 (57)



SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS TRN

CV-R

MSE

035

03

025

02

015

01

005

0




(119905) =01119909 (119905 minus 120591)

1 + 11990910 (119905 minus 120591)minus 01119909 (119905) (58)




5 Conclusions




SNR (dB)120591 = 60


0 03403 03714 NA 02388 02687 37 02027 02135 35 01495 01536 34 01244 01444 3110 01544 01714 NA 01312 01456 33 01069 01119 30 00805 00972 28 00532 00629 2520 00929 01007 NA 00708 00781 28 00566 00594 26 00424 00485 24 00342 00355 2130 00788 00847 NA 00526 00582 19 00411 00468 18 00309 00332 16 00227 00316 14Free 00749 00799 NA 00414 00477 10 00321 00353 9 00251 00319 6 00215 00293 4

SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS CHK

CV-R

MSE

04

035

03

025

02

015

01

005

0




Acknowledgments



References



















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X



Z(X

)

z1

z3

z2

8

7

6

5

4

3

2

1

0


1

08

06

04

02

0

Small Medium Large


120006|U

0 5 0 5 11100



8

7

6

5

4

3

2

1

0

f|r(X

)




1198911 (119909) + 1198912 (119909)

=

sum1198721

1198961=1sum1198722

1198962=1[1199081198961

11199081198962

2(1205721199111198961

1+ 1205741199111198962

2)]

1198631

1198971198632

119897

+

sum1198721

1198961=1sum1198722

1198962=1[1199081198961

11199081198962

2(1 minus 120572) 119911

1198961

1+ (1 minus 120574) 119911

1198962

2]

11986311199031198632119903

(29)

and that Φ1198961 1198962

= (1199081198961

11199081198962

2)(1198631

1199031198632

119903) and Φ1198961 1198962

= (1199081198961

11199081198962

2)

(1198631

1199031198632



1+1205741199111198962

2and (1minus120572)119911

1198961

1+(1minus120574)119911

1198962

2multiplied





1and 119911

1198962





1198911 (119909) 1198912 (119909)

=1

1198631

1198971198632

119897

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

21205721205741199111198961

11199111198962

2]

]

+1

1198631

1198971198632119903

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2120572 (1 minus 120574) 119911

1198961

11199111198962

2]

]

+1

11986311199031198632

119897

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2(1 minus 120572) 120574119911

1198961

11199111198962

2]

]

+1

11986311199031198632119903

[

[

1198721

sum

1198961=1

1198722

sum

1198962=1

1199081198961

11199081198962

2(1 minus 120572) (1 minus 120574) 119911

1198961

11199111198962

2]

]

(30)



11199111198962

2 120572(1minus120574)119911

1198961

11199111198962

2 (1minus

120572)1205741199111198961

11199111198962

2 and (1 minus 120572)(1 minus 120574)119911

1198961

11199111198962


FBFs (Φ11989711989711989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119897)Φ11989711990311989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119903)

Φ119903119897

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119897) and Φ

119903119903

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119903))



1and 119911

1198962

2


1and 119911

1198962


11199111198962

2is




119888119891 (119909) = 120572119888119910119897 (119909) + (1 minus 120572) 119888119910119903 (119909)

=

sum119872

119896=1[120572119863119903119908

119896(119909) + (1 minus 120572)119863119897119908

119896(119909)] 119888119911

119896(119909)

119863119897119863119903

(31)







0

1 1199090

2 119909

0

119899)

and 1199100= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2 and 119910

0

119894= (1199100

119897119894+

1199100

119903119894)2 with 119909

0

119894= 1199100


(1198651

119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(32)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(33)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(34)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(35)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



2 with 119911119896(119909) =

sum119899

119894=1119888119896

119894119909119894 + 119888

119896



119894 [1205831198651119894

1205831198651119894


119891 (1199090) = 120572 [120601

1(1199090) 1199111(1199090) + 1206012(1199090) 1199112(1199090)] + (1 minus 120572)

times [1206011

(1199090) 1199111(1199090) + (1 minus 120601

1

(1199090)) 1199112(1199090)]

(36)

where

1206011(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206011

(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

(37)

119891 (1199100) = 120572 [120601

1(1199100) 1199111(1199100) + 1206012(1199100) 1199112(1199100)] + (1 minus 120572)


(1199100)) 1199111(1199100) + 1206012

(1199100) 1199112(1199100)]

(38)

where

1206011(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1

(39)


119894 hence

we have prod119899

119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090



1199111= 0 and 119911

2= 1 then119891(119909

0) = 120572120601

2(1199090)+(1minus120572)[1minus120601

1

(1199090)] =

1205721206012(1199100) + (1 minus 120572)120601

2

(1199100) = 119891(119910






119896(119909) gt 0




119899rarr 119877 119909 = (1199091 1199092

119909119899) isin 119880 120583119865119896119894(119909)

isin [120583119865119896119894

(119909) 120583119865119896119894



1198941119898119896

119894

1119898119896


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909)

120601119896


119896

119897 119910119896

119903are


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909) 120601

119896

119903(119909) and let

119889infin(1198911 1198912) = sup119909isin119880



1198911 (119909) = 1205741199101

119897(119909) + (1 minus 120574) 119910

1

119903(119909) (40)

where

1199101

119897(119909) =

1198711

sum

1198961=1

11206011198961

119897(119909)11199111198961

119897(119909) +

1198721

sum

1198961=1198711+1

11206011198961

119897(119909)11199111198961

119897(119909)

=

sum1198711

1198961=11199081198961

1(119909)11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909)11199111198961

119897(119909)

1198631

119897

1199101

119903(119909) =

1198771

sum

1198961=1

11206011198961

119903(119909)11199111198961119903

(119909) +

1198721

sum

1198961=1198771+1

11206011198961

119903(119909)11199111198961119903

(119909)

=

sum1198771

1198961=11199081198961

1(119909)11199111198961119903

(119909) + sum1198721

1198961=1198771+11199081198961

1(119909)11199111198961119903

(119909)

1198631119903

(41)

where

1199081198961

1=

119899

prod

119894=1

12058311198651198961

119894

(119909) 1199081198961

1=

119899

prod

119894=1

120583 11198651198961

119894

(119909)

1198631

119897=

1198711

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198711+1

1199081198961

1

1198631

119903=

1198771

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198771+1

1199081198961

1

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1 1198711 (119909)

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1198711 (119909) + 1 1198721

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1 1198771 (119909)

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1198771 (119909) + 1 1198721

11199111198961

119897=

119899

sum

119894=1

11198881198961

119894119909119894 +1119888119896

0minus

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus11199041198961

0

1198961 = 1 1198721

11199111198961119903

=

119899

sum

119894=1

11198881198961

119894119909119894 +11198881198961

0+

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 +11199041198961

0

1198961 = 1 1198721

1198912 (119909) = 1205741199102

119897(119909) + (1 minus 120574) 119910

2

119903(119909)

(42)

where

1199102

119897(119909)

=

1198712

sum

1198962=1

21206011198962

119897(119909)21199111198962

119897(119909) +

1198722

sum

1198962=1198712+1

21206011198962

119897(119909)21199111198962

119897(119909)

=

sum1198712

1198962=11199081198962

2(119909)21199111198962

119897(119909) + sum

1198722

1198962=1198712+11199081198962

2(119909)21199111198962

119897(119909)

1198632

119897

(43)


1199102

119903(119909)

=

1198772

sum

1198962=1

21206011198962

119903(119909)21199111198962119903

(119909) +

1198722

sum

1198962=1198772+1

21206011198962

119903(119909)21199111198962119903

(119909)

=

sum11987721

1198962=11199081198962

2(119909)21199111198962119903

(119909) + sum1198722

1198962=1198772+11199081198962

2(119909)21199111198962119903

(119909)

1198631119903

(44)

where

1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909) 1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909)

1198632

119897=

1198712

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198712+1

1199081198962

2

1198632

119903=

1198772

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198772+1

1199081198962

2

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1 1198712 (119909)

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1198712 (119909) + 1 1198722

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1 1198772 (119909)

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1198772 (119909) + 1 1198722

21199111198962

119897=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0minus

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus21199041198962

0

1198962 = 1 1198722

21199111198962119903

=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0+

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 +21199041198962

0

1198962 = 1 1198722

(45)



1198911 (119909) + 1198912 (119909)

= ((

1198711

sum

1198961=1

1198712

sum

1198962=1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909) + 120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909)+120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198771

sum

1198961=1

1198772

sum

1198962=1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])times (1198631

1199031198632

119903)minus1

)

+ ((

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])

times (1198631

1199031198632

119903)minus1

)


119897+12057421199111198962

119897and (1minus120572)

11199111198961119903+(1minus120574)

21199111198962119903


119909isin119880(|119892(119909) minus 119891(119909)|) lt 120576



1and 119911

1198962





=120572120574

1198631

1198971198632

119897

times [

[

1198711

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198711+1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)


12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X


z1

z3

z2


8

7

6

5

4

3

2

1

0

Z|r(X

)


1

08

06

04

02

0

Small Medium Large


120006|U



8

7

6

5

4

3

2

1

0

f|r(X

)



+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)]

]

+120572 (1 minus 120574)

1198631

1198971198632119903

times [

[

1198711

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)]

]

+(1 minus 120572) 120574

11986311199031198632

119897

times [

[

1198771

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198712

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)]

]

+(1 minus 120572) (1 minus 120574)

11986311199031198632119903

times [

[

1198771

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)


+

1198721

sum

1198961=1198771+1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)]

]

(47)


119897

21199111198962

119897 120572(1 minus 120574)

11199111198961

119897

21199111198962119903

(1 minus 120572)12057411199111198961119903

21199111198962

119897 and (1 minus 120572)(1 minus 120574)

11199111198961119903

21199111198962119903




1and 119911

1198962



1198961

1and 119911

1198962

2were



(see Figure 10)



1198881198911 (119909)

= 1205721198881199101

119897(119909) + (1 minus 120572) 119888119910

1

119903(119909)

=

sum1198711

1198961=11199081198961

1(119909) 120572119888

11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909) 120572119888

11199111198961

119897(119909)

1198631

119897

+

sum1198771

1198961=11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

+

sum1198721

1198961=1198771+11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

(48)


11199111198961

119897(119909) and (1 minus






(1199090

1 1199090

2 119909

0

119899) and 119910

0= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2

and 1199100

119894= (1199100

119897119894+ 1199100

119903119894)2 with 119909

0

119894= 1199100


type-2 fuzzy sets (1198651119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(49)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(50)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(51)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(52)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



119897 1199111

119903] and 119911

2isin

[1199112

119897 1199112


except [119911119896119897 119911119896



119894 [120583119865119896119894

120583119865119896119894


119891 (1199090) = 120572 [120601

1

119897(1199090) 1199111

119897(1199090) + (1 minus 120601

1

119897(1199090)) 1199112

119897(1199090)]

+ (1 minus 120572) [1206011

119903(1199090) 1199111

119903(1199090) + 1206012

119903(1199090) 1199112

119903(1199090)]

(53)

where

1206011

119897(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

1206011

119903(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012

119903(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

119891 (1199100) = 120572 [120601

1

119897(1199100) 1199111

119897(1199100) + 1206012

119897(1199100) 1199112

119897(1199100)] + (1 minus 120572)


119903(1199100)) 1199111

119903(1199100) + 1206012

119903(1199100) 1199112

119903(1199100)]

(54)

where

1206012

119903(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1


1206011

119897(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

119897(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

(55)


119894= 1199100

119894


119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090

119894) = 1 If we

choose 1199111

119897119903= 0 and 119911

2

119897119903= 1 then 119891(119909

0) = 120572(1 minus 120601

1

119897(1199090)) +

(1 minus 120572)1206012

119903(1199090) = 120572120601

2

119897(1199100) + (1 minus 120572)120601

2

119903(1199100) = 119891(119910

0) Therefore





required 119891









100

10minus1

10minus2

10minus3

10minus4CV

-RM

SE

SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3





119891 (119906) = 06 sin (120587119906) + 03 sin (3120587119906) + 01 sin (5120587119906) + 120578

(56)





SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3

100

10minus02

10minus04

10minus06

10minus08

CV-R

MSE




119891 (1199091 1199092 1199093) = (1 + radic1199091 +1

1199092

+1

radic1199093

3

)

2

+ 120578 (57)



SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS TRN

CV-R

MSE

035

03

025

02

015

01

005

0




(119905) =01119909 (119905 minus 120591)

1 + 11990910 (119905 minus 120591)minus 01119909 (119905) (58)




5 Conclusions




SNR (dB)120591 = 60


0 03403 03714 NA 02388 02687 37 02027 02135 35 01495 01536 34 01244 01444 3110 01544 01714 NA 01312 01456 33 01069 01119 30 00805 00972 28 00532 00629 2520 00929 01007 NA 00708 00781 28 00566 00594 26 00424 00485 24 00342 00355 2130 00788 00847 NA 00526 00582 19 00411 00468 18 00309 00332 16 00227 00316 14Free 00749 00799 NA 00414 00477 10 00321 00353 9 00251 00319 6 00215 00293 4

SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS CHK

CV-R

MSE

04

035

03

025

02

015

01

005

0




Acknowledgments



References



















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





11199111198962

2 120572(1minus120574)119911

1198961

11199111198962

2 (1minus

120572)1205741199111198961

11199111198962

2 and (1 minus 120572)(1 minus 120574)119911

1198961

11199111198962


FBFs (Φ11989711989711989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119897)Φ11989711990311989611198962

= (1199081198961

11199081198962

2)(1198631

1198971198632

119903)

Φ119903119897

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119897) and Φ

119903119903

11989611198962= (1199081198961

11199081198962

2)(1198631

1199031198632

119903))



1and 119911

1198962

2


1and 119911

1198962


11199111198962

2is




119888119891 (119909) = 120572119888119910119897 (119909) + (1 minus 120572) 119888119910119903 (119909)

=

sum119872

119896=1[120572119863119903119908

119896(119909) + (1 minus 120572)119863119897119908

119896(119909)] 119888119911

119896(119909)

119863119897119863119903

(31)







0

1 1199090

2 119909

0

119899)

and 1199100= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2 and 119910

0

119894= (1199100

119897119894+

1199100

119903119894)2 with 119909

0

119894= 1199100


(1198651

119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(32)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(33)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(34)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(35)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



2 with 119911119896(119909) =

sum119899

119894=1119888119896

119894119909119894 + 119888

119896



119894 [1205831198651119894

1205831198651119894


119891 (1199090) = 120572 [120601

1(1199090) 1199111(1199090) + 1206012(1199090) 1199112(1199090)] + (1 minus 120572)

times [1206011

(1199090) 1199111(1199090) + (1 minus 120601

1

(1199090)) 1199112(1199090)]

(36)

where

1206011(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206011

(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

(37)

119891 (1199100) = 120572 [120601

1(1199100) 1199111(1199100) + 1206012(1199100) 1199112(1199100)] + (1 minus 120572)


(1199100)) 1199111(1199100) + 1206012

(1199100) 1199112(1199100)]

(38)

where

1206011(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1

(39)


119894 hence

we have prod119899

119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090



1199111= 0 and 119911

2= 1 then119891(119909

0) = 120572120601

2(1199090)+(1minus120572)[1minus120601

1

(1199090)] =

1205721206012(1199100) + (1 minus 120572)120601

2

(1199100) = 119891(119910






119896(119909) gt 0




119899rarr 119877 119909 = (1199091 1199092

119909119899) isin 119880 120583119865119896119894(119909)

isin [120583119865119896119894

(119909) 120583119865119896119894



1198941119898119896

119894

1119898119896


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909)

120601119896


119896

119897 119910119896

119903are


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909) 120601

119896

119903(119909) and let

119889infin(1198911 1198912) = sup119909isin119880



1198911 (119909) = 1205741199101

119897(119909) + (1 minus 120574) 119910

1

119903(119909) (40)

where

1199101

119897(119909) =

1198711

sum

1198961=1

11206011198961

119897(119909)11199111198961

119897(119909) +

1198721

sum

1198961=1198711+1

11206011198961

119897(119909)11199111198961

119897(119909)

=

sum1198711

1198961=11199081198961

1(119909)11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909)11199111198961

119897(119909)

1198631

119897

1199101

119903(119909) =

1198771

sum

1198961=1

11206011198961

119903(119909)11199111198961119903

(119909) +

1198721

sum

1198961=1198771+1

11206011198961

119903(119909)11199111198961119903

(119909)

=

sum1198771

1198961=11199081198961

1(119909)11199111198961119903

(119909) + sum1198721

1198961=1198771+11199081198961

1(119909)11199111198961119903

(119909)

1198631119903

(41)

where

1199081198961

1=

119899

prod

119894=1

12058311198651198961

119894

(119909) 1199081198961

1=

119899

prod

119894=1

120583 11198651198961

119894

(119909)

1198631

119897=

1198711

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198711+1

1199081198961

1

1198631

119903=

1198771

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198771+1

1199081198961

1

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1 1198711 (119909)

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1198711 (119909) + 1 1198721

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1 1198771 (119909)

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1198771 (119909) + 1 1198721

11199111198961

119897=

119899

sum

119894=1

11198881198961

119894119909119894 +1119888119896

0minus

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus11199041198961

0

1198961 = 1 1198721

11199111198961119903

=

119899

sum

119894=1

11198881198961

119894119909119894 +11198881198961

0+

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 +11199041198961

0

1198961 = 1 1198721

1198912 (119909) = 1205741199102

119897(119909) + (1 minus 120574) 119910

2

119903(119909)

(42)

where

1199102

119897(119909)

=

1198712

sum

1198962=1

21206011198962

119897(119909)21199111198962

119897(119909) +

1198722

sum

1198962=1198712+1

21206011198962

119897(119909)21199111198962

119897(119909)

=

sum1198712

1198962=11199081198962

2(119909)21199111198962

119897(119909) + sum

1198722

1198962=1198712+11199081198962

2(119909)21199111198962

119897(119909)

1198632

119897

(43)


1199102

119903(119909)

=

1198772

sum

1198962=1

21206011198962

119903(119909)21199111198962119903

(119909) +

1198722

sum

1198962=1198772+1

21206011198962

119903(119909)21199111198962119903

(119909)

=

sum11987721

1198962=11199081198962

2(119909)21199111198962119903

(119909) + sum1198722

1198962=1198772+11199081198962

2(119909)21199111198962119903

(119909)

1198631119903

(44)

where

1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909) 1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909)

1198632

119897=

1198712

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198712+1

1199081198962

2

1198632

119903=

1198772

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198772+1

1199081198962

2

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1 1198712 (119909)

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1198712 (119909) + 1 1198722

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1 1198772 (119909)

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1198772 (119909) + 1 1198722

21199111198962

119897=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0minus

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus21199041198962

0

1198962 = 1 1198722

21199111198962119903

=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0+

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 +21199041198962

0

1198962 = 1 1198722

(45)



1198911 (119909) + 1198912 (119909)

= ((

1198711

sum

1198961=1

1198712

sum

1198962=1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909) + 120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909)+120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198771

sum

1198961=1

1198772

sum

1198962=1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])times (1198631

1199031198632

119903)minus1

)

+ ((

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])

times (1198631

1199031198632

119903)minus1

)


119897+12057421199111198962

119897and (1minus120572)

11199111198961119903+(1minus120574)

21199111198962119903


119909isin119880(|119892(119909) minus 119891(119909)|) lt 120576



1and 119911

1198962





=120572120574

1198631

1198971198632

119897

times [

[

1198711

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198711+1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)


12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X


z1

z3

z2


8

7

6

5

4

3

2

1

0

Z|r(X

)


1

08

06

04

02

0

Small Medium Large


120006|U



8

7

6

5

4

3

2

1

0

f|r(X

)



+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)]

]

+120572 (1 minus 120574)

1198631

1198971198632119903

times [

[

1198711

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)]

]

+(1 minus 120572) 120574

11986311199031198632

119897

times [

[

1198771

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198712

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)]

]

+(1 minus 120572) (1 minus 120574)

11986311199031198632119903

times [

[

1198771

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)


+

1198721

sum

1198961=1198771+1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)]

]

(47)


119897

21199111198962

119897 120572(1 minus 120574)

11199111198961

119897

21199111198962119903

(1 minus 120572)12057411199111198961119903

21199111198962

119897 and (1 minus 120572)(1 minus 120574)

11199111198961119903

21199111198962119903




1and 119911

1198962



1198961

1and 119911

1198962

2were



(see Figure 10)



1198881198911 (119909)

= 1205721198881199101

119897(119909) + (1 minus 120572) 119888119910

1

119903(119909)

=

sum1198711

1198961=11199081198961

1(119909) 120572119888

11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909) 120572119888

11199111198961

119897(119909)

1198631

119897

+

sum1198771

1198961=11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

+

sum1198721

1198961=1198771+11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

(48)


11199111198961

119897(119909) and (1 minus






(1199090

1 1199090

2 119909

0

119899) and 119910

0= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2

and 1199100

119894= (1199100

119897119894+ 1199100

119903119894)2 with 119909

0

119894= 1199100


type-2 fuzzy sets (1198651119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(49)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(50)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(51)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(52)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



119897 1199111

119903] and 119911

2isin

[1199112

119897 1199112


except [119911119896119897 119911119896



119894 [120583119865119896119894

120583119865119896119894


119891 (1199090) = 120572 [120601

1

119897(1199090) 1199111

119897(1199090) + (1 minus 120601

1

119897(1199090)) 1199112

119897(1199090)]

+ (1 minus 120572) [1206011

119903(1199090) 1199111

119903(1199090) + 1206012

119903(1199090) 1199112

119903(1199090)]

(53)

where

1206011

119897(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

1206011

119903(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012

119903(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

119891 (1199100) = 120572 [120601

1

119897(1199100) 1199111

119897(1199100) + 1206012

119897(1199100) 1199112

119897(1199100)] + (1 minus 120572)


119903(1199100)) 1199111

119903(1199100) + 1206012

119903(1199100) 1199112

119903(1199100)]

(54)

where

1206012

119903(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1


1206011

119897(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

119897(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

(55)


119894= 1199100

119894


119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090

119894) = 1 If we

choose 1199111

119897119903= 0 and 119911

2

119897119903= 1 then 119891(119909

0) = 120572(1 minus 120601

1

119897(1199090)) +

(1 minus 120572)1206012

119903(1199090) = 120572120601

2

119897(1199100) + (1 minus 120572)120601

2

119903(1199100) = 119891(119910

0) Therefore





required 119891









100

10minus1

10minus2

10minus3

10minus4CV

-RM

SE

SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3





119891 (119906) = 06 sin (120587119906) + 03 sin (3120587119906) + 01 sin (5120587119906) + 120578

(56)





SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3

100

10minus02

10minus04

10minus06

10minus08

CV-R

MSE




119891 (1199091 1199092 1199093) = (1 + radic1199091 +1

1199092

+1

radic1199093

3

)

2

+ 120578 (57)



SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS TRN

CV-R

MSE

035

03

025

02

015

01

005

0




(119905) =01119909 (119905 minus 120591)

1 + 11990910 (119905 minus 120591)minus 01119909 (119905) (58)




5 Conclusions




SNR (dB)120591 = 60


0 03403 03714 NA 02388 02687 37 02027 02135 35 01495 01536 34 01244 01444 3110 01544 01714 NA 01312 01456 33 01069 01119 30 00805 00972 28 00532 00629 2520 00929 01007 NA 00708 00781 28 00566 00594 26 00424 00485 24 00342 00355 2130 00788 00847 NA 00526 00582 19 00411 00468 18 00309 00332 16 00227 00316 14Free 00749 00799 NA 00414 00477 10 00321 00353 9 00251 00319 6 00215 00293 4

SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS CHK

CV-R

MSE

04

035

03

025

02

015

01

005

0




Acknowledgments



References



















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




1199111= 0 and 119911

2= 1 then119891(119909

0) = 120572120601

2(1199090)+(1minus120572)[1minus120601

1

(1199090)] =

1205721206012(1199100) + (1 minus 120572)120601

2

(1199100) = 119891(119910






119896(119909) gt 0




119899rarr 119877 119909 = (1199091 1199092

119909119899) isin 119880 120583119865119896119894(119909)

isin [120583119865119896119894

(119909) 120583119865119896119894



1198941119898119896

119894

1119898119896


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909)

120601119896


119896

119897 119910119896

119903are


119896

119897(119909) 120601

119896

119897(119909) 120601119896

119903(119909) 120601

119896

119903(119909) and let

119889infin(1198911 1198912) = sup119909isin119880



1198911 (119909) = 1205741199101

119897(119909) + (1 minus 120574) 119910

1

119903(119909) (40)

where

1199101

119897(119909) =

1198711

sum

1198961=1

11206011198961

119897(119909)11199111198961

119897(119909) +

1198721

sum

1198961=1198711+1

11206011198961

119897(119909)11199111198961

119897(119909)

=

sum1198711

1198961=11199081198961

1(119909)11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909)11199111198961

119897(119909)

1198631

119897

1199101

119903(119909) =

1198771

sum

1198961=1

11206011198961

119903(119909)11199111198961119903

(119909) +

1198721

sum

1198961=1198771+1

11206011198961

119903(119909)11199111198961119903

(119909)

=

sum1198771

1198961=11199081198961

1(119909)11199111198961119903

(119909) + sum1198721

1198961=1198771+11199081198961

1(119909)11199111198961119903

(119909)

1198631119903

(41)

where

1199081198961

1=

119899

prod

119894=1

12058311198651198961

119894

(119909) 1199081198961

1=

119899

prod

119894=1

120583 11198651198961

119894

(119909)

1198631

119897=

1198711

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198711+1

1199081198961

1

1198631

119903=

1198771

sum

1198961=1

1199081198961

1+

1198721

sum

1198961=1198771+1

1199081198961

1

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1 1198711 (119909)

11206011198961

119897(119909) =

1199081198961

1

1198631

119897

forall1198961 = 1198711 (119909) + 1 1198721

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1 1198771 (119909)

11206011198961

119903(119909) =

1199081198961

1

1198631119903

forall1198961 = 1198771 (119909) + 1 1198721

11199111198961

119897=

119899

sum

119894=1

11198881198961

119894119909119894 +1119888119896

0minus

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus11199041198961

0

1198961 = 1 1198721

11199111198961119903

=

119899

sum

119894=1

11198881198961

119894119909119894 +11198881198961

0+

119899

sum

119894=1

11199041198961

119894

10038161003816100381610038161199091198941003816100381610038161003816 +11199041198961

0

1198961 = 1 1198721

1198912 (119909) = 1205741199102

119897(119909) + (1 minus 120574) 119910

2

119903(119909)

(42)

where

1199102

119897(119909)

=

1198712

sum

1198962=1

21206011198962

119897(119909)21199111198962

119897(119909) +

1198722

sum

1198962=1198712+1

21206011198962

119897(119909)21199111198962

119897(119909)

=

sum1198712

1198962=11199081198962

2(119909)21199111198962

119897(119909) + sum

1198722

1198962=1198712+11199081198962

2(119909)21199111198962

119897(119909)

1198632

119897

(43)


1199102

119903(119909)

=

1198772

sum

1198962=1

21206011198962

119903(119909)21199111198962119903

(119909) +

1198722

sum

1198962=1198772+1

21206011198962

119903(119909)21199111198962119903

(119909)

=

sum11987721

1198962=11199081198962

2(119909)21199111198962119903

(119909) + sum1198722

1198962=1198772+11199081198962

2(119909)21199111198962119903

(119909)

1198631119903

(44)

where

1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909) 1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909)

1198632

119897=

1198712

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198712+1

1199081198962

2

1198632

119903=

1198772

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198772+1

1199081198962

2

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1 1198712 (119909)

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1198712 (119909) + 1 1198722

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1 1198772 (119909)

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1198772 (119909) + 1 1198722

21199111198962

119897=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0minus

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus21199041198962

0

1198962 = 1 1198722

21199111198962119903

=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0+

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 +21199041198962

0

1198962 = 1 1198722

(45)



1198911 (119909) + 1198912 (119909)

= ((

1198711

sum

1198961=1

1198712

sum

1198962=1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909) + 120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909)+120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198771

sum

1198961=1

1198772

sum

1198962=1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])times (1198631

1199031198632

119903)minus1

)

+ ((

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])

times (1198631

1199031198632

119903)minus1

)


119897+12057421199111198962

119897and (1minus120572)

11199111198961119903+(1minus120574)

21199111198962119903


119909isin119880(|119892(119909) minus 119891(119909)|) lt 120576



1and 119911

1198962





=120572120574

1198631

1198971198632

119897

times [

[

1198711

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198711+1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)


12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X


z1

z3

z2


8

7

6

5

4

3

2

1

0

Z|r(X

)


1

08

06

04

02

0

Small Medium Large


120006|U



8

7

6

5

4

3

2

1

0

f|r(X

)



+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)]

]

+120572 (1 minus 120574)

1198631

1198971198632119903

times [

[

1198711

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)]

]

+(1 minus 120572) 120574

11986311199031198632

119897

times [

[

1198771

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198712

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)]

]

+(1 minus 120572) (1 minus 120574)

11986311199031198632119903

times [

[

1198771

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)


+

1198721

sum

1198961=1198771+1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)]

]

(47)


119897

21199111198962

119897 120572(1 minus 120574)

11199111198961

119897

21199111198962119903

(1 minus 120572)12057411199111198961119903

21199111198962

119897 and (1 minus 120572)(1 minus 120574)

11199111198961119903

21199111198962119903




1and 119911

1198962



1198961

1and 119911

1198962

2were



(see Figure 10)



1198881198911 (119909)

= 1205721198881199101

119897(119909) + (1 minus 120572) 119888119910

1

119903(119909)

=

sum1198711

1198961=11199081198961

1(119909) 120572119888

11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909) 120572119888

11199111198961

119897(119909)

1198631

119897

+

sum1198771

1198961=11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

+

sum1198721

1198961=1198771+11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

(48)


11199111198961

119897(119909) and (1 minus






(1199090

1 1199090

2 119909

0

119899) and 119910

0= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2

and 1199100

119894= (1199100

119897119894+ 1199100

119903119894)2 with 119909

0

119894= 1199100


type-2 fuzzy sets (1198651119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(49)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(50)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(51)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(52)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



119897 1199111

119903] and 119911

2isin

[1199112

119897 1199112


except [119911119896119897 119911119896



119894 [120583119865119896119894

120583119865119896119894


119891 (1199090) = 120572 [120601

1

119897(1199090) 1199111

119897(1199090) + (1 minus 120601

1

119897(1199090)) 1199112

119897(1199090)]

+ (1 minus 120572) [1206011

119903(1199090) 1199111

119903(1199090) + 1206012

119903(1199090) 1199112

119903(1199090)]

(53)

where

1206011

119897(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

1206011

119903(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012

119903(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

119891 (1199100) = 120572 [120601

1

119897(1199100) 1199111

119897(1199100) + 1206012

119897(1199100) 1199112

119897(1199100)] + (1 minus 120572)


119903(1199100)) 1199111

119903(1199100) + 1206012

119903(1199100) 1199112

119903(1199100)]

(54)

where

1206012

119903(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1


1206011

119897(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

119897(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

(55)


119894= 1199100

119894


119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090

119894) = 1 If we

choose 1199111

119897119903= 0 and 119911

2

119897119903= 1 then 119891(119909

0) = 120572(1 minus 120601

1

119897(1199090)) +

(1 minus 120572)1206012

119903(1199090) = 120572120601

2

119897(1199100) + (1 minus 120572)120601

2

119903(1199100) = 119891(119910

0) Therefore





required 119891









100

10minus1

10minus2

10minus3

10minus4CV

-RM

SE

SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3





119891 (119906) = 06 sin (120587119906) + 03 sin (3120587119906) + 01 sin (5120587119906) + 120578

(56)





SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3

100

10minus02

10minus04

10minus06

10minus08

CV-R

MSE




119891 (1199091 1199092 1199093) = (1 + radic1199091 +1

1199092

+1

radic1199093

3

)

2

+ 120578 (57)



SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS TRN

CV-R

MSE

035

03

025

02

015

01

005

0




(119905) =01119909 (119905 minus 120591)

1 + 11990910 (119905 minus 120591)minus 01119909 (119905) (58)




5 Conclusions




SNR (dB)120591 = 60


0 03403 03714 NA 02388 02687 37 02027 02135 35 01495 01536 34 01244 01444 3110 01544 01714 NA 01312 01456 33 01069 01119 30 00805 00972 28 00532 00629 2520 00929 01007 NA 00708 00781 28 00566 00594 26 00424 00485 24 00342 00355 2130 00788 00847 NA 00526 00582 19 00411 00468 18 00309 00332 16 00227 00316 14Free 00749 00799 NA 00414 00477 10 00321 00353 9 00251 00319 6 00215 00293 4

SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS CHK

CV-R

MSE

04

035

03

025

02

015

01

005

0




Acknowledgments



References



















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




1199102

119903(119909)

=

1198772

sum

1198962=1

21206011198962

119903(119909)21199111198962119903

(119909) +

1198722

sum

1198962=1198772+1

21206011198962

119903(119909)21199111198962119903

(119909)

=

sum11987721

1198962=11199081198962

2(119909)21199111198962119903

(119909) + sum1198722

1198962=1198772+11199081198962

2(119909)21199111198962119903

(119909)

1198631119903

(44)

where

1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909) 1199081198962

2=

119899

prod

119894=1

12058321198651198962

119894

(119909)

1198632

119897=

1198712

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198712+1

1199081198962

2

1198632

119903=

1198772

sum

1198962=1

1199081198962

2+

1198722

sum

1198962=1198772+1

1199081198962

2

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1 1198712 (119909)

21206011198962

119897(119909) =

1199081198962

2

1198632

119897

forall1198962 = 1198712 (119909) + 1 1198722

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1 1198772 (119909)

21206011198962

119903(119909) =

1199081198962

2

1198632119903

forall1198962 = 1198772 (119909) + 1 1198722

21199111198962

119897=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0minus

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 minus21199041198962

0

1198962 = 1 1198722

21199111198962119903

=

119899

sum

119894=1

21198881198962

119894119909119894 +21198881198962

0+

119899

sum

119894=1

21199041198962

119894

10038161003816100381610038161199091198941003816100381610038161003816 +21199041198962

0

1198962 = 1 1198722

(45)



1198911 (119909) + 1198912 (119909)

= ((

1198711

sum

1198961=1

1198712

sum

1198962=1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909) + 120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

[1205721198632

1198971199081198961

1

11199111198961

119897(119909)+120574119863

1

1198971199081198962

2

21199111198962

119897(119909)])

times (1198631

1198971198632

119897)minus1

)

+ ((

1198771

sum

1198961=1

1198772

sum

1198962=1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])times (1198631

1199031198632

119903)minus1

)

+ ((

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

[(1minus120572)1198632

1199031199081198961

1

11199111198961119903

(119909)

+(1minus120574)1198631

1199031199081198962

2

21199111198962119903

(119909)])

times (1198631

1199031198632

119903)minus1

)


119897+12057421199111198962

119897and (1minus120572)

11199111198961119903+(1minus120574)

21199111198962119903


119909isin119880(|119892(119909) minus 119891(119909)|) lt 120576



1and 119911

1198962





=120572120574

1198631

1198971198632

119897

times [

[

1198711

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198711+1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)


12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X


z1

z3

z2


8

7

6

5

4

3

2

1

0

Z|r(X

)


1

08

06

04

02

0

Small Medium Large


120006|U



8

7

6

5

4

3

2

1

0

f|r(X

)



+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)]

]

+120572 (1 minus 120574)

1198631

1198971198632119903

times [

[

1198711

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)]

]

+(1 minus 120572) 120574

11986311199031198632

119897

times [

[

1198771

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198712

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)]

]

+(1 minus 120572) (1 minus 120574)

11986311199031198632119903

times [

[

1198771

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)


+

1198721

sum

1198961=1198771+1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)]

]

(47)


119897

21199111198962

119897 120572(1 minus 120574)

11199111198961

119897

21199111198962119903

(1 minus 120572)12057411199111198961119903

21199111198962

119897 and (1 minus 120572)(1 minus 120574)

11199111198961119903

21199111198962119903




1and 119911

1198962



1198961

1and 119911

1198962

2were



(see Figure 10)



1198881198911 (119909)

= 1205721198881199101

119897(119909) + (1 minus 120572) 119888119910

1

119903(119909)

=

sum1198711

1198961=11199081198961

1(119909) 120572119888

11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909) 120572119888

11199111198961

119897(119909)

1198631

119897

+

sum1198771

1198961=11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

+

sum1198721

1198961=1198771+11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

(48)


11199111198961

119897(119909) and (1 minus






(1199090

1 1199090

2 119909

0

119899) and 119910

0= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2

and 1199100

119894= (1199100

119897119894+ 1199100

119903119894)2 with 119909

0

119894= 1199100


type-2 fuzzy sets (1198651119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(49)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(50)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(51)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(52)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



119897 1199111

119903] and 119911

2isin

[1199112

119897 1199112


except [119911119896119897 119911119896



119894 [120583119865119896119894

120583119865119896119894


119891 (1199090) = 120572 [120601

1

119897(1199090) 1199111

119897(1199090) + (1 minus 120601

1

119897(1199090)) 1199112

119897(1199090)]

+ (1 minus 120572) [1206011

119903(1199090) 1199111

119903(1199090) + 1206012

119903(1199090) 1199112

119903(1199090)]

(53)

where

1206011

119897(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

1206011

119903(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012

119903(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

119891 (1199100) = 120572 [120601

1

119897(1199100) 1199111

119897(1199100) + 1206012

119897(1199100) 1199112

119897(1199100)] + (1 minus 120572)


119903(1199100)) 1199111

119903(1199100) + 1206012

119903(1199100) 1199112

119903(1199100)]

(54)

where

1206012

119903(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1


1206011

119897(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

119897(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

(55)


119894= 1199100

119894


119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090

119894) = 1 If we

choose 1199111

119897119903= 0 and 119911

2

119897119903= 1 then 119891(119909

0) = 120572(1 minus 120601

1

119897(1199090)) +

(1 minus 120572)1206012

119903(1199090) = 120572120601

2

119897(1199100) + (1 minus 120572)120601

2

119903(1199100) = 119891(119910

0) Therefore





required 119891









100

10minus1

10minus2

10minus3

10minus4CV

-RM

SE

SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3





119891 (119906) = 06 sin (120587119906) + 03 sin (3120587119906) + 01 sin (5120587119906) + 120578

(56)





SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3

100

10minus02

10minus04

10minus06

10minus08

CV-R

MSE




119891 (1199091 1199092 1199093) = (1 + radic1199091 +1

1199092

+1

radic1199093

3

)

2

+ 120578 (57)



SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS TRN

CV-R

MSE

035

03

025

02

015

01

005

0




(119905) =01119909 (119905 minus 120591)

1 + 11990910 (119905 minus 120591)minus 01119909 (119905) (58)




5 Conclusions




SNR (dB)120591 = 60


0 03403 03714 NA 02388 02687 37 02027 02135 35 01495 01536 34 01244 01444 3110 01544 01714 NA 01312 01456 33 01069 01119 30 00805 00972 28 00532 00629 2520 00929 01007 NA 00708 00781 28 00566 00594 26 00424 00485 24 00342 00355 2130 00788 00847 NA 00526 00582 19 00411 00468 18 00309 00332 16 00227 00316 14Free 00749 00799 NA 00414 00477 10 00321 00353 9 00251 00319 6 00215 00293 4

SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS CHK

CV-R

MSE

04

035

03

025

02

015

01

005

0




Acknowledgments



References



















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




12

1

08

06

04

02

0

Small Medium Large


Mem

bers

hip

grad

es

X


z1

z3

z2


8

7

6

5

4

3

2

1

0

Z|r(X

)


1

08

06

04

02

0

Small Medium Large


120006|U



8

7

6

5

4

3

2

1

0

f|r(X

)



+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962

119897(119909)]

]

+120572 (1 minus 120574)

1198631

1198971198632119903

times [

[

1198711

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198711

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198772

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198711+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961

119897(119909)21199111198962119903

(119909)]

]

+(1 minus 120572) 120574

11986311199031198632

119897

times [

[

1198771

sum

1198961=1

1198712

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198712+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198712

sum

1198962=1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962

119897(119909)]

]

+(1 minus 120572) (1 minus 120574)

11986311199031198632119903

times [

[

1198771

sum

1198961=1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198771

sum

1198961=1

1198722

sum

1198962=1198772+1

1199081198961

1(119909) 1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)


+

1198721

sum

1198961=1198771+1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)]

]

(47)


119897

21199111198962

119897 120572(1 minus 120574)

11199111198961

119897

21199111198962119903

(1 minus 120572)12057411199111198961119903

21199111198962

119897 and (1 minus 120572)(1 minus 120574)

11199111198961119903

21199111198962119903




1and 119911

1198962



1198961

1and 119911

1198962

2were



(see Figure 10)



1198881198911 (119909)

= 1205721198881199101

119897(119909) + (1 minus 120572) 119888119910

1

119903(119909)

=

sum1198711

1198961=11199081198961

1(119909) 120572119888

11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909) 120572119888

11199111198961

119897(119909)

1198631

119897

+

sum1198771

1198961=11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

+

sum1198721

1198961=1198771+11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

(48)


11199111198961

119897(119909) and (1 minus






(1199090

1 1199090

2 119909

0

119899) and 119910

0= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2

and 1199100

119894= (1199100

119897119894+ 1199100

119903119894)2 with 119909

0

119894= 1199100


type-2 fuzzy sets (1198651119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(49)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(50)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(51)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(52)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



119897 1199111

119903] and 119911

2isin

[1199112

119897 1199112


except [119911119896119897 119911119896



119894 [120583119865119896119894

120583119865119896119894


119891 (1199090) = 120572 [120601

1

119897(1199090) 1199111

119897(1199090) + (1 minus 120601

1

119897(1199090)) 1199112

119897(1199090)]

+ (1 minus 120572) [1206011

119903(1199090) 1199111

119903(1199090) + 1206012

119903(1199090) 1199112

119903(1199090)]

(53)

where

1206011

119897(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

1206011

119903(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012

119903(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

119891 (1199100) = 120572 [120601

1

119897(1199100) 1199111

119897(1199100) + 1206012

119897(1199100) 1199112

119897(1199100)] + (1 minus 120572)


119903(1199100)) 1199111

119903(1199100) + 1206012

119903(1199100) 1199112

119903(1199100)]

(54)

where

1206012

119903(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1


1206011

119897(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

119897(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

(55)


119894= 1199100

119894


119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090

119894) = 1 If we

choose 1199111

119897119903= 0 and 119911

2

119897119903= 1 then 119891(119909

0) = 120572(1 minus 120601

1

119897(1199090)) +

(1 minus 120572)1206012

119903(1199090) = 120572120601

2

119897(1199100) + (1 minus 120572)120601

2

119903(1199100) = 119891(119910

0) Therefore





required 119891









100

10minus1

10minus2

10minus3

10minus4CV

-RM

SE

SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3





119891 (119906) = 06 sin (120587119906) + 03 sin (3120587119906) + 01 sin (5120587119906) + 120578

(56)





SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3

100

10minus02

10minus04

10minus06

10minus08

CV-R

MSE




119891 (1199091 1199092 1199093) = (1 + radic1199091 +1

1199092

+1

radic1199093

3

)

2

+ 120578 (57)



SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS TRN

CV-R

MSE

035

03

025

02

015

01

005

0




(119905) =01119909 (119905 minus 120591)

1 + 11990910 (119905 minus 120591)minus 01119909 (119905) (58)




5 Conclusions




SNR (dB)120591 = 60


0 03403 03714 NA 02388 02687 37 02027 02135 35 01495 01536 34 01244 01444 3110 01544 01714 NA 01312 01456 33 01069 01119 30 00805 00972 28 00532 00629 2520 00929 01007 NA 00708 00781 28 00566 00594 26 00424 00485 24 00342 00355 2130 00788 00847 NA 00526 00582 19 00411 00468 18 00309 00332 16 00227 00316 14Free 00749 00799 NA 00414 00477 10 00321 00353 9 00251 00319 6 00215 00293 4

SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS CHK

CV-R

MSE

04

035

03

025

02

015

01

005

0




Acknowledgments



References



















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




+

1198721

sum

1198961=1198771+1

1198772

sum

1198962=1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)

+

1198721

sum

1198961=1198771+1

1198722

sum

1198962=1198772+1

1199081198961

1(119909)1199081198962

2(119909)11199111198961119903

(119909)21199111198962119903

(119909)]

]

(47)


119897

21199111198962

119897 120572(1 minus 120574)

11199111198961

119897

21199111198962119903

(1 minus 120572)12057411199111198961119903

21199111198962

119897 and (1 minus 120572)(1 minus 120574)

11199111198961119903

21199111198962119903




1and 119911

1198962



1198961

1and 119911

1198962

2were



(see Figure 10)



1198881198911 (119909)

= 1205721198881199101

119897(119909) + (1 minus 120572) 119888119910

1

119903(119909)

=

sum1198711

1198961=11199081198961

1(119909) 120572119888

11199111198961

119897(119909) + sum

1198721

119896=1198711+11199081198961

1(119909) 120572119888

11199111198961

119897(119909)

1198631

119897

+

sum1198771

1198961=11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

+

sum1198721

1198961=1198771+11199081198961

1(119909) (1 minus 120572) 119888

11199111198961119903

(119909)

1198631119903

(48)


11199111198961

119897(119909) and (1 minus






(1199090

1 1199090

2 119909

0

119899) and 119910

0= (1199100

1 1199100

2 119910

0

119899) If 1199090119894= (1199090

119897119894+ 1199090

119903119894)2

and 1199100

119894= (1199100

119897119894+ 1199100

119903119894)2 with 119909

0

119894= 1199100


type-2 fuzzy sets (1198651119894 [1205831198651119894

1205831198651119894

]) and (1198652

119894 [1205831198652119894

1205831198652119894

]) with

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 lt 1199090

119897119894

1 1199090

119897119894le 119909119894 le 119909

0

119903119894

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 gt 1199090

119897119894

(49)

1205831198651119894

(119909119894) =

exp [minus1

2(119909119894 minus 119909

0

119903119894)2

] 119909119894 le 1199090

119897119894

exp [minus1

2(119909119894 minus 119909

0

119897119894)2

] 119909119894 gt 1199090

119903119894

(50)

1205831198652119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 lt 1199100

119897119894

1 1199100

119897119894le 119909119894 le 119910

0

119903119894

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 gt 1199100

119897119894

(51)

120583119865119896119894

(119909119894) =

exp [minus1

2(119909119894 minus 119910

0

119903119894)2

] 119909119894 le 1199100

119897119894

exp [minus1

2(119909119894 minus 119910

0

119897119894)2

] 119909119894 gt 1199100

119903119894

(52)

If 1199090119894= 1199100

119894 then 119865

1

119894= 1198652

119894and 120583

1198651119894

(1199090

119894) = 120583

1198652119894

(1199100

119894) 1205831198651119894

(1199090

119894) =

1205831198652119894

(1199100



119897 1199111

119903] and 119911

2isin

[1199112

119897 1199112


except [119911119896119897 119911119896



119894 [120583119865119896119894

120583119865119896119894


119891 (1199090) = 120572 [120601

1

119897(1199090) 1199111

119897(1199090) + (1 minus 120601

1

119897(1199090)) 1199112

119897(1199090)]

+ (1 minus 120572) [1206011

119903(1199090) 1199111

119903(1199090) + 1206012

119903(1199090) 1199112

119903(1199090)]

(53)

where

1206011

119897(1199090) =

1

1 + prod119899

119894=11205831198652119894

(1199090

119894)

1206011

119903(1199090) =

prod119899

119894=11205831198651119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

1206012

119903(1199090) =

prod119899

119894=11205831198652119894

(1199090

119894)

prod119899

119894=11205831198651119894

(1199090

119894) + prod

119899

119894=11205831198652119894

(1199090

119894)

119891 (1199100) = 120572 [120601

1

119897(1199100) 1199111

119897(1199100) + 1206012

119897(1199100) 1199112

119897(1199100)] + (1 minus 120572)


119903(1199100)) 1199111

119903(1199100) + 1206012

119903(1199100) 1199112

119903(1199100)]

(54)

where

1206012

119903(1199100) =

1

prod119899

119894=11205831198651119894

(1199100

119894) + 1


1206011

119897(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

119897(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

(55)


119894= 1199100

119894


119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090

119894) = 1 If we

choose 1199111

119897119903= 0 and 119911

2

119897119903= 1 then 119891(119909

0) = 120572(1 minus 120601

1

119897(1199090)) +

(1 minus 120572)1206012

119903(1199090) = 120572120601

2

119897(1199100) + (1 minus 120572)120601

2

119903(1199100) = 119891(119910

0) Therefore





required 119891









100

10minus1

10minus2

10minus3

10minus4CV

-RM

SE

SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3





119891 (119906) = 06 sin (120587119906) + 03 sin (3120587119906) + 01 sin (5120587119906) + 120578

(56)





SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3

100

10minus02

10minus04

10minus06

10minus08

CV-R

MSE




119891 (1199091 1199092 1199093) = (1 + radic1199091 +1

1199092

+1

radic1199093

3

)

2

+ 120578 (57)



SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS TRN

CV-R

MSE

035

03

025

02

015

01

005

0




(119905) =01119909 (119905 minus 120591)

1 + 11990910 (119905 minus 120591)minus 01119909 (119905) (58)




5 Conclusions




SNR (dB)120591 = 60


0 03403 03714 NA 02388 02687 37 02027 02135 35 01495 01536 34 01244 01444 3110 01544 01714 NA 01312 01456 33 01069 01119 30 00805 00972 28 00532 00629 2520 00929 01007 NA 00708 00781 28 00566 00594 26 00424 00485 24 00342 00355 2130 00788 00847 NA 00526 00582 19 00411 00468 18 00309 00332 16 00227 00316 14Free 00749 00799 NA 00414 00477 10 00321 00353 9 00251 00319 6 00215 00293 4

SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS CHK

CV-R

MSE

04

035

03

025

02

015

01

005

0




Acknowledgments



References



















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




1206011

119897(1199100) =

prod119899

119894=11205831198651119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

1206012

119897(1199100) =

prod119899

119894=11205831198652119894

(1199100

119894)

prod119899

119894=11205831198651119894

(1199100

119894) + prod

119899

119894=11205831198652119894

(1199100

119894)

(55)


119894= 1199100

119894


119894=11205831198651119894

(1199090

119894) = 1 and prod

119899

119894=11205831198652119894

(1199090

119894) = 1 If we

choose 1199111

119897119903= 0 and 119911

2

119897119903= 1 then 119891(119909

0) = 120572(1 minus 120601

1

119897(1199090)) +

(1 minus 120572)1206012

119903(1199090) = 120572120601

2

119897(1199100) + (1 minus 120572)120601

2

119903(1199100) = 119891(119910

0) Therefore





required 119891









100

10minus1

10minus2

10minus3

10minus4CV

-RM

SE

SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3





119891 (119906) = 06 sin (120587119906) + 03 sin (3120587119906) + 01 sin (5120587119906) + 120578

(56)





SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3

100

10minus02

10minus04

10minus06

10minus08

CV-R

MSE




119891 (1199091 1199092 1199093) = (1 + radic1199091 +1

1199092

+1

radic1199093

3

)

2

+ 120578 (57)



SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS TRN

CV-R

MSE

035

03

025

02

015

01

005

0




(119905) =01119909 (119905 minus 120591)

1 + 11990910 (119905 minus 120591)minus 01119909 (119905) (58)




5 Conclusions




SNR (dB)120591 = 60


0 03403 03714 NA 02388 02687 37 02027 02135 35 01495 01536 34 01244 01444 3110 01544 01714 NA 01312 01456 33 01069 01119 30 00805 00972 28 00532 00629 2520 00929 01007 NA 00708 00781 28 00566 00594 26 00424 00485 24 00342 00355 2130 00788 00847 NA 00526 00582 19 00411 00468 18 00309 00332 16 00227 00316 14Free 00749 00799 NA 00414 00477 10 00321 00353 9 00251 00319 6 00215 00293 4

SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS CHK

CV-R

MSE

04

035

03

025

02

015

01

005

0




Acknowledgments



References



















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






SNR (dB)0 10 20 30 40 50 60 70 80


IT2FNN-2IT2FNN-3

100

10minus02

10minus04

10minus06

10minus08

CV-R

MSE




119891 (1199091 1199092 1199093) = (1 + radic1199091 +1

1199092

+1

radic1199093

3

)

2

+ 120578 (57)



SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS TRN

CV-R

MSE

035

03

025

02

015

01

005

0




(119905) =01119909 (119905 minus 120591)

1 + 11990910 (119905 minus 120591)minus 01119909 (119905) (58)




5 Conclusions




SNR (dB)120591 = 60


0 03403 03714 NA 02388 02687 37 02027 02135 35 01495 01536 34 01244 01444 3110 01544 01714 NA 01312 01456 33 01069 01119 30 00805 00972 28 00532 00629 2520 00929 01007 NA 00708 00781 28 00566 00594 26 00424 00485 24 00342 00355 2130 00788 00847 NA 00526 00582 19 00411 00468 18 00309 00332 16 00227 00316 14Free 00749 00799 NA 00414 00477 10 00321 00353 9 00251 00319 6 00215 00293 4

SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS CHK

CV-R

MSE

04

035

03

025

02

015

01

005

0




Acknowledgments



References



















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





SNR (dB)120591 = 60


0 03403 03714 NA 02388 02687 37 02027 02135 35 01495 01536 34 01244 01444 3110 01544 01714 NA 01312 01456 33 01069 01119 30 00805 00972 28 00532 00629 2520 00929 01007 NA 00708 00781 28 00566 00594 26 00424 00485 24 00342 00355 2130 00788 00847 NA 00526 00582 19 00411 00468 18 00309 00332 16 00227 00316 14Free 00749 00799 NA 00414 00477 10 00321 00353 9 00251 00319 6 00215 00293 4

SNR (dB)0 10 20 30 40 50 60 70 80



ANFIS CHK

CV-R

MSE

04

035

03

025

02

015

01

005

0




Acknowledgments



References



















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



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