A LEAST ABSOLUTE APPROACH TO MULTIPLE FUZZY REGRESSION USING Tw- NORM BASED OPERATIONS

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

DOI : 10.5121/ijfls.2013.3206 73

A LEAST ABSOLUTE APPROACH TO

MULTIPLE FUZZY REGRESSION USING Tw-

NORM BASED OPERATIONS

B. Pushpa1 and R. Vasuki

2

1Manonmaniam Sundaranar Universty, Tirunelveli, India

1Panimalar Institute of Technology, Poonamallee, Chennai, India.

[email protected] 2SIVET College, Gowrivakkam, Chennai, India.

[email protected]

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


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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

triangular fuzzy number ( , , )a a α β=% is defined by

1 ,

( ) 1 ,

0 ,

a tif a t a

a ta t if a t a

otherwise

αα

ββ

−− − ≤ ≤

−

= − ≤ ≤ +

%

where a ∈� is the center, 0α > is the left spread and 0β > 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 ( , , )LRa a α β=% of type L-R is a

function from real number into the interval ( 0 , 1) satisfying

,

( ) ,

0 ,

t aR a t a

a ta t L a t a

otherwise

ββ

αα

−≤ ≤ +

−

= − ≤ ≤

%

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 [0,1]x ∈ . Moreover, every t-norm satisfies the inequality,

( , ) ( , ) ( , ) min( , )w MT a b T x y T a b a b≤ ≤ = where

, 1

( , ) , 1

0 ,

w

a if b

T a b b if a

otherwise

=

= =

The critical importance of min( , ), , max(0, 1) ( , )wa b a b a b and T a b+ −� is emphasized from

a mathematical point of view in Ling [9]. The usual arithmetic operations on real numbers can be


75

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 ⊕ :

( )x,y,x y z

A B (z) sup T A(x), B(y) .+ =

⊕ = % %% %

Fuzzy number multiplication ⊗ :

( )x,y,x.y z

A B (z) sup T A(x),B(y) .=

⊗ = % %% %

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 LR B B LRA a B b= α β = α β% % . Then using TM in the above definition,

( , , )M A B A B LR

A B a b α α β β⊕ = + + +% %

It is also known that the wT 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 L-

R fuzzy numbers. In this section, we consider wT based multiplication of L-R fuzzy numbers.

Let T= wT be the weakest t-norm and let ( , , ) , ( , , )LR LRA A B BA a B b= α β = α β% % be two L-R

fuzzy numbers, then the addition and multiplication of ( , , ) , ( , , )LRA A LR B BA a B b= α β = α β% %

is defined as[15],

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

( )

( )

( ,max( , ),max( , )) , 0

( ,max( , ),max( , )) , , 0

( ,max( , ),max( , )) , 0, 0

0, , , 0, 0

0, , , 0, 0

(0,0,0) , 0, 0

A B A B LR

A B A B RL

A B A B LL

W

A A LR

A A RL

LR

ab b a b a for a b

ab b a b a for a b

ab b a b a for a bA B

b b for a b

b b for a b

for a b

α α β β

β β α β

α β β α

α β

β α

>

<

< >⊗ =

= >

− − = <

= =

% %

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

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


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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].

),( αα BAdH is the Hausdorff metric between crisp sets αα BandA given by,

−−=∈∈∈∈

babaBAdBbAaAaBb

Hα

ααα

αα infsup,infsupmax),(

If [ ] [ ]212211 ,, bbIandaaI == are two intervals, then

( ) { } 2121221121 ,max, IsprIsprImidImidbabaIIdH −+−=−−=

Where 2

,2

121

211

aasprI

aaImid

−=

+= [16].

The generalized Hausdorff metric between TT bBaA ),(~

,),(~

βα == is then,

βαβα −+−=−+−= ∞ baBADbaBAD )~

,~

(,5.0)~

,~

(1

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 { }( , ), 1,...,i iX Y i n=% % where ( , )ii iX x= γ% and ( , )i i iY y e=%

are symmetric fuzzy numbers. Our aim is to fit a fuzzy regression model with fuzzy

coefficients to the aforementioned data set as follows:

0 1 1i w w i w w p w ip w i

ˆY A ( A X ) ..... ( A X ) A X= ⊕ ⊗ ⊕ ⊕ ⊗ = ⊗% % % %% % % %

, 1,....=i n (1)

where ( ), , 1,...j j jA a j p= α =% are symmetric fuzzy numbers and the arithmetic

operations are based on the weakest Tw norm.

Consider the least absolute optimization problem as follows:

Minimizes 1( , )

w jD Y A X⊗%% % i.e.,

Min ( )1 0 1 0

0.5 max ,n k n k

i ij j i j ij ij j

i j i j

y x a e a x= = = =

− + − γ α∑ ∑ ∑ ∑ (2)

subject to


77

( )

( )

( )

1 1

10 1

1 1

10 1

1

( ) max , ( )

( ) max , ( )

0 1,

max , 0, 1,2,...

n n

j ij j ij ij j i ij pj

n n

j ij j ij ij j i ij pj

j ij ij jj p

a x L h a x y L h e

a x L h a x y L h e

h

a x i m

− −

≤ ≤=

− −

≤ ≤=

≤ ≤

+ γ α ≥ −

− γ α ≤ +

< <

γ α ≥ ∀ =

∑ ∑

∑ ∑

(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 box-

and-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

( )i

M M

i

M

J Jr

J

−= is a synthetic evaluation of the impact of the i

th observation on the

value of the objective function. Observations with large ir 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

Chen[21] 1

( , )1 ( , )

S A Bd A B

=+

% %% %

, ( , ) ( ) ( )d A B P A P B= −% %% % , ( )P A% and ( )P B% are the

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

2

( , ) e x pa b

A Bσ τ

− = −

+

% %

is the goodness of fit of observed and estimated fuzzy

numbers A% and B% .


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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].

S.No.

Explanatory Variables response variable ( )i

M M

i

M

J Jr

J

−=

x1 x2 x3 y eps

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,

1 2 3(0,3.8212) (0,0) (0,0) (0.3013,0)

w w w w w wY X X X= ⊕ ⊗ ⊕ ⊗ ⊕ ⊗%

with

optimum value h=0.215 and the graph is given in Fig.1. In table 1, third data is an


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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 1 2 3

(2.482,0) (0,0) (0,0) (0.2042,0)w w w w w w

Y X X X= ⊕ ⊗ ⊕ ⊗ ⊕ ⊗%

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

1 2 32.8273 0.3878 1.0125 0.6185 (0,1.0696, 2.0042)Y x x x= − ⊕ ⊗ ⊕ ⊗ ⊕ ⊗ ⊕%

Chen Hseuh [23] proposed a least square approach to fuzzy regression models with crisp

coefficients. 1 2 316.7956 1.0989 1.1798 1.8559 (0, 2.8888)Y x x x= − ⊕ ⊗ ⊕ ⊗ ⊕ ⊗ ⊕%

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,

1 2 3( 2.8273,0.0000) (0.3877,0.0000) (1.0125,0.000) (0.6185,0.1790)Y x x x= − ⊕ ⊗ ⊕ ⊗ ⊕ ⊗%

Using the graded mean integration representation, the similarity measure for the proposed and

above existing models is given in table 2.


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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 i

s are taken from [25]. The

values are assumed by R.Xu et al. [26].

Obs.No. ( , )i i iy y s=%

1i

x 2i

x 3i

x

( , )i i iY Y σ=%

Goodness

of fit

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

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

By using the proposed method, we have

1 2 3(47.299,0.7093) (1.6963,0) (0.6914,0) (0.196,0)w w w w w wY x x x= ⊕ ⊗ ⊕ ⊗ ⊕ ⊗%

with h=0.675 (shown in Fig.3)


81

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.

The results are given in table3, the goodness of fit of everyi

y% and i

Y% are all greater than 0.9,

which indicates that the fitted result of the model is very good.

EXAMPLE-3

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

( ; ) ((8.0,1.8) ;1), ((6.4, 2.2) ;2), ((9.5, 2.6) ;3), ((13.5,2.6) ;4), ((13.0,2.4) ;5),T T T T Ty x =%

By applying the proposed approach described in section 3, the fuzzy regression model is derived

as (4.079,2.1) (1.8458,0)w w

Y 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.


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

Methods Goodness of fit

measure

Similarity

measure

Proposed method using Tw-norm based operations

(4.079,2.1) (1.8458,0)w wY x= ⊕ ⊗% with

h=0.468

0.9068 0.61116

Tanaka et al. method (THW)[27]

(0.385,7.7) 2.1Y x= ⊕ ⊗% 0.9424 0.43625

Kim and bishu model 1988 (KB)[28]

(3.11, 4.95,6.84) (1.55,1.71,1.82)Y x= ⊕ ⊗% 0.9125 0.4987

Nasrabedi and Nasrabedi (2006) method (NN)[32]

(2.36, 4.86,7) 1.73Y x= ⊕ ⊗% 0.9149 0.51554

Diamond (1988)(DM)[29]

(3.11, 4.95,6.79) (1.55,1.71,1.87)Y x= ⊕ ⊗% 0.9135 0.4865

Wu and Tseng method (WT)[30]

(3.11, 4.95,6.79) (1.55,1.71,1.87)Y x= ⊕ ⊗% 0.9135 0.4865

Hojati et al 2005(HBS)[31]

(5.1,6.75,8.4) (1.1,1.25,1.4)Y x= ⊕ ⊗% 0.8915 0.62467

Chen and Hseuh method 2009(CH)[33]

1.71 (2.63, 4.95,7.27)Y x= ⊗ ⊕% 0.9175 0.4865

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


83

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. 37 705-

712.

[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. 3-

28.

[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. 73-

78.


84

[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.

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