Research ArticleIntuitionistic Fuzzy Weighted Linear Regression Model withFuzzy Entropy under Linear Restrictions

Gaurav Kumar1 and Rakesh Kumar Bajaj2

1 Singhania University Pacheri Bari Jhunjhunu Rajasthan 333515 India2 Jaypee University of Information Technology Waknaghat 173234 India

Correspondence should be addressed to Rakesh Kumar Bajaj rakeshbajajgmailcom

Received 18 April 2014 Revised 6 August 2014 Accepted 23 August 2014 Published 30 October 2014

Academic Editor Bijan Davvaz

Copyright copy 2014 G Kumar and R K Bajaj This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In fuzzy set theory it is well known that a triangular fuzzy number can be uniquely determined through its position and entropiesIn the present communication we extend this concept on triangular intuitionistic fuzzy number for its one-to-one correspondencewith its position and entropies Using the concept of fuzzy entropy the estimators of the intuitionistic fuzzy regression coefficientshave been estimated in the unrestricted regression model An intuitionistic fuzzy weighted linear regression (IFWLR) model withsome restrictions in the form of prior information has been considered Further the estimators of regression coefficients have beenobtained with the help of fuzzy entropy for the restrictedunrestricted IFWLR model by assigning some weights in the distancefunction

1 Introduction

In statistical analysis regression is used to explore the rela-tionship between 119896 input variables x

1 x2 x

119896(also known

as independent variables or explanatory variables) and theoutput variable y (also called dependent variable or responsevariable) from 119899 sets of observations In linear regression themethod of least-squares is applied to find the regression coef-ficients 120573

119895 119895 = 0 1 119896 which describe the contribution

of the corresponding independent variable x119895in explaining

the dependent variable y The aim of regression analysis isto estimate the parameters on the basis of availableobservedempirical data Traditional studies on regression assumethe observations to have crisp values In the crisp linearregression model the parameters (regression coefficients arecrisp) appear in a linear form that is

y = 1205730+ 1205731x1+ 1205732x2+ sdot sdot sdot + 120573

119896x119896+ random error (1)

Once the coefficients 1205730 1205731 1205732 120573

119896are determined from

the observed samples the responses are estimated from anygiven sets of x

1 x2 x

119896values

Fuzzy set theory developed by Zadeh [1] has capability todescribe the uncertain situations containing ambiguity andvagueness It may be recalled that a fuzzy set 119860 defined on

a universe of discourse 119883 is characterized by a membershipfunction 120583

119860(119909) which takes values in the interval [0 1] (ie

120583119860

119883 rarr [0 1]) The value 120583119860(119909) represents the grade of

membership of 119909 isin 119883 in 119860 This grade corresponds to thedegree to which that element or individual is similar or com-patible with the concept represented by the fuzzy set Thusthe elements may belong in the fuzzy set to a greater or lesserdegree as indicated by a larger or smaller membership grade

Tanaka et al [2 3] initiated the research in the area of lin-ear regression analysis in a fuzzy environment where a fuzzylinear system is used as a regression model They consider aregression model in which the relations of the variables aresubject to fuzziness that is the model with crisp input andfuzzy parameters In general fuzzy regression can be classi-fied into two categories

(i) when the relations of the variables are subject to fuzzi-ness

(ii) when the variables themselves are fuzzy

There exist several conceptual and methodologicalapproaches to fuzzy regression with respect to the characteri-zationmentioned above Tanaka andWatada [4] Tanaka et al[5] and Tanaka and Ishibuchi [6] considered more general

Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2014 Article ID 358439 10 pageshttpdxdoiorg1011552014358439

2 International Scholarly Research Notices

models in fuzzy regression In the approaches of Tanaka etal they considered the L-R fuzzy data and minimized theindex of fuzziness of the fuzzy linear regression model Asdescribed by Tanaka and Watada [4] ldquoA fuzzy number is afuzzy subset of the real line whose highest membership valuesare clustered around a given real number called the meanvalue the membership function is monotonic on both sides ofthis mean valuerdquo Hence fuzzy number can be decomposedinto position and fuzziness where the position is representedby the element with the highest membership value and thefuzziness of a fuzzy number is represented by themembershipfunction The comparison among various fuzzy regressionmodels and the difference between the approaches of fuzzyregression analysis and conventional regression analysishave been presented by Redden and Woodall [7] Changand Lee [8] and Redden and Woodall [7] pointed out someweaknesses of the approaches proposed by Tanaka et al Afuzzy linear regression model based on Tanakarsquos approach byconsidering the fuzzy linear programming problem has alsobeen introduced by Peters [9]

In fuzzy set theory the entropy is a measure of degreeof fuzziness which expresses the amount of average ambi-guitydifficulty in making a decision whether an elementbelongs to a set or not The following are the four propertiesintroduced in de Luca and Termini [10] which are widelyaccepted as a criterion for defining any new fuzzy entropymeasure119867(sdot) of the fuzzy set 119860

(i) P1 (sharpness)119867(119860) is minimum if and only if 119860 is acrisp set that is 120583

119860(119909) = 0 or 1 for all 119909

(ii) P2 (maximality) 119867(119860) is maximum if and only if120583119860(119909) = 05 for all 119909

(iii) P3 (resolution) 119867(119860) ge 119867(119860lowast) where 119860

lowast is sharp-ened version of 119860

(iv) P4 (symmetry) 119867(119860) = 119867(119860) where 119860 is the com-plement of 119860 that is 120583

119860(119909) = 1 minus 120583

119860(119909)

Dubosis and Prade [11 12] interpreted the measure of fuzzi-ness 119867(119860) as quantity of information which is being lost ingoing from a crisp number to a fuzzy number Itmay be notedthat the entropy of an element with a givenmembership func-tion 120583

119860(119909) is increasing if 120583

119860(119909) is in [0 05] and decreasing if

120583119860(119909) is in [05 1] We accept the definition of fuzzy number

given by Tanaka andWatada [4] where themean value is alsocalled apex

Let 119883 = (1199091 1199092 119909

119899) be a discrete random variable

with probability distribution 119875 = (1199011 1199012 119901

119899) in an exper-

iment then according to Shannon [13] the informationcontained in this experiment is given by

119867(119875) = minus

119899

sum

119894=1

119901119894log119901119894 (2)

Based on this famous Shannonrsquos entropy de Luca andTermini[10] indicated the following measure of fuzzy entropy

119867(119860) = minus 119870int119909isin119883

[120583119860(119909) log120583

119860(119909)

+ (1 minus 120583119860(119909)) log (1 minus 120583

119860(119909))] 119889119909

(3)

Kumar et al [14] studied fuzzy linear regression (FLR) modelwith some restrictions in the form of prior information andobtained the estimators of regression coefficients with thehelp of fuzzy entropy for the restricted FLR model Herewe propose an intuitionistic fuzzy regression model and itsgeneral form in triangular intuitionistic fuzzy setup is givenby

y = 1205730+ 1205731x1+ sdot sdot sdot + 120573

119896x119896+ random error (4)

where the value of the output variable y defined by (4) is atriangular intuitionistic fuzzy number 120573

0 1205731 120573

119896is a vec-

tor of intuitionistic fuzzy parameters where 120573119895= (119898119895 120572119895 120573119895

1205721015840

119895 1205731015840

119895) is a triangular intuitionistic fuzzy number for 119895 =

0 1 119896 and x1 x2 x

119896are triangular intuitionistic fuzzy

(explanatory) variables

11 Intuitionistic Fuzzy Sets Basic Definitions and NotationsIt may be recalled that a fuzzy set 119860 in119883 given by Zadeh [1]is as follows

119860 = (119909 120583119860(119909)) 119909 isin 119883 (5)

where 120583119860

119883 rarr [0 1] is the membership function of thefuzzy set 119860 and 120583

119860(119909) is the grade of belongingness of 119909 into

119860 Thus in fuzzy set theory the grade of nonbelongingnessof an element 119909 into 119860 is equal to 1 minus 120583

119860(119909) However

while expressing the degree of membership of an element ina fuzzy set the corresponding degree of nonmembership isnot always equal to one minus the degree of belongingnessThe fact is that in real life the linguistic negation does notalways identify with logical negation Therefore Atanassov[15ndash18] suggested a generalization of classical fuzzy set calledintuitionistic fuzzy set (IFS)

Atanassovrsquos IFS 119860 under the universal set119883 is defined as

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ 119909 isin 119883 (6)

where 120583119860 ]119860

119883 rarr [0 1] are the membership andnonmembership functions such that 0 le 120583

119860+ 120583119860

le 1 forall 119909 isin 119883 The numbers 120583

119860(119909) and ]

119860(119909) denote the degree

of membership and nonmembership of an element 119909 isin 119883

to the set 119860 sub 119883 respectively For each element 119909 isin 119883 theamount 120587

119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) is called the degree of

indeterminacy (hesitation part) It is the degree of uncertaintywhether 119909 belongs to 119860 or not

12 Intuitionistic Fuzzy Numbers (IFNs) In literature Burilloand Bustince [19] Lee [20] Liu and Shi [21] and Grze-gorzewski [22] proposed various research works on intu-itionistic fuzzy numbers In this section the notion of IFNshas been studied and presented by the taking care of theseresearch works

Definition 1 An intuitionistic fuzzy subset 119860 = ⟨119909 120583119860(119909)

]119860(119909)⟩ 119909 isin 119883 of the real line R is called an intuitionistic

fuzzy number if the following axioms hold

(i) 119860 is normal that is there exist 119898 isin R (sometimescalled the mean value of 119860) such that 120583

119860(119898) = 1 and

]119860(119898) = 0

International Scholarly Research Notices 3

(ii) the membership function 120583119860is fuzzy-convex that is

120583119860(120582 sdot 1199091+ (1 minus 120582) sdot 119909

2) ge min 120583

119860(1199091) 120583119860(1199092)

forall1199091 1199092isin 119883 120582 isin [0 1]

(7)

(iii) the nonmembership function ]119860is fuzzy-concave

that is

]119860(120582 sdot 1199091+ (1 minus 120582) sdot 119909

2) le max ]

119860(1199091) ]119860(1199092)

forall1199091 1199092isin 119883 120582 isin [0 1]

(8)

(iv) the membership and the nonmembership functionsof 119860 satisfying the conditions 0 le 119891

1(119909) + 119892

1(119909) le 1

and 0 le 1198912(119909) + 119892

2(119909) le 1 have the following form

120583119860(119909) =

1198911(119909) for 119898 minus 120572 le 119909 le 119898

1 for 119909 = 119898

1198912(119909) for 119898 le 119909 le 119898 + 120573

0 otherwise

(9)

where the functions 1198911(119909) and 119891

2(119909) are strictly

increasing and decreasing functions in [119898minus120572119898] and[119898119898 + 120573] respectively and

]119860(119909) =

1198921(119909) for 119898 minus 120572

1015840le 119909 le 119898

0 for 119909 = 119898

1198922(119909) for 119898 le 119909 le 119898 + 120573

1015840

1 otherwise

(10)

where the functions 1198921(119909) and 119892

2(119909) are strictly

decreasing and increasing functions in [119898 minus 1205721015840 119898]

and [119898119898 + 1205731015840] respectively Here 120572 and 120573 are

called the left and right spreads of the membershipfunction 120583

119860 respectively 1205721015840 and 120573

1015840 are called the leftand right spreads of the nonmembership function]119860(119909) Symbolically an intuitionistic fuzzy number is

represented as 119860 IFN = (119898 120572 120573 1205721015840 1205731015840)

Definition 2 An IFN 119860 IFN = (119898 120572 120573 1205721015840 1205731015840)may be defined

as a triangular intuitionistic fuzzy number (TIFN) if and onlyif its membership and nonmembership functions take thefollowing form

120583119860(119909) =

1 minus119898 minus 119909

120572 for 119898 minus 120572 le 119909 le 119898

1 for 119909 = 119898

1 minus119909 minus 119898

120573 for 119898 le 119909 le 119898 + 120573

0 otherwise

(11)

]119860(119909) =

119898 minus 119909

1205721015840 for 119898 minus 120572

1015840le 119909 le 119898

0 for 119909 = 119898

119909 minus 119898

1205731015840 for 119898 le 119909 le 119898 + 120573

1015840

1 otherwise

(12)

It may be noted that a TIFN 119860 = (119898 120572 120573 1205721015840 1205731015840) degen-

erate to a triangular fuzzy number 119860 = (119898 120572 120573) if 120572 = 1205721015840

120573 = 1205731015840 and ]

119860(119909) = 1 minus 120583

119860(119909) forall119909 isin R Further an TIFN

119860 = ⟨119909 120583119860(119909) ]119860(119909)⟩ 119909 isin R that is 119860 = (119898 120572 120573 120572

1015840 1205731015840)

is a conjunction of two fuzzy numbers119860+ = (119898 120572 120573)with themembership function 120583

119860+(119909) = 120583

119860(119909) and 119860

minus= (119898 120572

1015840 1205731015840)

with the membership function 120583119860(119909) = 1 minus ]

119860(119909)

The entropy calculated using (3) from the membershipfunction of TIFN given by (11) can be expressed as followssize

119867(119860) = minus 119870[int119909isin[119898minus120572119898]

[120583119860(119909) log 120583

119860(119909) + (1 minus 120583

119860(119909))

times log (1 minus 120583119860(119909))] 119889119909

+ int119909isin[119898119898+120573]

[120583119860(119909) log120583

119860(119909) + (1 minus 120583

119860(119909))

times log (1 minus 120583119860(119909))] 119889119909]

= 119867119871(119860) + 119867

119877(119860)

(13)

where 119867119871(119860) = 1198701205722 and 119867

119877(119860) = 1198701205732 It follows that

119867(119860) = 119870(120572 + 120573)2 which does not depend on119898 It may beobserved that in the case of symmetrical TIFN the left andthe right entropies are identical On the other hand in case ofnonsymmetric TIFN the left entropy is a function of 120572 andthe right entropy is a function of 120573 Similarly the left entropyand the right entropy from the nonmembership function(which we called left to left and right to right entropies) ofthe TIFN are the functions of 1205721015840 and 120573

1015840 respectively Hencea triangular intuitionistic fuzzy number can be characterizedby five attributes the position parameter 119898 the left entropy120572 the right entropy 120573 left to left entropy 1205721015840 and right to rightentropy 1205731015840 There is a one-to-one correspondence between atriangular intuitionistic fuzzy number and its entropies Inother words given a triangular intuitionistic fuzzy numberone can determine the unique position and entropies Con-versely given a position and entropies one can construct aunique triangular intuitionistic fuzzy number

Sometimes experimenterrsquos past experiences may be avail-able as prior information about unknown regression coeffi-cients to estimate more efficient estimators Here we assumethat such prior information is provided in the form of exactlinear restrictions on regression coefficients In the presentwork we first find the unrestricted estimators of regressioncoefficients with the help of fuzzy entropy Next we introducethe restricted intuitionistic fuzzy linear regression modelwith fuzzy entropy Further the restricted estimators of theregression coefficients are obtained by incorporating theprior information in the form of linear restrictions

2 Restricted IFWLR Model withFuzzy Entropy

Without loss of generality suppose that all observations(y119894 x1198941 x1198942 x

119894119896) 119894 = 1 119899 in the regression analysis

are triangular intuitionistic fuzzy numbers The notion of

4 International Scholarly Research Notices

regression using fuzzy entropy is to construct five conven-tional regression equations (one for apex one for left entropyof the membership function one for right entropy of themembership function one for left entropy of the nonmem-bership function and one for right entropy of the non-membership function) for the response variable y using thecorresponding attributes of the 119896 fuzzy explanatory variablesx119895 In order to be specific we denote ya xa

1 xa2 xa

119896by the

apexes of y x1 x2 x

119896 respectively ely e

lx1

elx2

elx119896

bythe left entropy of y x

1 x2 x

119896 respectively ery e

rx1

erx2

erx119896

by the right entropy of y x1 x2 x

119896 respectively

el1015840

y el1015840x1

el1015840

x2

el1015840

x119896

by the left to left entropy of y x1 x2 x

119896

respectively and er1015840

y er1015840x1

er1015840

x2

er1015840

x119896

by the right to rightentropy of y x

1 x2 x

119896 respectively Therefore the five

fundamental regression equations in a nonrecursive (non-adaptive) setup may be written as

ya = 119860119886

0+

119896

sum

119894=1

(119860119886

119894xa119894+ 119861119886

119894elx119894

+ 119862119886

119894erx119894

+ 119863119886

119894el1015840

x119894

+ 119864119886

119894er1015840

x119894

) + 120576ya

ely = 119860119897

0+

119896

sum

119894=1

(119860119897

119894xa119894+ 119861119897

119894elx119894

+ 119862119897

119894erx119894

+ 119863119897

119894el1015840

x119894

+ 119864119897

119894er1015840

x119894

) + 120576ely

ery = 119860119903

0+

119896

sum

119894=1

(119860119903

119894xa119894+ 119861119903

119894elx119894

+ 119862119903

119894erx119894

+ 119863119903

119894el1015840

x119894

+ 119864119903

119894er1015840

x119894

) + 120576ery

el1015840

y = 1198601198971015840

0+

119896

sum

119894=1

(1198601198971015840

119894xa119894+ 1198611198971015840

119894elx119894

+ 1198621198971015840

119894erx119894

+ 1198631198971015840

119894el1015840

x119894

+ 1198641198971015840

119894er1015840

x119894

)

+ 120576el1015840y

er1015840

y = 1198601199031015840

0+

119896

sum

119894=1

(1198601199031015840

119894xa119894+ 1198611199031015840

119894elx119894

+ 1198621199031015840

119894erx119894

+ 1198631199031015840

119894el1015840

x119894

+ 1198641199031015840

119894er1015840

x119894

)

+ 120576er1015840y

(14)

where 120576ya 120576ely 120576ery 120576el1015840y and 120576er1015840y are the error vectors ofdimension 119899 times 1 The compact form of the above mentionednonrecursive or nonadaptive equations is given by

ya = X120573 + 120576ya

ely = X120572 + 120576ely

ery = X120574 + 120576ery

el1015840

y = X1205721015840 + 120576el1015840y

er1015840

y = X1205741015840 + 120576er1015840y

(15)

where

X= (1 xa1 xa2 xa

119896

elx1

elx2

elx119896

erx1

erx2

erx119896

el1015840

x1

el1015840

x2

el1015840

x119896

er1015840

x1

er1015840

x2

er1015840

x119896

)

119899times(5119896+1)

120573= (119860119886

0

1198601198861 119860119886

2 119860

119886

119896

1198611198861 119861119886

2 119861

119886

119896

1198621198861 119862119886

2

119862119886

119896

1198631198861 119863119886

2 119863

119886

119896

1198641198861 119864119886

2 119864

119886

119896)

119879

(5119896+1)times1

120572= (119860119897

0

1198601198971 119860119897

2 119860

119897

119896

1198611198971 119861119897

2 119861

119897

119896

1198621198971 119862119897

2

119862119897

119896

1198631198971 119863119897

2 119863

119897

119896

1198641198971 119864119897

2 119864

119897

119896)

119879

(5119896+1)times1

120574= (119860119903

0

1198601199031 119860119903

2 119860

119903

119896

1198611199031 119861119903

2 119861

119903

119896

1198621199031 119862119903

2

119862119903

119896

1198631199031 119863119903

2 119863

119903

119896

1198641199031 119864119903

2 119864

119903

119896)

119879

(5119896+1)times1

1205721015840= (119860

1198971015840

0

1198601198971015840

1 1198601198971015840

2 119860

1198971015840

119896

1198611198971015840

1 1198611198971015840

2 119861

1198971015840

119896

1198621198971015840

1 1198621198971015840

2

1198621198971015840

119896

1198631198971015840

1 1198631198971015840

2 119863

1198971015840

119896

1198641198971015840

1 1198641198971015840

2 119864

1198971015840

119896)

119879

(5119896+1)times1

1205741015840= (119860

1199031015840

0

1198601199031015840

1 1198601199031015840

2 119860

1199031015840

119896

1198611199031015840

1 1198611199031015840

2 119861

1199031015840

119896

1198621199031015840

1 1198621199031015840

2

1198621199031015840

119896

1198631199031015840

1 1198631199031015840

2 119863

1199031015840

119896

1198641199031015840

1 1198641199031015840

2 119864

1199031015840

119896)

119879

(5119896+1)times1

(16)

In many real life situations where the measurements arecarried out (for example car speed astronomical distance)it is natural to think that the spread (vagueness) in themeasure of a phenomenon is proportional to its intensityDrsquoUrso and Gastaldi [23] have done several simulations andobserved that even if we consider an adaptive or recursiveregression model along with nonadaptive or nonrecursiveregression model they yield identical solutions when thereis only one independent variable But if there are morethan one independent variable then the estimated values ofthe left entropies and right entropies obtained through therecursive fuzzy regression model will have less variance ascompared to the nonrecursive fuzzy regression model Withthis consideration we rewrite the proposed intuitionisticfuzzy linear regression model (15) in a recursiveadaptive

International Scholarly Research Notices 5

setup where dynamic of the entropies is dependent on themagnitude of the estimated apexes as follows

ya = yalowast

+ 120576ya where yalowast

= X120573

ely = ellowast

y + 120576lowast

ely where el

lowast

y = X120573119887 + 1119889

ery = erlowast

y + 120576lowast

ery where er

lowast

y = X120573119891 + 1119892

el1015840

y = el1015840lowast

y + 120576lowast

el1015840y where el

1015840lowast

y = X120573119901 + 1119902

er1015840

y = er1015840lowast

y + 120576lowast

er1015840y where er

1015840lowast

y = X120573119906 + 1V

(17)

where X is the 119899 times (5119896 + 1)-matrix containing the valuesof the input variables (data matrix) 120573 is a column 5119896 + 1-vector containing the regression coefficients for the apexesof the first model (referred to as core regression model)ya and yalowast are the vector of the observed apexes and thevector of the interpolated apexes respectively both havingdimension 119899 times 1 ely and ely

lowast

are the vector of the observedleft entropies and the vector of the interpolated left entropiesrespectively both having dimension 119899 times 1 ery and ery

lowast are thevector of the observed right entropies and the vector ofthe interpolated right entropies respectively both havingdimension 119899 times 1 el

1015840

y and el1015840

ylowast

are the vector of the observedleft to left entropies and the vector of the interpolated left toleft entropies respectively both having dimension 119899 times 1 er

1015840

y

and er1015840

ylowast

are the vector of the observed right to right entropiesand the vector of the interpolated right to right entropiesrespectively both having dimension 119899 times 1 and 1 is a (119899 times 1)-vector of all 11015840s 119887 and 119889 are regression parameters for thesecond regression equation model (referred to as left entropyregression model) 119891 and 119892 are regression parameters for thethird regressionmodel (referred to as right entropy regressionmodel) 119901 and 119902 are regression parameters for the fourthregression equation model (referred to as left to left entropyregression model) and 119906 and V are regression parametersfor the fifth regression equation model (referred to as rightto right entropy regression model) The error term in theregression equation of apexes will remain the same while theerror terms in the regression equations of entropies may bedifferent The error vectors 120576lowastely and 120576

lowast

eryin the left and right

entropies are of the dimension (119899 times 1) and the error vectors120576lowast

elyand 120576lowastery in the left to left and right to right entropies are of

the dimension (119899 times 1)If some prior information about unknown regression

coefficients is available on the basis of past experiences thenit may be used to estimate more efficient estimators Weassume that such prior information is in the form of exactlinear restrictions on regression coefficients In the presentmodel we associate such restrictions in the equations forthe estimation of regression coefficients in the intuitionisticfuzzy linear regression model with fuzzy entropy Thereforewe make the model capable of taking into account possible

linear relations between the size of the entropies and themagnitude of the estimated apexesMoreover we assume thatthe regression coefficients 120573 are subjected to the 119895 (119895 lt 5119896+1)exact linear restrictions which are given by

h = H120573 (18)

whereh andH are known and thematrixH is of full row rank

3 Estimation of Regression Coefficients

In many applications it is possible that the values of the vari-ables are on completely different scales ofmeasurement Alsothe possible larger variations in the values will have largerintersample differences so they will dominate in the calcu-lation of Euclidean distances Therefore some form of stan-dardization is necessary to balance out the individual con-tributions Consider the Euclidean distance between two tri-angular intuitionistic fuzzy numbers 119910

119894= (119910119886

119894 119890119897

119910119894

119890119903

119910119894

1198901198971015840

119910119894

1198901199031015840

119910119894

)

and 119910lowast

119894= (119910119886lowast

119894 119890119897lowast

119910119894

119890119903lowast

119910119894

1198901198971015840lowast

119910119894

1198901199031015840lowast

119910119894

) along with weights 1199081 1199082

1199083 1199084 and 119908

5as follows

120575119894equiv 120575 (119910

119894 119910lowast

119894)

= (1199081(119910119886

119894minus 119910119886lowast

119894)2

+ 1199082(119890119897

119910119894

minus 119890119897lowast

119910119894

)2

+ 1199083(119890119903

119910119894

minus 119890119903lowast

119910119894

)2

+1199084(1198901198971015840

119910119894

minus 1198901198971015840lowast

119910119894

)

2

+ 1199085(1198901199031015840

119910119894

minus 1198901199031015840lowast

119910119894

)

2

)

12

(19)

It may be observed that we compute the usual squared differ-ences between the values of variables on their original scalesas in the usual Euclidean distance but then multiply thesesquared differences by their corresponding weights

Next similar to common linear regression (based on crispdata) the regression parameters are estimated byminimizingthe following sum of square errors (we use a compact matrixnotation)

120593 (120573 119887 119889 119891 119892 119901 119902 119906 V)

=

119899

sum

119894=1

1199081(119910119886

119894minus 119910119886lowast

119894)2

+

119899

sum

119894=1

1199082(119890119897

119910119894

minus 119890119897lowast

119910119894

)2

+

119899

sum

119894=1

1199083(119890119903

119910119894

minus 119890119903lowast

119910119894

)2

+

119899

sum

119894=1

1199084(1198901198971015840

119910119894

minus 1198901198971015840lowast

119910119894

)2

+

119899

sum

119894=1

1199085(1198901199031015840

119910119894

minus 1198901199031015840lowast

119910119894

)2

= 1199081(ya minus yalowast)

T(ya minus yalowast) + 119908

2(ely minus ely

lowast

)T(ely minus ely

lowast

)

+ 1199083(ery minus er

lowast

y )T(ery minus er

lowast

y )

+ 1199084(el1015840

y minus el1015840

ylowast

)

T(el1015840

y minus el1015840

ylowast

)

+ 1199085(er1015840

y minus er1015840

ylowast

)

T(er1015840

y minus er1015840

ylowast

)

6 International Scholarly Research Notices

= 1199081((ya)Tya minus 2(ya)Tyalowast + (yalowast)

Tyalowast)

+ 1199082((ely)

Tely minus 2(ely)

Telylowast

+ (elylowast

)Telylowast

)

+ 1199083((ery)

Tery minus 2(ery)

Terylowast

+ (erylowast

)Terylowast

)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )Tel1015840

ylowast

+ (el1015840

ylowast

)

Tel1015840

ylowast

)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )Ter1015840

ylowast

+ (er1015840

ylowast

)

Ter1015840

ylowast

)

= 1199081((ya)Tya minus 2(ya)TX120573 + 120573TXTX120573)

+ 1199082((ely)

Tely minus 2(ely)

T(X120573119887 + 1119889))

+ 1199082((X120573119887 + 1119889)T (X120573119887 + 1119889))

+ 1199083((ery)

Tery minus 2(ery)

T(X120573119891 + 1119892))

+ 1199083((X120573119891 + 1119892)T (X120573119891 + 1119892))

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )T(X120573119901 + 1119902))

+ 1199084((X120573119901 + 1119902)T (X120573119901 + 1119902))

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )T(X120573119906 + 1V))

+ 1199085((X120573119906 + 1V)T (X120573119906 + 1V))

= 1199081((ya)Tya minus 2(ya)TX120573)

+ 120573TXTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

+ 1199082((ely)

Tely minus 2(ely)

TX120573119887 minus 2(ely)

T1119889)

+ 1199083((ery)

Tery minus 2(ery)

TX120573119891 minus 2(ery)

T1119892)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )TX120573119901 minus 2(el

1015840

y )T1119902)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )TX120573119906 minus 2(er

1015840

y )T1V)

+ 2120573TXT1 (119908

2119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V)

+ 119899 (11990821198892+ 11990831198922+ 11990841199022+ 1199085V2)

(20)

Differentiating 120593(120573 119887 119889 119891 119892 119901 119902 119906 V) that is (20) partiallywith respect to 120573 and equating it to zero we get

120597120593 (120573 119887 119889 119891 119892 119901 119902 119906 V)120597120573

= 0

997904rArr minus1199081XTya + XTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

minus 1199082XTely119887 minus 119908

3XTery119891 minus 119908

4XTel

1015840

y119901 minus 1199085XTer

1015840

y 119906

+ XT1 (1199082119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V) = 0

997904rArr 120573 = ((XTX)minus1

XT[1199081ya + 119908

2ely119887 + 119908

3ery119891

+ 1199084el1015840

y119901 + 1199085er1015840

y 119906

minus1 (1199082119887119889+119908

3119891119892+119908

4119901119902+119908

5119906V)])

times (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)minus1

(21)

Similarly differentiating (20) partially with respect to 119887 119889 119891119892 119901 119902 119906 and V we get

119887 = (120573TXTX120573)

minus1

[(ely)TX120573 minus 120573TXT1119889] (22)

119889 =1

119899[(ely)

T1 minus 120573

TXT1119887] (23)

119891 = (120573TXTX120573)

minus1

[(ery)TX120573 minus 120573TXT1119892] (24)

119892 =1

119899[(er

1015840

y )T1 minus 120573

TXT1119891] (25)

119901 = (120573TXTX120573)

minus1

[(el1015840

y )TX120573 minus 120573TXT1119902] (26)

119902 =1

119899[(el

1015840

y )T1 minus 120573

TXT1119901] (27)

119906 = (120573TXTX120573)

minus1

[(er1015840

y )TX120573 minus 120573TXT1V] (28)

V =1

119899[(er

1015840

y )T1 minus 120573

TXT1119906] (29)

respectively

International Scholarly Research Notices 7

Equations (21)ndash(29) are recursive solutions for the prob-lem of least square estimation with intuitionistic fuzzy dataTherefore we rewrite the system of equations explicitly in arecursive way as follows

120573119894+1

= ((XTX)minus1

XT[1199081ya + 119908

2ely119887119894 + 119908

3ery119891119894 + 119908

4el1015840

y119901119894

+ 1199085er1015840

y 119906119894 minus 1 (1199082119887119894119889119894+ 1199083119891119894119892119894

+1199084119901119894119902119894+ 1199085119906119894V119894) ] )

times (1199081+ 11990821198872

119894+ 11990831198912

119894+ 11990841199012

119894+ 11990851199062

119894)minus1

119887119894+1

= (120573T119894+1

XTX120573i+1)minus1

[(ely)TX120573i+1 minus 120573

Ti+1X

T1119889119894]

119889119894+1

=1

119899[(ely)

T1 minus 120573

Ti+1X

T1119887119894]

119891119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(ery)TX120573i+1 minus 120573

Ti+1X

T1119892119894]

119892119894+1

=1

119899[(er

1015840

y )T1 minus 120573

Ti+1X

T1119891119894]

119901119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(el1015840

y )TX120573i+1 minus 120573

Ti+1X

T1119902119894]

119902119894+1

=1

119899[(el

1015840

y )T1 minus 120573

Ti+1X

T1119901119894]

119906119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(er1015840

y )TX120573i+1 minus 120573

Ti+1X

T1V119894]

V119894+1

=1

119899[(er

1015840

y )T1 minus 120573

Ti+1X

T1119906119894]

(30)

In order to initiate the recursive process of obtaining theestimators we take some initial values for 119887 119889 119891 119892 119901 119902119906 V and 120573 After several numbers of iterations the valuesof estimators get corrected to a predefined error of toleranceWe denote these values by 119889 119891 119892 119901 119902 V and in orderto differentiate them from the eventually obtained restrictedestimator in the next commutation

In a more general setup if in the linear regression model(17) we consider 119896

1crisp and 119896

2intuitionistic fuzzy input

variables then the dimensions of X and 120573 will be 119899 times (1198961+

51198962+ 1) and (119896

1+ 51198962+ 1) times 1 respectively It may further be

noted that the core of the solutionrsquos structure will remain thesame and we will have similar kind of estimators

Remark If a TIFN 119860 = (119898 120572 120573 1205721015840 1205731015840) degenerate to a trian-

gular fuzzy number 119860 = (119898 120572 120573) then our nonsymmetric

intuitionistic fuzzy weighted linear regression model reducesto nonsymmetric fuzzy linear regression model defined byKumar et al [24]

Next we assume that the regression coefficients aresubjected to the linear restrictions which are given by (18) Itmay be noted that the unrestricted estimator obtained abovein (21) does not satisfy the given restrictions (18) We aimto obtain the restricted estimator which satisfies the givenrestrictions under the regression model (17) For this wepropose to minimize the following score function

119878 (120582120573 119887 119889 119891 119892 119901 119902 119906 V)

= 120593 (120573 119887 119889 119891 119892 119901 119902 119906 V) minus 2120582 (H120573 minus h)

= 1199081((ya)Tya minus 2(ya)TX120573)

+ 120573TXTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

+ 1199082((ely)

Tely minus 2(ely)

TX120573119887 minus 2(ely)

T1119889)

+ 1199083((ery)

Tery minus 2(ery)

TX120573119891 minus 2(ery)

T1119892)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )TX120573119901 minus 2(el

1015840

y )T1119902)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )TX120573119906 minus 2(er

1015840

y )T1V)

+ 2120573TXT1 (119908

2119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V)

+ 119899 (11990821198892+ 11990831198922+ 11990841199022+ 1199085V2)

minus 2120582 (H120573 minus h)

(31)

where 2120582 is the vector of Lagrangersquos Multiplier

Differentiating 119878(120582120573 119887 119889 119891 119892 119901 119902 119906 V) partially withrespect to 120573 and equating it to zero we get

997904rArr minus1199081XTya + XTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

minus 1199082XTely119887 minus 119908

3XTery119891 minus 119908

4XTel

1015840

y119901 minus 1199085XTer

1015840

y 119906

+ XT1 (1199082119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V) minusH1015840120582 = 0

(32)

8 International Scholarly Research Notices

Here we again relabel the computed restricted estimator by Therefore in view of (21) and (32) we get size

997904rArr = ((XTX)minus1

XT[1199081ya + 119908

2ely119887 + 119908

3ery119891 + 119908

4el1015840

y119901

+ 1199085er1015840

y 119906

minus1 (1199082119887119889+119908

3119891119892+119908

4119901119902+119908

5119906V) ] )

times (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)minus1

+(XTX)

minus1

HT120582

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

997904rArr = +1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

times (XTX)minus1

HT120582

(33)

Similarly differentiating 119878(120582120573 119887 119889 119891 119892 119901 119902 119906 V) par-tially with respect to 120582 and equating it to zero we get

997904rArr H = h

997904rArr H + 1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

timesH(XTX)minus1

HT120582 = h

997904rArr = (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

times [H(XTX)minus1

HT]minus1

(h minusH)

(34)

From (33) and (34) we have

997904rArr = + (XTX)minus1

HT[H(XTX)

minus1

HT]minus1

(h minusH) (35)

Also differentiating (31) partially with respect to 119887 119889 119891 119892 119901119902 119906 and V and equating all to zero we get

= 119889 = 119889 119891 = 119891 119892 = 119892

119901 = 119901 119902 = 119902 = V = V(36)

respectively From (35) we see that

997904rArr H = H + [H(XTX)minus1

HT] [H(XTX)

minus1

HT]minus1

times (h minusH)

997904rArr H = H + (h minusH) = h(37)

Therefore the estimator satisfies the given restrictions (18)

4 Numerical Examples

We consider the following numerical examples to illustratethe proposed model

Example 1 We apply our procedure to estimate the intuition-istic fuzzy output value for a data consisting of the crisp inputand intuitionistic fuzzy output (where left entropy and rightentropy are equal) and tabulate the data in Table 1

We obtain = (minus44026 35733 73786 56858)1015840 =

02942 119889 = 147144 119891 = 02942 119892 = 147144 119901 = 02909119902 = 174487 = 02909 and V = 174487 where the numberof iterations required is 125

Example 2 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp input andintuitionistic fuzzy output (where left and right entropy arenot equal) and tabulate the data in Table 2

We obtain = (minus47697 35933 72030 59152)1015840 =

02952 119889 = 145871 119891 = 02646 119892 = 203429 119901 = 03052119902 = 157050 = 02717 and V = 231201 where the numberof iterations required is 113

Example 3 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp inputintuitionistic fuzzy input and intuitionistic fuzzy output(where left and right entropy are not equal) and tabulate thedata in Table 3

We obtain = (minus32352 06811 05314 minus09164 00846

minus31631 2953)1015840 = 04225 119889 = 05478 119891 = 04307 119892 =

01637 119901 = 03231 119902 = 38723 = 04985 and V = 18659

where the number of iterations required is 51

Example 4 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of intuitionistic fuzzyinput and intuitionistic fuzzy output (where left and rightentropy are not equal) and tabulate the data in Table 4

We obtain = (118141 minus02161 16104 minus18254 05687

minus01879)1015840 = 03880 119889 = 03674 119891 = 03880 119892 = 03674

119901 = 03547 119902 = 22108 = 03547 and V = 32108 wherethe number of iterations required is 255

5 Conclusions

An intuitionistic fuzzy weighted linear regression (IFWLR)model with and without some linear restrictions in theform of prior information has been studied The estimators

International Scholarly Research Notices 9

Table 1 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y y

a ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 44 42 96 42 44 449018 424850 943828 424850 4490182 14 8 3 48 47 120 47 48 528505 505256 1217099 505256 5285053 7 1 4 35 33 52 33 35 322052 296416 507324 296416 3220524 11 7 3 50 45 106 45 50 475861 452004 1036114 452004 4758615 7 12 15 80 79 189 79 80 740058 719256 1944413 719256 7400586 8 15 10 68 65 194 65 68 732147 711253 1917213 711253 7321477 3 9 6 45 42 107 42 45 485252 461503 1068398 461503 4852528 12 15 11 80 78 216 78 80 790260 770038 2117003 770038 7902609 10 5 8 55 52 108 52 55 505235 481717 1137100 481717 50523510 9 7 4 45 44 103 44 45 471612 447706 1021507 447706 471612

Table 2 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y ya ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 45 42 96 47 48 447743 427104 952620 455472 4900532 14 8 3 48 47 120 43 45 525995 502809 120905 523320 5597343 7 1 4 35 33 52 50 55 313430 297162 512469 339018 3704524 11 7 3 46 45 106 45 47 471120 449720 102922 475741 5108705 7 12 15 82 79 189 80 85 753765 723166 195547 720805 7625556 8 15 10 70 65 194 60 67 740419 710254 191173 709234 7506717 3 9 6 45 42 107 40 46 481512 459774 106328 484752 5201248 12 15 11 80 78 216 88 90 802328 770149 211461 762912 8057999 10 5 8 55 52 108 50 55 506447 483897 114499 506370 54232710 9 7 4 45 44 103 42 44 467241 445967 101651 472377 507415

Table 3 Crisp and int fuzzy input-int fuzzy output data

Object 119894Crisp and int fuzzy inputX = (x1 e1

1015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy outputylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 6 7 6 10 5 7 6 3 5 2 5 54158 25665 47776 22215 424732 7 6 5 12 4 6 7 5 4 5 7 58827 31771 62226 28438 496763 8 4 2 15 3 5 8 3 9 4 6 67973 43734 90537 40632 637884 9 7 5 20 8 10 7 3 10 2 4 64422 39089 79544 35897 583085 10 8 6 5 2 5 8 5 12 5 7 74885 52774 11193 49846 744516 11 10 8 15 5 7 3 2 8 4 6 65757 40835 83678 37678 603697 12 20 15 25 12 14 6 5 7 3 5 62393 36436 73265 33193 551798 13 12 7 30 15 18 8 7 14 6 8 83421 63937 13835 61226 876209 14 16 12 20 10 15 11 9 16 10 12 97423 82250 18169 79893 10922310 15 17 13 22 8 12 8 7 18 5 10 90740 73509 16100 70982 98912

10 International Scholarly Research Notices

Table 4 Intuitionistic fuzzy input-intuitionistic fuzzy output data

Object iInt fuzzy input

X = (e11015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy output

ylowast = (e1lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

ye1lowast

y yalowast er

lowast

y erlowast1015840

y

1 5 3 4 5 6 5 4 12 4 6 57505 42398 99797 42398 675052 7 6 7 8 9 7 5 7 5 8 57737 42652 10045 42652 677373 5 3 6 8 9 5 3 9 3 6 48608 32665 74714 32665 586084 4 2 7 9 11 3 1 4 1 4 37872 20920 44446 20920 478725 3 2 5 7 8 4 2 6 2 5 49552 33698 77377 33698 595526 6 3 6 7 10 5 4 8 4 6 45158 28891 64987 28891 551587 5 2 4 9 12 4 3 9 3 5 55863 40602 95168 40602 658638 6 5 8 13 15 7 5 10 5 8 52404 36818 85417 36818 624049 8 7 12 15 17 4 3 5 3 5 39099 22262 47905 22262 4909910 15 10 15 20 25 4 2 3 2 5 36202 19092 39736 19092 46202

of regression coefficients have also been obtained with thehelp of fuzzy entropy for the restrictedunrestricted IFWLRmodel by assigning some weights in the distance functionIt has been observed that the restricted estimator is betterthan unrestricted estimator in some sense Thus wheneversome prior information is available in terms of exact linearrestrictions on regression coefficients it is advised to userestricted estimator in place of unrestricted estimator

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] H Tanaka S Uejima and K Asai ldquoFuzzy linear regressionmodelrdquo IEEE Transactions on Systems Man and Cyberneticsvol 10 pp 2933ndash2938 1980

[3] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982

[4] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988

[5] H Tanaka I Hayashi and JWatada ldquoPossibilistic linear regres-sion analysis for fuzzy datardquo European Journal of OperationalResearch vol 40 no 3 pp 389ndash396 1989

[6] H Tanaka and H Ishibuchi ldquoIdentification of possibilisticlinear systems by quadratic membership functions of fuzzyparametersrdquo Fuzzy Sets and Systems vol 41 no 2 pp 145ndash1601991

[7] D T Redden and W H Woodall ldquoProperties of certain fuzzylinear regression methodsrdquo Fuzzy Sets and Systems vol 64 no3 pp 361ndash375 1994

[8] P-T Chang and E S Lee ldquoFuzzy linear regression with spreadsunrestricted in signrdquoComputers andMathematics with Applica-tions vol 28 no 4 pp 61ndash70 1994

[9] G Peters ldquoFuzzy linear regression with fuzzy intervalsrdquo FuzzySets and Systems vol 63 no 1 pp 45ndash55 1994

[10] A de Luca and S Termini ldquoA definition of a nonprobabilisticentropy in the setting of fuzzy sets theoryrdquo Information andControl vol 20 no 4 pp 301ndash312 1972

[11] D Dubosis and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[12] D Dubois and H Prade Fuzzy Sets and Statistical PossibilityTheory Plenum Press New York NY USA 1988

[13] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash656 1948

[14] T Kumar N Gupta and R K Bajaj ldquoFuzzy entropy onrestricted fuzzy linear regression model with cross validationand applicationsrdquo in Proceedings of the International Conferenceon Advances in Computing and Communications (ICACC rsquo12)pp 5ndash8 August 2012

[15] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[16] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[17] K T Atanassov Intuitionistic Fuzzy Sets Theory and Applica-tions vol 35 of Studies in Fuzziness and SoftComputing Physica1999

[18] K T Atanassov ldquoNew operations defined over the intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 61 no 2 pp 137ndash1421994

[19] P Burillo and H Bustince ldquoSome definitions of intuitionisticfuzzy numberrdquo in Proceedings of the 3rd Conference of theEuropean Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[20] F Lee Fuzzy Information Processing System Peking UniversityPress Beijing China 1998

[21] H Liu and K Shi ldquoIntuitionistic fuzzy numbers and intuition-istic distribution numbersrdquo Journal of Fuzzy Mathematics vol8 no 4 pp 909ndash918 2000

[22] P Grzegorzewski ldquoDistances and orderings in a family of intu-itionistic fuzzy numbersrdquo in Proceedings of the 3rd Conference ofthe European Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[23] P DrsquoUrso and T Gastaldi ldquoA least-squares approach to fuzzylinear regression analysisrdquo Computational Statistics and DataAnalysis vol 34 no 4 pp 427ndash440 2000

[24] T Kumar R K Bajaj and N Gupta ldquoFuzzy entropy in fuzzyweighted linear regression model under linear restrictions withsimulation studyrdquo International Journal of General Systems vol43 no 2 pp 135ndash148 2014

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

2 International Scholarly Research Notices

models in fuzzy regression In the approaches of Tanaka etal they considered the L-R fuzzy data and minimized theindex of fuzziness of the fuzzy linear regression model Asdescribed by Tanaka and Watada [4] ldquoA fuzzy number is afuzzy subset of the real line whose highest membership valuesare clustered around a given real number called the meanvalue the membership function is monotonic on both sides ofthis mean valuerdquo Hence fuzzy number can be decomposedinto position and fuzziness where the position is representedby the element with the highest membership value and thefuzziness of a fuzzy number is represented by themembershipfunction The comparison among various fuzzy regressionmodels and the difference between the approaches of fuzzyregression analysis and conventional regression analysishave been presented by Redden and Woodall [7] Changand Lee [8] and Redden and Woodall [7] pointed out someweaknesses of the approaches proposed by Tanaka et al Afuzzy linear regression model based on Tanakarsquos approach byconsidering the fuzzy linear programming problem has alsobeen introduced by Peters [9]

In fuzzy set theory the entropy is a measure of degreeof fuzziness which expresses the amount of average ambi-guitydifficulty in making a decision whether an elementbelongs to a set or not The following are the four propertiesintroduced in de Luca and Termini [10] which are widelyaccepted as a criterion for defining any new fuzzy entropymeasure119867(sdot) of the fuzzy set 119860

(i) P1 (sharpness)119867(119860) is minimum if and only if 119860 is acrisp set that is 120583

119860(119909) = 0 or 1 for all 119909

(ii) P2 (maximality) 119867(119860) is maximum if and only if120583119860(119909) = 05 for all 119909

(iii) P3 (resolution) 119867(119860) ge 119867(119860lowast) where 119860

lowast is sharp-ened version of 119860

(iv) P4 (symmetry) 119867(119860) = 119867(119860) where 119860 is the com-plement of 119860 that is 120583

119860(119909) = 1 minus 120583

119860(119909)

Dubosis and Prade [11 12] interpreted the measure of fuzzi-ness 119867(119860) as quantity of information which is being lost ingoing from a crisp number to a fuzzy number Itmay be notedthat the entropy of an element with a givenmembership func-tion 120583

119860(119909) is increasing if 120583

119860(119909) is in [0 05] and decreasing if

120583119860(119909) is in [05 1] We accept the definition of fuzzy number

given by Tanaka andWatada [4] where themean value is alsocalled apex

Let 119883 = (1199091 1199092 119909

119899) be a discrete random variable

with probability distribution 119875 = (1199011 1199012 119901

119899) in an exper-

iment then according to Shannon [13] the informationcontained in this experiment is given by

119867(119875) = minus

119899

sum

119894=1

119901119894log119901119894 (2)

Based on this famous Shannonrsquos entropy de Luca andTermini[10] indicated the following measure of fuzzy entropy

119867(119860) = minus 119870int119909isin119883

[120583119860(119909) log120583

119860(119909)

+ (1 minus 120583119860(119909)) log (1 minus 120583

119860(119909))] 119889119909

(3)

Kumar et al [14] studied fuzzy linear regression (FLR) modelwith some restrictions in the form of prior information andobtained the estimators of regression coefficients with thehelp of fuzzy entropy for the restricted FLR model Herewe propose an intuitionistic fuzzy regression model and itsgeneral form in triangular intuitionistic fuzzy setup is givenby

y = 1205730+ 1205731x1+ sdot sdot sdot + 120573

119896x119896+ random error (4)

where the value of the output variable y defined by (4) is atriangular intuitionistic fuzzy number 120573

0 1205731 120573

119896is a vec-

tor of intuitionistic fuzzy parameters where 120573119895= (119898119895 120572119895 120573119895

1205721015840

119895 1205731015840

119895) is a triangular intuitionistic fuzzy number for 119895 =

0 1 119896 and x1 x2 x

119896are triangular intuitionistic fuzzy

(explanatory) variables

11 Intuitionistic Fuzzy Sets Basic Definitions and NotationsIt may be recalled that a fuzzy set 119860 in119883 given by Zadeh [1]is as follows

119860 = (119909 120583119860(119909)) 119909 isin 119883 (5)

where 120583119860

119883 rarr [0 1] is the membership function of thefuzzy set 119860 and 120583

119860(119909) is the grade of belongingness of 119909 into

119860 Thus in fuzzy set theory the grade of nonbelongingnessof an element 119909 into 119860 is equal to 1 minus 120583

119860(119909) However

while expressing the degree of membership of an element ina fuzzy set the corresponding degree of nonmembership isnot always equal to one minus the degree of belongingnessThe fact is that in real life the linguistic negation does notalways identify with logical negation Therefore Atanassov[15ndash18] suggested a generalization of classical fuzzy set calledintuitionistic fuzzy set (IFS)

Atanassovrsquos IFS 119860 under the universal set119883 is defined as

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ 119909 isin 119883 (6)

where 120583119860 ]119860

119883 rarr [0 1] are the membership andnonmembership functions such that 0 le 120583

119860+ 120583119860

le 1 forall 119909 isin 119883 The numbers 120583

119860(119909) and ]

119860(119909) denote the degree

of membership and nonmembership of an element 119909 isin 119883

to the set 119860 sub 119883 respectively For each element 119909 isin 119883 theamount 120587

119860(119909) = 1 minus 120583

119860(119909) minus ]

119860(119909) is called the degree of

indeterminacy (hesitation part) It is the degree of uncertaintywhether 119909 belongs to 119860 or not

12 Intuitionistic Fuzzy Numbers (IFNs) In literature Burilloand Bustince [19] Lee [20] Liu and Shi [21] and Grze-gorzewski [22] proposed various research works on intu-itionistic fuzzy numbers In this section the notion of IFNshas been studied and presented by the taking care of theseresearch works

Definition 1 An intuitionistic fuzzy subset 119860 = ⟨119909 120583119860(119909)

]119860(119909)⟩ 119909 isin 119883 of the real line R is called an intuitionistic

fuzzy number if the following axioms hold

(i) 119860 is normal that is there exist 119898 isin R (sometimescalled the mean value of 119860) such that 120583

119860(119898) = 1 and

]119860(119898) = 0

International Scholarly Research Notices 3

(ii) the membership function 120583119860is fuzzy-convex that is

120583119860(120582 sdot 1199091+ (1 minus 120582) sdot 119909

2) ge min 120583

119860(1199091) 120583119860(1199092)

forall1199091 1199092isin 119883 120582 isin [0 1]

(7)

(iii) the nonmembership function ]119860is fuzzy-concave

that is

]119860(120582 sdot 1199091+ (1 minus 120582) sdot 119909

2) le max ]

119860(1199091) ]119860(1199092)

forall1199091 1199092isin 119883 120582 isin [0 1]

(8)

(iv) the membership and the nonmembership functionsof 119860 satisfying the conditions 0 le 119891

1(119909) + 119892

1(119909) le 1

and 0 le 1198912(119909) + 119892

2(119909) le 1 have the following form

120583119860(119909) =

1198911(119909) for 119898 minus 120572 le 119909 le 119898

1 for 119909 = 119898

1198912(119909) for 119898 le 119909 le 119898 + 120573

0 otherwise

(9)

where the functions 1198911(119909) and 119891

2(119909) are strictly

increasing and decreasing functions in [119898minus120572119898] and[119898119898 + 120573] respectively and

]119860(119909) =

1198921(119909) for 119898 minus 120572

1015840le 119909 le 119898

0 for 119909 = 119898

1198922(119909) for 119898 le 119909 le 119898 + 120573

1015840

1 otherwise

(10)

where the functions 1198921(119909) and 119892

2(119909) are strictly

decreasing and increasing functions in [119898 minus 1205721015840 119898]

and [119898119898 + 1205731015840] respectively Here 120572 and 120573 are

called the left and right spreads of the membershipfunction 120583

119860 respectively 1205721015840 and 120573

1015840 are called the leftand right spreads of the nonmembership function]119860(119909) Symbolically an intuitionistic fuzzy number is

represented as 119860 IFN = (119898 120572 120573 1205721015840 1205731015840)

Definition 2 An IFN 119860 IFN = (119898 120572 120573 1205721015840 1205731015840)may be defined

as a triangular intuitionistic fuzzy number (TIFN) if and onlyif its membership and nonmembership functions take thefollowing form

120583119860(119909) =

1 minus119898 minus 119909

120572 for 119898 minus 120572 le 119909 le 119898

1 for 119909 = 119898

1 minus119909 minus 119898

120573 for 119898 le 119909 le 119898 + 120573

0 otherwise

(11)

]119860(119909) =

119898 minus 119909

1205721015840 for 119898 minus 120572

1015840le 119909 le 119898

0 for 119909 = 119898

119909 minus 119898

1205731015840 for 119898 le 119909 le 119898 + 120573

1015840

1 otherwise

(12)

It may be noted that a TIFN 119860 = (119898 120572 120573 1205721015840 1205731015840) degen-

erate to a triangular fuzzy number 119860 = (119898 120572 120573) if 120572 = 1205721015840

120573 = 1205731015840 and ]

119860(119909) = 1 minus 120583

119860(119909) forall119909 isin R Further an TIFN

119860 = ⟨119909 120583119860(119909) ]119860(119909)⟩ 119909 isin R that is 119860 = (119898 120572 120573 120572

1015840 1205731015840)

is a conjunction of two fuzzy numbers119860+ = (119898 120572 120573)with themembership function 120583

119860+(119909) = 120583

119860(119909) and 119860

minus= (119898 120572

1015840 1205731015840)

with the membership function 120583119860(119909) = 1 minus ]

119860(119909)

The entropy calculated using (3) from the membershipfunction of TIFN given by (11) can be expressed as followssize

119867(119860) = minus 119870[int119909isin[119898minus120572119898]

[120583119860(119909) log 120583

119860(119909) + (1 minus 120583

119860(119909))

times log (1 minus 120583119860(119909))] 119889119909

+ int119909isin[119898119898+120573]

[120583119860(119909) log120583

119860(119909) + (1 minus 120583

119860(119909))

times log (1 minus 120583119860(119909))] 119889119909]

= 119867119871(119860) + 119867

119877(119860)

(13)

where 119867119871(119860) = 1198701205722 and 119867

119877(119860) = 1198701205732 It follows that

119867(119860) = 119870(120572 + 120573)2 which does not depend on119898 It may beobserved that in the case of symmetrical TIFN the left andthe right entropies are identical On the other hand in case ofnonsymmetric TIFN the left entropy is a function of 120572 andthe right entropy is a function of 120573 Similarly the left entropyand the right entropy from the nonmembership function(which we called left to left and right to right entropies) ofthe TIFN are the functions of 1205721015840 and 120573

1015840 respectively Hencea triangular intuitionistic fuzzy number can be characterizedby five attributes the position parameter 119898 the left entropy120572 the right entropy 120573 left to left entropy 1205721015840 and right to rightentropy 1205731015840 There is a one-to-one correspondence between atriangular intuitionistic fuzzy number and its entropies Inother words given a triangular intuitionistic fuzzy numberone can determine the unique position and entropies Con-versely given a position and entropies one can construct aunique triangular intuitionistic fuzzy number

Sometimes experimenterrsquos past experiences may be avail-able as prior information about unknown regression coeffi-cients to estimate more efficient estimators Here we assumethat such prior information is provided in the form of exactlinear restrictions on regression coefficients In the presentwork we first find the unrestricted estimators of regressioncoefficients with the help of fuzzy entropy Next we introducethe restricted intuitionistic fuzzy linear regression modelwith fuzzy entropy Further the restricted estimators of theregression coefficients are obtained by incorporating theprior information in the form of linear restrictions

2 Restricted IFWLR Model withFuzzy Entropy

Without loss of generality suppose that all observations(y119894 x1198941 x1198942 x

119894119896) 119894 = 1 119899 in the regression analysis

are triangular intuitionistic fuzzy numbers The notion of

4 International Scholarly Research Notices

regression using fuzzy entropy is to construct five conven-tional regression equations (one for apex one for left entropyof the membership function one for right entropy of themembership function one for left entropy of the nonmem-bership function and one for right entropy of the non-membership function) for the response variable y using thecorresponding attributes of the 119896 fuzzy explanatory variablesx119895 In order to be specific we denote ya xa

1 xa2 xa

119896by the

apexes of y x1 x2 x

119896 respectively ely e

lx1

elx2

elx119896

bythe left entropy of y x

1 x2 x

119896 respectively ery e

rx1

erx2

erx119896

by the right entropy of y x1 x2 x

119896 respectively

el1015840

y el1015840x1

el1015840

x2

el1015840

x119896

by the left to left entropy of y x1 x2 x

119896

respectively and er1015840

y er1015840x1

er1015840

x2

er1015840

x119896

by the right to rightentropy of y x

1 x2 x

119896 respectively Therefore the five

fundamental regression equations in a nonrecursive (non-adaptive) setup may be written as

ya = 119860119886

0+

119896

sum

119894=1

(119860119886

119894xa119894+ 119861119886

119894elx119894

+ 119862119886

119894erx119894

+ 119863119886

119894el1015840

x119894

+ 119864119886

119894er1015840

x119894

) + 120576ya

ely = 119860119897

0+

119896

sum

119894=1

(119860119897

119894xa119894+ 119861119897

119894elx119894

+ 119862119897

119894erx119894

+ 119863119897

119894el1015840

x119894

+ 119864119897

119894er1015840

x119894

) + 120576ely

ery = 119860119903

0+

119896

sum

119894=1

(119860119903

119894xa119894+ 119861119903

119894elx119894

+ 119862119903

119894erx119894

+ 119863119903

119894el1015840

x119894

+ 119864119903

119894er1015840

x119894

) + 120576ery

el1015840

y = 1198601198971015840

0+

119896

sum

119894=1

(1198601198971015840

119894xa119894+ 1198611198971015840

119894elx119894

+ 1198621198971015840

119894erx119894

+ 1198631198971015840

119894el1015840

x119894

+ 1198641198971015840

119894er1015840

x119894

)

+ 120576el1015840y

er1015840

y = 1198601199031015840

0+

119896

sum

119894=1

(1198601199031015840

119894xa119894+ 1198611199031015840

119894elx119894

+ 1198621199031015840

119894erx119894

+ 1198631199031015840

119894el1015840

x119894

+ 1198641199031015840

119894er1015840

x119894

)

+ 120576er1015840y

(14)

where 120576ya 120576ely 120576ery 120576el1015840y and 120576er1015840y are the error vectors ofdimension 119899 times 1 The compact form of the above mentionednonrecursive or nonadaptive equations is given by

ya = X120573 + 120576ya

ely = X120572 + 120576ely

ery = X120574 + 120576ery

el1015840

y = X1205721015840 + 120576el1015840y

er1015840

y = X1205741015840 + 120576er1015840y

(15)

where

X= (1 xa1 xa2 xa

119896

elx1

elx2

elx119896

erx1

erx2

erx119896

el1015840

x1

el1015840

x2

el1015840

x119896

er1015840

x1

er1015840

x2

er1015840

x119896

)

119899times(5119896+1)

120573= (119860119886

0

1198601198861 119860119886

2 119860

119886

119896

1198611198861 119861119886

2 119861

119886

119896

1198621198861 119862119886

2

119862119886

119896

1198631198861 119863119886

2 119863

119886

119896

1198641198861 119864119886

2 119864

119886

119896)

119879

(5119896+1)times1

120572= (119860119897

0

1198601198971 119860119897

2 119860

119897

119896

1198611198971 119861119897

2 119861

119897

119896

1198621198971 119862119897

2

119862119897

119896

1198631198971 119863119897

2 119863

119897

119896

1198641198971 119864119897

2 119864

119897

119896)

119879

(5119896+1)times1

120574= (119860119903

0

1198601199031 119860119903

2 119860

119903

119896

1198611199031 119861119903

2 119861

119903

119896

1198621199031 119862119903

2

119862119903

119896

1198631199031 119863119903

2 119863

119903

119896

1198641199031 119864119903

2 119864

119903

119896)

119879

(5119896+1)times1

1205721015840= (119860

1198971015840

0

1198601198971015840

1 1198601198971015840

2 119860

1198971015840

119896

1198611198971015840

1 1198611198971015840

2 119861

1198971015840

119896

1198621198971015840

1 1198621198971015840

2

1198621198971015840

119896

1198631198971015840

1 1198631198971015840

2 119863

1198971015840

119896

1198641198971015840

1 1198641198971015840

2 119864

1198971015840

119896)

119879

(5119896+1)times1

1205741015840= (119860

1199031015840

0

1198601199031015840

1 1198601199031015840

2 119860

1199031015840

119896

1198611199031015840

1 1198611199031015840

2 119861

1199031015840

119896

1198621199031015840

1 1198621199031015840

2

1198621199031015840

119896

1198631199031015840

1 1198631199031015840

2 119863

1199031015840

119896

1198641199031015840

1 1198641199031015840

2 119864

1199031015840

119896)

119879

(5119896+1)times1

(16)

In many real life situations where the measurements arecarried out (for example car speed astronomical distance)it is natural to think that the spread (vagueness) in themeasure of a phenomenon is proportional to its intensityDrsquoUrso and Gastaldi [23] have done several simulations andobserved that even if we consider an adaptive or recursiveregression model along with nonadaptive or nonrecursiveregression model they yield identical solutions when thereis only one independent variable But if there are morethan one independent variable then the estimated values ofthe left entropies and right entropies obtained through therecursive fuzzy regression model will have less variance ascompared to the nonrecursive fuzzy regression model Withthis consideration we rewrite the proposed intuitionisticfuzzy linear regression model (15) in a recursiveadaptive

International Scholarly Research Notices 5

setup where dynamic of the entropies is dependent on themagnitude of the estimated apexes as follows

ya = yalowast

+ 120576ya where yalowast

= X120573

ely = ellowast

y + 120576lowast

ely where el

lowast

y = X120573119887 + 1119889

ery = erlowast

y + 120576lowast

ery where er

lowast

y = X120573119891 + 1119892

el1015840

y = el1015840lowast

y + 120576lowast

el1015840y where el

1015840lowast

y = X120573119901 + 1119902

er1015840

y = er1015840lowast

y + 120576lowast

er1015840y where er

1015840lowast

y = X120573119906 + 1V

(17)

where X is the 119899 times (5119896 + 1)-matrix containing the valuesof the input variables (data matrix) 120573 is a column 5119896 + 1-vector containing the regression coefficients for the apexesof the first model (referred to as core regression model)ya and yalowast are the vector of the observed apexes and thevector of the interpolated apexes respectively both havingdimension 119899 times 1 ely and ely

lowast

are the vector of the observedleft entropies and the vector of the interpolated left entropiesrespectively both having dimension 119899 times 1 ery and ery

lowast are thevector of the observed right entropies and the vector ofthe interpolated right entropies respectively both havingdimension 119899 times 1 el

1015840

y and el1015840

ylowast

are the vector of the observedleft to left entropies and the vector of the interpolated left toleft entropies respectively both having dimension 119899 times 1 er

1015840

y

and er1015840

ylowast

are the vector of the observed right to right entropiesand the vector of the interpolated right to right entropiesrespectively both having dimension 119899 times 1 and 1 is a (119899 times 1)-vector of all 11015840s 119887 and 119889 are regression parameters for thesecond regression equation model (referred to as left entropyregression model) 119891 and 119892 are regression parameters for thethird regressionmodel (referred to as right entropy regressionmodel) 119901 and 119902 are regression parameters for the fourthregression equation model (referred to as left to left entropyregression model) and 119906 and V are regression parametersfor the fifth regression equation model (referred to as rightto right entropy regression model) The error term in theregression equation of apexes will remain the same while theerror terms in the regression equations of entropies may bedifferent The error vectors 120576lowastely and 120576

lowast

eryin the left and right

entropies are of the dimension (119899 times 1) and the error vectors120576lowast

elyand 120576lowastery in the left to left and right to right entropies are of

the dimension (119899 times 1)If some prior information about unknown regression

coefficients is available on the basis of past experiences thenit may be used to estimate more efficient estimators Weassume that such prior information is in the form of exactlinear restrictions on regression coefficients In the presentmodel we associate such restrictions in the equations forthe estimation of regression coefficients in the intuitionisticfuzzy linear regression model with fuzzy entropy Thereforewe make the model capable of taking into account possible

linear relations between the size of the entropies and themagnitude of the estimated apexesMoreover we assume thatthe regression coefficients 120573 are subjected to the 119895 (119895 lt 5119896+1)exact linear restrictions which are given by

h = H120573 (18)

whereh andH are known and thematrixH is of full row rank

3 Estimation of Regression Coefficients

In many applications it is possible that the values of the vari-ables are on completely different scales ofmeasurement Alsothe possible larger variations in the values will have largerintersample differences so they will dominate in the calcu-lation of Euclidean distances Therefore some form of stan-dardization is necessary to balance out the individual con-tributions Consider the Euclidean distance between two tri-angular intuitionistic fuzzy numbers 119910

119894= (119910119886

119894 119890119897

119910119894

119890119903

119910119894

1198901198971015840

119910119894

1198901199031015840

119910119894

)

and 119910lowast

119894= (119910119886lowast

119894 119890119897lowast

119910119894

119890119903lowast

119910119894

1198901198971015840lowast

119910119894

1198901199031015840lowast

119910119894

) along with weights 1199081 1199082

1199083 1199084 and 119908

5as follows

120575119894equiv 120575 (119910

119894 119910lowast

119894)

= (1199081(119910119886

119894minus 119910119886lowast

119894)2

+ 1199082(119890119897

119910119894

minus 119890119897lowast

119910119894

)2

+ 1199083(119890119903

119910119894

minus 119890119903lowast

119910119894

)2

+1199084(1198901198971015840

119910119894

minus 1198901198971015840lowast

119910119894

)

2

+ 1199085(1198901199031015840

119910119894

minus 1198901199031015840lowast

119910119894

)

2

)

12

(19)

It may be observed that we compute the usual squared differ-ences between the values of variables on their original scalesas in the usual Euclidean distance but then multiply thesesquared differences by their corresponding weights

Next similar to common linear regression (based on crispdata) the regression parameters are estimated byminimizingthe following sum of square errors (we use a compact matrixnotation)

120593 (120573 119887 119889 119891 119892 119901 119902 119906 V)

=

119899

sum

119894=1

1199081(119910119886

119894minus 119910119886lowast

119894)2

+

119899

sum

119894=1

1199082(119890119897

119910119894

minus 119890119897lowast

119910119894

)2

+

119899

sum

119894=1

1199083(119890119903

119910119894

minus 119890119903lowast

119910119894

)2

+

119899

sum

119894=1

1199084(1198901198971015840

119910119894

minus 1198901198971015840lowast

119910119894

)2

+

119899

sum

119894=1

1199085(1198901199031015840

119910119894

minus 1198901199031015840lowast

119910119894

)2

= 1199081(ya minus yalowast)

T(ya minus yalowast) + 119908

2(ely minus ely

lowast

)T(ely minus ely

lowast

)

+ 1199083(ery minus er

lowast

y )T(ery minus er

lowast

y )

+ 1199084(el1015840

y minus el1015840

ylowast

)

T(el1015840

y minus el1015840

ylowast

)

+ 1199085(er1015840

y minus er1015840

ylowast

)

T(er1015840

y minus er1015840

ylowast

)

6 International Scholarly Research Notices

= 1199081((ya)Tya minus 2(ya)Tyalowast + (yalowast)

Tyalowast)

+ 1199082((ely)

Tely minus 2(ely)

Telylowast

+ (elylowast

)Telylowast

)

+ 1199083((ery)

Tery minus 2(ery)

Terylowast

+ (erylowast

)Terylowast

)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )Tel1015840

ylowast

+ (el1015840

ylowast

)

Tel1015840

ylowast

)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )Ter1015840

ylowast

+ (er1015840

ylowast

)

Ter1015840

ylowast

)

= 1199081((ya)Tya minus 2(ya)TX120573 + 120573TXTX120573)

+ 1199082((ely)

Tely minus 2(ely)

T(X120573119887 + 1119889))

+ 1199082((X120573119887 + 1119889)T (X120573119887 + 1119889))

+ 1199083((ery)

Tery minus 2(ery)

T(X120573119891 + 1119892))

+ 1199083((X120573119891 + 1119892)T (X120573119891 + 1119892))

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )T(X120573119901 + 1119902))

+ 1199084((X120573119901 + 1119902)T (X120573119901 + 1119902))

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )T(X120573119906 + 1V))

+ 1199085((X120573119906 + 1V)T (X120573119906 + 1V))

= 1199081((ya)Tya minus 2(ya)TX120573)

+ 120573TXTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

+ 1199082((ely)

Tely minus 2(ely)

TX120573119887 minus 2(ely)

T1119889)

+ 1199083((ery)

Tery minus 2(ery)

TX120573119891 minus 2(ery)

T1119892)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )TX120573119901 minus 2(el

1015840

y )T1119902)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )TX120573119906 minus 2(er

1015840

y )T1V)

+ 2120573TXT1 (119908

2119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V)

+ 119899 (11990821198892+ 11990831198922+ 11990841199022+ 1199085V2)

(20)

Differentiating 120593(120573 119887 119889 119891 119892 119901 119902 119906 V) that is (20) partiallywith respect to 120573 and equating it to zero we get

120597120593 (120573 119887 119889 119891 119892 119901 119902 119906 V)120597120573

= 0

997904rArr minus1199081XTya + XTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

minus 1199082XTely119887 minus 119908

3XTery119891 minus 119908

4XTel

1015840

y119901 minus 1199085XTer

1015840

y 119906

+ XT1 (1199082119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V) = 0

997904rArr 120573 = ((XTX)minus1

XT[1199081ya + 119908

2ely119887 + 119908

3ery119891

+ 1199084el1015840

y119901 + 1199085er1015840

y 119906

minus1 (1199082119887119889+119908

3119891119892+119908

4119901119902+119908

5119906V)])

times (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)minus1

(21)

Similarly differentiating (20) partially with respect to 119887 119889 119891119892 119901 119902 119906 and V we get

119887 = (120573TXTX120573)

minus1

[(ely)TX120573 minus 120573TXT1119889] (22)

119889 =1

119899[(ely)

T1 minus 120573

TXT1119887] (23)

119891 = (120573TXTX120573)

minus1

[(ery)TX120573 minus 120573TXT1119892] (24)

119892 =1

119899[(er

1015840

y )T1 minus 120573

TXT1119891] (25)

119901 = (120573TXTX120573)

minus1

[(el1015840

y )TX120573 minus 120573TXT1119902] (26)

119902 =1

119899[(el

1015840

y )T1 minus 120573

TXT1119901] (27)

119906 = (120573TXTX120573)

minus1

[(er1015840

y )TX120573 minus 120573TXT1V] (28)

V =1

119899[(er

1015840

y )T1 minus 120573

TXT1119906] (29)

respectively

International Scholarly Research Notices 7

Equations (21)ndash(29) are recursive solutions for the prob-lem of least square estimation with intuitionistic fuzzy dataTherefore we rewrite the system of equations explicitly in arecursive way as follows

120573119894+1

= ((XTX)minus1

XT[1199081ya + 119908

2ely119887119894 + 119908

3ery119891119894 + 119908

4el1015840

y119901119894

+ 1199085er1015840

y 119906119894 minus 1 (1199082119887119894119889119894+ 1199083119891119894119892119894

+1199084119901119894119902119894+ 1199085119906119894V119894) ] )

times (1199081+ 11990821198872

119894+ 11990831198912

119894+ 11990841199012

119894+ 11990851199062

119894)minus1

119887119894+1

= (120573T119894+1

XTX120573i+1)minus1

[(ely)TX120573i+1 minus 120573

Ti+1X

T1119889119894]

119889119894+1

=1

119899[(ely)

T1 minus 120573

Ti+1X

T1119887119894]

119891119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(ery)TX120573i+1 minus 120573

Ti+1X

T1119892119894]

119892119894+1

=1

119899[(er

1015840

y )T1 minus 120573

Ti+1X

T1119891119894]

119901119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(el1015840

y )TX120573i+1 minus 120573

Ti+1X

T1119902119894]

119902119894+1

=1

119899[(el

1015840

y )T1 minus 120573

Ti+1X

T1119901119894]

119906119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(er1015840

y )TX120573i+1 minus 120573

Ti+1X

T1V119894]

V119894+1

=1

119899[(er

1015840

y )T1 minus 120573

Ti+1X

T1119906119894]

(30)

In order to initiate the recursive process of obtaining theestimators we take some initial values for 119887 119889 119891 119892 119901 119902119906 V and 120573 After several numbers of iterations the valuesof estimators get corrected to a predefined error of toleranceWe denote these values by 119889 119891 119892 119901 119902 V and in orderto differentiate them from the eventually obtained restrictedestimator in the next commutation

In a more general setup if in the linear regression model(17) we consider 119896

1crisp and 119896

2intuitionistic fuzzy input

variables then the dimensions of X and 120573 will be 119899 times (1198961+

51198962+ 1) and (119896

1+ 51198962+ 1) times 1 respectively It may further be

noted that the core of the solutionrsquos structure will remain thesame and we will have similar kind of estimators

Remark If a TIFN 119860 = (119898 120572 120573 1205721015840 1205731015840) degenerate to a trian-

gular fuzzy number 119860 = (119898 120572 120573) then our nonsymmetric

intuitionistic fuzzy weighted linear regression model reducesto nonsymmetric fuzzy linear regression model defined byKumar et al [24]

Next we assume that the regression coefficients aresubjected to the linear restrictions which are given by (18) Itmay be noted that the unrestricted estimator obtained abovein (21) does not satisfy the given restrictions (18) We aimto obtain the restricted estimator which satisfies the givenrestrictions under the regression model (17) For this wepropose to minimize the following score function

119878 (120582120573 119887 119889 119891 119892 119901 119902 119906 V)

= 120593 (120573 119887 119889 119891 119892 119901 119902 119906 V) minus 2120582 (H120573 minus h)

= 1199081((ya)Tya minus 2(ya)TX120573)

+ 120573TXTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

+ 1199082((ely)

Tely minus 2(ely)

TX120573119887 minus 2(ely)

T1119889)

+ 1199083((ery)

Tery minus 2(ery)

TX120573119891 minus 2(ery)

T1119892)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )TX120573119901 minus 2(el

1015840

y )T1119902)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )TX120573119906 minus 2(er

1015840

y )T1V)

+ 2120573TXT1 (119908

2119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V)

+ 119899 (11990821198892+ 11990831198922+ 11990841199022+ 1199085V2)

minus 2120582 (H120573 minus h)

(31)

where 2120582 is the vector of Lagrangersquos Multiplier

Differentiating 119878(120582120573 119887 119889 119891 119892 119901 119902 119906 V) partially withrespect to 120573 and equating it to zero we get

997904rArr minus1199081XTya + XTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

minus 1199082XTely119887 minus 119908

3XTery119891 minus 119908

4XTel

1015840

y119901 minus 1199085XTer

1015840

y 119906

+ XT1 (1199082119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V) minusH1015840120582 = 0

(32)

8 International Scholarly Research Notices

Here we again relabel the computed restricted estimator by Therefore in view of (21) and (32) we get size

997904rArr = ((XTX)minus1

XT[1199081ya + 119908

2ely119887 + 119908

3ery119891 + 119908

4el1015840

y119901

+ 1199085er1015840

y 119906

minus1 (1199082119887119889+119908

3119891119892+119908

4119901119902+119908

5119906V) ] )

times (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)minus1

+(XTX)

minus1

HT120582

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

997904rArr = +1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

times (XTX)minus1

HT120582

(33)

Similarly differentiating 119878(120582120573 119887 119889 119891 119892 119901 119902 119906 V) par-tially with respect to 120582 and equating it to zero we get

997904rArr H = h

997904rArr H + 1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

timesH(XTX)minus1

HT120582 = h

997904rArr = (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

times [H(XTX)minus1

HT]minus1

(h minusH)

(34)

From (33) and (34) we have

997904rArr = + (XTX)minus1

HT[H(XTX)

minus1

HT]minus1

(h minusH) (35)

Also differentiating (31) partially with respect to 119887 119889 119891 119892 119901119902 119906 and V and equating all to zero we get

= 119889 = 119889 119891 = 119891 119892 = 119892

119901 = 119901 119902 = 119902 = V = V(36)

respectively From (35) we see that

997904rArr H = H + [H(XTX)minus1

HT] [H(XTX)

minus1

HT]minus1

times (h minusH)

997904rArr H = H + (h minusH) = h(37)

Therefore the estimator satisfies the given restrictions (18)

4 Numerical Examples

We consider the following numerical examples to illustratethe proposed model

Example 1 We apply our procedure to estimate the intuition-istic fuzzy output value for a data consisting of the crisp inputand intuitionistic fuzzy output (where left entropy and rightentropy are equal) and tabulate the data in Table 1

We obtain = (minus44026 35733 73786 56858)1015840 =

02942 119889 = 147144 119891 = 02942 119892 = 147144 119901 = 02909119902 = 174487 = 02909 and V = 174487 where the numberof iterations required is 125

Example 2 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp input andintuitionistic fuzzy output (where left and right entropy arenot equal) and tabulate the data in Table 2

We obtain = (minus47697 35933 72030 59152)1015840 =

02952 119889 = 145871 119891 = 02646 119892 = 203429 119901 = 03052119902 = 157050 = 02717 and V = 231201 where the numberof iterations required is 113

Example 3 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp inputintuitionistic fuzzy input and intuitionistic fuzzy output(where left and right entropy are not equal) and tabulate thedata in Table 3

We obtain = (minus32352 06811 05314 minus09164 00846

minus31631 2953)1015840 = 04225 119889 = 05478 119891 = 04307 119892 =

01637 119901 = 03231 119902 = 38723 = 04985 and V = 18659

where the number of iterations required is 51

Example 4 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of intuitionistic fuzzyinput and intuitionistic fuzzy output (where left and rightentropy are not equal) and tabulate the data in Table 4

We obtain = (118141 minus02161 16104 minus18254 05687

minus01879)1015840 = 03880 119889 = 03674 119891 = 03880 119892 = 03674

119901 = 03547 119902 = 22108 = 03547 and V = 32108 wherethe number of iterations required is 255

5 Conclusions

An intuitionistic fuzzy weighted linear regression (IFWLR)model with and without some linear restrictions in theform of prior information has been studied The estimators

International Scholarly Research Notices 9

Table 1 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y y

a ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 44 42 96 42 44 449018 424850 943828 424850 4490182 14 8 3 48 47 120 47 48 528505 505256 1217099 505256 5285053 7 1 4 35 33 52 33 35 322052 296416 507324 296416 3220524 11 7 3 50 45 106 45 50 475861 452004 1036114 452004 4758615 7 12 15 80 79 189 79 80 740058 719256 1944413 719256 7400586 8 15 10 68 65 194 65 68 732147 711253 1917213 711253 7321477 3 9 6 45 42 107 42 45 485252 461503 1068398 461503 4852528 12 15 11 80 78 216 78 80 790260 770038 2117003 770038 7902609 10 5 8 55 52 108 52 55 505235 481717 1137100 481717 50523510 9 7 4 45 44 103 44 45 471612 447706 1021507 447706 471612

Table 2 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y ya ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 45 42 96 47 48 447743 427104 952620 455472 4900532 14 8 3 48 47 120 43 45 525995 502809 120905 523320 5597343 7 1 4 35 33 52 50 55 313430 297162 512469 339018 3704524 11 7 3 46 45 106 45 47 471120 449720 102922 475741 5108705 7 12 15 82 79 189 80 85 753765 723166 195547 720805 7625556 8 15 10 70 65 194 60 67 740419 710254 191173 709234 7506717 3 9 6 45 42 107 40 46 481512 459774 106328 484752 5201248 12 15 11 80 78 216 88 90 802328 770149 211461 762912 8057999 10 5 8 55 52 108 50 55 506447 483897 114499 506370 54232710 9 7 4 45 44 103 42 44 467241 445967 101651 472377 507415

Table 3 Crisp and int fuzzy input-int fuzzy output data

Object 119894Crisp and int fuzzy inputX = (x1 e1

1015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy outputylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 6 7 6 10 5 7 6 3 5 2 5 54158 25665 47776 22215 424732 7 6 5 12 4 6 7 5 4 5 7 58827 31771 62226 28438 496763 8 4 2 15 3 5 8 3 9 4 6 67973 43734 90537 40632 637884 9 7 5 20 8 10 7 3 10 2 4 64422 39089 79544 35897 583085 10 8 6 5 2 5 8 5 12 5 7 74885 52774 11193 49846 744516 11 10 8 15 5 7 3 2 8 4 6 65757 40835 83678 37678 603697 12 20 15 25 12 14 6 5 7 3 5 62393 36436 73265 33193 551798 13 12 7 30 15 18 8 7 14 6 8 83421 63937 13835 61226 876209 14 16 12 20 10 15 11 9 16 10 12 97423 82250 18169 79893 10922310 15 17 13 22 8 12 8 7 18 5 10 90740 73509 16100 70982 98912

10 International Scholarly Research Notices

Table 4 Intuitionistic fuzzy input-intuitionistic fuzzy output data

Object iInt fuzzy input

X = (e11015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy output

ylowast = (e1lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

ye1lowast

y yalowast er

lowast

y erlowast1015840

y

1 5 3 4 5 6 5 4 12 4 6 57505 42398 99797 42398 675052 7 6 7 8 9 7 5 7 5 8 57737 42652 10045 42652 677373 5 3 6 8 9 5 3 9 3 6 48608 32665 74714 32665 586084 4 2 7 9 11 3 1 4 1 4 37872 20920 44446 20920 478725 3 2 5 7 8 4 2 6 2 5 49552 33698 77377 33698 595526 6 3 6 7 10 5 4 8 4 6 45158 28891 64987 28891 551587 5 2 4 9 12 4 3 9 3 5 55863 40602 95168 40602 658638 6 5 8 13 15 7 5 10 5 8 52404 36818 85417 36818 624049 8 7 12 15 17 4 3 5 3 5 39099 22262 47905 22262 4909910 15 10 15 20 25 4 2 3 2 5 36202 19092 39736 19092 46202

of regression coefficients have also been obtained with thehelp of fuzzy entropy for the restrictedunrestricted IFWLRmodel by assigning some weights in the distance functionIt has been observed that the restricted estimator is betterthan unrestricted estimator in some sense Thus wheneversome prior information is available in terms of exact linearrestrictions on regression coefficients it is advised to userestricted estimator in place of unrestricted estimator

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] H Tanaka S Uejima and K Asai ldquoFuzzy linear regressionmodelrdquo IEEE Transactions on Systems Man and Cyberneticsvol 10 pp 2933ndash2938 1980

[3] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982

[4] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988

[5] H Tanaka I Hayashi and JWatada ldquoPossibilistic linear regres-sion analysis for fuzzy datardquo European Journal of OperationalResearch vol 40 no 3 pp 389ndash396 1989

[6] H Tanaka and H Ishibuchi ldquoIdentification of possibilisticlinear systems by quadratic membership functions of fuzzyparametersrdquo Fuzzy Sets and Systems vol 41 no 2 pp 145ndash1601991

[7] D T Redden and W H Woodall ldquoProperties of certain fuzzylinear regression methodsrdquo Fuzzy Sets and Systems vol 64 no3 pp 361ndash375 1994

[8] P-T Chang and E S Lee ldquoFuzzy linear regression with spreadsunrestricted in signrdquoComputers andMathematics with Applica-tions vol 28 no 4 pp 61ndash70 1994

[9] G Peters ldquoFuzzy linear regression with fuzzy intervalsrdquo FuzzySets and Systems vol 63 no 1 pp 45ndash55 1994

[10] A de Luca and S Termini ldquoA definition of a nonprobabilisticentropy in the setting of fuzzy sets theoryrdquo Information andControl vol 20 no 4 pp 301ndash312 1972

[11] D Dubosis and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[12] D Dubois and H Prade Fuzzy Sets and Statistical PossibilityTheory Plenum Press New York NY USA 1988

[13] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash656 1948

[14] T Kumar N Gupta and R K Bajaj ldquoFuzzy entropy onrestricted fuzzy linear regression model with cross validationand applicationsrdquo in Proceedings of the International Conferenceon Advances in Computing and Communications (ICACC rsquo12)pp 5ndash8 August 2012

[15] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[16] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[17] K T Atanassov Intuitionistic Fuzzy Sets Theory and Applica-tions vol 35 of Studies in Fuzziness and SoftComputing Physica1999

[18] K T Atanassov ldquoNew operations defined over the intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 61 no 2 pp 137ndash1421994

[19] P Burillo and H Bustince ldquoSome definitions of intuitionisticfuzzy numberrdquo in Proceedings of the 3rd Conference of theEuropean Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[20] F Lee Fuzzy Information Processing System Peking UniversityPress Beijing China 1998

[21] H Liu and K Shi ldquoIntuitionistic fuzzy numbers and intuition-istic distribution numbersrdquo Journal of Fuzzy Mathematics vol8 no 4 pp 909ndash918 2000

[22] P Grzegorzewski ldquoDistances and orderings in a family of intu-itionistic fuzzy numbersrdquo in Proceedings of the 3rd Conference ofthe European Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[23] P DrsquoUrso and T Gastaldi ldquoA least-squares approach to fuzzylinear regression analysisrdquo Computational Statistics and DataAnalysis vol 34 no 4 pp 427ndash440 2000

[24] T Kumar R K Bajaj and N Gupta ldquoFuzzy entropy in fuzzyweighted linear regression model under linear restrictions withsimulation studyrdquo International Journal of General Systems vol43 no 2 pp 135ndash148 2014

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Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 3

(ii) the membership function 120583119860is fuzzy-convex that is

120583119860(120582 sdot 1199091+ (1 minus 120582) sdot 119909

2) ge min 120583

119860(1199091) 120583119860(1199092)

forall1199091 1199092isin 119883 120582 isin [0 1]

(7)

(iii) the nonmembership function ]119860is fuzzy-concave

that is

]119860(120582 sdot 1199091+ (1 minus 120582) sdot 119909

2) le max ]

119860(1199091) ]119860(1199092)

forall1199091 1199092isin 119883 120582 isin [0 1]

(8)

(iv) the membership and the nonmembership functionsof 119860 satisfying the conditions 0 le 119891

1(119909) + 119892

1(119909) le 1

and 0 le 1198912(119909) + 119892

2(119909) le 1 have the following form

120583119860(119909) =

1198911(119909) for 119898 minus 120572 le 119909 le 119898

1 for 119909 = 119898

1198912(119909) for 119898 le 119909 le 119898 + 120573

0 otherwise

(9)

where the functions 1198911(119909) and 119891

2(119909) are strictly

increasing and decreasing functions in [119898minus120572119898] and[119898119898 + 120573] respectively and

]119860(119909) =

1198921(119909) for 119898 minus 120572

1015840le 119909 le 119898

0 for 119909 = 119898

1198922(119909) for 119898 le 119909 le 119898 + 120573

1015840

1 otherwise

(10)

where the functions 1198921(119909) and 119892

2(119909) are strictly

decreasing and increasing functions in [119898 minus 1205721015840 119898]

and [119898119898 + 1205731015840] respectively Here 120572 and 120573 are

called the left and right spreads of the membershipfunction 120583

119860 respectively 1205721015840 and 120573

1015840 are called the leftand right spreads of the nonmembership function]119860(119909) Symbolically an intuitionistic fuzzy number is

represented as 119860 IFN = (119898 120572 120573 1205721015840 1205731015840)

Definition 2 An IFN 119860 IFN = (119898 120572 120573 1205721015840 1205731015840)may be defined

as a triangular intuitionistic fuzzy number (TIFN) if and onlyif its membership and nonmembership functions take thefollowing form

120583119860(119909) =

1 minus119898 minus 119909

120572 for 119898 minus 120572 le 119909 le 119898

1 for 119909 = 119898

1 minus119909 minus 119898

120573 for 119898 le 119909 le 119898 + 120573

0 otherwise

(11)

]119860(119909) =

119898 minus 119909

1205721015840 for 119898 minus 120572

1015840le 119909 le 119898

0 for 119909 = 119898

119909 minus 119898

1205731015840 for 119898 le 119909 le 119898 + 120573

1015840

1 otherwise

(12)

It may be noted that a TIFN 119860 = (119898 120572 120573 1205721015840 1205731015840) degen-

erate to a triangular fuzzy number 119860 = (119898 120572 120573) if 120572 = 1205721015840

120573 = 1205731015840 and ]

119860(119909) = 1 minus 120583

119860(119909) forall119909 isin R Further an TIFN

119860 = ⟨119909 120583119860(119909) ]119860(119909)⟩ 119909 isin R that is 119860 = (119898 120572 120573 120572

1015840 1205731015840)

is a conjunction of two fuzzy numbers119860+ = (119898 120572 120573)with themembership function 120583

119860+(119909) = 120583

119860(119909) and 119860

minus= (119898 120572

1015840 1205731015840)

with the membership function 120583119860(119909) = 1 minus ]

119860(119909)

The entropy calculated using (3) from the membershipfunction of TIFN given by (11) can be expressed as followssize

119867(119860) = minus 119870[int119909isin[119898minus120572119898]

[120583119860(119909) log 120583

119860(119909) + (1 minus 120583

119860(119909))

times log (1 minus 120583119860(119909))] 119889119909

+ int119909isin[119898119898+120573]

[120583119860(119909) log120583

119860(119909) + (1 minus 120583

119860(119909))

times log (1 minus 120583119860(119909))] 119889119909]

= 119867119871(119860) + 119867

119877(119860)

(13)

where 119867119871(119860) = 1198701205722 and 119867

119877(119860) = 1198701205732 It follows that

119867(119860) = 119870(120572 + 120573)2 which does not depend on119898 It may beobserved that in the case of symmetrical TIFN the left andthe right entropies are identical On the other hand in case ofnonsymmetric TIFN the left entropy is a function of 120572 andthe right entropy is a function of 120573 Similarly the left entropyand the right entropy from the nonmembership function(which we called left to left and right to right entropies) ofthe TIFN are the functions of 1205721015840 and 120573

1015840 respectively Hencea triangular intuitionistic fuzzy number can be characterizedby five attributes the position parameter 119898 the left entropy120572 the right entropy 120573 left to left entropy 1205721015840 and right to rightentropy 1205731015840 There is a one-to-one correspondence between atriangular intuitionistic fuzzy number and its entropies Inother words given a triangular intuitionistic fuzzy numberone can determine the unique position and entropies Con-versely given a position and entropies one can construct aunique triangular intuitionistic fuzzy number

Sometimes experimenterrsquos past experiences may be avail-able as prior information about unknown regression coeffi-cients to estimate more efficient estimators Here we assumethat such prior information is provided in the form of exactlinear restrictions on regression coefficients In the presentwork we first find the unrestricted estimators of regressioncoefficients with the help of fuzzy entropy Next we introducethe restricted intuitionistic fuzzy linear regression modelwith fuzzy entropy Further the restricted estimators of theregression coefficients are obtained by incorporating theprior information in the form of linear restrictions

2 Restricted IFWLR Model withFuzzy Entropy

Without loss of generality suppose that all observations(y119894 x1198941 x1198942 x

119894119896) 119894 = 1 119899 in the regression analysis

are triangular intuitionistic fuzzy numbers The notion of

4 International Scholarly Research Notices

regression using fuzzy entropy is to construct five conven-tional regression equations (one for apex one for left entropyof the membership function one for right entropy of themembership function one for left entropy of the nonmem-bership function and one for right entropy of the non-membership function) for the response variable y using thecorresponding attributes of the 119896 fuzzy explanatory variablesx119895 In order to be specific we denote ya xa

1 xa2 xa

119896by the

apexes of y x1 x2 x

119896 respectively ely e

lx1

elx2

elx119896

bythe left entropy of y x

1 x2 x

119896 respectively ery e

rx1

erx2

erx119896

by the right entropy of y x1 x2 x

119896 respectively

el1015840

y el1015840x1

el1015840

x2

el1015840

x119896

by the left to left entropy of y x1 x2 x

119896

respectively and er1015840

y er1015840x1

er1015840

x2

er1015840

x119896

by the right to rightentropy of y x

1 x2 x

119896 respectively Therefore the five

fundamental regression equations in a nonrecursive (non-adaptive) setup may be written as

ya = 119860119886

0+

119896

sum

119894=1

(119860119886

119894xa119894+ 119861119886

119894elx119894

+ 119862119886

119894erx119894

+ 119863119886

119894el1015840

x119894

+ 119864119886

119894er1015840

x119894

) + 120576ya

ely = 119860119897

0+

119896

sum

119894=1

(119860119897

119894xa119894+ 119861119897

119894elx119894

+ 119862119897

119894erx119894

+ 119863119897

119894el1015840

x119894

+ 119864119897

119894er1015840

x119894

) + 120576ely

ery = 119860119903

0+

119896

sum

119894=1

(119860119903

119894xa119894+ 119861119903

119894elx119894

+ 119862119903

119894erx119894

+ 119863119903

119894el1015840

x119894

+ 119864119903

119894er1015840

x119894

) + 120576ery

el1015840

y = 1198601198971015840

0+

119896

sum

119894=1

(1198601198971015840

119894xa119894+ 1198611198971015840

119894elx119894

+ 1198621198971015840

119894erx119894

+ 1198631198971015840

119894el1015840

x119894

+ 1198641198971015840

119894er1015840

x119894

)

+ 120576el1015840y

er1015840

y = 1198601199031015840

0+

119896

sum

119894=1

(1198601199031015840

119894xa119894+ 1198611199031015840

119894elx119894

+ 1198621199031015840

119894erx119894

+ 1198631199031015840

119894el1015840

x119894

+ 1198641199031015840

119894er1015840

x119894

)

+ 120576er1015840y

(14)

where 120576ya 120576ely 120576ery 120576el1015840y and 120576er1015840y are the error vectors ofdimension 119899 times 1 The compact form of the above mentionednonrecursive or nonadaptive equations is given by

ya = X120573 + 120576ya

ely = X120572 + 120576ely

ery = X120574 + 120576ery

el1015840

y = X1205721015840 + 120576el1015840y

er1015840

y = X1205741015840 + 120576er1015840y

(15)

where

X= (1 xa1 xa2 xa

119896

elx1

elx2

elx119896

erx1

erx2

erx119896

el1015840

x1

el1015840

x2

el1015840

x119896

er1015840

x1

er1015840

x2

er1015840

x119896

)

119899times(5119896+1)

120573= (119860119886

0

1198601198861 119860119886

2 119860

119886

119896

1198611198861 119861119886

2 119861

119886

119896

1198621198861 119862119886

2

119862119886

119896

1198631198861 119863119886

2 119863

119886

119896

1198641198861 119864119886

2 119864

119886

119896)

119879

(5119896+1)times1

120572= (119860119897

0

1198601198971 119860119897

2 119860

119897

119896

1198611198971 119861119897

2 119861

119897

119896

1198621198971 119862119897

2

119862119897

119896

1198631198971 119863119897

2 119863

119897

119896

1198641198971 119864119897

2 119864

119897

119896)

119879

(5119896+1)times1

120574= (119860119903

0

1198601199031 119860119903

2 119860

119903

119896

1198611199031 119861119903

2 119861

119903

119896

1198621199031 119862119903

2

119862119903

119896

1198631199031 119863119903

2 119863

119903

119896

1198641199031 119864119903

2 119864

119903

119896)

119879

(5119896+1)times1

1205721015840= (119860

1198971015840

0

1198601198971015840

1 1198601198971015840

2 119860

1198971015840

119896

1198611198971015840

1 1198611198971015840

2 119861

1198971015840

119896

1198621198971015840

1 1198621198971015840

2

1198621198971015840

119896

1198631198971015840

1 1198631198971015840

2 119863

1198971015840

119896

1198641198971015840

1 1198641198971015840

2 119864

1198971015840

119896)

119879

(5119896+1)times1

1205741015840= (119860

1199031015840

0

1198601199031015840

1 1198601199031015840

2 119860

1199031015840

119896

1198611199031015840

1 1198611199031015840

2 119861

1199031015840

119896

1198621199031015840

1 1198621199031015840

2

1198621199031015840

119896

1198631199031015840

1 1198631199031015840

2 119863

1199031015840

119896

1198641199031015840

1 1198641199031015840

2 119864

1199031015840

119896)

119879

(5119896+1)times1

(16)

In many real life situations where the measurements arecarried out (for example car speed astronomical distance)it is natural to think that the spread (vagueness) in themeasure of a phenomenon is proportional to its intensityDrsquoUrso and Gastaldi [23] have done several simulations andobserved that even if we consider an adaptive or recursiveregression model along with nonadaptive or nonrecursiveregression model they yield identical solutions when thereis only one independent variable But if there are morethan one independent variable then the estimated values ofthe left entropies and right entropies obtained through therecursive fuzzy regression model will have less variance ascompared to the nonrecursive fuzzy regression model Withthis consideration we rewrite the proposed intuitionisticfuzzy linear regression model (15) in a recursiveadaptive

International Scholarly Research Notices 5

setup where dynamic of the entropies is dependent on themagnitude of the estimated apexes as follows

ya = yalowast

+ 120576ya where yalowast

= X120573

ely = ellowast

y + 120576lowast

ely where el

lowast

y = X120573119887 + 1119889

ery = erlowast

y + 120576lowast

ery where er

lowast

y = X120573119891 + 1119892

el1015840

y = el1015840lowast

y + 120576lowast

el1015840y where el

1015840lowast

y = X120573119901 + 1119902

er1015840

y = er1015840lowast

y + 120576lowast

er1015840y where er

1015840lowast

y = X120573119906 + 1V

(17)

where X is the 119899 times (5119896 + 1)-matrix containing the valuesof the input variables (data matrix) 120573 is a column 5119896 + 1-vector containing the regression coefficients for the apexesof the first model (referred to as core regression model)ya and yalowast are the vector of the observed apexes and thevector of the interpolated apexes respectively both havingdimension 119899 times 1 ely and ely

lowast

are the vector of the observedleft entropies and the vector of the interpolated left entropiesrespectively both having dimension 119899 times 1 ery and ery

lowast are thevector of the observed right entropies and the vector ofthe interpolated right entropies respectively both havingdimension 119899 times 1 el

1015840

y and el1015840

ylowast

are the vector of the observedleft to left entropies and the vector of the interpolated left toleft entropies respectively both having dimension 119899 times 1 er

1015840

y

and er1015840

ylowast

are the vector of the observed right to right entropiesand the vector of the interpolated right to right entropiesrespectively both having dimension 119899 times 1 and 1 is a (119899 times 1)-vector of all 11015840s 119887 and 119889 are regression parameters for thesecond regression equation model (referred to as left entropyregression model) 119891 and 119892 are regression parameters for thethird regressionmodel (referred to as right entropy regressionmodel) 119901 and 119902 are regression parameters for the fourthregression equation model (referred to as left to left entropyregression model) and 119906 and V are regression parametersfor the fifth regression equation model (referred to as rightto right entropy regression model) The error term in theregression equation of apexes will remain the same while theerror terms in the regression equations of entropies may bedifferent The error vectors 120576lowastely and 120576

lowast

eryin the left and right

entropies are of the dimension (119899 times 1) and the error vectors120576lowast

elyand 120576lowastery in the left to left and right to right entropies are of

the dimension (119899 times 1)If some prior information about unknown regression

coefficients is available on the basis of past experiences thenit may be used to estimate more efficient estimators Weassume that such prior information is in the form of exactlinear restrictions on regression coefficients In the presentmodel we associate such restrictions in the equations forthe estimation of regression coefficients in the intuitionisticfuzzy linear regression model with fuzzy entropy Thereforewe make the model capable of taking into account possible

linear relations between the size of the entropies and themagnitude of the estimated apexesMoreover we assume thatthe regression coefficients 120573 are subjected to the 119895 (119895 lt 5119896+1)exact linear restrictions which are given by

h = H120573 (18)

whereh andH are known and thematrixH is of full row rank

3 Estimation of Regression Coefficients

In many applications it is possible that the values of the vari-ables are on completely different scales ofmeasurement Alsothe possible larger variations in the values will have largerintersample differences so they will dominate in the calcu-lation of Euclidean distances Therefore some form of stan-dardization is necessary to balance out the individual con-tributions Consider the Euclidean distance between two tri-angular intuitionistic fuzzy numbers 119910

119894= (119910119886

119894 119890119897

119910119894

119890119903

119910119894

1198901198971015840

119910119894

1198901199031015840

119910119894

)

and 119910lowast

119894= (119910119886lowast

119894 119890119897lowast

119910119894

119890119903lowast

119910119894

1198901198971015840lowast

119910119894

1198901199031015840lowast

119910119894

) along with weights 1199081 1199082

1199083 1199084 and 119908

5as follows

120575119894equiv 120575 (119910

119894 119910lowast

119894)

= (1199081(119910119886

119894minus 119910119886lowast

119894)2

+ 1199082(119890119897

119910119894

minus 119890119897lowast

119910119894

)2

+ 1199083(119890119903

119910119894

minus 119890119903lowast

119910119894

)2

+1199084(1198901198971015840

119910119894

minus 1198901198971015840lowast

119910119894

)

2

+ 1199085(1198901199031015840

119910119894

minus 1198901199031015840lowast

119910119894

)

2

)

12

(19)

It may be observed that we compute the usual squared differ-ences between the values of variables on their original scalesas in the usual Euclidean distance but then multiply thesesquared differences by their corresponding weights

Next similar to common linear regression (based on crispdata) the regression parameters are estimated byminimizingthe following sum of square errors (we use a compact matrixnotation)

120593 (120573 119887 119889 119891 119892 119901 119902 119906 V)

=

119899

sum

119894=1

1199081(119910119886

119894minus 119910119886lowast

119894)2

+

119899

sum

119894=1

1199082(119890119897

119910119894

minus 119890119897lowast

119910119894

)2

+

119899

sum

119894=1

1199083(119890119903

119910119894

minus 119890119903lowast

119910119894

)2

+

119899

sum

119894=1

1199084(1198901198971015840

119910119894

minus 1198901198971015840lowast

119910119894

)2

+

119899

sum

119894=1

1199085(1198901199031015840

119910119894

minus 1198901199031015840lowast

119910119894

)2

= 1199081(ya minus yalowast)

T(ya minus yalowast) + 119908

2(ely minus ely

lowast

)T(ely minus ely

lowast

)

+ 1199083(ery minus er

lowast

y )T(ery minus er

lowast

y )

+ 1199084(el1015840

y minus el1015840

ylowast

)

T(el1015840

y minus el1015840

ylowast

)

+ 1199085(er1015840

y minus er1015840

ylowast

)

T(er1015840

y minus er1015840

ylowast

)

6 International Scholarly Research Notices

= 1199081((ya)Tya minus 2(ya)Tyalowast + (yalowast)

Tyalowast)

+ 1199082((ely)

Tely minus 2(ely)

Telylowast

+ (elylowast

)Telylowast

)

+ 1199083((ery)

Tery minus 2(ery)

Terylowast

+ (erylowast

)Terylowast

)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )Tel1015840

ylowast

+ (el1015840

ylowast

)

Tel1015840

ylowast

)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )Ter1015840

ylowast

+ (er1015840

ylowast

)

Ter1015840

ylowast

)

= 1199081((ya)Tya minus 2(ya)TX120573 + 120573TXTX120573)

+ 1199082((ely)

Tely minus 2(ely)

T(X120573119887 + 1119889))

+ 1199082((X120573119887 + 1119889)T (X120573119887 + 1119889))

+ 1199083((ery)

Tery minus 2(ery)

T(X120573119891 + 1119892))

+ 1199083((X120573119891 + 1119892)T (X120573119891 + 1119892))

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )T(X120573119901 + 1119902))

+ 1199084((X120573119901 + 1119902)T (X120573119901 + 1119902))

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )T(X120573119906 + 1V))

+ 1199085((X120573119906 + 1V)T (X120573119906 + 1V))

= 1199081((ya)Tya minus 2(ya)TX120573)

+ 120573TXTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

+ 1199082((ely)

Tely minus 2(ely)

TX120573119887 minus 2(ely)

T1119889)

+ 1199083((ery)

Tery minus 2(ery)

TX120573119891 minus 2(ery)

T1119892)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )TX120573119901 minus 2(el

1015840

y )T1119902)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )TX120573119906 minus 2(er

1015840

y )T1V)

+ 2120573TXT1 (119908

2119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V)

+ 119899 (11990821198892+ 11990831198922+ 11990841199022+ 1199085V2)

(20)

Differentiating 120593(120573 119887 119889 119891 119892 119901 119902 119906 V) that is (20) partiallywith respect to 120573 and equating it to zero we get

120597120593 (120573 119887 119889 119891 119892 119901 119902 119906 V)120597120573

= 0

997904rArr minus1199081XTya + XTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

minus 1199082XTely119887 minus 119908

3XTery119891 minus 119908

4XTel

1015840

y119901 minus 1199085XTer

1015840

y 119906

+ XT1 (1199082119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V) = 0

997904rArr 120573 = ((XTX)minus1

XT[1199081ya + 119908

2ely119887 + 119908

3ery119891

+ 1199084el1015840

y119901 + 1199085er1015840

y 119906

minus1 (1199082119887119889+119908

3119891119892+119908

4119901119902+119908

5119906V)])

times (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)minus1

(21)

Similarly differentiating (20) partially with respect to 119887 119889 119891119892 119901 119902 119906 and V we get

119887 = (120573TXTX120573)

minus1

[(ely)TX120573 minus 120573TXT1119889] (22)

119889 =1

119899[(ely)

T1 minus 120573

TXT1119887] (23)

119891 = (120573TXTX120573)

minus1

[(ery)TX120573 minus 120573TXT1119892] (24)

119892 =1

119899[(er

1015840

y )T1 minus 120573

TXT1119891] (25)

119901 = (120573TXTX120573)

minus1

[(el1015840

y )TX120573 minus 120573TXT1119902] (26)

119902 =1

119899[(el

1015840

y )T1 minus 120573

TXT1119901] (27)

119906 = (120573TXTX120573)

minus1

[(er1015840

y )TX120573 minus 120573TXT1V] (28)

V =1

119899[(er

1015840

y )T1 minus 120573

TXT1119906] (29)

respectively

International Scholarly Research Notices 7

Equations (21)ndash(29) are recursive solutions for the prob-lem of least square estimation with intuitionistic fuzzy dataTherefore we rewrite the system of equations explicitly in arecursive way as follows

120573119894+1

= ((XTX)minus1

XT[1199081ya + 119908

2ely119887119894 + 119908

3ery119891119894 + 119908

4el1015840

y119901119894

+ 1199085er1015840

y 119906119894 minus 1 (1199082119887119894119889119894+ 1199083119891119894119892119894

+1199084119901119894119902119894+ 1199085119906119894V119894) ] )

times (1199081+ 11990821198872

119894+ 11990831198912

119894+ 11990841199012

119894+ 11990851199062

119894)minus1

119887119894+1

= (120573T119894+1

XTX120573i+1)minus1

[(ely)TX120573i+1 minus 120573

Ti+1X

T1119889119894]

119889119894+1

=1

119899[(ely)

T1 minus 120573

Ti+1X

T1119887119894]

119891119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(ery)TX120573i+1 minus 120573

Ti+1X

T1119892119894]

119892119894+1

=1

119899[(er

1015840

y )T1 minus 120573

Ti+1X

T1119891119894]

119901119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(el1015840

y )TX120573i+1 minus 120573

Ti+1X

T1119902119894]

119902119894+1

=1

119899[(el

1015840

y )T1 minus 120573

Ti+1X

T1119901119894]

119906119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(er1015840

y )TX120573i+1 minus 120573

Ti+1X

T1V119894]

V119894+1

=1

119899[(er

1015840

y )T1 minus 120573

Ti+1X

T1119906119894]

(30)

In order to initiate the recursive process of obtaining theestimators we take some initial values for 119887 119889 119891 119892 119901 119902119906 V and 120573 After several numbers of iterations the valuesof estimators get corrected to a predefined error of toleranceWe denote these values by 119889 119891 119892 119901 119902 V and in orderto differentiate them from the eventually obtained restrictedestimator in the next commutation

In a more general setup if in the linear regression model(17) we consider 119896

1crisp and 119896

2intuitionistic fuzzy input

variables then the dimensions of X and 120573 will be 119899 times (1198961+

51198962+ 1) and (119896

1+ 51198962+ 1) times 1 respectively It may further be

noted that the core of the solutionrsquos structure will remain thesame and we will have similar kind of estimators

Remark If a TIFN 119860 = (119898 120572 120573 1205721015840 1205731015840) degenerate to a trian-

gular fuzzy number 119860 = (119898 120572 120573) then our nonsymmetric

intuitionistic fuzzy weighted linear regression model reducesto nonsymmetric fuzzy linear regression model defined byKumar et al [24]

Next we assume that the regression coefficients aresubjected to the linear restrictions which are given by (18) Itmay be noted that the unrestricted estimator obtained abovein (21) does not satisfy the given restrictions (18) We aimto obtain the restricted estimator which satisfies the givenrestrictions under the regression model (17) For this wepropose to minimize the following score function

119878 (120582120573 119887 119889 119891 119892 119901 119902 119906 V)

= 120593 (120573 119887 119889 119891 119892 119901 119902 119906 V) minus 2120582 (H120573 minus h)

= 1199081((ya)Tya minus 2(ya)TX120573)

+ 120573TXTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

+ 1199082((ely)

Tely minus 2(ely)

TX120573119887 minus 2(ely)

T1119889)

+ 1199083((ery)

Tery minus 2(ery)

TX120573119891 minus 2(ery)

T1119892)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )TX120573119901 minus 2(el

1015840

y )T1119902)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )TX120573119906 minus 2(er

1015840

y )T1V)

+ 2120573TXT1 (119908

2119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V)

+ 119899 (11990821198892+ 11990831198922+ 11990841199022+ 1199085V2)

minus 2120582 (H120573 minus h)

(31)

where 2120582 is the vector of Lagrangersquos Multiplier

Differentiating 119878(120582120573 119887 119889 119891 119892 119901 119902 119906 V) partially withrespect to 120573 and equating it to zero we get

997904rArr minus1199081XTya + XTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

minus 1199082XTely119887 minus 119908

3XTery119891 minus 119908

4XTel

1015840

y119901 minus 1199085XTer

1015840

y 119906

+ XT1 (1199082119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V) minusH1015840120582 = 0

(32)

8 International Scholarly Research Notices

Here we again relabel the computed restricted estimator by Therefore in view of (21) and (32) we get size

997904rArr = ((XTX)minus1

XT[1199081ya + 119908

2ely119887 + 119908

3ery119891 + 119908

4el1015840

y119901

+ 1199085er1015840

y 119906

minus1 (1199082119887119889+119908

3119891119892+119908

4119901119902+119908

5119906V) ] )

times (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)minus1

+(XTX)

minus1

HT120582

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

997904rArr = +1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

times (XTX)minus1

HT120582

(33)

Similarly differentiating 119878(120582120573 119887 119889 119891 119892 119901 119902 119906 V) par-tially with respect to 120582 and equating it to zero we get

997904rArr H = h

997904rArr H + 1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

timesH(XTX)minus1

HT120582 = h

997904rArr = (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

times [H(XTX)minus1

HT]minus1

(h minusH)

(34)

From (33) and (34) we have

997904rArr = + (XTX)minus1

HT[H(XTX)

minus1

HT]minus1

(h minusH) (35)

Also differentiating (31) partially with respect to 119887 119889 119891 119892 119901119902 119906 and V and equating all to zero we get

= 119889 = 119889 119891 = 119891 119892 = 119892

119901 = 119901 119902 = 119902 = V = V(36)

respectively From (35) we see that

997904rArr H = H + [H(XTX)minus1

HT] [H(XTX)

minus1

HT]minus1

times (h minusH)

997904rArr H = H + (h minusH) = h(37)

Therefore the estimator satisfies the given restrictions (18)

4 Numerical Examples

We consider the following numerical examples to illustratethe proposed model

Example 1 We apply our procedure to estimate the intuition-istic fuzzy output value for a data consisting of the crisp inputand intuitionistic fuzzy output (where left entropy and rightentropy are equal) and tabulate the data in Table 1

We obtain = (minus44026 35733 73786 56858)1015840 =

02942 119889 = 147144 119891 = 02942 119892 = 147144 119901 = 02909119902 = 174487 = 02909 and V = 174487 where the numberof iterations required is 125

Example 2 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp input andintuitionistic fuzzy output (where left and right entropy arenot equal) and tabulate the data in Table 2

We obtain = (minus47697 35933 72030 59152)1015840 =

02952 119889 = 145871 119891 = 02646 119892 = 203429 119901 = 03052119902 = 157050 = 02717 and V = 231201 where the numberof iterations required is 113

Example 3 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp inputintuitionistic fuzzy input and intuitionistic fuzzy output(where left and right entropy are not equal) and tabulate thedata in Table 3

We obtain = (minus32352 06811 05314 minus09164 00846

minus31631 2953)1015840 = 04225 119889 = 05478 119891 = 04307 119892 =

01637 119901 = 03231 119902 = 38723 = 04985 and V = 18659

where the number of iterations required is 51

Example 4 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of intuitionistic fuzzyinput and intuitionistic fuzzy output (where left and rightentropy are not equal) and tabulate the data in Table 4

We obtain = (118141 minus02161 16104 minus18254 05687

minus01879)1015840 = 03880 119889 = 03674 119891 = 03880 119892 = 03674

119901 = 03547 119902 = 22108 = 03547 and V = 32108 wherethe number of iterations required is 255

5 Conclusions

An intuitionistic fuzzy weighted linear regression (IFWLR)model with and without some linear restrictions in theform of prior information has been studied The estimators

International Scholarly Research Notices 9

Table 1 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y y

a ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 44 42 96 42 44 449018 424850 943828 424850 4490182 14 8 3 48 47 120 47 48 528505 505256 1217099 505256 5285053 7 1 4 35 33 52 33 35 322052 296416 507324 296416 3220524 11 7 3 50 45 106 45 50 475861 452004 1036114 452004 4758615 7 12 15 80 79 189 79 80 740058 719256 1944413 719256 7400586 8 15 10 68 65 194 65 68 732147 711253 1917213 711253 7321477 3 9 6 45 42 107 42 45 485252 461503 1068398 461503 4852528 12 15 11 80 78 216 78 80 790260 770038 2117003 770038 7902609 10 5 8 55 52 108 52 55 505235 481717 1137100 481717 50523510 9 7 4 45 44 103 44 45 471612 447706 1021507 447706 471612

Table 2 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y ya ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 45 42 96 47 48 447743 427104 952620 455472 4900532 14 8 3 48 47 120 43 45 525995 502809 120905 523320 5597343 7 1 4 35 33 52 50 55 313430 297162 512469 339018 3704524 11 7 3 46 45 106 45 47 471120 449720 102922 475741 5108705 7 12 15 82 79 189 80 85 753765 723166 195547 720805 7625556 8 15 10 70 65 194 60 67 740419 710254 191173 709234 7506717 3 9 6 45 42 107 40 46 481512 459774 106328 484752 5201248 12 15 11 80 78 216 88 90 802328 770149 211461 762912 8057999 10 5 8 55 52 108 50 55 506447 483897 114499 506370 54232710 9 7 4 45 44 103 42 44 467241 445967 101651 472377 507415

Table 3 Crisp and int fuzzy input-int fuzzy output data

Object 119894Crisp and int fuzzy inputX = (x1 e1

1015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy outputylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 6 7 6 10 5 7 6 3 5 2 5 54158 25665 47776 22215 424732 7 6 5 12 4 6 7 5 4 5 7 58827 31771 62226 28438 496763 8 4 2 15 3 5 8 3 9 4 6 67973 43734 90537 40632 637884 9 7 5 20 8 10 7 3 10 2 4 64422 39089 79544 35897 583085 10 8 6 5 2 5 8 5 12 5 7 74885 52774 11193 49846 744516 11 10 8 15 5 7 3 2 8 4 6 65757 40835 83678 37678 603697 12 20 15 25 12 14 6 5 7 3 5 62393 36436 73265 33193 551798 13 12 7 30 15 18 8 7 14 6 8 83421 63937 13835 61226 876209 14 16 12 20 10 15 11 9 16 10 12 97423 82250 18169 79893 10922310 15 17 13 22 8 12 8 7 18 5 10 90740 73509 16100 70982 98912

10 International Scholarly Research Notices

Table 4 Intuitionistic fuzzy input-intuitionistic fuzzy output data

Object iInt fuzzy input

X = (e11015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy output

ylowast = (e1lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

ye1lowast

y yalowast er

lowast

y erlowast1015840

y

1 5 3 4 5 6 5 4 12 4 6 57505 42398 99797 42398 675052 7 6 7 8 9 7 5 7 5 8 57737 42652 10045 42652 677373 5 3 6 8 9 5 3 9 3 6 48608 32665 74714 32665 586084 4 2 7 9 11 3 1 4 1 4 37872 20920 44446 20920 478725 3 2 5 7 8 4 2 6 2 5 49552 33698 77377 33698 595526 6 3 6 7 10 5 4 8 4 6 45158 28891 64987 28891 551587 5 2 4 9 12 4 3 9 3 5 55863 40602 95168 40602 658638 6 5 8 13 15 7 5 10 5 8 52404 36818 85417 36818 624049 8 7 12 15 17 4 3 5 3 5 39099 22262 47905 22262 4909910 15 10 15 20 25 4 2 3 2 5 36202 19092 39736 19092 46202

of regression coefficients have also been obtained with thehelp of fuzzy entropy for the restrictedunrestricted IFWLRmodel by assigning some weights in the distance functionIt has been observed that the restricted estimator is betterthan unrestricted estimator in some sense Thus wheneversome prior information is available in terms of exact linearrestrictions on regression coefficients it is advised to userestricted estimator in place of unrestricted estimator

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] H Tanaka S Uejima and K Asai ldquoFuzzy linear regressionmodelrdquo IEEE Transactions on Systems Man and Cyberneticsvol 10 pp 2933ndash2938 1980

[3] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982

[4] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988

[5] H Tanaka I Hayashi and JWatada ldquoPossibilistic linear regres-sion analysis for fuzzy datardquo European Journal of OperationalResearch vol 40 no 3 pp 389ndash396 1989

[6] H Tanaka and H Ishibuchi ldquoIdentification of possibilisticlinear systems by quadratic membership functions of fuzzyparametersrdquo Fuzzy Sets and Systems vol 41 no 2 pp 145ndash1601991

[7] D T Redden and W H Woodall ldquoProperties of certain fuzzylinear regression methodsrdquo Fuzzy Sets and Systems vol 64 no3 pp 361ndash375 1994

[8] P-T Chang and E S Lee ldquoFuzzy linear regression with spreadsunrestricted in signrdquoComputers andMathematics with Applica-tions vol 28 no 4 pp 61ndash70 1994

[9] G Peters ldquoFuzzy linear regression with fuzzy intervalsrdquo FuzzySets and Systems vol 63 no 1 pp 45ndash55 1994

[10] A de Luca and S Termini ldquoA definition of a nonprobabilisticentropy in the setting of fuzzy sets theoryrdquo Information andControl vol 20 no 4 pp 301ndash312 1972

[11] D Dubosis and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[12] D Dubois and H Prade Fuzzy Sets and Statistical PossibilityTheory Plenum Press New York NY USA 1988

[13] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash656 1948

[14] T Kumar N Gupta and R K Bajaj ldquoFuzzy entropy onrestricted fuzzy linear regression model with cross validationand applicationsrdquo in Proceedings of the International Conferenceon Advances in Computing and Communications (ICACC rsquo12)pp 5ndash8 August 2012

[15] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[16] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[17] K T Atanassov Intuitionistic Fuzzy Sets Theory and Applica-tions vol 35 of Studies in Fuzziness and SoftComputing Physica1999

[18] K T Atanassov ldquoNew operations defined over the intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 61 no 2 pp 137ndash1421994

[19] P Burillo and H Bustince ldquoSome definitions of intuitionisticfuzzy numberrdquo in Proceedings of the 3rd Conference of theEuropean Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[20] F Lee Fuzzy Information Processing System Peking UniversityPress Beijing China 1998

[21] H Liu and K Shi ldquoIntuitionistic fuzzy numbers and intuition-istic distribution numbersrdquo Journal of Fuzzy Mathematics vol8 no 4 pp 909ndash918 2000

[22] P Grzegorzewski ldquoDistances and orderings in a family of intu-itionistic fuzzy numbersrdquo in Proceedings of the 3rd Conference ofthe European Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[23] P DrsquoUrso and T Gastaldi ldquoA least-squares approach to fuzzylinear regression analysisrdquo Computational Statistics and DataAnalysis vol 34 no 4 pp 427ndash440 2000

[24] T Kumar R K Bajaj and N Gupta ldquoFuzzy entropy in fuzzyweighted linear regression model under linear restrictions withsimulation studyrdquo International Journal of General Systems vol43 no 2 pp 135ndash148 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Scholarly Research Notices

regression using fuzzy entropy is to construct five conven-tional regression equations (one for apex one for left entropyof the membership function one for right entropy of themembership function one for left entropy of the nonmem-bership function and one for right entropy of the non-membership function) for the response variable y using thecorresponding attributes of the 119896 fuzzy explanatory variablesx119895 In order to be specific we denote ya xa

1 xa2 xa

119896by the

apexes of y x1 x2 x

119896 respectively ely e

lx1

elx2

elx119896

bythe left entropy of y x

1 x2 x

119896 respectively ery e

rx1

erx2

erx119896

by the right entropy of y x1 x2 x

119896 respectively

el1015840

y el1015840x1

el1015840

x2

el1015840

x119896

by the left to left entropy of y x1 x2 x

119896

respectively and er1015840

y er1015840x1

er1015840

x2

er1015840

x119896

by the right to rightentropy of y x

1 x2 x

119896 respectively Therefore the five

fundamental regression equations in a nonrecursive (non-adaptive) setup may be written as

ya = 119860119886

0+

119896

sum

119894=1

(119860119886

119894xa119894+ 119861119886

119894elx119894

+ 119862119886

119894erx119894

+ 119863119886

119894el1015840

x119894

+ 119864119886

119894er1015840

x119894

) + 120576ya

ely = 119860119897

0+

119896

sum

119894=1

(119860119897

119894xa119894+ 119861119897

119894elx119894

+ 119862119897

119894erx119894

+ 119863119897

119894el1015840

x119894

+ 119864119897

119894er1015840

x119894

) + 120576ely

ery = 119860119903

0+

119896

sum

119894=1

(119860119903

119894xa119894+ 119861119903

119894elx119894

+ 119862119903

119894erx119894

+ 119863119903

119894el1015840

x119894

+ 119864119903

119894er1015840

x119894

) + 120576ery

el1015840

y = 1198601198971015840

0+

119896

sum

119894=1

(1198601198971015840

119894xa119894+ 1198611198971015840

119894elx119894

+ 1198621198971015840

119894erx119894

+ 1198631198971015840

119894el1015840

x119894

+ 1198641198971015840

119894er1015840

x119894

)

+ 120576el1015840y

er1015840

y = 1198601199031015840

0+

119896

sum

119894=1

(1198601199031015840

119894xa119894+ 1198611199031015840

119894elx119894

+ 1198621199031015840

119894erx119894

+ 1198631199031015840

119894el1015840

x119894

+ 1198641199031015840

119894er1015840

x119894

)

+ 120576er1015840y

(14)

where 120576ya 120576ely 120576ery 120576el1015840y and 120576er1015840y are the error vectors ofdimension 119899 times 1 The compact form of the above mentionednonrecursive or nonadaptive equations is given by

ya = X120573 + 120576ya

ely = X120572 + 120576ely

ery = X120574 + 120576ery

el1015840

y = X1205721015840 + 120576el1015840y

er1015840

y = X1205741015840 + 120576er1015840y

(15)

where

X= (1 xa1 xa2 xa

119896

elx1

elx2

elx119896

erx1

erx2

erx119896

el1015840

x1

el1015840

x2

el1015840

x119896

er1015840

x1

er1015840

x2

er1015840

x119896

)

119899times(5119896+1)

120573= (119860119886

0

1198601198861 119860119886

2 119860

119886

119896

1198611198861 119861119886

2 119861

119886

119896

1198621198861 119862119886

2

119862119886

119896

1198631198861 119863119886

2 119863

119886

119896

1198641198861 119864119886

2 119864

119886

119896)

119879

(5119896+1)times1

120572= (119860119897

0

1198601198971 119860119897

2 119860

119897

119896

1198611198971 119861119897

2 119861

119897

119896

1198621198971 119862119897

2

119862119897

119896

1198631198971 119863119897

2 119863

119897

119896

1198641198971 119864119897

2 119864

119897

119896)

119879

(5119896+1)times1

120574= (119860119903

0

1198601199031 119860119903

2 119860

119903

119896

1198611199031 119861119903

2 119861

119903

119896

1198621199031 119862119903

2

119862119903

119896

1198631199031 119863119903

2 119863

119903

119896

1198641199031 119864119903

2 119864

119903

119896)

119879

(5119896+1)times1

1205721015840= (119860

1198971015840

0

1198601198971015840

1 1198601198971015840

2 119860

1198971015840

119896

1198611198971015840

1 1198611198971015840

2 119861

1198971015840

119896

1198621198971015840

1 1198621198971015840

2

1198621198971015840

119896

1198631198971015840

1 1198631198971015840

2 119863

1198971015840

119896

1198641198971015840

1 1198641198971015840

2 119864

1198971015840

119896)

119879

(5119896+1)times1

1205741015840= (119860

1199031015840

0

1198601199031015840

1 1198601199031015840

2 119860

1199031015840

119896

1198611199031015840

1 1198611199031015840

2 119861

1199031015840

119896

1198621199031015840

1 1198621199031015840

2

1198621199031015840

119896

1198631199031015840

1 1198631199031015840

2 119863

1199031015840

119896

1198641199031015840

1 1198641199031015840

2 119864

1199031015840

119896)

119879

(5119896+1)times1

(16)

In many real life situations where the measurements arecarried out (for example car speed astronomical distance)it is natural to think that the spread (vagueness) in themeasure of a phenomenon is proportional to its intensityDrsquoUrso and Gastaldi [23] have done several simulations andobserved that even if we consider an adaptive or recursiveregression model along with nonadaptive or nonrecursiveregression model they yield identical solutions when thereis only one independent variable But if there are morethan one independent variable then the estimated values ofthe left entropies and right entropies obtained through therecursive fuzzy regression model will have less variance ascompared to the nonrecursive fuzzy regression model Withthis consideration we rewrite the proposed intuitionisticfuzzy linear regression model (15) in a recursiveadaptive

International Scholarly Research Notices 5

setup where dynamic of the entropies is dependent on themagnitude of the estimated apexes as follows

ya = yalowast

+ 120576ya where yalowast

= X120573

ely = ellowast

y + 120576lowast

ely where el

lowast

y = X120573119887 + 1119889

ery = erlowast

y + 120576lowast

ery where er

lowast

y = X120573119891 + 1119892

el1015840

y = el1015840lowast

y + 120576lowast

el1015840y where el

1015840lowast

y = X120573119901 + 1119902

er1015840

y = er1015840lowast

y + 120576lowast

er1015840y where er

1015840lowast

y = X120573119906 + 1V

(17)

where X is the 119899 times (5119896 + 1)-matrix containing the valuesof the input variables (data matrix) 120573 is a column 5119896 + 1-vector containing the regression coefficients for the apexesof the first model (referred to as core regression model)ya and yalowast are the vector of the observed apexes and thevector of the interpolated apexes respectively both havingdimension 119899 times 1 ely and ely

lowast

are the vector of the observedleft entropies and the vector of the interpolated left entropiesrespectively both having dimension 119899 times 1 ery and ery

lowast are thevector of the observed right entropies and the vector ofthe interpolated right entropies respectively both havingdimension 119899 times 1 el

1015840

y and el1015840

ylowast

are the vector of the observedleft to left entropies and the vector of the interpolated left toleft entropies respectively both having dimension 119899 times 1 er

1015840

y

and er1015840

ylowast

are the vector of the observed right to right entropiesand the vector of the interpolated right to right entropiesrespectively both having dimension 119899 times 1 and 1 is a (119899 times 1)-vector of all 11015840s 119887 and 119889 are regression parameters for thesecond regression equation model (referred to as left entropyregression model) 119891 and 119892 are regression parameters for thethird regressionmodel (referred to as right entropy regressionmodel) 119901 and 119902 are regression parameters for the fourthregression equation model (referred to as left to left entropyregression model) and 119906 and V are regression parametersfor the fifth regression equation model (referred to as rightto right entropy regression model) The error term in theregression equation of apexes will remain the same while theerror terms in the regression equations of entropies may bedifferent The error vectors 120576lowastely and 120576

lowast

eryin the left and right

entropies are of the dimension (119899 times 1) and the error vectors120576lowast

elyand 120576lowastery in the left to left and right to right entropies are of

the dimension (119899 times 1)If some prior information about unknown regression

coefficients is available on the basis of past experiences thenit may be used to estimate more efficient estimators Weassume that such prior information is in the form of exactlinear restrictions on regression coefficients In the presentmodel we associate such restrictions in the equations forthe estimation of regression coefficients in the intuitionisticfuzzy linear regression model with fuzzy entropy Thereforewe make the model capable of taking into account possible

linear relations between the size of the entropies and themagnitude of the estimated apexesMoreover we assume thatthe regression coefficients 120573 are subjected to the 119895 (119895 lt 5119896+1)exact linear restrictions which are given by

h = H120573 (18)

whereh andH are known and thematrixH is of full row rank

3 Estimation of Regression Coefficients

In many applications it is possible that the values of the vari-ables are on completely different scales ofmeasurement Alsothe possible larger variations in the values will have largerintersample differences so they will dominate in the calcu-lation of Euclidean distances Therefore some form of stan-dardization is necessary to balance out the individual con-tributions Consider the Euclidean distance between two tri-angular intuitionistic fuzzy numbers 119910

119894= (119910119886

119894 119890119897

119910119894

119890119903

119910119894

1198901198971015840

119910119894

1198901199031015840

119910119894

)

and 119910lowast

119894= (119910119886lowast

119894 119890119897lowast

119910119894

119890119903lowast

119910119894

1198901198971015840lowast

119910119894

1198901199031015840lowast

119910119894

) along with weights 1199081 1199082

1199083 1199084 and 119908

5as follows

120575119894equiv 120575 (119910

119894 119910lowast

119894)

= (1199081(119910119886

119894minus 119910119886lowast

119894)2

+ 1199082(119890119897

119910119894

minus 119890119897lowast

119910119894

)2

+ 1199083(119890119903

119910119894

minus 119890119903lowast

119910119894

)2

+1199084(1198901198971015840

119910119894

minus 1198901198971015840lowast

119910119894

)

2

+ 1199085(1198901199031015840

119910119894

minus 1198901199031015840lowast

119910119894

)

2

)

12

(19)

It may be observed that we compute the usual squared differ-ences between the values of variables on their original scalesas in the usual Euclidean distance but then multiply thesesquared differences by their corresponding weights

Next similar to common linear regression (based on crispdata) the regression parameters are estimated byminimizingthe following sum of square errors (we use a compact matrixnotation)

120593 (120573 119887 119889 119891 119892 119901 119902 119906 V)

=

119899

sum

119894=1

1199081(119910119886

119894minus 119910119886lowast

119894)2

+

119899

sum

119894=1

1199082(119890119897

119910119894

minus 119890119897lowast

119910119894

)2

+

119899

sum

119894=1

1199083(119890119903

119910119894

minus 119890119903lowast

119910119894

)2

+

119899

sum

119894=1

1199084(1198901198971015840

119910119894

minus 1198901198971015840lowast

119910119894

)2

+

119899

sum

119894=1

1199085(1198901199031015840

119910119894

minus 1198901199031015840lowast

119910119894

)2

= 1199081(ya minus yalowast)

T(ya minus yalowast) + 119908

2(ely minus ely

lowast

)T(ely minus ely

lowast

)

+ 1199083(ery minus er

lowast

y )T(ery minus er

lowast

y )

+ 1199084(el1015840

y minus el1015840

ylowast

)

T(el1015840

y minus el1015840

ylowast

)

+ 1199085(er1015840

y minus er1015840

ylowast

)

T(er1015840

y minus er1015840

ylowast

)

6 International Scholarly Research Notices

= 1199081((ya)Tya minus 2(ya)Tyalowast + (yalowast)

Tyalowast)

+ 1199082((ely)

Tely minus 2(ely)

Telylowast

+ (elylowast

)Telylowast

)

+ 1199083((ery)

Tery minus 2(ery)

Terylowast

+ (erylowast

)Terylowast

)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )Tel1015840

ylowast

+ (el1015840

ylowast

)

Tel1015840

ylowast

)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )Ter1015840

ylowast

+ (er1015840

ylowast

)

Ter1015840

ylowast

)

= 1199081((ya)Tya minus 2(ya)TX120573 + 120573TXTX120573)

+ 1199082((ely)

Tely minus 2(ely)

T(X120573119887 + 1119889))

+ 1199082((X120573119887 + 1119889)T (X120573119887 + 1119889))

+ 1199083((ery)

Tery minus 2(ery)

T(X120573119891 + 1119892))

+ 1199083((X120573119891 + 1119892)T (X120573119891 + 1119892))

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )T(X120573119901 + 1119902))

+ 1199084((X120573119901 + 1119902)T (X120573119901 + 1119902))

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )T(X120573119906 + 1V))

+ 1199085((X120573119906 + 1V)T (X120573119906 + 1V))

= 1199081((ya)Tya minus 2(ya)TX120573)

+ 120573TXTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

+ 1199082((ely)

Tely minus 2(ely)

TX120573119887 minus 2(ely)

T1119889)

+ 1199083((ery)

Tery minus 2(ery)

TX120573119891 minus 2(ery)

T1119892)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )TX120573119901 minus 2(el

1015840

y )T1119902)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )TX120573119906 minus 2(er

1015840

y )T1V)

+ 2120573TXT1 (119908

2119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V)

+ 119899 (11990821198892+ 11990831198922+ 11990841199022+ 1199085V2)

(20)

Differentiating 120593(120573 119887 119889 119891 119892 119901 119902 119906 V) that is (20) partiallywith respect to 120573 and equating it to zero we get

120597120593 (120573 119887 119889 119891 119892 119901 119902 119906 V)120597120573

= 0

997904rArr minus1199081XTya + XTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

minus 1199082XTely119887 minus 119908

3XTery119891 minus 119908

4XTel

1015840

y119901 minus 1199085XTer

1015840

y 119906

+ XT1 (1199082119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V) = 0

997904rArr 120573 = ((XTX)minus1

XT[1199081ya + 119908

2ely119887 + 119908

3ery119891

+ 1199084el1015840

y119901 + 1199085er1015840

y 119906

minus1 (1199082119887119889+119908

3119891119892+119908

4119901119902+119908

5119906V)])

times (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)minus1

(21)

Similarly differentiating (20) partially with respect to 119887 119889 119891119892 119901 119902 119906 and V we get

119887 = (120573TXTX120573)

minus1

[(ely)TX120573 minus 120573TXT1119889] (22)

119889 =1

119899[(ely)

T1 minus 120573

TXT1119887] (23)

119891 = (120573TXTX120573)

minus1

[(ery)TX120573 minus 120573TXT1119892] (24)

119892 =1

119899[(er

1015840

y )T1 minus 120573

TXT1119891] (25)

119901 = (120573TXTX120573)

minus1

[(el1015840

y )TX120573 minus 120573TXT1119902] (26)

119902 =1

119899[(el

1015840

y )T1 minus 120573

TXT1119901] (27)

119906 = (120573TXTX120573)

minus1

[(er1015840

y )TX120573 minus 120573TXT1V] (28)

V =1

119899[(er

1015840

y )T1 minus 120573

TXT1119906] (29)

respectively

International Scholarly Research Notices 7

Equations (21)ndash(29) are recursive solutions for the prob-lem of least square estimation with intuitionistic fuzzy dataTherefore we rewrite the system of equations explicitly in arecursive way as follows

120573119894+1

= ((XTX)minus1

XT[1199081ya + 119908

2ely119887119894 + 119908

3ery119891119894 + 119908

4el1015840

y119901119894

+ 1199085er1015840

y 119906119894 minus 1 (1199082119887119894119889119894+ 1199083119891119894119892119894

+1199084119901119894119902119894+ 1199085119906119894V119894) ] )

times (1199081+ 11990821198872

119894+ 11990831198912

119894+ 11990841199012

119894+ 11990851199062

119894)minus1

119887119894+1

= (120573T119894+1

XTX120573i+1)minus1

[(ely)TX120573i+1 minus 120573

Ti+1X

T1119889119894]

119889119894+1

=1

119899[(ely)

T1 minus 120573

Ti+1X

T1119887119894]

119891119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(ery)TX120573i+1 minus 120573

Ti+1X

T1119892119894]

119892119894+1

=1

119899[(er

1015840

y )T1 minus 120573

Ti+1X

T1119891119894]

119901119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(el1015840

y )TX120573i+1 minus 120573

Ti+1X

T1119902119894]

119902119894+1

=1

119899[(el

1015840

y )T1 minus 120573

Ti+1X

T1119901119894]

119906119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(er1015840

y )TX120573i+1 minus 120573

Ti+1X

T1V119894]

V119894+1

=1

119899[(er

1015840

y )T1 minus 120573

Ti+1X

T1119906119894]

(30)

In order to initiate the recursive process of obtaining theestimators we take some initial values for 119887 119889 119891 119892 119901 119902119906 V and 120573 After several numbers of iterations the valuesof estimators get corrected to a predefined error of toleranceWe denote these values by 119889 119891 119892 119901 119902 V and in orderto differentiate them from the eventually obtained restrictedestimator in the next commutation

In a more general setup if in the linear regression model(17) we consider 119896

1crisp and 119896

2intuitionistic fuzzy input

variables then the dimensions of X and 120573 will be 119899 times (1198961+

51198962+ 1) and (119896

1+ 51198962+ 1) times 1 respectively It may further be

noted that the core of the solutionrsquos structure will remain thesame and we will have similar kind of estimators

Remark If a TIFN 119860 = (119898 120572 120573 1205721015840 1205731015840) degenerate to a trian-

gular fuzzy number 119860 = (119898 120572 120573) then our nonsymmetric

intuitionistic fuzzy weighted linear regression model reducesto nonsymmetric fuzzy linear regression model defined byKumar et al [24]

Next we assume that the regression coefficients aresubjected to the linear restrictions which are given by (18) Itmay be noted that the unrestricted estimator obtained abovein (21) does not satisfy the given restrictions (18) We aimto obtain the restricted estimator which satisfies the givenrestrictions under the regression model (17) For this wepropose to minimize the following score function

119878 (120582120573 119887 119889 119891 119892 119901 119902 119906 V)

= 120593 (120573 119887 119889 119891 119892 119901 119902 119906 V) minus 2120582 (H120573 minus h)

= 1199081((ya)Tya minus 2(ya)TX120573)

+ 120573TXTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

+ 1199082((ely)

Tely minus 2(ely)

TX120573119887 minus 2(ely)

T1119889)

+ 1199083((ery)

Tery minus 2(ery)

TX120573119891 minus 2(ery)

T1119892)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )TX120573119901 minus 2(el

1015840

y )T1119902)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )TX120573119906 minus 2(er

1015840

y )T1V)

+ 2120573TXT1 (119908

2119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V)

+ 119899 (11990821198892+ 11990831198922+ 11990841199022+ 1199085V2)

minus 2120582 (H120573 minus h)

(31)

where 2120582 is the vector of Lagrangersquos Multiplier

Differentiating 119878(120582120573 119887 119889 119891 119892 119901 119902 119906 V) partially withrespect to 120573 and equating it to zero we get

997904rArr minus1199081XTya + XTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

minus 1199082XTely119887 minus 119908

3XTery119891 minus 119908

4XTel

1015840

y119901 minus 1199085XTer

1015840

y 119906

+ XT1 (1199082119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V) minusH1015840120582 = 0

(32)

8 International Scholarly Research Notices

Here we again relabel the computed restricted estimator by Therefore in view of (21) and (32) we get size

997904rArr = ((XTX)minus1

XT[1199081ya + 119908

2ely119887 + 119908

3ery119891 + 119908

4el1015840

y119901

+ 1199085er1015840

y 119906

minus1 (1199082119887119889+119908

3119891119892+119908

4119901119902+119908

5119906V) ] )

times (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)minus1

+(XTX)

minus1

HT120582

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

997904rArr = +1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

times (XTX)minus1

HT120582

(33)

Similarly differentiating 119878(120582120573 119887 119889 119891 119892 119901 119902 119906 V) par-tially with respect to 120582 and equating it to zero we get

997904rArr H = h

997904rArr H + 1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

timesH(XTX)minus1

HT120582 = h

997904rArr = (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

times [H(XTX)minus1

HT]minus1

(h minusH)

(34)

From (33) and (34) we have

997904rArr = + (XTX)minus1

HT[H(XTX)

minus1

HT]minus1

(h minusH) (35)

Also differentiating (31) partially with respect to 119887 119889 119891 119892 119901119902 119906 and V and equating all to zero we get

= 119889 = 119889 119891 = 119891 119892 = 119892

119901 = 119901 119902 = 119902 = V = V(36)

respectively From (35) we see that

997904rArr H = H + [H(XTX)minus1

HT] [H(XTX)

minus1

HT]minus1

times (h minusH)

997904rArr H = H + (h minusH) = h(37)

Therefore the estimator satisfies the given restrictions (18)

4 Numerical Examples

We consider the following numerical examples to illustratethe proposed model

Example 1 We apply our procedure to estimate the intuition-istic fuzzy output value for a data consisting of the crisp inputand intuitionistic fuzzy output (where left entropy and rightentropy are equal) and tabulate the data in Table 1

We obtain = (minus44026 35733 73786 56858)1015840 =

02942 119889 = 147144 119891 = 02942 119892 = 147144 119901 = 02909119902 = 174487 = 02909 and V = 174487 where the numberof iterations required is 125

Example 2 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp input andintuitionistic fuzzy output (where left and right entropy arenot equal) and tabulate the data in Table 2

We obtain = (minus47697 35933 72030 59152)1015840 =

02952 119889 = 145871 119891 = 02646 119892 = 203429 119901 = 03052119902 = 157050 = 02717 and V = 231201 where the numberof iterations required is 113

Example 3 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp inputintuitionistic fuzzy input and intuitionistic fuzzy output(where left and right entropy are not equal) and tabulate thedata in Table 3

We obtain = (minus32352 06811 05314 minus09164 00846

minus31631 2953)1015840 = 04225 119889 = 05478 119891 = 04307 119892 =

01637 119901 = 03231 119902 = 38723 = 04985 and V = 18659

where the number of iterations required is 51

Example 4 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of intuitionistic fuzzyinput and intuitionistic fuzzy output (where left and rightentropy are not equal) and tabulate the data in Table 4

We obtain = (118141 minus02161 16104 minus18254 05687

minus01879)1015840 = 03880 119889 = 03674 119891 = 03880 119892 = 03674

119901 = 03547 119902 = 22108 = 03547 and V = 32108 wherethe number of iterations required is 255

5 Conclusions

An intuitionistic fuzzy weighted linear regression (IFWLR)model with and without some linear restrictions in theform of prior information has been studied The estimators

International Scholarly Research Notices 9

Table 1 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y y

a ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 44 42 96 42 44 449018 424850 943828 424850 4490182 14 8 3 48 47 120 47 48 528505 505256 1217099 505256 5285053 7 1 4 35 33 52 33 35 322052 296416 507324 296416 3220524 11 7 3 50 45 106 45 50 475861 452004 1036114 452004 4758615 7 12 15 80 79 189 79 80 740058 719256 1944413 719256 7400586 8 15 10 68 65 194 65 68 732147 711253 1917213 711253 7321477 3 9 6 45 42 107 42 45 485252 461503 1068398 461503 4852528 12 15 11 80 78 216 78 80 790260 770038 2117003 770038 7902609 10 5 8 55 52 108 52 55 505235 481717 1137100 481717 50523510 9 7 4 45 44 103 44 45 471612 447706 1021507 447706 471612

Table 2 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y ya ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 45 42 96 47 48 447743 427104 952620 455472 4900532 14 8 3 48 47 120 43 45 525995 502809 120905 523320 5597343 7 1 4 35 33 52 50 55 313430 297162 512469 339018 3704524 11 7 3 46 45 106 45 47 471120 449720 102922 475741 5108705 7 12 15 82 79 189 80 85 753765 723166 195547 720805 7625556 8 15 10 70 65 194 60 67 740419 710254 191173 709234 7506717 3 9 6 45 42 107 40 46 481512 459774 106328 484752 5201248 12 15 11 80 78 216 88 90 802328 770149 211461 762912 8057999 10 5 8 55 52 108 50 55 506447 483897 114499 506370 54232710 9 7 4 45 44 103 42 44 467241 445967 101651 472377 507415

Table 3 Crisp and int fuzzy input-int fuzzy output data

Object 119894Crisp and int fuzzy inputX = (x1 e1

1015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy outputylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 6 7 6 10 5 7 6 3 5 2 5 54158 25665 47776 22215 424732 7 6 5 12 4 6 7 5 4 5 7 58827 31771 62226 28438 496763 8 4 2 15 3 5 8 3 9 4 6 67973 43734 90537 40632 637884 9 7 5 20 8 10 7 3 10 2 4 64422 39089 79544 35897 583085 10 8 6 5 2 5 8 5 12 5 7 74885 52774 11193 49846 744516 11 10 8 15 5 7 3 2 8 4 6 65757 40835 83678 37678 603697 12 20 15 25 12 14 6 5 7 3 5 62393 36436 73265 33193 551798 13 12 7 30 15 18 8 7 14 6 8 83421 63937 13835 61226 876209 14 16 12 20 10 15 11 9 16 10 12 97423 82250 18169 79893 10922310 15 17 13 22 8 12 8 7 18 5 10 90740 73509 16100 70982 98912

10 International Scholarly Research Notices

Table 4 Intuitionistic fuzzy input-intuitionistic fuzzy output data

Object iInt fuzzy input

X = (e11015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy output

ylowast = (e1lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

ye1lowast

y yalowast er

lowast

y erlowast1015840

y

1 5 3 4 5 6 5 4 12 4 6 57505 42398 99797 42398 675052 7 6 7 8 9 7 5 7 5 8 57737 42652 10045 42652 677373 5 3 6 8 9 5 3 9 3 6 48608 32665 74714 32665 586084 4 2 7 9 11 3 1 4 1 4 37872 20920 44446 20920 478725 3 2 5 7 8 4 2 6 2 5 49552 33698 77377 33698 595526 6 3 6 7 10 5 4 8 4 6 45158 28891 64987 28891 551587 5 2 4 9 12 4 3 9 3 5 55863 40602 95168 40602 658638 6 5 8 13 15 7 5 10 5 8 52404 36818 85417 36818 624049 8 7 12 15 17 4 3 5 3 5 39099 22262 47905 22262 4909910 15 10 15 20 25 4 2 3 2 5 36202 19092 39736 19092 46202

of regression coefficients have also been obtained with thehelp of fuzzy entropy for the restrictedunrestricted IFWLRmodel by assigning some weights in the distance functionIt has been observed that the restricted estimator is betterthan unrestricted estimator in some sense Thus wheneversome prior information is available in terms of exact linearrestrictions on regression coefficients it is advised to userestricted estimator in place of unrestricted estimator

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] H Tanaka S Uejima and K Asai ldquoFuzzy linear regressionmodelrdquo IEEE Transactions on Systems Man and Cyberneticsvol 10 pp 2933ndash2938 1980

[3] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982

[4] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988

[5] H Tanaka I Hayashi and JWatada ldquoPossibilistic linear regres-sion analysis for fuzzy datardquo European Journal of OperationalResearch vol 40 no 3 pp 389ndash396 1989

[6] H Tanaka and H Ishibuchi ldquoIdentification of possibilisticlinear systems by quadratic membership functions of fuzzyparametersrdquo Fuzzy Sets and Systems vol 41 no 2 pp 145ndash1601991

[7] D T Redden and W H Woodall ldquoProperties of certain fuzzylinear regression methodsrdquo Fuzzy Sets and Systems vol 64 no3 pp 361ndash375 1994

[8] P-T Chang and E S Lee ldquoFuzzy linear regression with spreadsunrestricted in signrdquoComputers andMathematics with Applica-tions vol 28 no 4 pp 61ndash70 1994

[9] G Peters ldquoFuzzy linear regression with fuzzy intervalsrdquo FuzzySets and Systems vol 63 no 1 pp 45ndash55 1994

[10] A de Luca and S Termini ldquoA definition of a nonprobabilisticentropy in the setting of fuzzy sets theoryrdquo Information andControl vol 20 no 4 pp 301ndash312 1972

[11] D Dubosis and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[12] D Dubois and H Prade Fuzzy Sets and Statistical PossibilityTheory Plenum Press New York NY USA 1988

[13] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash656 1948

[14] T Kumar N Gupta and R K Bajaj ldquoFuzzy entropy onrestricted fuzzy linear regression model with cross validationand applicationsrdquo in Proceedings of the International Conferenceon Advances in Computing and Communications (ICACC rsquo12)pp 5ndash8 August 2012

[15] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[16] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[17] K T Atanassov Intuitionistic Fuzzy Sets Theory and Applica-tions vol 35 of Studies in Fuzziness and SoftComputing Physica1999

[18] K T Atanassov ldquoNew operations defined over the intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 61 no 2 pp 137ndash1421994

[19] P Burillo and H Bustince ldquoSome definitions of intuitionisticfuzzy numberrdquo in Proceedings of the 3rd Conference of theEuropean Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[20] F Lee Fuzzy Information Processing System Peking UniversityPress Beijing China 1998

[21] H Liu and K Shi ldquoIntuitionistic fuzzy numbers and intuition-istic distribution numbersrdquo Journal of Fuzzy Mathematics vol8 no 4 pp 909ndash918 2000

[22] P Grzegorzewski ldquoDistances and orderings in a family of intu-itionistic fuzzy numbersrdquo in Proceedings of the 3rd Conference ofthe European Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[23] P DrsquoUrso and T Gastaldi ldquoA least-squares approach to fuzzylinear regression analysisrdquo Computational Statistics and DataAnalysis vol 34 no 4 pp 427ndash440 2000

[24] T Kumar R K Bajaj and N Gupta ldquoFuzzy entropy in fuzzyweighted linear regression model under linear restrictions withsimulation studyrdquo International Journal of General Systems vol43 no 2 pp 135ndash148 2014

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 5

setup where dynamic of the entropies is dependent on themagnitude of the estimated apexes as follows

ya = yalowast

+ 120576ya where yalowast

= X120573

ely = ellowast

y + 120576lowast

ely where el

lowast

y = X120573119887 + 1119889

ery = erlowast

y + 120576lowast

ery where er

lowast

y = X120573119891 + 1119892

el1015840

y = el1015840lowast

y + 120576lowast

el1015840y where el

1015840lowast

y = X120573119901 + 1119902

er1015840

y = er1015840lowast

y + 120576lowast

er1015840y where er

1015840lowast

y = X120573119906 + 1V

(17)

where X is the 119899 times (5119896 + 1)-matrix containing the valuesof the input variables (data matrix) 120573 is a column 5119896 + 1-vector containing the regression coefficients for the apexesof the first model (referred to as core regression model)ya and yalowast are the vector of the observed apexes and thevector of the interpolated apexes respectively both havingdimension 119899 times 1 ely and ely

lowast

are the vector of the observedleft entropies and the vector of the interpolated left entropiesrespectively both having dimension 119899 times 1 ery and ery

lowast are thevector of the observed right entropies and the vector ofthe interpolated right entropies respectively both havingdimension 119899 times 1 el

1015840

y and el1015840

ylowast

are the vector of the observedleft to left entropies and the vector of the interpolated left toleft entropies respectively both having dimension 119899 times 1 er

1015840

y

and er1015840

ylowast

are the vector of the observed right to right entropiesand the vector of the interpolated right to right entropiesrespectively both having dimension 119899 times 1 and 1 is a (119899 times 1)-vector of all 11015840s 119887 and 119889 are regression parameters for thesecond regression equation model (referred to as left entropyregression model) 119891 and 119892 are regression parameters for thethird regressionmodel (referred to as right entropy regressionmodel) 119901 and 119902 are regression parameters for the fourthregression equation model (referred to as left to left entropyregression model) and 119906 and V are regression parametersfor the fifth regression equation model (referred to as rightto right entropy regression model) The error term in theregression equation of apexes will remain the same while theerror terms in the regression equations of entropies may bedifferent The error vectors 120576lowastely and 120576

lowast

eryin the left and right

entropies are of the dimension (119899 times 1) and the error vectors120576lowast

elyand 120576lowastery in the left to left and right to right entropies are of

the dimension (119899 times 1)If some prior information about unknown regression

coefficients is available on the basis of past experiences thenit may be used to estimate more efficient estimators Weassume that such prior information is in the form of exactlinear restrictions on regression coefficients In the presentmodel we associate such restrictions in the equations forthe estimation of regression coefficients in the intuitionisticfuzzy linear regression model with fuzzy entropy Thereforewe make the model capable of taking into account possible

linear relations between the size of the entropies and themagnitude of the estimated apexesMoreover we assume thatthe regression coefficients 120573 are subjected to the 119895 (119895 lt 5119896+1)exact linear restrictions which are given by

h = H120573 (18)

whereh andH are known and thematrixH is of full row rank

3 Estimation of Regression Coefficients

In many applications it is possible that the values of the vari-ables are on completely different scales ofmeasurement Alsothe possible larger variations in the values will have largerintersample differences so they will dominate in the calcu-lation of Euclidean distances Therefore some form of stan-dardization is necessary to balance out the individual con-tributions Consider the Euclidean distance between two tri-angular intuitionistic fuzzy numbers 119910

119894= (119910119886

119894 119890119897

119910119894

119890119903

119910119894

1198901198971015840

119910119894

1198901199031015840

119910119894

)

and 119910lowast

119894= (119910119886lowast

119894 119890119897lowast

119910119894

119890119903lowast

119910119894

1198901198971015840lowast

119910119894

1198901199031015840lowast

119910119894

) along with weights 1199081 1199082

1199083 1199084 and 119908

5as follows

120575119894equiv 120575 (119910

119894 119910lowast

119894)

= (1199081(119910119886

119894minus 119910119886lowast

119894)2

+ 1199082(119890119897

119910119894

minus 119890119897lowast

119910119894

)2

+ 1199083(119890119903

119910119894

minus 119890119903lowast

119910119894

)2

+1199084(1198901198971015840

119910119894

minus 1198901198971015840lowast

119910119894

)

2

+ 1199085(1198901199031015840

119910119894

minus 1198901199031015840lowast

119910119894

)

2

)

12

(19)

It may be observed that we compute the usual squared differ-ences between the values of variables on their original scalesas in the usual Euclidean distance but then multiply thesesquared differences by their corresponding weights

Next similar to common linear regression (based on crispdata) the regression parameters are estimated byminimizingthe following sum of square errors (we use a compact matrixnotation)

120593 (120573 119887 119889 119891 119892 119901 119902 119906 V)

=

119899

sum

119894=1

1199081(119910119886

119894minus 119910119886lowast

119894)2

+

119899

sum

119894=1

1199082(119890119897

119910119894

minus 119890119897lowast

119910119894

)2

+

119899

sum

119894=1

1199083(119890119903

119910119894

minus 119890119903lowast

119910119894

)2

+

119899

sum

119894=1

1199084(1198901198971015840

119910119894

minus 1198901198971015840lowast

119910119894

)2

+

119899

sum

119894=1

1199085(1198901199031015840

119910119894

minus 1198901199031015840lowast

119910119894

)2

= 1199081(ya minus yalowast)

T(ya minus yalowast) + 119908

2(ely minus ely

lowast

)T(ely minus ely

lowast

)

+ 1199083(ery minus er

lowast

y )T(ery minus er

lowast

y )

+ 1199084(el1015840

y minus el1015840

ylowast

)

T(el1015840

y minus el1015840

ylowast

)

+ 1199085(er1015840

y minus er1015840

ylowast

)

T(er1015840

y minus er1015840

ylowast

)

6 International Scholarly Research Notices

= 1199081((ya)Tya minus 2(ya)Tyalowast + (yalowast)

Tyalowast)

+ 1199082((ely)

Tely minus 2(ely)

Telylowast

+ (elylowast

)Telylowast

)

+ 1199083((ery)

Tery minus 2(ery)

Terylowast

+ (erylowast

)Terylowast

)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )Tel1015840

ylowast

+ (el1015840

ylowast

)

Tel1015840

ylowast

)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )Ter1015840

ylowast

+ (er1015840

ylowast

)

Ter1015840

ylowast

)

= 1199081((ya)Tya minus 2(ya)TX120573 + 120573TXTX120573)

+ 1199082((ely)

Tely minus 2(ely)

T(X120573119887 + 1119889))

+ 1199082((X120573119887 + 1119889)T (X120573119887 + 1119889))

+ 1199083((ery)

Tery minus 2(ery)

T(X120573119891 + 1119892))

+ 1199083((X120573119891 + 1119892)T (X120573119891 + 1119892))

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )T(X120573119901 + 1119902))

+ 1199084((X120573119901 + 1119902)T (X120573119901 + 1119902))

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )T(X120573119906 + 1V))

+ 1199085((X120573119906 + 1V)T (X120573119906 + 1V))

= 1199081((ya)Tya minus 2(ya)TX120573)

+ 120573TXTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

+ 1199082((ely)

Tely minus 2(ely)

TX120573119887 minus 2(ely)

T1119889)

+ 1199083((ery)

Tery minus 2(ery)

TX120573119891 minus 2(ery)

T1119892)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )TX120573119901 minus 2(el

1015840

y )T1119902)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )TX120573119906 minus 2(er

1015840

y )T1V)

+ 2120573TXT1 (119908

2119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V)

+ 119899 (11990821198892+ 11990831198922+ 11990841199022+ 1199085V2)

(20)

Differentiating 120593(120573 119887 119889 119891 119892 119901 119902 119906 V) that is (20) partiallywith respect to 120573 and equating it to zero we get

120597120593 (120573 119887 119889 119891 119892 119901 119902 119906 V)120597120573

= 0

997904rArr minus1199081XTya + XTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

minus 1199082XTely119887 minus 119908

3XTery119891 minus 119908

4XTel

1015840

y119901 minus 1199085XTer

1015840

y 119906

+ XT1 (1199082119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V) = 0

997904rArr 120573 = ((XTX)minus1

XT[1199081ya + 119908

2ely119887 + 119908

3ery119891

+ 1199084el1015840

y119901 + 1199085er1015840

y 119906

minus1 (1199082119887119889+119908

3119891119892+119908

4119901119902+119908

5119906V)])

times (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)minus1

(21)

Similarly differentiating (20) partially with respect to 119887 119889 119891119892 119901 119902 119906 and V we get

119887 = (120573TXTX120573)

minus1

[(ely)TX120573 minus 120573TXT1119889] (22)

119889 =1

119899[(ely)

T1 minus 120573

TXT1119887] (23)

119891 = (120573TXTX120573)

minus1

[(ery)TX120573 minus 120573TXT1119892] (24)

119892 =1

119899[(er

1015840

y )T1 minus 120573

TXT1119891] (25)

119901 = (120573TXTX120573)

minus1

[(el1015840

y )TX120573 minus 120573TXT1119902] (26)

119902 =1

119899[(el

1015840

y )T1 minus 120573

TXT1119901] (27)

119906 = (120573TXTX120573)

minus1

[(er1015840

y )TX120573 minus 120573TXT1V] (28)

V =1

119899[(er

1015840

y )T1 minus 120573

TXT1119906] (29)

respectively

International Scholarly Research Notices 7

Equations (21)ndash(29) are recursive solutions for the prob-lem of least square estimation with intuitionistic fuzzy dataTherefore we rewrite the system of equations explicitly in arecursive way as follows

120573119894+1

= ((XTX)minus1

XT[1199081ya + 119908

2ely119887119894 + 119908

3ery119891119894 + 119908

4el1015840

y119901119894

+ 1199085er1015840

y 119906119894 minus 1 (1199082119887119894119889119894+ 1199083119891119894119892119894

+1199084119901119894119902119894+ 1199085119906119894V119894) ] )

times (1199081+ 11990821198872

119894+ 11990831198912

119894+ 11990841199012

119894+ 11990851199062

119894)minus1

119887119894+1

= (120573T119894+1

XTX120573i+1)minus1

[(ely)TX120573i+1 minus 120573

Ti+1X

T1119889119894]

119889119894+1

=1

119899[(ely)

T1 minus 120573

Ti+1X

T1119887119894]

119891119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(ery)TX120573i+1 minus 120573

Ti+1X

T1119892119894]

119892119894+1

=1

119899[(er

1015840

y )T1 minus 120573

Ti+1X

T1119891119894]

119901119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(el1015840

y )TX120573i+1 minus 120573

Ti+1X

T1119902119894]

119902119894+1

=1

119899[(el

1015840

y )T1 minus 120573

Ti+1X

T1119901119894]

119906119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(er1015840

y )TX120573i+1 minus 120573

Ti+1X

T1V119894]

V119894+1

=1

119899[(er

1015840

y )T1 minus 120573

Ti+1X

T1119906119894]

(30)

In order to initiate the recursive process of obtaining theestimators we take some initial values for 119887 119889 119891 119892 119901 119902119906 V and 120573 After several numbers of iterations the valuesof estimators get corrected to a predefined error of toleranceWe denote these values by 119889 119891 119892 119901 119902 V and in orderto differentiate them from the eventually obtained restrictedestimator in the next commutation

In a more general setup if in the linear regression model(17) we consider 119896

1crisp and 119896

2intuitionistic fuzzy input

variables then the dimensions of X and 120573 will be 119899 times (1198961+

51198962+ 1) and (119896

1+ 51198962+ 1) times 1 respectively It may further be

noted that the core of the solutionrsquos structure will remain thesame and we will have similar kind of estimators

Remark If a TIFN 119860 = (119898 120572 120573 1205721015840 1205731015840) degenerate to a trian-

gular fuzzy number 119860 = (119898 120572 120573) then our nonsymmetric

intuitionistic fuzzy weighted linear regression model reducesto nonsymmetric fuzzy linear regression model defined byKumar et al [24]

Next we assume that the regression coefficients aresubjected to the linear restrictions which are given by (18) Itmay be noted that the unrestricted estimator obtained abovein (21) does not satisfy the given restrictions (18) We aimto obtain the restricted estimator which satisfies the givenrestrictions under the regression model (17) For this wepropose to minimize the following score function

119878 (120582120573 119887 119889 119891 119892 119901 119902 119906 V)

= 120593 (120573 119887 119889 119891 119892 119901 119902 119906 V) minus 2120582 (H120573 minus h)

= 1199081((ya)Tya minus 2(ya)TX120573)

+ 120573TXTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

+ 1199082((ely)

Tely minus 2(ely)

TX120573119887 minus 2(ely)

T1119889)

+ 1199083((ery)

Tery minus 2(ery)

TX120573119891 minus 2(ery)

T1119892)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )TX120573119901 minus 2(el

1015840

y )T1119902)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )TX120573119906 minus 2(er

1015840

y )T1V)

+ 2120573TXT1 (119908

2119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V)

+ 119899 (11990821198892+ 11990831198922+ 11990841199022+ 1199085V2)

minus 2120582 (H120573 minus h)

(31)

where 2120582 is the vector of Lagrangersquos Multiplier

Differentiating 119878(120582120573 119887 119889 119891 119892 119901 119902 119906 V) partially withrespect to 120573 and equating it to zero we get

997904rArr minus1199081XTya + XTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

minus 1199082XTely119887 minus 119908

3XTery119891 minus 119908

4XTel

1015840

y119901 minus 1199085XTer

1015840

y 119906

+ XT1 (1199082119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V) minusH1015840120582 = 0

(32)

8 International Scholarly Research Notices

Here we again relabel the computed restricted estimator by Therefore in view of (21) and (32) we get size

997904rArr = ((XTX)minus1

XT[1199081ya + 119908

2ely119887 + 119908

3ery119891 + 119908

4el1015840

y119901

+ 1199085er1015840

y 119906

minus1 (1199082119887119889+119908

3119891119892+119908

4119901119902+119908

5119906V) ] )

times (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)minus1

+(XTX)

minus1

HT120582

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

997904rArr = +1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

times (XTX)minus1

HT120582

(33)

Similarly differentiating 119878(120582120573 119887 119889 119891 119892 119901 119902 119906 V) par-tially with respect to 120582 and equating it to zero we get

997904rArr H = h

997904rArr H + 1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

timesH(XTX)minus1

HT120582 = h

997904rArr = (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

times [H(XTX)minus1

HT]minus1

(h minusH)

(34)

From (33) and (34) we have

997904rArr = + (XTX)minus1

HT[H(XTX)

minus1

HT]minus1

(h minusH) (35)

Also differentiating (31) partially with respect to 119887 119889 119891 119892 119901119902 119906 and V and equating all to zero we get

= 119889 = 119889 119891 = 119891 119892 = 119892

119901 = 119901 119902 = 119902 = V = V(36)

respectively From (35) we see that

997904rArr H = H + [H(XTX)minus1

HT] [H(XTX)

minus1

HT]minus1

times (h minusH)

997904rArr H = H + (h minusH) = h(37)

Therefore the estimator satisfies the given restrictions (18)

4 Numerical Examples

We consider the following numerical examples to illustratethe proposed model

Example 1 We apply our procedure to estimate the intuition-istic fuzzy output value for a data consisting of the crisp inputand intuitionistic fuzzy output (where left entropy and rightentropy are equal) and tabulate the data in Table 1

We obtain = (minus44026 35733 73786 56858)1015840 =

02942 119889 = 147144 119891 = 02942 119892 = 147144 119901 = 02909119902 = 174487 = 02909 and V = 174487 where the numberof iterations required is 125

Example 2 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp input andintuitionistic fuzzy output (where left and right entropy arenot equal) and tabulate the data in Table 2

We obtain = (minus47697 35933 72030 59152)1015840 =

02952 119889 = 145871 119891 = 02646 119892 = 203429 119901 = 03052119902 = 157050 = 02717 and V = 231201 where the numberof iterations required is 113

Example 3 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp inputintuitionistic fuzzy input and intuitionistic fuzzy output(where left and right entropy are not equal) and tabulate thedata in Table 3

We obtain = (minus32352 06811 05314 minus09164 00846

minus31631 2953)1015840 = 04225 119889 = 05478 119891 = 04307 119892 =

01637 119901 = 03231 119902 = 38723 = 04985 and V = 18659

where the number of iterations required is 51

Example 4 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of intuitionistic fuzzyinput and intuitionistic fuzzy output (where left and rightentropy are not equal) and tabulate the data in Table 4

We obtain = (118141 minus02161 16104 minus18254 05687

minus01879)1015840 = 03880 119889 = 03674 119891 = 03880 119892 = 03674

119901 = 03547 119902 = 22108 = 03547 and V = 32108 wherethe number of iterations required is 255

5 Conclusions

An intuitionistic fuzzy weighted linear regression (IFWLR)model with and without some linear restrictions in theform of prior information has been studied The estimators

International Scholarly Research Notices 9

Table 1 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y y

a ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 44 42 96 42 44 449018 424850 943828 424850 4490182 14 8 3 48 47 120 47 48 528505 505256 1217099 505256 5285053 7 1 4 35 33 52 33 35 322052 296416 507324 296416 3220524 11 7 3 50 45 106 45 50 475861 452004 1036114 452004 4758615 7 12 15 80 79 189 79 80 740058 719256 1944413 719256 7400586 8 15 10 68 65 194 65 68 732147 711253 1917213 711253 7321477 3 9 6 45 42 107 42 45 485252 461503 1068398 461503 4852528 12 15 11 80 78 216 78 80 790260 770038 2117003 770038 7902609 10 5 8 55 52 108 52 55 505235 481717 1137100 481717 50523510 9 7 4 45 44 103 44 45 471612 447706 1021507 447706 471612

Table 2 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y ya ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 45 42 96 47 48 447743 427104 952620 455472 4900532 14 8 3 48 47 120 43 45 525995 502809 120905 523320 5597343 7 1 4 35 33 52 50 55 313430 297162 512469 339018 3704524 11 7 3 46 45 106 45 47 471120 449720 102922 475741 5108705 7 12 15 82 79 189 80 85 753765 723166 195547 720805 7625556 8 15 10 70 65 194 60 67 740419 710254 191173 709234 7506717 3 9 6 45 42 107 40 46 481512 459774 106328 484752 5201248 12 15 11 80 78 216 88 90 802328 770149 211461 762912 8057999 10 5 8 55 52 108 50 55 506447 483897 114499 506370 54232710 9 7 4 45 44 103 42 44 467241 445967 101651 472377 507415

Table 3 Crisp and int fuzzy input-int fuzzy output data

Object 119894Crisp and int fuzzy inputX = (x1 e1

1015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy outputylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 6 7 6 10 5 7 6 3 5 2 5 54158 25665 47776 22215 424732 7 6 5 12 4 6 7 5 4 5 7 58827 31771 62226 28438 496763 8 4 2 15 3 5 8 3 9 4 6 67973 43734 90537 40632 637884 9 7 5 20 8 10 7 3 10 2 4 64422 39089 79544 35897 583085 10 8 6 5 2 5 8 5 12 5 7 74885 52774 11193 49846 744516 11 10 8 15 5 7 3 2 8 4 6 65757 40835 83678 37678 603697 12 20 15 25 12 14 6 5 7 3 5 62393 36436 73265 33193 551798 13 12 7 30 15 18 8 7 14 6 8 83421 63937 13835 61226 876209 14 16 12 20 10 15 11 9 16 10 12 97423 82250 18169 79893 10922310 15 17 13 22 8 12 8 7 18 5 10 90740 73509 16100 70982 98912

10 International Scholarly Research Notices

Table 4 Intuitionistic fuzzy input-intuitionistic fuzzy output data

Object iInt fuzzy input

X = (e11015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy output

ylowast = (e1lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

ye1lowast

y yalowast er

lowast

y erlowast1015840

y

1 5 3 4 5 6 5 4 12 4 6 57505 42398 99797 42398 675052 7 6 7 8 9 7 5 7 5 8 57737 42652 10045 42652 677373 5 3 6 8 9 5 3 9 3 6 48608 32665 74714 32665 586084 4 2 7 9 11 3 1 4 1 4 37872 20920 44446 20920 478725 3 2 5 7 8 4 2 6 2 5 49552 33698 77377 33698 595526 6 3 6 7 10 5 4 8 4 6 45158 28891 64987 28891 551587 5 2 4 9 12 4 3 9 3 5 55863 40602 95168 40602 658638 6 5 8 13 15 7 5 10 5 8 52404 36818 85417 36818 624049 8 7 12 15 17 4 3 5 3 5 39099 22262 47905 22262 4909910 15 10 15 20 25 4 2 3 2 5 36202 19092 39736 19092 46202

of regression coefficients have also been obtained with thehelp of fuzzy entropy for the restrictedunrestricted IFWLRmodel by assigning some weights in the distance functionIt has been observed that the restricted estimator is betterthan unrestricted estimator in some sense Thus wheneversome prior information is available in terms of exact linearrestrictions on regression coefficients it is advised to userestricted estimator in place of unrestricted estimator

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] H Tanaka S Uejima and K Asai ldquoFuzzy linear regressionmodelrdquo IEEE Transactions on Systems Man and Cyberneticsvol 10 pp 2933ndash2938 1980

[3] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982

[4] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988

[5] H Tanaka I Hayashi and JWatada ldquoPossibilistic linear regres-sion analysis for fuzzy datardquo European Journal of OperationalResearch vol 40 no 3 pp 389ndash396 1989

[6] H Tanaka and H Ishibuchi ldquoIdentification of possibilisticlinear systems by quadratic membership functions of fuzzyparametersrdquo Fuzzy Sets and Systems vol 41 no 2 pp 145ndash1601991

[7] D T Redden and W H Woodall ldquoProperties of certain fuzzylinear regression methodsrdquo Fuzzy Sets and Systems vol 64 no3 pp 361ndash375 1994

[8] P-T Chang and E S Lee ldquoFuzzy linear regression with spreadsunrestricted in signrdquoComputers andMathematics with Applica-tions vol 28 no 4 pp 61ndash70 1994

[9] G Peters ldquoFuzzy linear regression with fuzzy intervalsrdquo FuzzySets and Systems vol 63 no 1 pp 45ndash55 1994

[10] A de Luca and S Termini ldquoA definition of a nonprobabilisticentropy in the setting of fuzzy sets theoryrdquo Information andControl vol 20 no 4 pp 301ndash312 1972

[11] D Dubosis and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[12] D Dubois and H Prade Fuzzy Sets and Statistical PossibilityTheory Plenum Press New York NY USA 1988

[13] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash656 1948

[14] T Kumar N Gupta and R K Bajaj ldquoFuzzy entropy onrestricted fuzzy linear regression model with cross validationand applicationsrdquo in Proceedings of the International Conferenceon Advances in Computing and Communications (ICACC rsquo12)pp 5ndash8 August 2012

[15] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[16] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[17] K T Atanassov Intuitionistic Fuzzy Sets Theory and Applica-tions vol 35 of Studies in Fuzziness and SoftComputing Physica1999

[18] K T Atanassov ldquoNew operations defined over the intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 61 no 2 pp 137ndash1421994

[19] P Burillo and H Bustince ldquoSome definitions of intuitionisticfuzzy numberrdquo in Proceedings of the 3rd Conference of theEuropean Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[20] F Lee Fuzzy Information Processing System Peking UniversityPress Beijing China 1998

[21] H Liu and K Shi ldquoIntuitionistic fuzzy numbers and intuition-istic distribution numbersrdquo Journal of Fuzzy Mathematics vol8 no 4 pp 909ndash918 2000

[22] P Grzegorzewski ldquoDistances and orderings in a family of intu-itionistic fuzzy numbersrdquo in Proceedings of the 3rd Conference ofthe European Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[23] P DrsquoUrso and T Gastaldi ldquoA least-squares approach to fuzzylinear regression analysisrdquo Computational Statistics and DataAnalysis vol 34 no 4 pp 427ndash440 2000

[24] T Kumar R K Bajaj and N Gupta ldquoFuzzy entropy in fuzzyweighted linear regression model under linear restrictions withsimulation studyrdquo International Journal of General Systems vol43 no 2 pp 135ndash148 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Scholarly Research Notices

= 1199081((ya)Tya minus 2(ya)Tyalowast + (yalowast)

Tyalowast)

+ 1199082((ely)

Tely minus 2(ely)

Telylowast

+ (elylowast

)Telylowast

)

+ 1199083((ery)

Tery minus 2(ery)

Terylowast

+ (erylowast

)Terylowast

)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )Tel1015840

ylowast

+ (el1015840

ylowast

)

Tel1015840

ylowast

)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )Ter1015840

ylowast

+ (er1015840

ylowast

)

Ter1015840

ylowast

)

= 1199081((ya)Tya minus 2(ya)TX120573 + 120573TXTX120573)

+ 1199082((ely)

Tely minus 2(ely)

T(X120573119887 + 1119889))

+ 1199082((X120573119887 + 1119889)T (X120573119887 + 1119889))

+ 1199083((ery)

Tery minus 2(ery)

T(X120573119891 + 1119892))

+ 1199083((X120573119891 + 1119892)T (X120573119891 + 1119892))

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )T(X120573119901 + 1119902))

+ 1199084((X120573119901 + 1119902)T (X120573119901 + 1119902))

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )T(X120573119906 + 1V))

+ 1199085((X120573119906 + 1V)T (X120573119906 + 1V))

= 1199081((ya)Tya minus 2(ya)TX120573)

+ 120573TXTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

+ 1199082((ely)

Tely minus 2(ely)

TX120573119887 minus 2(ely)

T1119889)

+ 1199083((ery)

Tery minus 2(ery)

TX120573119891 minus 2(ery)

T1119892)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )TX120573119901 minus 2(el

1015840

y )T1119902)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )TX120573119906 minus 2(er

1015840

y )T1V)

+ 2120573TXT1 (119908

2119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V)

+ 119899 (11990821198892+ 11990831198922+ 11990841199022+ 1199085V2)

(20)

Differentiating 120593(120573 119887 119889 119891 119892 119901 119902 119906 V) that is (20) partiallywith respect to 120573 and equating it to zero we get

120597120593 (120573 119887 119889 119891 119892 119901 119902 119906 V)120597120573

= 0

997904rArr minus1199081XTya + XTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

minus 1199082XTely119887 minus 119908

3XTery119891 minus 119908

4XTel

1015840

y119901 minus 1199085XTer

1015840

y 119906

+ XT1 (1199082119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V) = 0

997904rArr 120573 = ((XTX)minus1

XT[1199081ya + 119908

2ely119887 + 119908

3ery119891

+ 1199084el1015840

y119901 + 1199085er1015840

y 119906

minus1 (1199082119887119889+119908

3119891119892+119908

4119901119902+119908

5119906V)])

times (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)minus1

(21)

Similarly differentiating (20) partially with respect to 119887 119889 119891119892 119901 119902 119906 and V we get

119887 = (120573TXTX120573)

minus1

[(ely)TX120573 minus 120573TXT1119889] (22)

119889 =1

119899[(ely)

T1 minus 120573

TXT1119887] (23)

119891 = (120573TXTX120573)

minus1

[(ery)TX120573 minus 120573TXT1119892] (24)

119892 =1

119899[(er

1015840

y )T1 minus 120573

TXT1119891] (25)

119901 = (120573TXTX120573)

minus1

[(el1015840

y )TX120573 minus 120573TXT1119902] (26)

119902 =1

119899[(el

1015840

y )T1 minus 120573

TXT1119901] (27)

119906 = (120573TXTX120573)

minus1

[(er1015840

y )TX120573 minus 120573TXT1V] (28)

V =1

119899[(er

1015840

y )T1 minus 120573

TXT1119906] (29)

respectively

International Scholarly Research Notices 7

Equations (21)ndash(29) are recursive solutions for the prob-lem of least square estimation with intuitionistic fuzzy dataTherefore we rewrite the system of equations explicitly in arecursive way as follows

120573119894+1

= ((XTX)minus1

XT[1199081ya + 119908

2ely119887119894 + 119908

3ery119891119894 + 119908

4el1015840

y119901119894

+ 1199085er1015840

y 119906119894 minus 1 (1199082119887119894119889119894+ 1199083119891119894119892119894

+1199084119901119894119902119894+ 1199085119906119894V119894) ] )

times (1199081+ 11990821198872

119894+ 11990831198912

119894+ 11990841199012

119894+ 11990851199062

119894)minus1

119887119894+1

= (120573T119894+1

XTX120573i+1)minus1

[(ely)TX120573i+1 minus 120573

Ti+1X

T1119889119894]

119889119894+1

=1

119899[(ely)

T1 minus 120573

Ti+1X

T1119887119894]

119891119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(ery)TX120573i+1 minus 120573

Ti+1X

T1119892119894]

119892119894+1

=1

119899[(er

1015840

y )T1 minus 120573

Ti+1X

T1119891119894]

119901119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(el1015840

y )TX120573i+1 minus 120573

Ti+1X

T1119902119894]

119902119894+1

=1

119899[(el

1015840

y )T1 minus 120573

Ti+1X

T1119901119894]

119906119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(er1015840

y )TX120573i+1 minus 120573

Ti+1X

T1V119894]

V119894+1

=1

119899[(er

1015840

y )T1 minus 120573

Ti+1X

T1119906119894]

(30)

In order to initiate the recursive process of obtaining theestimators we take some initial values for 119887 119889 119891 119892 119901 119902119906 V and 120573 After several numbers of iterations the valuesof estimators get corrected to a predefined error of toleranceWe denote these values by 119889 119891 119892 119901 119902 V and in orderto differentiate them from the eventually obtained restrictedestimator in the next commutation

In a more general setup if in the linear regression model(17) we consider 119896

1crisp and 119896

2intuitionistic fuzzy input

variables then the dimensions of X and 120573 will be 119899 times (1198961+

51198962+ 1) and (119896

1+ 51198962+ 1) times 1 respectively It may further be

noted that the core of the solutionrsquos structure will remain thesame and we will have similar kind of estimators

Remark If a TIFN 119860 = (119898 120572 120573 1205721015840 1205731015840) degenerate to a trian-

gular fuzzy number 119860 = (119898 120572 120573) then our nonsymmetric

intuitionistic fuzzy weighted linear regression model reducesto nonsymmetric fuzzy linear regression model defined byKumar et al [24]

Next we assume that the regression coefficients aresubjected to the linear restrictions which are given by (18) Itmay be noted that the unrestricted estimator obtained abovein (21) does not satisfy the given restrictions (18) We aimto obtain the restricted estimator which satisfies the givenrestrictions under the regression model (17) For this wepropose to minimize the following score function

119878 (120582120573 119887 119889 119891 119892 119901 119902 119906 V)

= 120593 (120573 119887 119889 119891 119892 119901 119902 119906 V) minus 2120582 (H120573 minus h)

= 1199081((ya)Tya minus 2(ya)TX120573)

+ 120573TXTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

+ 1199082((ely)

Tely minus 2(ely)

TX120573119887 minus 2(ely)

T1119889)

+ 1199083((ery)

Tery minus 2(ery)

TX120573119891 minus 2(ery)

T1119892)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )TX120573119901 minus 2(el

1015840

y )T1119902)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )TX120573119906 minus 2(er

1015840

y )T1V)

+ 2120573TXT1 (119908

2119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V)

+ 119899 (11990821198892+ 11990831198922+ 11990841199022+ 1199085V2)

minus 2120582 (H120573 minus h)

(31)

where 2120582 is the vector of Lagrangersquos Multiplier

Differentiating 119878(120582120573 119887 119889 119891 119892 119901 119902 119906 V) partially withrespect to 120573 and equating it to zero we get

997904rArr minus1199081XTya + XTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

minus 1199082XTely119887 minus 119908

3XTery119891 minus 119908

4XTel

1015840

y119901 minus 1199085XTer

1015840

y 119906

+ XT1 (1199082119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V) minusH1015840120582 = 0

(32)

8 International Scholarly Research Notices

Here we again relabel the computed restricted estimator by Therefore in view of (21) and (32) we get size

997904rArr = ((XTX)minus1

XT[1199081ya + 119908

2ely119887 + 119908

3ery119891 + 119908

4el1015840

y119901

+ 1199085er1015840

y 119906

minus1 (1199082119887119889+119908

3119891119892+119908

4119901119902+119908

5119906V) ] )

times (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)minus1

+(XTX)

minus1

HT120582

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

997904rArr = +1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

times (XTX)minus1

HT120582

(33)

Similarly differentiating 119878(120582120573 119887 119889 119891 119892 119901 119902 119906 V) par-tially with respect to 120582 and equating it to zero we get

997904rArr H = h

997904rArr H + 1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

timesH(XTX)minus1

HT120582 = h

997904rArr = (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

times [H(XTX)minus1

HT]minus1

(h minusH)

(34)

From (33) and (34) we have

997904rArr = + (XTX)minus1

HT[H(XTX)

minus1

HT]minus1

(h minusH) (35)

Also differentiating (31) partially with respect to 119887 119889 119891 119892 119901119902 119906 and V and equating all to zero we get

= 119889 = 119889 119891 = 119891 119892 = 119892

119901 = 119901 119902 = 119902 = V = V(36)

respectively From (35) we see that

997904rArr H = H + [H(XTX)minus1

HT] [H(XTX)

minus1

HT]minus1

times (h minusH)

997904rArr H = H + (h minusH) = h(37)

Therefore the estimator satisfies the given restrictions (18)

4 Numerical Examples

We consider the following numerical examples to illustratethe proposed model

Example 1 We apply our procedure to estimate the intuition-istic fuzzy output value for a data consisting of the crisp inputand intuitionistic fuzzy output (where left entropy and rightentropy are equal) and tabulate the data in Table 1

We obtain = (minus44026 35733 73786 56858)1015840 =

02942 119889 = 147144 119891 = 02942 119892 = 147144 119901 = 02909119902 = 174487 = 02909 and V = 174487 where the numberof iterations required is 125

Example 2 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp input andintuitionistic fuzzy output (where left and right entropy arenot equal) and tabulate the data in Table 2

We obtain = (minus47697 35933 72030 59152)1015840 =

02952 119889 = 145871 119891 = 02646 119892 = 203429 119901 = 03052119902 = 157050 = 02717 and V = 231201 where the numberof iterations required is 113

Example 3 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp inputintuitionistic fuzzy input and intuitionistic fuzzy output(where left and right entropy are not equal) and tabulate thedata in Table 3

We obtain = (minus32352 06811 05314 minus09164 00846

minus31631 2953)1015840 = 04225 119889 = 05478 119891 = 04307 119892 =

01637 119901 = 03231 119902 = 38723 = 04985 and V = 18659

where the number of iterations required is 51

Example 4 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of intuitionistic fuzzyinput and intuitionistic fuzzy output (where left and rightentropy are not equal) and tabulate the data in Table 4

We obtain = (118141 minus02161 16104 minus18254 05687

minus01879)1015840 = 03880 119889 = 03674 119891 = 03880 119892 = 03674

119901 = 03547 119902 = 22108 = 03547 and V = 32108 wherethe number of iterations required is 255

5 Conclusions

An intuitionistic fuzzy weighted linear regression (IFWLR)model with and without some linear restrictions in theform of prior information has been studied The estimators

International Scholarly Research Notices 9

Table 1 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y y

a ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 44 42 96 42 44 449018 424850 943828 424850 4490182 14 8 3 48 47 120 47 48 528505 505256 1217099 505256 5285053 7 1 4 35 33 52 33 35 322052 296416 507324 296416 3220524 11 7 3 50 45 106 45 50 475861 452004 1036114 452004 4758615 7 12 15 80 79 189 79 80 740058 719256 1944413 719256 7400586 8 15 10 68 65 194 65 68 732147 711253 1917213 711253 7321477 3 9 6 45 42 107 42 45 485252 461503 1068398 461503 4852528 12 15 11 80 78 216 78 80 790260 770038 2117003 770038 7902609 10 5 8 55 52 108 52 55 505235 481717 1137100 481717 50523510 9 7 4 45 44 103 44 45 471612 447706 1021507 447706 471612

Table 2 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y ya ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 45 42 96 47 48 447743 427104 952620 455472 4900532 14 8 3 48 47 120 43 45 525995 502809 120905 523320 5597343 7 1 4 35 33 52 50 55 313430 297162 512469 339018 3704524 11 7 3 46 45 106 45 47 471120 449720 102922 475741 5108705 7 12 15 82 79 189 80 85 753765 723166 195547 720805 7625556 8 15 10 70 65 194 60 67 740419 710254 191173 709234 7506717 3 9 6 45 42 107 40 46 481512 459774 106328 484752 5201248 12 15 11 80 78 216 88 90 802328 770149 211461 762912 8057999 10 5 8 55 52 108 50 55 506447 483897 114499 506370 54232710 9 7 4 45 44 103 42 44 467241 445967 101651 472377 507415

Table 3 Crisp and int fuzzy input-int fuzzy output data

Object 119894Crisp and int fuzzy inputX = (x1 e1

1015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy outputylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 6 7 6 10 5 7 6 3 5 2 5 54158 25665 47776 22215 424732 7 6 5 12 4 6 7 5 4 5 7 58827 31771 62226 28438 496763 8 4 2 15 3 5 8 3 9 4 6 67973 43734 90537 40632 637884 9 7 5 20 8 10 7 3 10 2 4 64422 39089 79544 35897 583085 10 8 6 5 2 5 8 5 12 5 7 74885 52774 11193 49846 744516 11 10 8 15 5 7 3 2 8 4 6 65757 40835 83678 37678 603697 12 20 15 25 12 14 6 5 7 3 5 62393 36436 73265 33193 551798 13 12 7 30 15 18 8 7 14 6 8 83421 63937 13835 61226 876209 14 16 12 20 10 15 11 9 16 10 12 97423 82250 18169 79893 10922310 15 17 13 22 8 12 8 7 18 5 10 90740 73509 16100 70982 98912

10 International Scholarly Research Notices

Table 4 Intuitionistic fuzzy input-intuitionistic fuzzy output data

Object iInt fuzzy input

X = (e11015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy output

ylowast = (e1lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

ye1lowast

y yalowast er

lowast

y erlowast1015840

y

1 5 3 4 5 6 5 4 12 4 6 57505 42398 99797 42398 675052 7 6 7 8 9 7 5 7 5 8 57737 42652 10045 42652 677373 5 3 6 8 9 5 3 9 3 6 48608 32665 74714 32665 586084 4 2 7 9 11 3 1 4 1 4 37872 20920 44446 20920 478725 3 2 5 7 8 4 2 6 2 5 49552 33698 77377 33698 595526 6 3 6 7 10 5 4 8 4 6 45158 28891 64987 28891 551587 5 2 4 9 12 4 3 9 3 5 55863 40602 95168 40602 658638 6 5 8 13 15 7 5 10 5 8 52404 36818 85417 36818 624049 8 7 12 15 17 4 3 5 3 5 39099 22262 47905 22262 4909910 15 10 15 20 25 4 2 3 2 5 36202 19092 39736 19092 46202

of regression coefficients have also been obtained with thehelp of fuzzy entropy for the restrictedunrestricted IFWLRmodel by assigning some weights in the distance functionIt has been observed that the restricted estimator is betterthan unrestricted estimator in some sense Thus wheneversome prior information is available in terms of exact linearrestrictions on regression coefficients it is advised to userestricted estimator in place of unrestricted estimator

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] H Tanaka S Uejima and K Asai ldquoFuzzy linear regressionmodelrdquo IEEE Transactions on Systems Man and Cyberneticsvol 10 pp 2933ndash2938 1980

[3] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982

[4] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988

[5] H Tanaka I Hayashi and JWatada ldquoPossibilistic linear regres-sion analysis for fuzzy datardquo European Journal of OperationalResearch vol 40 no 3 pp 389ndash396 1989

[6] H Tanaka and H Ishibuchi ldquoIdentification of possibilisticlinear systems by quadratic membership functions of fuzzyparametersrdquo Fuzzy Sets and Systems vol 41 no 2 pp 145ndash1601991

[7] D T Redden and W H Woodall ldquoProperties of certain fuzzylinear regression methodsrdquo Fuzzy Sets and Systems vol 64 no3 pp 361ndash375 1994

[8] P-T Chang and E S Lee ldquoFuzzy linear regression with spreadsunrestricted in signrdquoComputers andMathematics with Applica-tions vol 28 no 4 pp 61ndash70 1994

[9] G Peters ldquoFuzzy linear regression with fuzzy intervalsrdquo FuzzySets and Systems vol 63 no 1 pp 45ndash55 1994

[10] A de Luca and S Termini ldquoA definition of a nonprobabilisticentropy in the setting of fuzzy sets theoryrdquo Information andControl vol 20 no 4 pp 301ndash312 1972

[11] D Dubosis and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[12] D Dubois and H Prade Fuzzy Sets and Statistical PossibilityTheory Plenum Press New York NY USA 1988

[13] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash656 1948

[14] T Kumar N Gupta and R K Bajaj ldquoFuzzy entropy onrestricted fuzzy linear regression model with cross validationand applicationsrdquo in Proceedings of the International Conferenceon Advances in Computing and Communications (ICACC rsquo12)pp 5ndash8 August 2012

[15] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[16] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[17] K T Atanassov Intuitionistic Fuzzy Sets Theory and Applica-tions vol 35 of Studies in Fuzziness and SoftComputing Physica1999

[18] K T Atanassov ldquoNew operations defined over the intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 61 no 2 pp 137ndash1421994

[19] P Burillo and H Bustince ldquoSome definitions of intuitionisticfuzzy numberrdquo in Proceedings of the 3rd Conference of theEuropean Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[20] F Lee Fuzzy Information Processing System Peking UniversityPress Beijing China 1998

[21] H Liu and K Shi ldquoIntuitionistic fuzzy numbers and intuition-istic distribution numbersrdquo Journal of Fuzzy Mathematics vol8 no 4 pp 909ndash918 2000

[22] P Grzegorzewski ldquoDistances and orderings in a family of intu-itionistic fuzzy numbersrdquo in Proceedings of the 3rd Conference ofthe European Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[23] P DrsquoUrso and T Gastaldi ldquoA least-squares approach to fuzzylinear regression analysisrdquo Computational Statistics and DataAnalysis vol 34 no 4 pp 427ndash440 2000

[24] T Kumar R K Bajaj and N Gupta ldquoFuzzy entropy in fuzzyweighted linear regression model under linear restrictions withsimulation studyrdquo International Journal of General Systems vol43 no 2 pp 135ndash148 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 7

Equations (21)ndash(29) are recursive solutions for the prob-lem of least square estimation with intuitionistic fuzzy dataTherefore we rewrite the system of equations explicitly in arecursive way as follows

120573119894+1

= ((XTX)minus1

XT[1199081ya + 119908

2ely119887119894 + 119908

3ery119891119894 + 119908

4el1015840

y119901119894

+ 1199085er1015840

y 119906119894 minus 1 (1199082119887119894119889119894+ 1199083119891119894119892119894

+1199084119901119894119902119894+ 1199085119906119894V119894) ] )

times (1199081+ 11990821198872

119894+ 11990831198912

119894+ 11990841199012

119894+ 11990851199062

119894)minus1

119887119894+1

= (120573T119894+1

XTX120573i+1)minus1

[(ely)TX120573i+1 minus 120573

Ti+1X

T1119889119894]

119889119894+1

=1

119899[(ely)

T1 minus 120573

Ti+1X

T1119887119894]

119891119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(ery)TX120573i+1 minus 120573

Ti+1X

T1119892119894]

119892119894+1

=1

119899[(er

1015840

y )T1 minus 120573

Ti+1X

T1119891119894]

119901119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(el1015840

y )TX120573i+1 minus 120573

Ti+1X

T1119902119894]

119902119894+1

=1

119899[(el

1015840

y )T1 minus 120573

Ti+1X

T1119901119894]

119906119894+1

= (120573Ti+1X

TX120573i+1)minus1

[(er1015840

y )TX120573i+1 minus 120573

Ti+1X

T1V119894]

V119894+1

=1

119899[(er

1015840

y )T1 minus 120573

Ti+1X

T1119906119894]

(30)

In order to initiate the recursive process of obtaining theestimators we take some initial values for 119887 119889 119891 119892 119901 119902119906 V and 120573 After several numbers of iterations the valuesof estimators get corrected to a predefined error of toleranceWe denote these values by 119889 119891 119892 119901 119902 V and in orderto differentiate them from the eventually obtained restrictedestimator in the next commutation

In a more general setup if in the linear regression model(17) we consider 119896

1crisp and 119896

2intuitionistic fuzzy input

variables then the dimensions of X and 120573 will be 119899 times (1198961+

51198962+ 1) and (119896

1+ 51198962+ 1) times 1 respectively It may further be

noted that the core of the solutionrsquos structure will remain thesame and we will have similar kind of estimators

Remark If a TIFN 119860 = (119898 120572 120573 1205721015840 1205731015840) degenerate to a trian-

gular fuzzy number 119860 = (119898 120572 120573) then our nonsymmetric

intuitionistic fuzzy weighted linear regression model reducesto nonsymmetric fuzzy linear regression model defined byKumar et al [24]

Next we assume that the regression coefficients aresubjected to the linear restrictions which are given by (18) Itmay be noted that the unrestricted estimator obtained abovein (21) does not satisfy the given restrictions (18) We aimto obtain the restricted estimator which satisfies the givenrestrictions under the regression model (17) For this wepropose to minimize the following score function

119878 (120582120573 119887 119889 119891 119892 119901 119902 119906 V)

= 120593 (120573 119887 119889 119891 119892 119901 119902 119906 V) minus 2120582 (H120573 minus h)

= 1199081((ya)Tya minus 2(ya)TX120573)

+ 120573TXTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

+ 1199082((ely)

Tely minus 2(ely)

TX120573119887 minus 2(ely)

T1119889)

+ 1199083((ery)

Tery minus 2(ery)

TX120573119891 minus 2(ery)

T1119892)

+ 1199084((el

1015840

y )Tel1015840

y minus 2(el1015840

y )TX120573119901 minus 2(el

1015840

y )T1119902)

+ 1199085((er

1015840

y )Ter1015840

y minus 2(er1015840

y )TX120573119906 minus 2(er

1015840

y )T1V)

+ 2120573TXT1 (119908

2119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V)

+ 119899 (11990821198892+ 11990831198922+ 11990841199022+ 1199085V2)

minus 2120582 (H120573 minus h)

(31)

where 2120582 is the vector of Lagrangersquos Multiplier

Differentiating 119878(120582120573 119887 119889 119891 119892 119901 119902 119906 V) partially withrespect to 120573 and equating it to zero we get

997904rArr minus1199081XTya + XTX120573 (119908

1+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

minus 1199082XTely119887 minus 119908

3XTery119891 minus 119908

4XTel

1015840

y119901 minus 1199085XTer

1015840

y 119906

+ XT1 (1199082119887119889 + 119908

3119891119892 + 119908

4119901119902 + 119908

5119906V) minusH1015840120582 = 0

(32)

8 International Scholarly Research Notices

Here we again relabel the computed restricted estimator by Therefore in view of (21) and (32) we get size

997904rArr = ((XTX)minus1

XT[1199081ya + 119908

2ely119887 + 119908

3ery119891 + 119908

4el1015840

y119901

+ 1199085er1015840

y 119906

minus1 (1199082119887119889+119908

3119891119892+119908

4119901119902+119908

5119906V) ] )

times (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)minus1

+(XTX)

minus1

HT120582

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

997904rArr = +1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

times (XTX)minus1

HT120582

(33)

Similarly differentiating 119878(120582120573 119887 119889 119891 119892 119901 119902 119906 V) par-tially with respect to 120582 and equating it to zero we get

997904rArr H = h

997904rArr H + 1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

timesH(XTX)minus1

HT120582 = h

997904rArr = (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

times [H(XTX)minus1

HT]minus1

(h minusH)

(34)

From (33) and (34) we have

997904rArr = + (XTX)minus1

HT[H(XTX)

minus1

HT]minus1

(h minusH) (35)

Also differentiating (31) partially with respect to 119887 119889 119891 119892 119901119902 119906 and V and equating all to zero we get

= 119889 = 119889 119891 = 119891 119892 = 119892

119901 = 119901 119902 = 119902 = V = V(36)

respectively From (35) we see that

997904rArr H = H + [H(XTX)minus1

HT] [H(XTX)

minus1

HT]minus1

times (h minusH)

997904rArr H = H + (h minusH) = h(37)

Therefore the estimator satisfies the given restrictions (18)

4 Numerical Examples

We consider the following numerical examples to illustratethe proposed model

Example 1 We apply our procedure to estimate the intuition-istic fuzzy output value for a data consisting of the crisp inputand intuitionistic fuzzy output (where left entropy and rightentropy are equal) and tabulate the data in Table 1

We obtain = (minus44026 35733 73786 56858)1015840 =

02942 119889 = 147144 119891 = 02942 119892 = 147144 119901 = 02909119902 = 174487 = 02909 and V = 174487 where the numberof iterations required is 125

Example 2 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp input andintuitionistic fuzzy output (where left and right entropy arenot equal) and tabulate the data in Table 2

We obtain = (minus47697 35933 72030 59152)1015840 =

02952 119889 = 145871 119891 = 02646 119892 = 203429 119901 = 03052119902 = 157050 = 02717 and V = 231201 where the numberof iterations required is 113

Example 3 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp inputintuitionistic fuzzy input and intuitionistic fuzzy output(where left and right entropy are not equal) and tabulate thedata in Table 3

We obtain = (minus32352 06811 05314 minus09164 00846

minus31631 2953)1015840 = 04225 119889 = 05478 119891 = 04307 119892 =

01637 119901 = 03231 119902 = 38723 = 04985 and V = 18659

where the number of iterations required is 51

Example 4 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of intuitionistic fuzzyinput and intuitionistic fuzzy output (where left and rightentropy are not equal) and tabulate the data in Table 4

We obtain = (118141 minus02161 16104 minus18254 05687

minus01879)1015840 = 03880 119889 = 03674 119891 = 03880 119892 = 03674

119901 = 03547 119902 = 22108 = 03547 and V = 32108 wherethe number of iterations required is 255

5 Conclusions

An intuitionistic fuzzy weighted linear regression (IFWLR)model with and without some linear restrictions in theform of prior information has been studied The estimators

International Scholarly Research Notices 9

Table 1 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y y

a ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 44 42 96 42 44 449018 424850 943828 424850 4490182 14 8 3 48 47 120 47 48 528505 505256 1217099 505256 5285053 7 1 4 35 33 52 33 35 322052 296416 507324 296416 3220524 11 7 3 50 45 106 45 50 475861 452004 1036114 452004 4758615 7 12 15 80 79 189 79 80 740058 719256 1944413 719256 7400586 8 15 10 68 65 194 65 68 732147 711253 1917213 711253 7321477 3 9 6 45 42 107 42 45 485252 461503 1068398 461503 4852528 12 15 11 80 78 216 78 80 790260 770038 2117003 770038 7902609 10 5 8 55 52 108 52 55 505235 481717 1137100 481717 50523510 9 7 4 45 44 103 44 45 471612 447706 1021507 447706 471612

Table 2 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y ya ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 45 42 96 47 48 447743 427104 952620 455472 4900532 14 8 3 48 47 120 43 45 525995 502809 120905 523320 5597343 7 1 4 35 33 52 50 55 313430 297162 512469 339018 3704524 11 7 3 46 45 106 45 47 471120 449720 102922 475741 5108705 7 12 15 82 79 189 80 85 753765 723166 195547 720805 7625556 8 15 10 70 65 194 60 67 740419 710254 191173 709234 7506717 3 9 6 45 42 107 40 46 481512 459774 106328 484752 5201248 12 15 11 80 78 216 88 90 802328 770149 211461 762912 8057999 10 5 8 55 52 108 50 55 506447 483897 114499 506370 54232710 9 7 4 45 44 103 42 44 467241 445967 101651 472377 507415

Table 3 Crisp and int fuzzy input-int fuzzy output data

Object 119894Crisp and int fuzzy inputX = (x1 e1

1015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy outputylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 6 7 6 10 5 7 6 3 5 2 5 54158 25665 47776 22215 424732 7 6 5 12 4 6 7 5 4 5 7 58827 31771 62226 28438 496763 8 4 2 15 3 5 8 3 9 4 6 67973 43734 90537 40632 637884 9 7 5 20 8 10 7 3 10 2 4 64422 39089 79544 35897 583085 10 8 6 5 2 5 8 5 12 5 7 74885 52774 11193 49846 744516 11 10 8 15 5 7 3 2 8 4 6 65757 40835 83678 37678 603697 12 20 15 25 12 14 6 5 7 3 5 62393 36436 73265 33193 551798 13 12 7 30 15 18 8 7 14 6 8 83421 63937 13835 61226 876209 14 16 12 20 10 15 11 9 16 10 12 97423 82250 18169 79893 10922310 15 17 13 22 8 12 8 7 18 5 10 90740 73509 16100 70982 98912

10 International Scholarly Research Notices

Table 4 Intuitionistic fuzzy input-intuitionistic fuzzy output data

Object iInt fuzzy input

X = (e11015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy output

ylowast = (e1lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

ye1lowast

y yalowast er

lowast

y erlowast1015840

y

1 5 3 4 5 6 5 4 12 4 6 57505 42398 99797 42398 675052 7 6 7 8 9 7 5 7 5 8 57737 42652 10045 42652 677373 5 3 6 8 9 5 3 9 3 6 48608 32665 74714 32665 586084 4 2 7 9 11 3 1 4 1 4 37872 20920 44446 20920 478725 3 2 5 7 8 4 2 6 2 5 49552 33698 77377 33698 595526 6 3 6 7 10 5 4 8 4 6 45158 28891 64987 28891 551587 5 2 4 9 12 4 3 9 3 5 55863 40602 95168 40602 658638 6 5 8 13 15 7 5 10 5 8 52404 36818 85417 36818 624049 8 7 12 15 17 4 3 5 3 5 39099 22262 47905 22262 4909910 15 10 15 20 25 4 2 3 2 5 36202 19092 39736 19092 46202

of regression coefficients have also been obtained with thehelp of fuzzy entropy for the restrictedunrestricted IFWLRmodel by assigning some weights in the distance functionIt has been observed that the restricted estimator is betterthan unrestricted estimator in some sense Thus wheneversome prior information is available in terms of exact linearrestrictions on regression coefficients it is advised to userestricted estimator in place of unrestricted estimator

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] H Tanaka S Uejima and K Asai ldquoFuzzy linear regressionmodelrdquo IEEE Transactions on Systems Man and Cyberneticsvol 10 pp 2933ndash2938 1980

[3] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982

[4] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988

[5] H Tanaka I Hayashi and JWatada ldquoPossibilistic linear regres-sion analysis for fuzzy datardquo European Journal of OperationalResearch vol 40 no 3 pp 389ndash396 1989

[6] H Tanaka and H Ishibuchi ldquoIdentification of possibilisticlinear systems by quadratic membership functions of fuzzyparametersrdquo Fuzzy Sets and Systems vol 41 no 2 pp 145ndash1601991

[7] D T Redden and W H Woodall ldquoProperties of certain fuzzylinear regression methodsrdquo Fuzzy Sets and Systems vol 64 no3 pp 361ndash375 1994

[8] P-T Chang and E S Lee ldquoFuzzy linear regression with spreadsunrestricted in signrdquoComputers andMathematics with Applica-tions vol 28 no 4 pp 61ndash70 1994

[9] G Peters ldquoFuzzy linear regression with fuzzy intervalsrdquo FuzzySets and Systems vol 63 no 1 pp 45ndash55 1994

[10] A de Luca and S Termini ldquoA definition of a nonprobabilisticentropy in the setting of fuzzy sets theoryrdquo Information andControl vol 20 no 4 pp 301ndash312 1972

[11] D Dubosis and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[12] D Dubois and H Prade Fuzzy Sets and Statistical PossibilityTheory Plenum Press New York NY USA 1988

[13] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash656 1948

[14] T Kumar N Gupta and R K Bajaj ldquoFuzzy entropy onrestricted fuzzy linear regression model with cross validationand applicationsrdquo in Proceedings of the International Conferenceon Advances in Computing and Communications (ICACC rsquo12)pp 5ndash8 August 2012

[15] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[16] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[17] K T Atanassov Intuitionistic Fuzzy Sets Theory and Applica-tions vol 35 of Studies in Fuzziness and SoftComputing Physica1999

[18] K T Atanassov ldquoNew operations defined over the intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 61 no 2 pp 137ndash1421994

[19] P Burillo and H Bustince ldquoSome definitions of intuitionisticfuzzy numberrdquo in Proceedings of the 3rd Conference of theEuropean Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[20] F Lee Fuzzy Information Processing System Peking UniversityPress Beijing China 1998

[21] H Liu and K Shi ldquoIntuitionistic fuzzy numbers and intuition-istic distribution numbersrdquo Journal of Fuzzy Mathematics vol8 no 4 pp 909ndash918 2000

[22] P Grzegorzewski ldquoDistances and orderings in a family of intu-itionistic fuzzy numbersrdquo in Proceedings of the 3rd Conference ofthe European Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[23] P DrsquoUrso and T Gastaldi ldquoA least-squares approach to fuzzylinear regression analysisrdquo Computational Statistics and DataAnalysis vol 34 no 4 pp 427ndash440 2000

[24] T Kumar R K Bajaj and N Gupta ldquoFuzzy entropy in fuzzyweighted linear regression model under linear restrictions withsimulation studyrdquo International Journal of General Systems vol43 no 2 pp 135ndash148 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Scholarly Research Notices

Here we again relabel the computed restricted estimator by Therefore in view of (21) and (32) we get size

997904rArr = ((XTX)minus1

XT[1199081ya + 119908

2ely119887 + 119908

3ery119891 + 119908

4el1015840

y119901

+ 1199085er1015840

y 119906

minus1 (1199082119887119889+119908

3119891119892+119908

4119901119902+119908

5119906V) ] )

times (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)minus1

+(XTX)

minus1

HT120582

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

997904rArr = +1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

times (XTX)minus1

HT120582

(33)

Similarly differentiating 119878(120582120573 119887 119889 119891 119892 119901 119902 119906 V) par-tially with respect to 120582 and equating it to zero we get

997904rArr H = h

997904rArr H + 1

(1199081+ 11990821198872 + 119908

31198912 + 119908

41199012 + 119908

51199062)

timesH(XTX)minus1

HT120582 = h

997904rArr = (1199081+ 11990821198872+ 11990831198912+ 11990841199012+ 11990851199062)

times [H(XTX)minus1

HT]minus1

(h minusH)

(34)

From (33) and (34) we have

997904rArr = + (XTX)minus1

HT[H(XTX)

minus1

HT]minus1

(h minusH) (35)

Also differentiating (31) partially with respect to 119887 119889 119891 119892 119901119902 119906 and V and equating all to zero we get

= 119889 = 119889 119891 = 119891 119892 = 119892

119901 = 119901 119902 = 119902 = V = V(36)

respectively From (35) we see that

997904rArr H = H + [H(XTX)minus1

HT] [H(XTX)

minus1

HT]minus1

times (h minusH)

997904rArr H = H + (h minusH) = h(37)

Therefore the estimator satisfies the given restrictions (18)

4 Numerical Examples

We consider the following numerical examples to illustratethe proposed model

Example 1 We apply our procedure to estimate the intuition-istic fuzzy output value for a data consisting of the crisp inputand intuitionistic fuzzy output (where left entropy and rightentropy are equal) and tabulate the data in Table 1

We obtain = (minus44026 35733 73786 56858)1015840 =

02942 119889 = 147144 119891 = 02942 119892 = 147144 119901 = 02909119902 = 174487 = 02909 and V = 174487 where the numberof iterations required is 125

Example 2 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp input andintuitionistic fuzzy output (where left and right entropy arenot equal) and tabulate the data in Table 2

We obtain = (minus47697 35933 72030 59152)1015840 =

02952 119889 = 145871 119891 = 02646 119892 = 203429 119901 = 03052119902 = 157050 = 02717 and V = 231201 where the numberof iterations required is 113

Example 3 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of crisp inputintuitionistic fuzzy input and intuitionistic fuzzy output(where left and right entropy are not equal) and tabulate thedata in Table 3

We obtain = (minus32352 06811 05314 minus09164 00846

minus31631 2953)1015840 = 04225 119889 = 05478 119891 = 04307 119892 =

01637 119901 = 03231 119902 = 38723 = 04985 and V = 18659

where the number of iterations required is 51

Example 4 We apply our procedure to estimate intuitionisticfuzzy output value for a data consisting of intuitionistic fuzzyinput and intuitionistic fuzzy output (where left and rightentropy are not equal) and tabulate the data in Table 4

We obtain = (118141 minus02161 16104 minus18254 05687

minus01879)1015840 = 03880 119889 = 03674 119891 = 03880 119892 = 03674

119901 = 03547 119902 = 22108 = 03547 and V = 32108 wherethe number of iterations required is 255

5 Conclusions

An intuitionistic fuzzy weighted linear regression (IFWLR)model with and without some linear restrictions in theform of prior information has been studied The estimators

International Scholarly Research Notices 9

Table 1 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y y

a ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 44 42 96 42 44 449018 424850 943828 424850 4490182 14 8 3 48 47 120 47 48 528505 505256 1217099 505256 5285053 7 1 4 35 33 52 33 35 322052 296416 507324 296416 3220524 11 7 3 50 45 106 45 50 475861 452004 1036114 452004 4758615 7 12 15 80 79 189 79 80 740058 719256 1944413 719256 7400586 8 15 10 68 65 194 65 68 732147 711253 1917213 711253 7321477 3 9 6 45 42 107 42 45 485252 461503 1068398 461503 4852528 12 15 11 80 78 216 78 80 790260 770038 2117003 770038 7902609 10 5 8 55 52 108 52 55 505235 481717 1137100 481717 50523510 9 7 4 45 44 103 44 45 471612 447706 1021507 447706 471612

Table 2 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y ya ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 45 42 96 47 48 447743 427104 952620 455472 4900532 14 8 3 48 47 120 43 45 525995 502809 120905 523320 5597343 7 1 4 35 33 52 50 55 313430 297162 512469 339018 3704524 11 7 3 46 45 106 45 47 471120 449720 102922 475741 5108705 7 12 15 82 79 189 80 85 753765 723166 195547 720805 7625556 8 15 10 70 65 194 60 67 740419 710254 191173 709234 7506717 3 9 6 45 42 107 40 46 481512 459774 106328 484752 5201248 12 15 11 80 78 216 88 90 802328 770149 211461 762912 8057999 10 5 8 55 52 108 50 55 506447 483897 114499 506370 54232710 9 7 4 45 44 103 42 44 467241 445967 101651 472377 507415

Table 3 Crisp and int fuzzy input-int fuzzy output data

Object 119894Crisp and int fuzzy inputX = (x1 e1

1015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy outputylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 6 7 6 10 5 7 6 3 5 2 5 54158 25665 47776 22215 424732 7 6 5 12 4 6 7 5 4 5 7 58827 31771 62226 28438 496763 8 4 2 15 3 5 8 3 9 4 6 67973 43734 90537 40632 637884 9 7 5 20 8 10 7 3 10 2 4 64422 39089 79544 35897 583085 10 8 6 5 2 5 8 5 12 5 7 74885 52774 11193 49846 744516 11 10 8 15 5 7 3 2 8 4 6 65757 40835 83678 37678 603697 12 20 15 25 12 14 6 5 7 3 5 62393 36436 73265 33193 551798 13 12 7 30 15 18 8 7 14 6 8 83421 63937 13835 61226 876209 14 16 12 20 10 15 11 9 16 10 12 97423 82250 18169 79893 10922310 15 17 13 22 8 12 8 7 18 5 10 90740 73509 16100 70982 98912

10 International Scholarly Research Notices

Table 4 Intuitionistic fuzzy input-intuitionistic fuzzy output data

Object iInt fuzzy input

X = (e11015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy output

ylowast = (e1lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

ye1lowast

y yalowast er

lowast

y erlowast1015840

y

1 5 3 4 5 6 5 4 12 4 6 57505 42398 99797 42398 675052 7 6 7 8 9 7 5 7 5 8 57737 42652 10045 42652 677373 5 3 6 8 9 5 3 9 3 6 48608 32665 74714 32665 586084 4 2 7 9 11 3 1 4 1 4 37872 20920 44446 20920 478725 3 2 5 7 8 4 2 6 2 5 49552 33698 77377 33698 595526 6 3 6 7 10 5 4 8 4 6 45158 28891 64987 28891 551587 5 2 4 9 12 4 3 9 3 5 55863 40602 95168 40602 658638 6 5 8 13 15 7 5 10 5 8 52404 36818 85417 36818 624049 8 7 12 15 17 4 3 5 3 5 39099 22262 47905 22262 4909910 15 10 15 20 25 4 2 3 2 5 36202 19092 39736 19092 46202

of regression coefficients have also been obtained with thehelp of fuzzy entropy for the restrictedunrestricted IFWLRmodel by assigning some weights in the distance functionIt has been observed that the restricted estimator is betterthan unrestricted estimator in some sense Thus wheneversome prior information is available in terms of exact linearrestrictions on regression coefficients it is advised to userestricted estimator in place of unrestricted estimator

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] H Tanaka S Uejima and K Asai ldquoFuzzy linear regressionmodelrdquo IEEE Transactions on Systems Man and Cyberneticsvol 10 pp 2933ndash2938 1980

[3] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982

[4] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988

[5] H Tanaka I Hayashi and JWatada ldquoPossibilistic linear regres-sion analysis for fuzzy datardquo European Journal of OperationalResearch vol 40 no 3 pp 389ndash396 1989

[6] H Tanaka and H Ishibuchi ldquoIdentification of possibilisticlinear systems by quadratic membership functions of fuzzyparametersrdquo Fuzzy Sets and Systems vol 41 no 2 pp 145ndash1601991

[7] D T Redden and W H Woodall ldquoProperties of certain fuzzylinear regression methodsrdquo Fuzzy Sets and Systems vol 64 no3 pp 361ndash375 1994

[8] P-T Chang and E S Lee ldquoFuzzy linear regression with spreadsunrestricted in signrdquoComputers andMathematics with Applica-tions vol 28 no 4 pp 61ndash70 1994

[9] G Peters ldquoFuzzy linear regression with fuzzy intervalsrdquo FuzzySets and Systems vol 63 no 1 pp 45ndash55 1994

[10] A de Luca and S Termini ldquoA definition of a nonprobabilisticentropy in the setting of fuzzy sets theoryrdquo Information andControl vol 20 no 4 pp 301ndash312 1972

[11] D Dubosis and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[12] D Dubois and H Prade Fuzzy Sets and Statistical PossibilityTheory Plenum Press New York NY USA 1988

[13] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash656 1948

[14] T Kumar N Gupta and R K Bajaj ldquoFuzzy entropy onrestricted fuzzy linear regression model with cross validationand applicationsrdquo in Proceedings of the International Conferenceon Advances in Computing and Communications (ICACC rsquo12)pp 5ndash8 August 2012

[15] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[16] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[17] K T Atanassov Intuitionistic Fuzzy Sets Theory and Applica-tions vol 35 of Studies in Fuzziness and SoftComputing Physica1999

[18] K T Atanassov ldquoNew operations defined over the intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 61 no 2 pp 137ndash1421994

[19] P Burillo and H Bustince ldquoSome definitions of intuitionisticfuzzy numberrdquo in Proceedings of the 3rd Conference of theEuropean Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[20] F Lee Fuzzy Information Processing System Peking UniversityPress Beijing China 1998

[21] H Liu and K Shi ldquoIntuitionistic fuzzy numbers and intuition-istic distribution numbersrdquo Journal of Fuzzy Mathematics vol8 no 4 pp 909ndash918 2000

[22] P Grzegorzewski ldquoDistances and orderings in a family of intu-itionistic fuzzy numbersrdquo in Proceedings of the 3rd Conference ofthe European Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[23] P DrsquoUrso and T Gastaldi ldquoA least-squares approach to fuzzylinear regression analysisrdquo Computational Statistics and DataAnalysis vol 34 no 4 pp 427ndash440 2000

[24] T Kumar R K Bajaj and N Gupta ldquoFuzzy entropy in fuzzyweighted linear regression model under linear restrictions withsimulation studyrdquo International Journal of General Systems vol43 no 2 pp 135ndash148 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 9

Table 1 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y y

a ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 44 42 96 42 44 449018 424850 943828 424850 4490182 14 8 3 48 47 120 47 48 528505 505256 1217099 505256 5285053 7 1 4 35 33 52 33 35 322052 296416 507324 296416 3220524 11 7 3 50 45 106 45 50 475861 452004 1036114 452004 4758615 7 12 15 80 79 189 79 80 740058 719256 1944413 719256 7400586 8 15 10 68 65 194 65 68 732147 711253 1917213 711253 7321477 3 9 6 45 42 107 42 45 485252 461503 1068398 461503 4852528 12 15 11 80 78 216 78 80 790260 770038 2117003 770038 7902609 10 5 8 55 52 108 52 55 505235 481717 1137100 481717 50523510 9 7 4 45 44 103 44 45 471612 447706 1021507 447706 471612

Table 2 Crisp input-int fuzzy output data

Object119894

Crisp inputX = (x1 x2 x3)

Int fuzzy output y = (e11015840

y e1y ya ery e

r1015840y ) Estimated int fuzzy output ylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 x2 x3 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 3 5 9 45 42 96 47 48 447743 427104 952620 455472 4900532 14 8 3 48 47 120 43 45 525995 502809 120905 523320 5597343 7 1 4 35 33 52 50 55 313430 297162 512469 339018 3704524 11 7 3 46 45 106 45 47 471120 449720 102922 475741 5108705 7 12 15 82 79 189 80 85 753765 723166 195547 720805 7625556 8 15 10 70 65 194 60 67 740419 710254 191173 709234 7506717 3 9 6 45 42 107 40 46 481512 459774 106328 484752 5201248 12 15 11 80 78 216 88 90 802328 770149 211461 762912 8057999 10 5 8 55 52 108 50 55 506447 483897 114499 506370 54232710 9 7 4 45 44 103 42 44 467241 445967 101651 472377 507415

Table 3 Crisp and int fuzzy input-int fuzzy output data

Object 119894Crisp and int fuzzy inputX = (x1 e1

1015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy outputylowast = (e1

lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

x1 e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

y e1lowast

y yalowast er

lowast

y erlowast1015840

y

1 6 7 6 10 5 7 6 3 5 2 5 54158 25665 47776 22215 424732 7 6 5 12 4 6 7 5 4 5 7 58827 31771 62226 28438 496763 8 4 2 15 3 5 8 3 9 4 6 67973 43734 90537 40632 637884 9 7 5 20 8 10 7 3 10 2 4 64422 39089 79544 35897 583085 10 8 6 5 2 5 8 5 12 5 7 74885 52774 11193 49846 744516 11 10 8 15 5 7 3 2 8 4 6 65757 40835 83678 37678 603697 12 20 15 25 12 14 6 5 7 3 5 62393 36436 73265 33193 551798 13 12 7 30 15 18 8 7 14 6 8 83421 63937 13835 61226 876209 14 16 12 20 10 15 11 9 16 10 12 97423 82250 18169 79893 10922310 15 17 13 22 8 12 8 7 18 5 10 90740 73509 16100 70982 98912

10 International Scholarly Research Notices

Table 4 Intuitionistic fuzzy input-intuitionistic fuzzy output data

Object iInt fuzzy input

X = (e11015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy output

ylowast = (e1lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

ye1lowast

y yalowast er

lowast

y erlowast1015840

y

1 5 3 4 5 6 5 4 12 4 6 57505 42398 99797 42398 675052 7 6 7 8 9 7 5 7 5 8 57737 42652 10045 42652 677373 5 3 6 8 9 5 3 9 3 6 48608 32665 74714 32665 586084 4 2 7 9 11 3 1 4 1 4 37872 20920 44446 20920 478725 3 2 5 7 8 4 2 6 2 5 49552 33698 77377 33698 595526 6 3 6 7 10 5 4 8 4 6 45158 28891 64987 28891 551587 5 2 4 9 12 4 3 9 3 5 55863 40602 95168 40602 658638 6 5 8 13 15 7 5 10 5 8 52404 36818 85417 36818 624049 8 7 12 15 17 4 3 5 3 5 39099 22262 47905 22262 4909910 15 10 15 20 25 4 2 3 2 5 36202 19092 39736 19092 46202

of regression coefficients have also been obtained with thehelp of fuzzy entropy for the restrictedunrestricted IFWLRmodel by assigning some weights in the distance functionIt has been observed that the restricted estimator is betterthan unrestricted estimator in some sense Thus wheneversome prior information is available in terms of exact linearrestrictions on regression coefficients it is advised to userestricted estimator in place of unrestricted estimator

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] H Tanaka S Uejima and K Asai ldquoFuzzy linear regressionmodelrdquo IEEE Transactions on Systems Man and Cyberneticsvol 10 pp 2933ndash2938 1980

[3] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982

[4] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988

[5] H Tanaka I Hayashi and JWatada ldquoPossibilistic linear regres-sion analysis for fuzzy datardquo European Journal of OperationalResearch vol 40 no 3 pp 389ndash396 1989

[6] H Tanaka and H Ishibuchi ldquoIdentification of possibilisticlinear systems by quadratic membership functions of fuzzyparametersrdquo Fuzzy Sets and Systems vol 41 no 2 pp 145ndash1601991

[7] D T Redden and W H Woodall ldquoProperties of certain fuzzylinear regression methodsrdquo Fuzzy Sets and Systems vol 64 no3 pp 361ndash375 1994

[8] P-T Chang and E S Lee ldquoFuzzy linear regression with spreadsunrestricted in signrdquoComputers andMathematics with Applica-tions vol 28 no 4 pp 61ndash70 1994

[9] G Peters ldquoFuzzy linear regression with fuzzy intervalsrdquo FuzzySets and Systems vol 63 no 1 pp 45ndash55 1994

[10] A de Luca and S Termini ldquoA definition of a nonprobabilisticentropy in the setting of fuzzy sets theoryrdquo Information andControl vol 20 no 4 pp 301ndash312 1972

[11] D Dubosis and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[12] D Dubois and H Prade Fuzzy Sets and Statistical PossibilityTheory Plenum Press New York NY USA 1988

[13] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash656 1948

[14] T Kumar N Gupta and R K Bajaj ldquoFuzzy entropy onrestricted fuzzy linear regression model with cross validationand applicationsrdquo in Proceedings of the International Conferenceon Advances in Computing and Communications (ICACC rsquo12)pp 5ndash8 August 2012

[15] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[16] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[17] K T Atanassov Intuitionistic Fuzzy Sets Theory and Applica-tions vol 35 of Studies in Fuzziness and SoftComputing Physica1999

[18] K T Atanassov ldquoNew operations defined over the intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 61 no 2 pp 137ndash1421994

[19] P Burillo and H Bustince ldquoSome definitions of intuitionisticfuzzy numberrdquo in Proceedings of the 3rd Conference of theEuropean Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[20] F Lee Fuzzy Information Processing System Peking UniversityPress Beijing China 1998

[21] H Liu and K Shi ldquoIntuitionistic fuzzy numbers and intuition-istic distribution numbersrdquo Journal of Fuzzy Mathematics vol8 no 4 pp 909ndash918 2000

[22] P Grzegorzewski ldquoDistances and orderings in a family of intu-itionistic fuzzy numbersrdquo in Proceedings of the 3rd Conference ofthe European Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[23] P DrsquoUrso and T Gastaldi ldquoA least-squares approach to fuzzylinear regression analysisrdquo Computational Statistics and DataAnalysis vol 34 no 4 pp 427ndash440 2000

[24] T Kumar R K Bajaj and N Gupta ldquoFuzzy entropy in fuzzyweighted linear regression model under linear restrictions withsimulation studyrdquo International Journal of General Systems vol43 no 2 pp 135ndash148 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 International Scholarly Research Notices

Table 4 Intuitionistic fuzzy input-intuitionistic fuzzy output data

Object iInt fuzzy input

X = (e11015840

x1 e1x1 x

a1 e

rx1 e

r1015840x1 )

Int fuzzy outputy = (e1

1015840

y e1y ya ery e

r1015840y )

Estimated int fuzzy output

ylowast = (e1lowast1015840

y e1lowast

y yalowast

erlowast

y erlowast1015840

y )

e11015840

x1e1x1 xa1 erx1 er

1015840

x1 e11015840

y e1y ya ery er1015840

y e1lowast1015840

ye1lowast

y yalowast er

lowast

y erlowast1015840

y

1 5 3 4 5 6 5 4 12 4 6 57505 42398 99797 42398 675052 7 6 7 8 9 7 5 7 5 8 57737 42652 10045 42652 677373 5 3 6 8 9 5 3 9 3 6 48608 32665 74714 32665 586084 4 2 7 9 11 3 1 4 1 4 37872 20920 44446 20920 478725 3 2 5 7 8 4 2 6 2 5 49552 33698 77377 33698 595526 6 3 6 7 10 5 4 8 4 6 45158 28891 64987 28891 551587 5 2 4 9 12 4 3 9 3 5 55863 40602 95168 40602 658638 6 5 8 13 15 7 5 10 5 8 52404 36818 85417 36818 624049 8 7 12 15 17 4 3 5 3 5 39099 22262 47905 22262 4909910 15 10 15 20 25 4 2 3 2 5 36202 19092 39736 19092 46202

of regression coefficients have also been obtained with thehelp of fuzzy entropy for the restrictedunrestricted IFWLRmodel by assigning some weights in the distance functionIt has been observed that the restricted estimator is betterthan unrestricted estimator in some sense Thus wheneversome prior information is available in terms of exact linearrestrictions on regression coefficients it is advised to userestricted estimator in place of unrestricted estimator

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] H Tanaka S Uejima and K Asai ldquoFuzzy linear regressionmodelrdquo IEEE Transactions on Systems Man and Cyberneticsvol 10 pp 2933ndash2938 1980

[3] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982

[4] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988

[5] H Tanaka I Hayashi and JWatada ldquoPossibilistic linear regres-sion analysis for fuzzy datardquo European Journal of OperationalResearch vol 40 no 3 pp 389ndash396 1989

[6] H Tanaka and H Ishibuchi ldquoIdentification of possibilisticlinear systems by quadratic membership functions of fuzzyparametersrdquo Fuzzy Sets and Systems vol 41 no 2 pp 145ndash1601991

[7] D T Redden and W H Woodall ldquoProperties of certain fuzzylinear regression methodsrdquo Fuzzy Sets and Systems vol 64 no3 pp 361ndash375 1994

[8] P-T Chang and E S Lee ldquoFuzzy linear regression with spreadsunrestricted in signrdquoComputers andMathematics with Applica-tions vol 28 no 4 pp 61ndash70 1994

[9] G Peters ldquoFuzzy linear regression with fuzzy intervalsrdquo FuzzySets and Systems vol 63 no 1 pp 45ndash55 1994

[10] A de Luca and S Termini ldquoA definition of a nonprobabilisticentropy in the setting of fuzzy sets theoryrdquo Information andControl vol 20 no 4 pp 301ndash312 1972

[11] D Dubosis and H Prade Fuzzy Sets and Systems Theory andApplications Academic Press New York NY USA 1980

[12] D Dubois and H Prade Fuzzy Sets and Statistical PossibilityTheory Plenum Press New York NY USA 1988

[13] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash656 1948

[14] T Kumar N Gupta and R K Bajaj ldquoFuzzy entropy onrestricted fuzzy linear regression model with cross validationand applicationsrdquo in Proceedings of the International Conferenceon Advances in Computing and Communications (ICACC rsquo12)pp 5ndash8 August 2012

[15] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets and Sys-tems vol 20 no 1 pp 87ndash96 1986

[16] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[17] K T Atanassov Intuitionistic Fuzzy Sets Theory and Applica-tions vol 35 of Studies in Fuzziness and SoftComputing Physica1999

[18] K T Atanassov ldquoNew operations defined over the intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 61 no 2 pp 137ndash1421994

[19] P Burillo and H Bustince ldquoSome definitions of intuitionisticfuzzy numberrdquo in Proceedings of the 3rd Conference of theEuropean Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[20] F Lee Fuzzy Information Processing System Peking UniversityPress Beijing China 1998

[21] H Liu and K Shi ldquoIntuitionistic fuzzy numbers and intuition-istic distribution numbersrdquo Journal of Fuzzy Mathematics vol8 no 4 pp 909ndash918 2000

[22] P Grzegorzewski ldquoDistances and orderings in a family of intu-itionistic fuzzy numbersrdquo in Proceedings of the 3rd Conference ofthe European Society for Fuzzy Logic and Technology (EUSFLATrsquo03) pp 223ndash227 Zittau Germany September 2003

[23] P DrsquoUrso and T Gastaldi ldquoA least-squares approach to fuzzylinear regression analysisrdquo Computational Statistics and DataAnalysis vol 34 no 4 pp 427ndash440 2000

[24] T Kumar R K Bajaj and N Gupta ldquoFuzzy entropy in fuzzyweighted linear regression model under linear restrictions withsimulation studyrdquo International Journal of General Systems vol43 no 2 pp 135ndash148 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of