A FUZZY MINMAD PROCEDURE

A FUZZY MINMAD PROCEDURE

CIPRIAN COSTIN POPESCU

We present a new approach to parameter estimation on fuzzy sets. The model combines fuzzy number theory with the minmad (minimizing mean of absolute deviations) method.

AMS 2000 Subject Classification: 90C05.

Key words: fuzzy number, minmad, convex function.

1. INTRODUCTION

We present a new mixed method for parameter estimation when the input data are fuzzy numbers. In Section 2 we transfer fuzzy numbers to intervals bounded by functions with some special properties while the norm in this space of fuzzy numbers is the distance between the images of these functions [2, 3, 12]. In Section 3 we discuss the regression model based on the norm previously defined. Many authors, for example Ming et al [12], have used variants of the least squares method for determining the solution. We suggest a new approach, a generalized minmad technique [1]. We prove the equivalence between our model and the minimization of a special function.

Various methods are available for this minimization, for example the tangents method [16].

2. PRELIMINARIES

LetF be a fuzzy number space [2, 12]. For a fuzzy numberX∈ F and a real numberr∈[0,1] we define [3, 7, 9, 12]: [X]^r={t/X(t)≥r}if 0< r≤1 and [X]^r ={t/X(t)>0} ifr= 0.

If X(r), X(r), X(r) ≤ X(r) are the end points of the interval [X]^r, then these two functions with some special properties [5,12], define a unique fuzzy numberX. Thus we may introduce a metricD₂ over F [3,5,7,12] as

D₂(X, Y) = ₁

0 [X(r)−Y (r)]²dr+ ₁

0

X(r)−Y (r)₂ dr.

MATH. REPORTS9(59),2 (2007), 211–221

To simplify the writing, in the next sections we write X, X instead of X(r), X(r).

3. THE MODEL

Let the input data (X_i, Y_i), i = 1, n, where X_i, Y_i are fuzzy numbers in F. The goal is to find real numbers α, β such that Y = α+βX give the fit line to the above data. We choose a minmad technique (minimizing mean of absolute deviations) which consist in finding the parametersα,β such that ¹_n

n

i=1D₂(α+βX_i, Y_i) is minimal. Thus, Ω (α, β) = n

i=1D₂(α+βX_i, Y_i) should be minimal. We consider the model with an additional constraint:

the regression line contains the point (X₀, Y₀) i.e. Y₀ = α+βX₀, where X₀ =

X₀, X₀

,Y₀ = Y₀, Y₀

, withX₀(r) =X₀(r) = const.,Y₀(r) =Y₀(r) = const. for all r∈[0,1].

Theorem 1. The minimization of Ω (α, β) is equivalent to the min- imization of a certain function ω : R^∗ → R₊, ω(x) =

ω₁(x), x >0, ω₂(x), x <0, where ω₁(x), ω₂(x) are convex, continuous functions of general form n

i=1 p⁽²⁾_i x²+p⁽¹⁾_i x+p⁽⁰⁾_i with p⁽²⁾_i ,p⁽¹⁾_i , p⁽⁰⁾_i real numbers for i= 1, n. Fur- ther, under an additional condition which is given below,Ω (α, β) has a unique minimizing point.

Proof. Since the sign of the parameter β influences the form of Ω (α, β), two cases should be discussed: β >0 or β <0.

Case1: β >0. In this case Ω (α, β) has the form:

Ω (α, β) = Ω₁(α, β) = n

i=1

₁

0

α+βX_i−Y_i2 dr+

₁

0

α+βX_i−Y_i2 dr.

We make the substitutions

ζ_i=X_i−X₀ ξ_i=Y_i−Y₀ ;

ζ_i =X_i−X₀ ξ_i=Y_i−Y₀ . Obviously,Y₀ =α+βX₀. Thus

Y_i−ξ_i=Y₀ =α+βX₀ =α+β

X_i−ζ_i

=α+βX_i−βζ_i⇒

⇒α+βX_i−Y_i=βζ_i−ξ_i. Similarly,α+βX_i−Y_i =βζ_i−ξ_i.

Now, the problem is equivalent to determining the minimum of the func- tion

ω₁(β) = n

i=1

₁

0

βζ_i−ξ_i2 dr+

₁

0

βζ_i−ξ_i2 dr.

We have ω₁(β) =

n i=1

β²

₁

0

ζ_i²+ζ_i²

dr−2β ₁

0

ζ_iξ_i+ζ_iξ_i dr+

₁

0

ξ_i²+ξ_i²

dr or

ω₁(x) = n

i=1

a_ix²+b_ix+c_i = n i=1

v_i(x) = n

i=1

f_i(x), where β = x, a_i = ₁

0

ζ_i²+ζ_i²

dr > 0, b_i = −2₁

0

ζ_iξ_i+ζ_iξ_i

dr, c_i = ₁

0

ξ_i²+ξ_i²

dr.Obviously, in this case,ω(x) =ω₁(x) andp⁽²⁾_i =a_i,p⁽¹⁾_i =b_i, p⁽⁰⁾_i =c_i.

We also have

∆_vi =b²_i −4a_ic_i=

= 4 ₁

0

ζ_iξ_i+ζ_iξ_i dr

₂

− ₁

0

ζ_i²+ζ_i²

dr

1 0

ξ_i²+ξ_i²

dr

. By using twice the Cauchy-Buniakowski-Schwarz inequality we obtain

₁

0

ζ_i²+ζ_i²

dr

1 0

ξ_i²+ξ_i²

dr

=

= ₁

0 ζ_i²+ζ_i² ₂

dr

1

0 ξ_i²+ξ_i² ₂

dr

≥

≥ ₁

0

ζ_i²+ζ_i² ξ_i²+ξ_i²

dr ₂

≥ ₁

0

ζ_iξ_i+ζ_iξ_i dr

₂

which gives ∆_vi ≤0; without very much restricting generality, we may consider the case when ∆_vi <0. Then ω₁ is differentiable in its domain and

ω₁(x) = n

i=1

f_i(x) = n i=1

2a_ix+b_i 2√

a_ix²+b_ix+c_i, ω₁(x) =

n i=1

f_i(x) = n i=1

−∆_vi

4 (a_ix²+b_ix+c_i)^3/2 >0.

Thereforeω₁(x) is a convex function.

Since lim

x→∞

f_i(x)

x = √

a_i, lim

x→∞

f_i(x)− √a_ix

= ₂√^bi

ai, the function ω₁(x) admits the asymptotes

− n

i=1

√a_i

x− n

i=1

b_i 2√ a_i

,

n i=1

√a_i

x+ n

i=1

b_i 2√ a_i

at −∞and +∞, respectively.

The previous allow the conclusion that ω₁(x) has a unique minimizing point,x, which is situated in the closed interval

mini

−_2a^bi_i ,max

i

−_2a^bi_i

or, more precisely, in [A₁, B₁],where A₁ = max

i=1,n

− b_i 2a_iω₁

− b_i 2a_i

≤0

,

B₁ = min

i=1,n

− b_i 2a_iω₁

− b_i 2a_i

≥0

. Case2: β <0. Now, we obtain

Ω (α, β) = Ω₂(α, β) =

= n

i=1

₁

0

α+βX_i−Y_i2 dr+

₁

0

α+βX_i−Y_i2 dr.

We make the same substitutions as in Case 1:

ζ_i=X_i−X₀ ξ_i=Y_i−Y₀ ;

ζ_i =X_i−X₀ ξ_i=Y_i−Y₀ . It follows that

Y_i−ξ_i=Y₀ =α+βX₀ =α+β

X_i−ζ_i

=α+βX_i−βζ_i⇒

⇒α+βX_i−Y_i =βζ_i−ξ_i and

Y_i−ξ_i=Y₀ =α+βX₀ =α+β

X_i−ζ_i

=α+βX_i−βζ_i⇒

⇒α+βX_i−Y_i=βζ_i−ξ_i. The function which should be minimized is

ω₂(β) = n

i=1

₁

0

βζ_i−ξ_i2 dr+

₁

0

βζ_i−ξ_i2 dr.

Thus ω₂(β) =

n i=1

β²

₁

0

ζ_i²+ζ_i²

dr−2β ₁

0

ζ_iξ_i+ζ_iξ_i dr+

₁

0

ξ_i²+ξ_i²

dr,

or

ω₂(x) = n

i=1

a_ix²+d_ix+c_i = n

i=1

w_i(x) = n

i=1

h_i(x), where β = x, a_i = ₁

0

ζ_i²+ζ_i²

dr, d_i = −2₁

0

ζ_iξ_i+ζ_iξ_i

dr, c_i = ₁

0

ξ_i²+ξ_i²

dr. In this case,ω(x) =ω₂(x) andp⁽²⁾_i =a_i,p⁽¹⁾_i =d_i,p⁽⁰⁾_i =c_i. We next have

∆_wi =d²_i −4a_ic_i =

= 4 ₁

0

ζ_iξ_i+ζ_iξ_i dr

₂

− ₁

0

ζ_i²+ζ_i²

dr

1 0

ξ_i²+ξ_i²

dr

. Using the Cauchy-Buniakowski-Schwarz inequality, we obtain

₁

0

ζ_i²+ζ_i²

dr

1 0

ξ_i²+ξ_i²

dr

=

= ₁

0

ζ_i²+ζ_i²

dr

1 0

ξ_i²+ξ_i²

dr

=

= ₁

0 ζ_i²+ζ_i² ₂

dr

1

0 ξ_i²+ξ_i² ₂

dr

≥

≥ ₁

0

ζ_i²+ζ_i² ξ_i²+ξ_i²

dr ₂

≥ ₁

0

ζ_iξ_i+ζ_iξ_i dr

₂

which gives ∆_wi ≤0; as above, in practice the most common case is ∆_wi <0.

Now,

ω₂(x) = n

i=1

h_i(x) = n

i=1

2a_ix+d_i 2√

a_ix²+d_ix+c_i; ω₂(x) =

n i=1

h_i(x) = n

i=1

−∆_wi

4 (a_ix²+d_ix+c_i)^3/2 >0.

Thereforeω₂ is a convex function.

It is easy to calculate lim

x→∞

h_i(x)

x =√a_i and lim

x→∞

h_i(x)− √a_ix

= ₂√^di a_i. Then the functionω₂(x) has the asymptotes

− _n

i=1

√a_i

x− _n

i=1

d_i 2√a_i

,

_n

i=1

√a_i

x+ _n

i=1

d_i 2√a_i

at −∞and +∞, respectively.

The conclusion for this case is thatω₂(x) has a unique minimizing point,

x which belongs to the closed interval

mini

−_2a^di_i ,max

i

−_2a^di_i

or, more precisely, to [C₁, D₁],where

C₁= max

i=1,n

− d_i 2a_iω₂

− d_i 2a_i

≤0

,

D₁= min

i=1,n

− d_i 2a_iω₂

− d_i 2a_i

≥0

.

To conclude, a few remarks. For all i= 1, . . . , n we have

− b_i 2a_i =

₁

0

ζ_iξ_i+ζ_iξ_{i 1} dr

0

ζ_i²+ζ_i²

dr

, −d_i 2a_i =

₁

0

ζ_iξ_i+ζ_iξ_{i 1} dr

0

ζ_i²+ζ_i²

dr and

− b_i 2a_i

−

− d_i 2a_i

= ₁

0

ζ_iξ_i+ζ_iξ_i

dr−₁

0

ζ_iξ_i+ζ_iξ_{i 1} dr

0

ζ_i²+ζ_i²

dr

=

= ₁

0

ζ_i−ζ_i ξ_i−ξ_{i 1} dr

0

ζ_i²+ζ_i²

dr ≥0.

Thus D₁ ≤A₁ and x≤x. As we have seen, ω(x) = ω₁(x) ifx > 0 and ω(x) =ω₂(x) if x <0.

If x > 0, x <0 and minω₁(x) =ω₁(x) ≤ ω₂(x) = minω₂(x), then the solution is the minimizing point forω₁(x), and minω(x) =ω₁(x). Thus, xis the estimator forβ.

If x > 0, x <0 and minω₂(x) =ω₂(x) ≤ ω₁(x) = min ω₁(x), then the solution is the minimizing point for ω₂(x) and minω(x) = ω₂(x). Now, an estimator forβ isx.

In both cases, the solution for α results from the initial condition Y₀ = α+βX₀.

The theorem is proved.

The only remaining thing is to find the minimum points for ω₁(x) and ω₂(x), which provides a concrete answer to initial problem. We may obtain this result by various approaches which are described in literature, for example the tangents method, illustrated below.

4. THE FINAL STEPS

We look for the minimum of the convex function ω₁ in [A₁, B₁]. To this purpose, the method which use the tangents gives us an useful algorithm [16].

First, we compute ω₁ (A₁),ω₁(B₁). If ω₁(A₁) = 0 or ω₁ (B₁) = 0, stop.

Otherwise, let u₀ ∈[A₁, B₁] arbitrary. The tangent to the graph of ω₁ at the point u₀ has the equation t(x, u₀) =y=ω₁(u₀) +ω₁(u₀) (x−u₀).

For j = 1, . . . , m we consecutively, look for u_j,the solution of the opti- mization problem:

(∗) minz subject to z≥ max

l=0,j−1t(x, u_l), A₁≤x≤B₁. Let T₁(x) = max{t(x, u₀), t(x, u₁)}, T₂(x) = max

t(x, u₀), t(x, u₁), t(x, u₂) , . . . , T_m−1(x) = max{t(x, u₀), t(x, u₁), . . . , t(x, u_m−1)}.

Thus T_m−1(x) = max{T_m−2(x), t(x, u_m−1)}.

If ω₁(u_m−1) = 0 then (∗) is equivalent to finding u_m ∈ [A₁, B₁] for which T_m−1(u_m) = min

x∈[A1,B₁]T_m−1(x) [4, 16].

ω₁ is a convex continuous differentiable function on [A₁, B₁]. We have proved in Section 3 that it has a unique minimizing pointx in [A₁, B₁].

Thus [10, 16] lim

m→∞u_m =x. In other words, the algorithm converges to

x, which is the required solution.

Similar arguments and calculations lead us to the minimizing pointxof ω₂(x) in [C₁, D₁].

5. A FEW REMARKS

In this section we discuss the problem without the constraint

∆_vi = 0, ∆_wi = 0 for all i= 1, n.

Case 1. First, suppose that β >0. Consider a partition {1,2, . . . , n} = {j₁, j₂, . . . , j_p} ∪ {j_p+1, j_p+2, . . . , j_n} such that ∆_vjk <0 fork= 1, p, ∆_vjk = 0 fork=p+ 1, nand

− b_j1

2a_j1 ≤ − b_j2

2a_j2 ≤ · · · ≤ − b_jp

2a_jp, − b_jp+1

2a_jp+1 ≤ − b_jp+2

2a_jp+2 ≤ · · · ≤ − b_jn 2a_jn. Then card{j₁, j₂, . . . , j_p} =p, card{j_p+1, j_p+2, . . . , j_n} =n−p. Three possi- bilities appears.

Case1.1. Ifp= 0 then ∆_vi = 0 for alli= 1, . . . , nand the functionω₁(x) has the form

n i=1

!!!√

a_ix+ ₂^√bi_a

i

!!!. According to Arthanary and Dodge [1], this

type of piecewise linear convex function is minimized atx=x_(r) =−_2a^bjr_jr for which−ⁿ

i=1

√a_i+ 2^r−1

k=1

√a_jk is negative and−ⁿ

i=1

√a_i+ 2 ^r

k=1

√a_jk is nonnega- tive (forr = 1 only the second condition remains). If−ⁿ

i=1

√a_i+ 2 ^r

k=1

√a_jk = 0, the minimum is obtained for all x_(r)≤x≤x_(r+1).

Case1.2. If 0< p < n we have ω₁(x) =

p k=1

a_jkx²+b_jkx+c_jk+ n k=p+1

!!!!√a_jkx+ b_jk 2√a_jk

!!!!, ω₁(x) =

n i=1

f_i(x) = p k=1

2a_jkx+b_jk 2

a_jkx²+b_jkx+c_jk + n k=p+1

f_jk(x) =

= p k=1

2a_jkx+b_jk 2

a_jkx²+b_jkx+c_jk + n k=p+1

√a_jk·sgn (g_jk(x)) ,

wherex=−_2a^bjp+1

jp+1, . . . ,−_2a^bjn_jn and f_jk(x) =|gj_k|=!!

!!√

a_jkx+ b_jk 2√a_jk

!!!! fork=p+ 1, n.

Forl= 1, . . . , n−p we have (ω₁)_S

− b_jp+1 2a_jp+1

= p k=1

a_jp+1b_jk−a_jkb_jp+1

4c_jka²_jp+1−2b_jka_jp+1b_jp+1+a_jkb²_jp+1

− n k=p+1

√a_jk,

(ω₁)_S

− b_jp+l 2a_jp+l

= p k=1

a_jp+lb_jk−a_jkb_jp+l 4c_jka²_j

p+l−2b_jka_jp+lb_jp+l+a_jkb²_j

p+l

−

− n k=p+1

√a_jk + 2

p+l−1

k=p+1

√a_jk, l= 1,

(ω₁)_D

− b_jp+l 2a_jp+l

= p k=1

a_jp+lb_jk−a_jkb_jp+l 4c_jka²_j

p+l−2b_jka_jp+lb_jp+l+a_jkb²_j

p+l

−

− n k=p+1

√a_jk + 2 p+l k=p+1

√a_jk,

(ω₁)_D

− b_jn 2a_jn

= p k=1

a_jnb_jk −a_jkb_jn 4c_jka²_j

n−2b_jka_jnb_jn+a_jkb²_n +

n k=p+1

√a_jk,

ω₁(x) = n i=1

f_i(x) = p k=1

−∆_vjk

4 (a_jkx²+b_jkx+c_jk)^3/2 >0 forx=−_2a^bjp+1

jp+1, . . . ,−_2a^bjn_jn. Therefore,ω₁(x) is a piecewise convex function.

For the points−_2a^bjk_jk,k=p+ 1, n we have (ω₁)_S

− b_jk 2a_jk

+ 2√

a_jk = (ω₁)_D

− b_jk 2a_jk

. Thus,

− n k=1

√a_jk <(ω₁)_S

− b_jp+1 2a_jp+1

<(ω₁)_D

− b_jp+1 2a_jp+1

<· · ·

· · ·<(ω₁)_S

− b_jn 2a_jn

<(ω₁)_D

− b_jn 2a_jn

<

n k=1

√a_jk and

(ω₁)_D

− b_jk 2a_jk

< ω₁(x)<(ω₁)_S

− b_jk+1 2a_jk+1

for allx∈

−_2a^bjk_jk,−_2a^bjk+1_jk+1

,k=p+ 1, n−1.

If there exists a numberr ∈ {p+ 1, . . . , n} such that

(ω₁)_S

− b_jr 2a_jr

·

(ω₁)_D

− b_jr 2a_jr

≤0 then the unique minimizing point isx=−_2a^bjr_jr.

If

"

(ω₁)_S

−_2a^bjk_jk#

·"

(ω₁)_D

−_2a^bjk_jk#

> 0 for all k = p+ 1, n, we look for the minimizing point in one of the intervals I₀ =

−∞,−_2a^bjp+1_jp+1#

; I_k =

"

−_2a^bjk_jk,−_2a^bjk+1_jk+1#

,k=p+ 1, . . . , n−1;I_n−p =

"

−_2a^bjn_jn,+∞

, as we shall show below.

If (ω₁)_S

−_2a^bjp+1_jp+1

>0 thenx∈I₀; if (ω₁)_D

−_2a^bjn_jn

<0 thenx∈I_n−p. If (ω₁)_S

−_2a^bjp+1_jp+1

<0 and (ω₁)_D

−_2a^bjn_jn

>0 we look for the minimiz- ing point in the unique interval I_r =

"

−_2a^bjr_jr,−_2a^bjr+1

jr+1

#

,r ∈ {p+ 1, . . . , n−1} for which (ω₁)_D

−_2a^bjr_jr

<0, (ω₁)_S

−_2a^bjr+1

jr+1

>0.

The search on I_r is more precise when

− b_jr

2a_jr,− b_jr+1 2a_jr+1

∩

− b_jk

2a_jk/k= 1, p

=

− b_js

2a_js,− b_js+1

2a_js+1, . . . ,− b_js+t 2a_js+t

. Then

− b_jr

2a_jr ≤ b_js

2a_js ≤ · · · ≤ − b_js+t

2a_js+t ≤ − b_jr+1 2a_jr+1 and

I_r =

− b_jr

2a_jr,− b_js 2a_js

∪

− b_js

2a_js,− b_js+1 2a_js+1

∪ · · · ∪

− b_js+t

2a_js+t,− b_jr+1 2a_jr+1

. Consequently, the minimizing pointxis situated in a smaller interval as follows:

x∈I_r,s=

− b_jr

2a_jr,− b_js 2a_js

ifω₁

− b_js 2a_js

≥0,

x∈I_r,s+t+1 =

− b_js+t

2a_js+t,− b_jr+1 2a_jr+1

ifω₁

− b_js+t 2a_js+t

≤0,

x∈I_r,s+q+1=

− b_js+q

2a_js+q,− b_js+q+1 2a_js+q+1

, 0≤q≤t−1 ifω₁

− b_js+q 2a_js+q

≤0, ω₁

− b_js+q+1 2a_js+q+1

≥0.

The conclusion is that the unique minimizing point x forω₁ belongs to only one of the intervals I₀, I_r,I_r,s, I_r,s+q+1,I_r,s+t+1, I_n−p. For determining it in one of these six intervals, we can use the algorithm given in Section 4.

Case1.3. Finally, forp=nwe obtain ∆_vi <0 for alli= 1, n. This kind of problem was discussed in Section 3.

Forβ <0 we have analogous situations and calculations.

The final assertion from Theorem 1 keeps its validity, too.

REFERENCES

[1] T.S. Arthanari and Yadolah Dodge, Mathematical Programming in Statistics. Wiley, New York, 1980.

[2] Wu Cong-Xin and Ma Ming,Embedding problem of fuzzy number space: Part I. Fuzzy Sets and Systems44(1991), 33–38.

[3] Wu Cong-Xin and Ma Ming,Embedding problem of fuzzy number space: PartIII. Fuzzy Sets and Systems46(1992), 281–286.

[4] M. Curtillot, Deux algorithmes de programmation math´ematique. RAIRO7(1973).

[5] P. Diamond and P. Kloeden, Metric spaces of fuzzy sets. Fuzzy Sets and Systems35 (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 Systems45(1992), 123.

[8] R. Goetschel and W. Voxman,Elementary Calculus. Fuzzy Sets and Systems18(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] V.G. Karmanov,Matimaticescoe programmirovanie. Nauka, Moscova, 1975.

[11] A. Katsaras and D.B. Liu,Fuzzy vector spaces and fuzzy topological vector spaces. J.

Math. Anal. Appl.58(1977), 135–146.

[12] Ma Ming, M. Friedman and A. Kandel, General fuzzy least squares. Fuzzy Sets and Systems88(1997), 107–118.

[13] C.V. Negoit¸˘a and D. A. Ralescu,Applications of Fuzzy Sets to Systems Analysis. Wiley, New York, 1975.

[14] 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.

pp. 115–169. Plenum, New York, 1980.

[15] M.L. Puri and D.A. Ralescu,Differentials for fuzzy functions. J. Math. Anal. Appl.91 (1983), 552–558.

[16] Radu S¸erban,Optimizare cu aplicat¸ie ˆın economie. Matrix Rom., Bucure¸sti, 1999.

[17] H. Tanaka, H. Isibuchi and S. Yoshikawa, Exponential possibility regression analysis.

Fuzzy Sets and Systems69(1995), 305–318.

[18] H. Tanaka, S. Uejima and K. Asai,Linear regression analysis with fuzzy model. IEEE Trans. Systems Man Cybernet.SMC-12(1982), 903–907.

[19] H.J. Zimmermann,Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems1(1978), 45–55.

[20] R.R. Yager, Fuzzy prediction based upon regression models. Inform. Sci. 26 (1982), 45–63.

Received 3 May 2006 Academy of Economic Studies

Department of Mathematics Calea Dorobant¸ilor 15-17, Sector 1

Bucharest, Romania [email protected]