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. Vasuki2
1
Manonmaniam Sundaranar Universty, Tirunelveli, India
1
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

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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 t
if a t a
a t
a t if a t a
otherwise
α
α
β
β
 −
− − ≤ ≤

 −
= − ≤ ≤ +




%
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 a
R a t a
a t
a t L a t a
otherwise
β
β
α
α
  −
≤ ≤ +  
 
 −  
= − ≤ ≤  
 



%
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 LRA 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 b
A 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⊗ =% %

76
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,






−−=
∈∈∈∈
babaBAd
BbAaAaBb
H
α
αα
α
αα infsup,infsupmax),(
If [ ] [ ]212211 ,, bbIandaaI == are two intervals, then
( ) { } 2121221121 ,max, IsprIsprImidImidbabaIIdH −+−=−−=
Where
2
,
2
12
1
21
1
aa
sprI
aa
Imid
−
=
+
= [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
ˆ
Y 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 i
j pj
n n
j ij j ij ij j i i
j pj
j ij ij j
j 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 J
r
J
−
= is a synthetic evaluation of the impact of the ith
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 B
d 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 p
a b
A B
σ τ
 − 
= −  
+  
% %
is the goodness of fit of observed and estimated fuzzy
numbers A% and B% .

78
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 J
r
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 w
Y X X X= ⊕ ⊗ ⊕ ⊗ ⊕ ⊗% with
optimum value h=0.215 and the graph is given in Fig.1. In table 1, third data is an

79
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 every i
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.

82
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.
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