A LEAST ABSOLUTE APPROACH TO MULTIPLE FUZZY REGRESSION USING TwNORM BASED OPERATIONS

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 2013

A LEAST ABSOLUTE APPROACH TO MULTIPLE FUZZY REGRESSION USING TwNORM BASED OPERATIONS B. Pushpa1 and R. Vasuki2 1

1

Manonmaniam Sundaranar Universty, Tirunelveli, India Panimalar Institute of Technology, Poonamallee, Chennai, India. pushpajuly14@gmail.com 2

SIVET College, Gowrivakkam, Chennai, India. vasukidevi06@gmail.com

ABSTRACT A least absolute approach to multiple fuzzy regression using Tw-norm based arithmetic operations is discussed by using the generalized Hausdorff metric and it is investigated for the crisp input- fuzzy output data. A comparative study based on two data sets are presented using the proposed method using shape preserving operations with other existing method.

KEYWORDS Fuzzy Regression, Hausdorff- metric, Fuzzy Linear Programming, Fuzzy parameters, Crisp input data, Fuzzy output data, shape preserving operations.

1. INTRODUCTION Regression analysis has a wide spread applications in various fields such as business, engineering, agriculture, health sciences, biology and economics to explore the statistical relationship between input (independent or explanatory) and output (dependent or response) variables. Fuzzy regression models were proposed to model the relationship between the variables, when the data available are imprecise (fuzzy) quantities and/or the relationship between the variables are fuzzy. Regression analysis based on the method of least -absolute deviation has been used as a robust method. When outlier exists in the response variable, the least absolute deviation is more robust than the least square deviations estimators. Some recent works on this topic are as follows: Chang and Lee [1] studied the fuzzy least absolute deviation regression based on the ranking method for fuzzy numbers. Kim et al. [2] proposed a two stage method to construct the fuzzy linear regression models, using a least absolutes deviations method. Torabi and Behboodian [3] investigated the usage of ordinary least absolute deviation method to estimate the fuzzy coefficients in a linear regression model with fuzzy input – fuzzy output observations.

Considering a certain fuzzy regression model, Chen and Hsueh [4] developed a mathematical programming method to determine the crisp coefficients as well as an adjusted term for a fuzzy regression model, based on L1 norm (absolute norm) criteria. Choi and Buckley [5] suggested two methods to obtain the least absolute deviation estimators for common fuzzy linear regression models using TM based arithmetic operations. Taheri and Kelkinnama [6,7] introduced some least absolute regression models, based on crisp input- fuzzy output and fuzzy input-fuzzy output data respectively.

In a regression model, multiplication of fuzzy numbers are done by arithmetic operations such as

α -levels of multiplication of fuzzy numbers and the approximate formula for multiplication of fuzzy numbers. Apart from these two, we know that using the weakest T – norm (Tw), the shape DOI : 10.5121/ijfls.2013.3206

73

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 2013

of fuzzy numbers in multiplication will be preserved. In this regard, Hong et al. [8] presented a method to evaluate fuzzy linear regression models based on a possibilistic approach, using Tw norm based arithmetic operations.

The objective of this study is to develop a least absolute multiple fuzzy regression model to handle the functional dependence of crisp inputs-fuzzy output variables using the generalized Hausdorff- metric between fuzzy numbers as well as linear programming approach.

In this paper, section 2 focuses on some important preliminary definitions and basics on fuzzy arithmetic operations based on the weakest T-norm. In section 3, the new approach based on Hausdorff metric is presented using the shape preserving operations on fuzzy numbers and it is analyzed with crisp input and fuzzy output and discussed the goodness of fit of the proposed model. In section 4, by using numerical examples we provide some comparative studies to show the performance of the proposed method.

2. PRELIMINARIES A fuzzy number is a convex subset of the real line

with a normalized membership function. A  a −t , if a − α ≤ t ≤ a 1 − α  a −t triangular fuzzy number a% = ( a, α , β ) is defined by a% (t ) =  , if a ≤ t ≤ a + β 1 − β  0 , otherwise  

where a ∈ is the center, α > 0 is the left spread and β > 0 is the right spread of a% .

If α = β , then the triangular fuzzy number is called a symmetric triangular fuzzy number and it is denoted by ( a, α ) . A fuzzy number a% = (a, α , β ) LR of type L-R is a  t −a R  , a ≤ t ≤ a + β   β  function from real number into the interval ( 0 , 1) satisfying   a − t  a% (t ) =  L  , a −α ≤ t ≤ a   α  0 , otherwise  

where L and R are non increasing and continuous functions from (0,1) to (0,1) satisfying L(0)=R(0)=1 and L(1)=R(1)=0.A binary operation T on the unit interval is said to be a triangular norm [9] (t-norm) if and only if f T is associative, commutative, non-decreasing and T(x,1)=x for each x ∈ [0,1] . Moreover, every t-norm satisfies the inequality,

a , if b = 1

Tw (a, b) ≤ T ( x, y ) ≤ TM (a, b) = min( a, b) where Tw (a, b) = b , if a = 1

0 , otherwise 

The critical importance of min(a, b), a b, max(0, a + b − 1) and Tw ( a, b) is emphasized from a mathematical point of view in Ling [9]. The usual arithmetic operations on real numbers can be 74

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 2013

extended to the arithmetic operations on fuzzy numbers by means of extension principle by Zadeh [10], which is based on a triangular norm T. Let A% and B% be fuzzy numbers of the real line .

The fuzzy number arithmetic operations are summarized as follows: Fuzzy number addition ⊕ :

( A% ⊕ B% ) (z) =

% % . sup T  A(x), B(y) 

x ,y,x + y = z

Fuzzy number multiplication ⊗ :

( A% ⊗ B% ) (z) =

% % . sup T  A(x), B(y) 

x ,y,x.y = z

The addition (subtraction) rule for L-R fuzzy number is well known in case of TM based addition, in which the resulting sum is again an L-R fuzzy number, i.e., the shape is preserved. Let

A% = (a, α A , β A ) LR , B% = (b, α B , β B ) LR . Then using TM in the above definition, A% ⊕ B% = (a + b, α + α , β + β ) M

A

B

A

B LR

It is also known that the Tw based addition and multiplication preserves the shape of L-R fuzzy numbers[11,12,13,14]. We know that TM based multiplication does not preserve the shape of LR fuzzy numbers. In this section, we consider Tw based multiplication of L-R fuzzy numbers.

Let T= Tw be the weakest t-norm and let A% = (a, α A , β A ) LR , B% = (b, α B , β B ) LR be two L-R fuzzy numbers, then the addition and multiplication of A% = (a, α , β ) , B% = (b, α , β ) A

A LR

B

B

LR

is defined as[15],

A% ⊕W B% = (a + b, max(α A , α B ), max( β A , β B )) LR

(ab, max(α Ab, α B a ), max( β Ab, β B a)) LR for a, b > 0  (ab, max( β A b , β B a ), max(α A b , β B a )) RL , for a, b < 0 (ab, max(α b, β a ), max( β b, α a )) , for a < 0, b > 0 A B A B LL %A ⊗ B% =  W ( 0, α Ab, β Ab ) LR , for a = 0, b > 0  0, − β b, −α b , for a = 0, b < 0 A A ) RL ( (0, 0,0) LR , for a = 0, b = 0

In particular, if A% = (a, α A ), B% = (b, α B ) are symmetric fuzzy numbers, then the multiplication of

A% and B% is written as, A% ⊗w B% = (ab, max(α A b , α B a )) LL

75

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 2013

A distance between fuzzy numbers: Several metrics are defined on the family of all fuzzy numbers[16]. The generalized Hausdorff metric fulfil many good properties, also easy to compute in terms of generalized mid and spread functions. The concept of generalized mid and spread may be useful in working with (fuzzy) convex compact sets [17]. d H ( Aα , Bα ) is the Hausdorff metric between crisp sets Aα and Bα given by,

  d H ( Aα , Bα ) = max sup inf a − b , sup inf a − b  a ∈ A α b ∈ B α a∈ Aα b∈Bα  If I1 = [a1 , a2 ] and I 2 = [b1 , b2 ]are two intervals, then

d H (I1 , I 2 ) = max{a1 − b1 , a2 − b2 } = mid I1 − mid I 2 + spr I1 − spr I 2 a1 + a2 a −a , sprI1 = 2 1 [16].

2 2 ~ ~ The generalized Hausdorff metric between A = (a,α )T , B = (b, β )T is then, ~ ~ ~ ~ D1 ( A, B ) = a − b + 0.5 α − β , D∞ ( A, B ) = a − b + α − β

Where mid I1 =

3. FUZZY LINEAR REGRESSION USING THE PROPOSED APPROACH In this section, we are discussing fuzzy linear regression about the proposed approach based on Hausdorff metric using Tw norm based operations with crisp input- fuzzy output data, in which the coefficients of the models are also considered as fuzzy numbers.

{

}

Consider the set of observed data ( X% i , Y%i ), i = 1,..., n where X% i = ( xi , γ i ) and Y%i = ( yi , ei ) are symmetric fuzzy numbers. Our aim is to fit a fuzzy regression model with fuzzy coefficients to the aforementioned data set as follows: ˆ Y% = A% ⊕ ( A% ⊗ X% ) ⊕ ..... ⊕ ( A% ⊗ X% ) = A% ⊗ X% i = 1,....n i

0

1

w

i1

w

(

w

)

where A% j = a j , α j , j = 1,... p

w

p

w

ip

w

i

,

(1)

are symmetric fuzzy numbers and the arithmetic

operations are based on the weakest Tw norm.

Consider the least absolute optimization problem as follows:

Minimizes

n

Min

D1 (Y% , A% ⊗ w X% j ) i.e.,

k

∑ y −∑x a i

i =1

ij

j =0

j

n

k

i =1

j =0

+ 0.5∑ ei − ∑ max a j γ ij , xij α j

(

)

(2)

subject to

76

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 2013 n

n

j =0

1

n

n

j =0

1

( a j γij , 1≤ j ≤ p

xij α j ≥ yi − L−1 (h)ei

( a j γij , 1≤ j ≤ p

xij α j ≤ yi + L−1 (h)ei

∑ a j xij + L−1(h)∑ max ∑ a j xij − L−1(h)∑ max

)

)

0 < h < 1, max 1≤ j ≤ p

( a j γij ,

)

xij α j ≥ 0, ∀i = 1, 2,...m (4)

Solving this optimization problem using LINGO, we can estimate the fuzzy coefficients of the model. Several methods have been proposed to detect the presence of outliers, relying on graphical representation and/or on analytical procedures. The box plot or boxand-whisker plot describes the key features of data through the five-number summaries: the smallest observation, lower quartile (Q1), median (Q2), upper quartile (Q3), and the largest observation (sample maximum). A box plot indicates the abnormal observations.

Usually, outlier cut offs are set at 1.5 times the inter-quartile range [18, 19]. In a fuzzy framework, we can draw box plots side by side to detect outliers in the distributions of the centers, of the spreads and of the input variables. To overcome limitations in previous approaches, Hung and Yang [20] consider the effect of each observation on the value of the objective function in Tanaka’s approach. Let JM be the optimal value of the objective function and JM(i) is the corresponding value obtained deleting the ith observation. The ratio ri =

J M − J M (i ) JM

is a synthetic evaluation of the impact of the ith observation on the

value of the objective function. Observations with large ri value are more likely to be anomalous. Combining these ratios with box plots, or with other suitable graphical representations, provides an effective way to detect a single outlier, with respect to the input variables and/or to the centers or the spreads of the fuzzy response variable. This approach could be generalized to the inspection of multiple outliers, but the process becomes more computing demanding as the sample size and/or the number of outliers increases.

3.1. Evaluation of Regression models

To investigate the performance of the fuzzy regression models, we use similarity measure based on the Graded mean integration representation of distance proposed by Hsieh and 1 Chen[21] S ( A% , B% ) = , d ( A% , B% ) = P( A% ) − P( B% ) , P ( A% ) and P( B% ) are the 1 + d ( A% , B% ) graded mean integration representation distance.

Also to evaluate the goodness of fit between the observed and estimated values from [22], if

A% = (a, σ ) and B% = (b,τ ) be two normal fuzzy numbers, then   a − b  ( A% , B% ) = e x p  −     σ + τ 

2

  

is the goodness of fit of observed and estimated fuzzy

numbers A% and B% .

77

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 2013

4. EXAMPLES In this section, we discuss our proposed method with multiple inputs in the following examples.

For the example 1, we study the sensitivity of the proposed approach with respect to outlier data points with Hung and Yang [20] outlier treatment.

EXAMPLE-1 Consider the dataset in Table1 in which the observations are crisp multiple inputs and fuzzy outputs without modifying the outlier in the spread using Hung and Yang omission approach [20].

Explanatory Variables S.No.

response variable

J M − J M (i )

ri = x1

x2

x3

y

eps

JM

1

2

0.5

15.25

5.83

3.56

0.095626

2

0

5

14.13

0.85

0.52

0.163109

3

1.13

1.5

14.13

13.93*

8.5*

0.445439

4

2

1.25

13.63

4

2.44

-0.30979

5

2.19

3.75

14.75

1.65

1.01

-0.22509

6

0.25

3.5

13.75

1.58

0.96

0.077927

7

0.75

5.25

15.25

8.18

4.99

0.198405

8

4.25

2

13.5

1.85

1.13

-0.25154

* indicates outlier Table 1. Dataset with crisp input and fuzzy output with outlier

Figure 1. Fuzzy regression model with outlier for the dataset in Table 1 using the proposed approach.

The fuzzy regression model obtained by the proposed approach,

Y% = (0,3.8212) ⊕ w (0,0) ⊗w X 1 ⊕ w (0,0) ⊗ w X 2 ⊕ w (0.3013,0) ⊗w X 3 with optimum value h=0.215 and the graph is given in Fig.1. In table 1, third data is an 78

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 2013

abnormal data, after identifying the abnormal data using Hung and Yang [20] method, deleting the data which yields a better result. The fuzzy regression model obtained by proposed approach is

Y% = (2.482,0) ⊕w (0,0) ⊗w X 1 ⊕w (0,0) ⊗w X 2 ⊕w (0.2042,0) ⊗w X 3

with optimum level h= 0.322 (shown in Figure 2.)

Figure 2. Using Hung and Yang [20] method of omission approach for the dataset with outlier in table1 using the proposed approach

Comparing the performance of the proposed with the some other existing methods, using the Choi and Buckley’s [5] method, the optimal model is

Y% = −2.8273 ⊕ 0.3878 ⊗ x1 ⊕ 1.0125 ⊗ x2 ⊕ 0.6185 ⊗ x3 ⊕ (0,1.0696, 2.0042) Chen Hseuh [23] proposed a least square approach to fuzzy regression models with crisp coefficients. Y% = −16.7956 ⊕ 1.0989 ⊗ x1 ⊕ 1.1798 ⊗ x2 ⊕ 1.8559 ⊗ x3 ⊕ (0, 2.8888) Hassanpour et al. [24] proposed least absolute regression method that minimizes the difference between centers of the observed and estimated fuzzy responses and also between the spreads of them, using goal programming approach. They took into account fuzzy coefficients for crisp inputs in their model. Employing their model for the given example yields the following model,

Y% = ( −2.8273, 0.0000) ⊕ (0.3877, 0.0000) ⊗ x1 ⊕ (1.0125, 0.000) ⊗ x2 ⊕ (0.6185, 0.1790) ⊗ x3 Using the graded mean integration representation, the similarity measure for the proposed and above existing models is given in table 2.

79

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 2013

Methods

Proposed

Choi and Buckley[5]

Chen and Hseuh [23]

Hassanpour [24]

Mean Similarity measure between the observed and the existing methods

0.326389

0.307915

0.122202

0.209148

Table 2. Similarity measure between various models with multiple inputs

The table 2 illustrates that the mean similarity measure for the proposed model is 0.3264 which has effective performance with other existing methods using Hung and Yang method of omission approach with outlier.

EXAMPLE-2 We now apply this model to analyse the effect of the composition of Portland cement on heat evolved during hardening. The data shown in Table 3 except si are taken from [25]. The values are assumed by R.Xu et al. [26].

Obs.No.

y% i = ( yi , si )

xi1

xi 2

xi 3

Y%i = (Yi , σ i )

Goodness

1

(78.5,6.9)

7

26

6

(78.33,0.709)

0.9995

2

(74.3,6.4)

1

29

15

(71.99,0.709)

0.8997

3

(104.3,9.4)

11

56

8

(106.25,0.709)

0.9364

4

(87.6,7.8)

11

31

8

(88.96,0.709)

0.9748

5

(95.5,8.6)

7

52

6

(96.30,0.709)

0.9926

6

(109.2,9.9)

11

55

9

(105.75,0.709)

0.8997

7

(102.7,9.3)

3

71

17

(104.81,0.709)

0.9565

8

(72.5,6.2)

1

31

22

(74.75,0.709)

0.8994

9

(93.1,8.3)

2

54

18

(91.56,0.709)

0.9712

10

(115.9,10.6)

21

47

4

(116.20,0.709)

0.9993

11

(83.8,7.4)

1

40

23

(81.16,0.709)

0.8994

12

(113.3,10.6)

11

66

9

(113.35,0.709)

0.9999

13

(109.4,9.9)

10

68

8

(112.85,0.709)

0.8996

of fit

Table 3. Performance of the proposed model in Example 2.

By using the proposed method, we have

Y% = (47.299, 0.7093) ⊕ w (1.6963, 0) ⊗ w x1 ⊕ w (0.6914, 0) ⊗ w x2 ⊕ w (0.196, 0) ⊗ w x3 with h=0.675 (shown in Fig.3)

80

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 2013

Figure 3. Fuzzy regression model with crisp multiple input fuzzy output in Example 2.

Furthermore, we calculate the goodness of fit using the fitted equation and the observed values.

% i and The results are given in table3, the goodness of fit of every y which indicates that the fitted result of the model is very good.

Y%i

are all greater than 0.9,

EXAMPLE-3 Consider the following crisp input-fuzzy output data given by Tanaka et al. [27]

( y% ; x) = ((8.0,1.8)T ;1), ((6.4, 2.2)T ; 2), ((9.5, 2.6)T ;3), ((13.5, 2.6)T ; 4), ((13.0, 2.4)T ;5), By applying the proposed approach described in section 3, the fuzzy regression model is derived as Y% = (4.079, 2.1) ⊕ w (1.8458,0) ⊗ w x with h = 0.468. A Summary of results of some other techniques, including their models as well as their performances, are given in Table 4.

To show the performance of fuzzy regression models we considered these five pairs of observations listed above. The dataset do not have the level of detail and complexity than those used in other studies, but in the literature these data have been considered by many researchers for the experimental evaluation and comparison of their proposed methodology. Table 4 lists the regression models obtained by the methods proposed by other authors based on the five pairs of observations considered above. The first fuzzy regression model based on fuzzy observations was proposed by Tanaka et al.[27] (THW) (1982). Several authors pointed out that this method has several disadvantages and modified it or developed their own methodologies to prevent the problems. THW[26] regression model estimates the fuzzy regression coefficients by linear programming. This model has a numeric slope and a fuzzy intercept. KB[28], DM[29], WT[30] and HBS[31] models have a fuzzy slope and intercept. If the explanatory variables are numeric values and dependent variables are symmetric fuzzy numbers, WT[30] model is the same as DM[29] approach. NN[32] and CH[33]model have a numeric slope. In this case the proposed regression model is given in Figure 4.

81

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 2013

Figure 4. Fuzzy regression model for the data set in Example 3.

Goodness of fit

Similarity

measure

measure

0.9068

0.61116

0.9424

0.43625

0.9125

0.4987

0.9149

0.51554

0.9135

0.4865

0.9135

0.4865

0.8915

0.62467

0.9175

0.4865

Methods Proposed method using Tw-norm based operations

Y% = (4.079, 2.1) ⊕ w (1.8458, 0) ⊗ w x with h=0.468 Tanaka et al. method (THW)[27]

Y% = (0.385, 7.7) ⊕ 2.1 ⊗ x Kim and bishu model 1988 (KB)[28]

Y% = (3.11, 4.95, 6.84) ⊕ (1.55,1.71,1.82) ⊗ x Nasrabedi and Nasrabedi (2006) method (NN)[32]

Y% = (2.36, 4.86, 7) ⊕ 1.73 ⊗ x Diamond (1988)(DM)[29]

Y% = (3.11, 4.95, 6.79) ⊕ (1.55,1.71,1.87) ⊗ x Wu and Tseng method (WT)[30]

Y% = (3.11, 4.95, 6.79) ⊕ (1.55,1.71,1.87) ⊗ x Hojati et al 2005(HBS)[31]

Y% = (5.1, 6.75,8.4) ⊕ (1.1,1.25,1.4) ⊗ x Chen and Hseuh method 2009(CH)[33]

Y% = 1.71 ⊗ x ⊕ (2.63, 4.95, 7.27)

Table 4.Performance of different models for crisp input and fuzzy output given in Example 3.

Analyzing these results, we can observe that our technique is very easy to compute in practice and gives a first and easy approach for the problem of analyzing regression problems with crisp input and triangular fuzzy output data.

5. CONCLUSION Based on the distance between the centers and spreads, a new method is proposed for fuzzy simple regression using Tw -norms. The models which are used here have the input and the 82

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 2013

output data as well as the coefficients are assumed to be fuzzy. The arithmetic operations based on the weakest t-norm are employed to derive the exact results for estimation of parameters. The efficiency of the proposed approach is studied by similarity measure based on the graded mean integration representation distance of fuzzy numbers. In addition the effect of the outlier is discussed for the proposed approach. By comparing the proposed approach with some well known methods, applied to three data sets, it is shown that the proposed approach using shape preserving operations was more effective. Studying the effect of outlier in center value of response variable using proposed method with TW- norm operation is our future work.

REFERENCES [1]

Chang P.T. & Lee E.S, (1994) “fuzzy least absolute deviations regression based on the ranking of fuzzy numbers”, in Proc. IEEE world congress on computational intelligence, pp.1365-1369.

[2]

Kim K.J, Kim D.H & Choi S.H, (2005) “Least absolute deviation estimator in fuzzy regression”, J. Appl. Math. Comput. Vol.18, pp. 649-656.

[3]

Torabi H, & Behboodian J, (2007) “Fuzzy least absolutes estimates in linear regression models”, Communi. Stat. – Theory methods, Vol. 36, pp. 1935-1944.

[4]

Chen L. H. & Hsueh C.C, (2007) “A mathematical programming method for formuation a fuzzy regression model based on distance criterion”, IEEE Trans. Syst. Man Cybernet. Vol.B, pp. 37 705712.

[5]

Choi S. H. & Buckley J. J, (2008) “Fuzzy regression using least absolute deviation estimators”, Soft Comput. Vol. 12, pp. 257-263.

[6]

Taheri S. M. & Kelkinnama M, (2008) “Least absolute regression”, in Proc. 4th international IEEE conference on intelligent systems, varna Bulgaria, , Vol. 11, pp. 55-58.

[7]

Taheri S. M. & Kelkinnama M, (2012) “Fuzzy linear regression based on least absolute deviations”, Iranian J. fuzzy system Vol. 9, pp. 121-140.

[8]

Hong D.H., Lee S. & Do D.Y, (2001) “Fuzzy linear regression analysis for fuzzy input-output data using shape preserving operations”, Fuzzy sets Syst. Vol.122, pp. 513-526.

[9]

Ling C.H, (1965) “Representation of associative functions”, Publications Mathematicae – Debrecen, Vol. 12, pp. 189-212.

[10]

Zadeh L.A, (1978) “Fuzzy sets as a basis for a theory of possibility”, Fuzzy sets syst. Vol. 1, pp. 328.

[11]

Hong D.H, (2001) “Shape preserving multiplication of fuzzy intervals”, Fuzzy sets and Syst. Vol. 123, pp. 81-84.

[12]

Hong D.H,(2002) “On shape preserving addition of fuzzy intervals”, Journal of Mathematical Analysis and applications, Vol. 267, pp. 369-376.

[13] Kolsevera A, (1994) “Additive preserving the linearity of fuzzy intervals”, Tata mountains Math.

Publi. Vol. 6 pp.75-81.

[14] Mesiar R, (1997) “Shape preserving addition of fuzzy intervals”, Fuzzy sets and Syst. Vol. 86 pp. 7378.

83

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 2013 [15] Hong D. H. & Do H.Y, (1997) “Fuzzy system reliability analysis by the use TW (the weakest norm) on fuzzy arithmetic operations”, Fuzzy Sets and Systems, Vol. 90, pp. 307-316.

[16] Trutsching W, Gonzalez- Rodriguez G, Colubi A. & Angeles Gil A, (2009) “A new family of metrics for compact, convex(fuzzy) sets based on a generalized concept of mid and spread”, Inf. Sci. Vol. 179, pp. 3964-3972.

[17]

Diamond P. & Kloeden P, (1994) “Metric spaces of fuzzy sets”, World scientific, Singapore.

[18] Emerson J.D. & Strenio J, (1983) Boxplots and batch comparisons, in: D.C. Hoaglin, F. Mosteller, J.W. Tukey (Eds.), “Understanding Robust and Exploratory Data Analysis”, Wiley, New York, pp.

58–96.

[19]

Tukey J.W, (1977) “Exploratory Data Analysis”, Addison-Wesley.

[20] Hung W.L. & Yang M.S, (2006) “An omission approach for detecting outliers in fuzzy regressions models”, Fuzzy Sets and Systems Vol. 157, pp. 3109–3122.

[21]

Hsieh C.H. & Chen. S.H, (1999) “Similarity of generalized fuzzy numbers with graded mean integration representation”, in Proceedings of Eighth international fuzzy system association world congress, vol2, Taipei, Taiwan, Republic of China, pp. 551-555.

[22] Xu R. & Li C, (2001) “Multidimensional least- squares fitting with a fuzzy model”, Fuzzy sets and systems, Vol. 119, pp. 215-223.

[23] Chen L. H. & Hseuh C.C, (2009) “Fuzzy regression models using the least squares method based on the concept of distance”, IEEE Transactions on fuzzy systems, Vol. 17, pp. 1259-1272.

[24]

Hassanpour H, Maleki H. R. & Yaghoobi M.A, (2009) “A goal programmingto fuzzy linear regression with non-fuzzy input and fuzzy output data”, Asia Pacific Journal of Operational research, Vol. 26, pp. 587-604.

[25] Draper N. R. & Smith H, (1980) “Applied Regression analysis”, Wiley, Newyork.

[26] Xu R. & Li C, (2001) “Multidimensional least squares fitting with a fuzzy model”, Fuzzy sets and systems, Vol. 119, pp. 215-223.

[27] Tanaka H, Hayashi I. & Watada J, (1980) “Possibilistic linear regression analysis for fuzzy data”, Eur. J. Opera. Res. Vol. 40, pp. 389-396.

[28]

Kim B. & Bishu R. R, (1998) “Evalaution of fuzzy linear regresssion models by comparing membership functions”, Fuzzy Sets and Systems, Vol. 100, pp. 343-352.

[29]

Diamond P, (1988) “Fuzzy least squares”, Information Sciences, Vol. 46, pp. 141–157.

[30] Wu B. & Tseng N. F, (2002) “A new approach to fuzzy regression models with application to business cycle analysis”, Fuzzy Sets Syst, Vol. 130, pp. 33–42.

[31]

Hojati M, Bector C. R. & Smimou K, (2005) “A simple method for computation of fuzzy linear regression”, Eur J Oper Res. Vol. 166, pp. 172–184.

[32]

Nasrabadi M. M, & Nasrabadi E, (2004) “A mathematical-programming approach to fuzzy linear regression analysis”, Appl Math Comput. 155:873–881.

[33] Chen L. H, Hsueh C. C, (2009) “Fuzzy regression models using the least squares method based on the concept of distance”, IEEE Trans Fuzzy Syst. Vol. 17, pp. 1259–1272.

84