About Two Manners Of Use Weighted Least Squares Approach In Fuzzy Statistics.
ABOUT TWO MANNERS OF USE WEIGHTED LEAST
SQUARES APPROACH IN FUZZY STATISTICS
1
1
Ciprian Costin POPESCU, 2 Supian SUDRADJAT
Department of Mathematics, Academy of Economic Studies Bucharest
Calea Dorobanţilor, nr. 15-17, Sector 1, Bucharest, Romania
e-mail: cippx@yahoo.com
2
Faculty of Mathematics and Natural Sciences, Padjanjaran University,
Bandung, Indonesia
e-mail: adjat03@yahoo.com
Abstract
Two fuzzy statistical models are generalized and extended to the weighted
models. The new regression problems are solved and compared with the originals.
Key Words: fuzzy number; regression; weighted distance; weighted least squares.
1. INTRODUCTION
This work tackles the parameter estimation for regression problems with
fuzzy data. Generally, the estimation problems consist in choosing and minimizing an objective function (see Arthanary and Dodge [1]). One of the most
used methods in this domain is fuzzy least squares (Diamond [5,6,7] and Coppi
[4] for triangular fuzzy numbers; Ming, Friedman and Kandel [11] in a more general case). This paper brings together two models, consequently two approaches
for this subject: …rst, proposed by Coppi et al. [4] and the second, given by
Ming et al. [11]. We generalize the two norms in each case and obtain the
weighted models. This kind of weighted models describes with more accuracy
various phenomenons (for example a lot of economic models). The estimations
for parameters and the regression lines in each distinct situation are given.
2. THE WEIGHTED COPPI MODEL
We generalize and test the viability of the algorithm developed by R. Coppi
et al. [4] for estimation problems which implies fuzzy data.
Let the input fuzzy variables X1 ; :::; Xm and a fuzzy output variable, Y , on
a size n sample [4]. The data will be denoted by (yei ; xi ), i = 1; :::; n, where
xTi = (xi1 ; :::; xim ). We work with LR fuzzy variables: Ye = (m; l; u)LR , where
1
m
L
y
l
(y) =
y
R
m
u
;y
m; (l > 0)
;y
m; (u > 0)
.
We consider the theoretical values , L , U and the errors ", "L , "U . Thus
we may write
m= +"
m l=
L + "L
m + u = u + U + "U (2.1)
or, after a reparametrization:
=F
L = L + "L 1
U = U + "U 1 (2.2)
where F is a matrix, whose rows have the form fiT = [f1 (xi ) ; :::; fp (xi )].
Thus the theoretical values of the output variable are yei =
i ; Li ; Ui
LR
,
i = 1; n.
De…nition 2.1.
We introduce the following weighted distance which is a generalization of a
metric proposed by Coppi and d’Urso [4]: for w = (w1 ; w2 ; w3 ) we have
d2 (w; ye; ye ) = d2 w; (m; l; u)LR ; ; L ; U LR = 2LR (w; ) =
= w1 km
2
k + w2 (m
+w3 (m + u)
2
+
2
l)
U
L
Theorem 2.1.
The following relation holds: h
d2 (w; ye; ye ) = (w1 + w2 + w3 ) (m
2w2 (m
T
)
l
T
+2w3 (m
)
Proof:
d2 (w; ye; ye ) =
+w3 (m
= hw1 (m
+ w2 (m
2
+w2
l
h
+ w3 (m
+w3
2
u
L
u
2
LR
U
+ w2
+ w3
2
2
i
)
T
) (m
T
l
l
L
u
T
L
u
U
+
U
.
2
k + w2 (m
(w; ) = w1 km
)
2
=
)+ u
U
T
) (m
)+
T
) (m
) 2w2 (m
i
T
+
l
L
L
T
) (m
T
u
h
= (w1 + w2 + w3 ) (m
2w2 (m
+
.
U
T
)
l
) + 2w3 (m
i
=
U
T
) (m
L
+ w2
2
l
T
)
T
)
i
)
T
L
2
l
L
u
U
l
L
+
+
+
l
2
L
+
T
T
2
u
u
+2w3 (m
) u
U
U .
U + w3
For particular case w1 = w2 = w3 = 1 we obtain the results from Coppi and
d’Urso [4].
Remark 2.1.
The algorithm for …nding , L , U , L , U is reducing to minimizing the
weighted distance d2w between the experimental measurements of the response
variable yei , i = 1; n and the theoretical values yei .
In other words, we have to solve the problem:
;
Theorem 2.2.
The problem
;
2
LR
min
L; U ; L; U
2
LR
min
L; U ; L; U
w; ;
w; ;
L; U ; L; U
L; U ; L; U
.
admitts local solutions
(local minimum) which may be improved using an iterative estimation algorithm.
Proof:
We have:
2
LR w; ; L ; U ;h L ; U =
i
T
= (w1 + w2 + w3 ) m F
m F
2w2
+w2
m
2
+2w3
1
m
2
F
F
F
T
L
T
1
1
u
F
T
L
F
L
1
1
F
U
1
+
L
1
L
U
L
+
+
T
u F U 1 U =
+w3
u F U 1 U
T
= (w1 + w2 + w3 ) m m 2mT F + T F T F
T T
F l + TFTF L + TFT1 L +
2w2 mT l mT F L mT 1 L
2 T
+w2
l l 2lT F L 2lT 1 L + T F T F 2L + 2 T F T 1 L L + n 2L +
T T
F u + TFTF U + TFT1 U +
+2w3 mT u mT F U mT 1 U
T
T
T
2
T T
u u 2u F U 2u 1 U + F F 2U + 2 T F T 1 U U + n 2U .
+w3
We equate to zero the partial derivatives of
2
LR
w;
L; U ; L; U ;
:
T T
T T
T T
T T
w2 T F T m
F F
F l
F F L
F 1 L = 0 (2.1)
T T
T T
T T
T T
T T
F m+ F F
F u
F F U
F 1 U = 0 (2.2)
w3
T T
T
T
T T
w2
F 1 m 1+ l 1
F 1 L n L = 0 (2.3)
T T
uT 1
F 1 U n U = 0 (2.4)
w3 T F T 1 mT 1
T
T
(w1 + w2 + w3 ) 2w2 L + w2 2 2L + 2w3 U + w3 2 2U =
F F
= (w1 + w2 + w3 ) F T m w2 F T m L + F T l F T 1 L +
+w2 2 F T l L F T 1 L L + w3 F T m U + F T u F T 1 U +
+w3 2 F T u U F T 1 U U (2.5)
3
An iterative solution is given by relations (2.6)-(2.10):
1
1
T T
T T
T T
T T
F F
F l
F 1 L
F m
L =
1
T
T
1
T
T
T
=
FTF
(2.6)
T
F F
F u
F 1 U +
T T
FTm
F F
(2.7)
1
(2.8)
1T m F
1T l F L
L = (n )
1
T
T
(2.9)
1 u F U +1 m F
U = (n )
= [(w1 + w2 + w3 ) w2 L (2
L ) + w3
U (2 +
U
+
T
T
U )]
1
1
FTF
FT
(w1 + w2 + w3 ) m w2 (m L + l 1 L ) + w2 2 (l L 1 L L ) +
+w3 (m U + u 1 U ) + w3 2 (u U 1 U U ) (2.10)
We don’t have the certainty that the equations (2.6)-(2.10) give a global
solution. It’s necessary to resort to an iterative algorithm. The routine for
…nding the iterative solution with help of a computer is available in literature
(see Coppi et al.).
Remark 2.2.
After some calculation, we observe that the properties of the solution obtained
from the weighted model are the same as in the Coppi’s nonweighted model (see
Coppi et al. [4], Proposition 1,2,3):
T
i) w1 1T (m b) = 0; w2 1T l c
u c
L = 0; w3 1
U = 0;
T
ii) w1 (m b) b = 0;
T
c
iii) l c
L = 0, u
L
c
U
T
c
U = 0;
where b,dL ,d
U are the iterative solutions obtained from the normal equations system (2.6)-(2.10).
3. AN EXTENSION FOR THE MING APPROACH
In this section, a weighted model especially based on Ming approach in the
…eld of fuzzy optimization, is developed.
De…nition 3.1.
We consider a fuzzy space F 1 as a function space with the following properties [2,11]:
i) the elements of F 1 are the functions f : R ! [0; 1] , which are called fuzzy
numbers;
ii) f (x0 ) = 1 for some x0 2 R;
iii) f ( x + (1
) y) min ff (x) ; f (y)g ; x; y 2 R; 2 [0; 1] ;
iv) lim supf (x) = f (t) ; t 2 R;
x!t
0
v) [f ] =closureft=t 2 R; f (t) > 0g is compact.
4
De…nition 3.2.
If f; g 2 F 1 , 2 R, r 2 [0; 1] we may introduce [10,12]:
ft=f (t) rg ; 0 < r 1
r
[f ] =
; consequently, the operations with fuzzy
ft=f (t) > 0g ; r = 0
numbers are:
r
r
r
r
i) [f ] + [g] = fa + b=a 2 [f ] ; b 2 [g] g;
r
r
ii) [f ] = f a=a 2 [f ] g.
Remark 3.1.
r
[f ] = f (r) ; f (r) is a closed interval and [8,11]:
i)f (r) is a bounded left continuous nondecreasing function over [0; 1] ;
ii)f (r) is a bounded left continuous nonincreasing function over [0; 1] ;
iii)f (r) f (r) ; for all r 2 [0; 1] ;
iv)the functions f (r) ; f (r) de…ne a unique fuzzy number f 2 F 1 :
Remark 3.2.
Since Xi = Xi (r) ; Xi (r) 2 F 1 then the triangular form for Xi is:
Xi = xi ; ui ; ui where Xi (r) = xi ui + ui r and Xi (r) = xi + ui ui r [11].
De…nition 3.3.
Now we consider a generalization of metric D2 [2,3,8,9,11], in fact a weighted
metric:
for f; g; w 2 F 1 we de…ne the weighted distance between f and g as follows:
D2 (w; f; g) =
R1
0
w (r) f (r)
g (r)
2
dr +
R1
0
w (r) f (r)
g (r)
2
dr:
For Xi = Xi (r) ; Xi (r) 2 F 1 input independent variables,
Yi = Yi (r) ; Yi (r) 2 F 1 response variables and wi = wi (r) ; wi (r) 2 F 1 ,
i = 1; n we have:
Xi = xi ; ui ; ui , Yi = yi ; vi ; vi , wi = zi ; ti ; ti where
Xi (r) = xi ui + ui r, Xi (r) = xi + ui ui r,
Yi (r) = yi vi + vi r, Yi (r) = yi + vi vi r,
wi (r) = zi ti + ti r, wi (r) = zi + ti ti r.
We assume that the dependence between X and Y is given by Y = a + bX;
a, b are unknown real parameters and b 6= 0. It is necessary to estimate a, b
using weighted least squares method.
Thus we must minimize the sum of squared deviations between theoretical
and experimental values:
S (a; b) =
n
P
D2 (wi ; a + bXi ; Yi ).
i=1
Theorem 3.1.
The sum of squared deviations S (a; b) depends on sign of real parameter b.
5
Proof:
R1
f (0) + f (1)
(see [6,11]).
We approximate 0 f (r) dr '
2
If b > 0 we have
n hR
P
2
1
S1 (a; b) =
wi (r) a + bXi (r) Yi (r) dr+
0
i=1
i
R1
2
+ 0 wi (r) a + bXi (r) Yi (r) dr =
i
n hR
P
2
1
=
a
+
b
x
u
+
t
+
u
r
y
+
v
v
r
dr
+
z
t
i
i
i
i
i
i
i
i
i
0
i=1
i
h
n
P R1
2
+
zi + ti ti r [a + b (xi + ui ui r) yi vi + vi r] dr '
0
i=1
n n
P
2
zi ti a + b xi ui
' 12
yi vi
+
i=1
+ zi + ti [a + b (xi o
+ ui )
2
(yi + vi )] +
2
+2zi (a + bxi yi ) .
If b < 0 we obtain
n hR
P
1
S2 (a; b) =
wi (r) a + bXi (r) Yi (r)
0
i=1
i
R1
2
+ 0 wi (r) a + bXi (r) Yi (r) dr =
n hR
P
1
=
zi ti + ti a + b (xi + ui ui r)
0
i=1
h
n R
P
1
+
zi + ti ti r a + b xi ui + ui r
0
i=1
n
n
P
' 21
yi vi
zi ti a + b (xi + ui )
2
dr+
i
dr +
i
2
vi + vi r dr '
yi + vi
yi
2
vi r
2
+
i=1
2
(yi + vi ) +
a + b xi o ui
2
+ 2zi (a + bxi yi ) .
In conclusion, if b > 0 then S (a; b) = S1 (a; b); if b < 0 then S (a; b) =
S2 (a; b).
+ zi + t i
Theorem 3.2.
The problem min
a2R;b2R
S (a; b) has a unique solution
a; b .
Consequently,
there exists a single line, the best line Y = a + bX which …t the given data
(Xi ; Yi ).
Proof:
We have the problem
min
a2R;b2R
S (a; b).
Accordingly to Theorem 3.1 we must discuss two possibilites:
1) b > 0. After equate with zero the partial derivatives of S (a; b) we obtain
the system:
n
P
4zi + ti ti +
a
i=1
6
+b
=
n
P
i=1
n
P
a
i=1
n
P
+b
i=1
=
zi
i=1
n
P
n
P
zi
ti
xi
yi
ti
ui + zi + ti (xi + ui ) + 2zi xi =
vi + zi + ti (yi + vi ) + 2zi yi
zi ti xi ui + zi + ti (xi + ui ) + 2zi xi +
h
i
2
2
zi ti xi ui + zi + ti (xi + ui ) + 2zi x2i =
zi
ti
xi
ui
yi
vi +
i=1
+ zi + ti (xi + ui ) (yi + vi ) + 2zi xi yi
The determinant is:
n
P
zi ti + zi + ti + 2zi
1 =
i=1
i
h
n
P
2
2
zi ti xi ui + zi + ti (xi + ui ) + 2zi x2i
i=1
n
P
zi
ti
xi
2
ui + zi + ti (xi + ui ) + 2zi xi
0; from Cauchy-
i=1
Schwarz inequality. We may choose 1 > 0 (because Xi are independent variables), and obtain a unique minimization point (a1 ; b1 ) for S1 (a; b):
a1 =
a1
, b1 =
b1
1
=
n h
P
a1
i=1
n
P
where
1
n
P
zi
ti
i=1
zi
zi
ti
ti
xi
xi
yi
vi + zi + ti (yi + vi ) + 2zi yi
i
2
2
ui + zi + ti (xi + ui ) + 2zi x2i
ui
yi
vi +
i=1
+ zi + ti (xi + ui ) (yi + vi ) + 2zi xi yi
n
P
zi ti xi ui + zi + ti (xi + ui ) + 2zi xi
i=1
b1
=
n
P
n
P
4zi + ti
and
ti
i=1
zi
ti
xi
ui
yi
vi +
i=1
+ zi + ti (xi + ui ) (yi + vi ) + 2zi xi yi
n
P
zi ti xi ui + zi + ti (xi + ui ) + 2zi xi
i=1
n
P
zi
i=1
n
P
zi
ti
yi
vi + zi + ti (yi + vi ) + 2zi yi
.
i=1
2) b < 0. We have:
n
P
4zi + ti ti +
a
+b
ti (xi + ui ) + zi + ti
xi
i=1
7
ui + 2zi xi =
=
a
n
P
i=1
n
P
i=1
n
P
+b
i=1
=
n
P
zi
ti
yi
vi + zi + ti (yi + vi ) + 2zi yi
zi ti (xi + ui ) + zi + ti xi ui + 2zi xi +
i
h
2
2
zi ti (xi + ui ) + zi + ti xi ui + 2zi x2i =
zi
ti (xi + ui ) yi
vi +
i=1
+ z i + ti
Now,
n h
P
i=1
n
P
2
xi ui (yi + vi ) + 2zi xi yi
n
P
4zi + ti ti
=
i=1
zi
zi
2
ti (xi + ui ) + zi + ti
2
xi
ui
ti (xi + ui ) + zi + ti
xi
ui + 2zi xi
+ 2zi x2i
i
2
0.
i=1
As in above case we obtain that this system has a unique minimization point
(a2 ; b2 ).
If S (a1 ; b1 ) < S (a2 ; b2 ) then a; b = (a1 ; b1 ).
If S (a1 ; b1 ) > S (a2 ; b2 ) then a; b = (a2 ; b2 ).
Thus we obtain a unique solution for our estimation problem. The theorem
was proved.
References
[1] T. S. Arthanari and Yadolah Dodge, Mathematical Programming in Statistics, John Wiley and Sons, New York, 1980.
[2] Wu Cong-Xin and Ma Ming, Embedding problem of fuzzy number space:
Part I, Fuzzy Sets and Systems 44 (1991) 33-38.
[3] Wu Cong-Xin and Ma Ming, Embedding problem of fuzzy number space:
Part III, Fuzzy Sets and Systems 46 (1992) 281-286.
[4] Renato Coppi, Pierpaolo D’Urso, Paolo Giordani, Adriana Santoro, Least
squares estimation of a linear regression model with LR fuzzy response, Computational Statistics and Data Analysis 51 (2006), 267-286.
[5] P. Diamond and P. Kloeden, Metric spaces of fuzzy sets, Fuzzy Sets and
Systems 35 (1990) 241-249.
[6] P. Diamond, Fuzzy least squares, Inform. Sci. 46 (1988) 141-157.
[7] P. Diamond and P. Kloeden, Metric spaces of fuzzy sets, Corrigendum,
Fuzzy Sets and Systems 45 (1992) 123.
[8] R. Goetschel, W. Voxman, Elementary Calculus, Fuzzy Sets and Systems
18 (1986) 31-43.
[9] I.M. Hammerbacher and R. R. Yager, Predicting television revenues using
fuzzy subsets, TIMS Stud. Management Sci. 20 (1984) 469-477.
[10] A. Katsaras and D. B. Liu, Fuzzy vector spaces and fuzzy topological
vector spaces, J. Math. Anal. Appl. 58 (1977) 135-146.
8
[11] Ma Ming, M. Friedman, A. Kandel, General fuzzy least squares, Fuzzy
Sets and Systems 88 (1997) 107-118.
[12] C. V. Negoiţ¼
a and D. A. Ralescu, Applications of Fuzzy Sets to Systems
Analysis, Wiley, New York, 1975.
[13] H. Prade, Operations research with fuzzy data, in: P. P. Wang and
S. K. Chang, Eds., Fuzzy sets: Theory and Application to Policy Analysis and
Information Systems (plenum, New York, 1980) 115-169.
[14] V. Preda and V. Craiu, Inferenţ¼a statistic¼a pentru valori medii condiţionate, Editura Universit¼
aţii Bucureşti (1992).
[15] M. L. Puri and D. A. Ralescu, Di¤erentials for fuzzy functions, J. Math.
Anal. Appl. 91 (1983) 552-558.
[16] H. Tanaka, H. Isibuchi and S. Yoshikawa, Exponential possibility regression analysis, Fuzzy Sets and Systems 69 (1995) 305-318.
[17] H. Tanaka, S. Uejima and K. Asai, Linear regression analysis with fuzzy
model, IEEE Trans. Systems Man Cybernet. SMC-12 (1982) 903-907.
[18] H. J. Zimmermann, Fuzzy programming and linear programming with
several objective functions, Fuzzy Sets and Systems 1 (1978) 45-55.
[19] R. R. Yager, Fuzzy prediction based upon regression models, Inform.
Sci. 26 (1982) 45-63.
9
SQUARES APPROACH IN FUZZY STATISTICS
1
1
Ciprian Costin POPESCU, 2 Supian SUDRADJAT
Department of Mathematics, Academy of Economic Studies Bucharest
Calea Dorobanţilor, nr. 15-17, Sector 1, Bucharest, Romania
e-mail: cippx@yahoo.com
2
Faculty of Mathematics and Natural Sciences, Padjanjaran University,
Bandung, Indonesia
e-mail: adjat03@yahoo.com
Abstract
Two fuzzy statistical models are generalized and extended to the weighted
models. The new regression problems are solved and compared with the originals.
Key Words: fuzzy number; regression; weighted distance; weighted least squares.
1. INTRODUCTION
This work tackles the parameter estimation for regression problems with
fuzzy data. Generally, the estimation problems consist in choosing and minimizing an objective function (see Arthanary and Dodge [1]). One of the most
used methods in this domain is fuzzy least squares (Diamond [5,6,7] and Coppi
[4] for triangular fuzzy numbers; Ming, Friedman and Kandel [11] in a more general case). This paper brings together two models, consequently two approaches
for this subject: …rst, proposed by Coppi et al. [4] and the second, given by
Ming et al. [11]. We generalize the two norms in each case and obtain the
weighted models. This kind of weighted models describes with more accuracy
various phenomenons (for example a lot of economic models). The estimations
for parameters and the regression lines in each distinct situation are given.
2. THE WEIGHTED COPPI MODEL
We generalize and test the viability of the algorithm developed by R. Coppi
et al. [4] for estimation problems which implies fuzzy data.
Let the input fuzzy variables X1 ; :::; Xm and a fuzzy output variable, Y , on
a size n sample [4]. The data will be denoted by (yei ; xi ), i = 1; :::; n, where
xTi = (xi1 ; :::; xim ). We work with LR fuzzy variables: Ye = (m; l; u)LR , where
1
m
L
y
l
(y) =
y
R
m
u
;y
m; (l > 0)
;y
m; (u > 0)
.
We consider the theoretical values , L , U and the errors ", "L , "U . Thus
we may write
m= +"
m l=
L + "L
m + u = u + U + "U (2.1)
or, after a reparametrization:
=F
L = L + "L 1
U = U + "U 1 (2.2)
where F is a matrix, whose rows have the form fiT = [f1 (xi ) ; :::; fp (xi )].
Thus the theoretical values of the output variable are yei =
i ; Li ; Ui
LR
,
i = 1; n.
De…nition 2.1.
We introduce the following weighted distance which is a generalization of a
metric proposed by Coppi and d’Urso [4]: for w = (w1 ; w2 ; w3 ) we have
d2 (w; ye; ye ) = d2 w; (m; l; u)LR ; ; L ; U LR = 2LR (w; ) =
= w1 km
2
k + w2 (m
+w3 (m + u)
2
+
2
l)
U
L
Theorem 2.1.
The following relation holds: h
d2 (w; ye; ye ) = (w1 + w2 + w3 ) (m
2w2 (m
T
)
l
T
+2w3 (m
)
Proof:
d2 (w; ye; ye ) =
+w3 (m
= hw1 (m
+ w2 (m
2
+w2
l
h
+ w3 (m
+w3
2
u
L
u
2
LR
U
+ w2
+ w3
2
2
i
)
T
) (m
T
l
l
L
u
T
L
u
U
+
U
.
2
k + w2 (m
(w; ) = w1 km
)
2
=
)+ u
U
T
) (m
)+
T
) (m
) 2w2 (m
i
T
+
l
L
L
T
) (m
T
u
h
= (w1 + w2 + w3 ) (m
2w2 (m
+
.
U
T
)
l
) + 2w3 (m
i
=
U
T
) (m
L
+ w2
2
l
T
)
T
)
i
)
T
L
2
l
L
u
U
l
L
+
+
+
l
2
L
+
T
T
2
u
u
+2w3 (m
) u
U
U .
U + w3
For particular case w1 = w2 = w3 = 1 we obtain the results from Coppi and
d’Urso [4].
Remark 2.1.
The algorithm for …nding , L , U , L , U is reducing to minimizing the
weighted distance d2w between the experimental measurements of the response
variable yei , i = 1; n and the theoretical values yei .
In other words, we have to solve the problem:
;
Theorem 2.2.
The problem
;
2
LR
min
L; U ; L; U
2
LR
min
L; U ; L; U
w; ;
w; ;
L; U ; L; U
L; U ; L; U
.
admitts local solutions
(local minimum) which may be improved using an iterative estimation algorithm.
Proof:
We have:
2
LR w; ; L ; U ;h L ; U =
i
T
= (w1 + w2 + w3 ) m F
m F
2w2
+w2
m
2
+2w3
1
m
2
F
F
F
T
L
T
1
1
u
F
T
L
F
L
1
1
F
U
1
+
L
1
L
U
L
+
+
T
u F U 1 U =
+w3
u F U 1 U
T
= (w1 + w2 + w3 ) m m 2mT F + T F T F
T T
F l + TFTF L + TFT1 L +
2w2 mT l mT F L mT 1 L
2 T
+w2
l l 2lT F L 2lT 1 L + T F T F 2L + 2 T F T 1 L L + n 2L +
T T
F u + TFTF U + TFT1 U +
+2w3 mT u mT F U mT 1 U
T
T
T
2
T T
u u 2u F U 2u 1 U + F F 2U + 2 T F T 1 U U + n 2U .
+w3
We equate to zero the partial derivatives of
2
LR
w;
L; U ; L; U ;
:
T T
T T
T T
T T
w2 T F T m
F F
F l
F F L
F 1 L = 0 (2.1)
T T
T T
T T
T T
T T
F m+ F F
F u
F F U
F 1 U = 0 (2.2)
w3
T T
T
T
T T
w2
F 1 m 1+ l 1
F 1 L n L = 0 (2.3)
T T
uT 1
F 1 U n U = 0 (2.4)
w3 T F T 1 mT 1
T
T
(w1 + w2 + w3 ) 2w2 L + w2 2 2L + 2w3 U + w3 2 2U =
F F
= (w1 + w2 + w3 ) F T m w2 F T m L + F T l F T 1 L +
+w2 2 F T l L F T 1 L L + w3 F T m U + F T u F T 1 U +
+w3 2 F T u U F T 1 U U (2.5)
3
An iterative solution is given by relations (2.6)-(2.10):
1
1
T T
T T
T T
T T
F F
F l
F 1 L
F m
L =
1
T
T
1
T
T
T
=
FTF
(2.6)
T
F F
F u
F 1 U +
T T
FTm
F F
(2.7)
1
(2.8)
1T m F
1T l F L
L = (n )
1
T
T
(2.9)
1 u F U +1 m F
U = (n )
= [(w1 + w2 + w3 ) w2 L (2
L ) + w3
U (2 +
U
+
T
T
U )]
1
1
FTF
FT
(w1 + w2 + w3 ) m w2 (m L + l 1 L ) + w2 2 (l L 1 L L ) +
+w3 (m U + u 1 U ) + w3 2 (u U 1 U U ) (2.10)
We don’t have the certainty that the equations (2.6)-(2.10) give a global
solution. It’s necessary to resort to an iterative algorithm. The routine for
…nding the iterative solution with help of a computer is available in literature
(see Coppi et al.).
Remark 2.2.
After some calculation, we observe that the properties of the solution obtained
from the weighted model are the same as in the Coppi’s nonweighted model (see
Coppi et al. [4], Proposition 1,2,3):
T
i) w1 1T (m b) = 0; w2 1T l c
u c
L = 0; w3 1
U = 0;
T
ii) w1 (m b) b = 0;
T
c
iii) l c
L = 0, u
L
c
U
T
c
U = 0;
where b,dL ,d
U are the iterative solutions obtained from the normal equations system (2.6)-(2.10).
3. AN EXTENSION FOR THE MING APPROACH
In this section, a weighted model especially based on Ming approach in the
…eld of fuzzy optimization, is developed.
De…nition 3.1.
We consider a fuzzy space F 1 as a function space with the following properties [2,11]:
i) the elements of F 1 are the functions f : R ! [0; 1] , which are called fuzzy
numbers;
ii) f (x0 ) = 1 for some x0 2 R;
iii) f ( x + (1
) y) min ff (x) ; f (y)g ; x; y 2 R; 2 [0; 1] ;
iv) lim supf (x) = f (t) ; t 2 R;
x!t
0
v) [f ] =closureft=t 2 R; f (t) > 0g is compact.
4
De…nition 3.2.
If f; g 2 F 1 , 2 R, r 2 [0; 1] we may introduce [10,12]:
ft=f (t) rg ; 0 < r 1
r
[f ] =
; consequently, the operations with fuzzy
ft=f (t) > 0g ; r = 0
numbers are:
r
r
r
r
i) [f ] + [g] = fa + b=a 2 [f ] ; b 2 [g] g;
r
r
ii) [f ] = f a=a 2 [f ] g.
Remark 3.1.
r
[f ] = f (r) ; f (r) is a closed interval and [8,11]:
i)f (r) is a bounded left continuous nondecreasing function over [0; 1] ;
ii)f (r) is a bounded left continuous nonincreasing function over [0; 1] ;
iii)f (r) f (r) ; for all r 2 [0; 1] ;
iv)the functions f (r) ; f (r) de…ne a unique fuzzy number f 2 F 1 :
Remark 3.2.
Since Xi = Xi (r) ; Xi (r) 2 F 1 then the triangular form for Xi is:
Xi = xi ; ui ; ui where Xi (r) = xi ui + ui r and Xi (r) = xi + ui ui r [11].
De…nition 3.3.
Now we consider a generalization of metric D2 [2,3,8,9,11], in fact a weighted
metric:
for f; g; w 2 F 1 we de…ne the weighted distance between f and g as follows:
D2 (w; f; g) =
R1
0
w (r) f (r)
g (r)
2
dr +
R1
0
w (r) f (r)
g (r)
2
dr:
For Xi = Xi (r) ; Xi (r) 2 F 1 input independent variables,
Yi = Yi (r) ; Yi (r) 2 F 1 response variables and wi = wi (r) ; wi (r) 2 F 1 ,
i = 1; n we have:
Xi = xi ; ui ; ui , Yi = yi ; vi ; vi , wi = zi ; ti ; ti where
Xi (r) = xi ui + ui r, Xi (r) = xi + ui ui r,
Yi (r) = yi vi + vi r, Yi (r) = yi + vi vi r,
wi (r) = zi ti + ti r, wi (r) = zi + ti ti r.
We assume that the dependence between X and Y is given by Y = a + bX;
a, b are unknown real parameters and b 6= 0. It is necessary to estimate a, b
using weighted least squares method.
Thus we must minimize the sum of squared deviations between theoretical
and experimental values:
S (a; b) =
n
P
D2 (wi ; a + bXi ; Yi ).
i=1
Theorem 3.1.
The sum of squared deviations S (a; b) depends on sign of real parameter b.
5
Proof:
R1
f (0) + f (1)
(see [6,11]).
We approximate 0 f (r) dr '
2
If b > 0 we have
n hR
P
2
1
S1 (a; b) =
wi (r) a + bXi (r) Yi (r) dr+
0
i=1
i
R1
2
+ 0 wi (r) a + bXi (r) Yi (r) dr =
i
n hR
P
2
1
=
a
+
b
x
u
+
t
+
u
r
y
+
v
v
r
dr
+
z
t
i
i
i
i
i
i
i
i
i
0
i=1
i
h
n
P R1
2
+
zi + ti ti r [a + b (xi + ui ui r) yi vi + vi r] dr '
0
i=1
n n
P
2
zi ti a + b xi ui
' 12
yi vi
+
i=1
+ zi + ti [a + b (xi o
+ ui )
2
(yi + vi )] +
2
+2zi (a + bxi yi ) .
If b < 0 we obtain
n hR
P
1
S2 (a; b) =
wi (r) a + bXi (r) Yi (r)
0
i=1
i
R1
2
+ 0 wi (r) a + bXi (r) Yi (r) dr =
n hR
P
1
=
zi ti + ti a + b (xi + ui ui r)
0
i=1
h
n R
P
1
+
zi + ti ti r a + b xi ui + ui r
0
i=1
n
n
P
' 21
yi vi
zi ti a + b (xi + ui )
2
dr+
i
dr +
i
2
vi + vi r dr '
yi + vi
yi
2
vi r
2
+
i=1
2
(yi + vi ) +
a + b xi o ui
2
+ 2zi (a + bxi yi ) .
In conclusion, if b > 0 then S (a; b) = S1 (a; b); if b < 0 then S (a; b) =
S2 (a; b).
+ zi + t i
Theorem 3.2.
The problem min
a2R;b2R
S (a; b) has a unique solution
a; b .
Consequently,
there exists a single line, the best line Y = a + bX which …t the given data
(Xi ; Yi ).
Proof:
We have the problem
min
a2R;b2R
S (a; b).
Accordingly to Theorem 3.1 we must discuss two possibilites:
1) b > 0. After equate with zero the partial derivatives of S (a; b) we obtain
the system:
n
P
4zi + ti ti +
a
i=1
6
+b
=
n
P
i=1
n
P
a
i=1
n
P
+b
i=1
=
zi
i=1
n
P
n
P
zi
ti
xi
yi
ti
ui + zi + ti (xi + ui ) + 2zi xi =
vi + zi + ti (yi + vi ) + 2zi yi
zi ti xi ui + zi + ti (xi + ui ) + 2zi xi +
h
i
2
2
zi ti xi ui + zi + ti (xi + ui ) + 2zi x2i =
zi
ti
xi
ui
yi
vi +
i=1
+ zi + ti (xi + ui ) (yi + vi ) + 2zi xi yi
The determinant is:
n
P
zi ti + zi + ti + 2zi
1 =
i=1
i
h
n
P
2
2
zi ti xi ui + zi + ti (xi + ui ) + 2zi x2i
i=1
n
P
zi
ti
xi
2
ui + zi + ti (xi + ui ) + 2zi xi
0; from Cauchy-
i=1
Schwarz inequality. We may choose 1 > 0 (because Xi are independent variables), and obtain a unique minimization point (a1 ; b1 ) for S1 (a; b):
a1 =
a1
, b1 =
b1
1
=
n h
P
a1
i=1
n
P
where
1
n
P
zi
ti
i=1
zi
zi
ti
ti
xi
xi
yi
vi + zi + ti (yi + vi ) + 2zi yi
i
2
2
ui + zi + ti (xi + ui ) + 2zi x2i
ui
yi
vi +
i=1
+ zi + ti (xi + ui ) (yi + vi ) + 2zi xi yi
n
P
zi ti xi ui + zi + ti (xi + ui ) + 2zi xi
i=1
b1
=
n
P
n
P
4zi + ti
and
ti
i=1
zi
ti
xi
ui
yi
vi +
i=1
+ zi + ti (xi + ui ) (yi + vi ) + 2zi xi yi
n
P
zi ti xi ui + zi + ti (xi + ui ) + 2zi xi
i=1
n
P
zi
i=1
n
P
zi
ti
yi
vi + zi + ti (yi + vi ) + 2zi yi
.
i=1
2) b < 0. We have:
n
P
4zi + ti ti +
a
+b
ti (xi + ui ) + zi + ti
xi
i=1
7
ui + 2zi xi =
=
a
n
P
i=1
n
P
i=1
n
P
+b
i=1
=
n
P
zi
ti
yi
vi + zi + ti (yi + vi ) + 2zi yi
zi ti (xi + ui ) + zi + ti xi ui + 2zi xi +
i
h
2
2
zi ti (xi + ui ) + zi + ti xi ui + 2zi x2i =
zi
ti (xi + ui ) yi
vi +
i=1
+ z i + ti
Now,
n h
P
i=1
n
P
2
xi ui (yi + vi ) + 2zi xi yi
n
P
4zi + ti ti
=
i=1
zi
zi
2
ti (xi + ui ) + zi + ti
2
xi
ui
ti (xi + ui ) + zi + ti
xi
ui + 2zi xi
+ 2zi x2i
i
2
0.
i=1
As in above case we obtain that this system has a unique minimization point
(a2 ; b2 ).
If S (a1 ; b1 ) < S (a2 ; b2 ) then a; b = (a1 ; b1 ).
If S (a1 ; b1 ) > S (a2 ; b2 ) then a; b = (a2 ; b2 ).
Thus we obtain a unique solution for our estimation problem. The theorem
was proved.
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8
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9