ON FUZZY LEAST SQUARES

CIPRIAN COSTIN POPESCU

An improvement an optimization algorithm with application to LR and LL fuzzy numbers, is given. Two situations are presented: the case of LR fuzzy numbers and the special LL case.

AMS 2000 Subject Classification: 90C05.

Key words: LR-fuzzy number, LL-fuzzy number, weighted least squares.

1. INTRODUCTION

We discuss and extend an optimization method [2] based on the fuzzy least squares approach. The improvement consists in the introduction of the weighted model which substantially enlarge the range of the applications. In Section 2 we recall some basic definitions and results about fuzzy sets. Sec- tion 3 contains the general model setted up on n statistical units. In fact, our aim is to discuss the problems which occur when the weighted algorithm is applied to the particular case of LL fuzzy numbers, as in Section 5. Till then it is necessary to recall some results on more generall LR fuzzy numbers.

These preliminary results are presented in Section 4.

2. SOME DEFINITIONS AND PRELIMINARIES

We start with

Definition 2.1. Let X be a universal set [9, 11], a collection of objects denoted by x [1]. Then a fuzzy set A, A ⊂X is defined as a set of ordered pairs, namely, A = {(x, µ_A(x))|x∈X}. Here, µ_A(x) is a function called membership function of A inX with the properties

(i)µ_A(x) :X →[0,1];

(ii)µ_A assigns to each elementx∈X a numberµ_A(x)∈[0,1] called the grade of membership (or “degree of compatibility” or “degree of truth”, see [1]) of x inA.

MATH. REPORTS10(60),2 (2008), 197–203

Definition 2.2. Denote by Aα ⊂ A the subset of elements x ∈A whose degree of membership is at least α. Then A_α is called the α-level set [9] (or the α-level cut [1] and, clearly, Aα={x∈X |µA(x)≥α}.

The setA⁰_α ={x∈X |µ_A(x)> α} is called the strongα-level cut [1].

Definition 2.3. A fuzzy setA⊂X is said to be convex if µA(λx1+ (1−λ)x2)≥min{µ_A(x1), µA(x2)}, for any x1, x2 ∈X and λ∈[0,1] [1].

Definition 2.4. IfA andB are two fuzzy sets, we say thatA⊂B if and only if µA(x) ≤ µB(x) for all x ∈ X. For a fuzzy set A, its complement A is defined as a fuzzy set with membership function µ_A(x) = 1−µ_A(x) for all x∈X see [9]).

Definition 2.5. A fuzzy number M is of LR type if there exist two functions L (left), R (right) and the scalars a, b > 0 such that µ_M(x) = L(m−x/a) if x ≤m, and µM(x) = L(x−m/b) if x ≥m, where m ∈R is the mean value ofM anda, bare called the left and right spreads respectively [1]. A triangular fuzzy number is a particular type of semisymmetricLRfuzzy number [1, 8, 13].

3. THE MODEL

We start with the model given by Coppi et al. [2]. Consider input crisp variables X1, . . . , Xm and a fuzzy output variable, Ye, on a sample of size n.

The data will be denoted by (ye_i,x_i),i= 1, . . . , n.

Assume thatYe is anLRfuzzy variable: Ye = (m, a, b)_LR,a, b >0, where µYe(y) =L(m−y/a) if y ≤m, and µ

Ye(y) =L(y−m/b) if y ≥m. Consider the theoretical values m_t, a_t, b_t and the errors ε, ε_A, ε_B. Thus, see [2], we can write

m=mt+ε, m−a= (mt−at) +εA, m+b= (mt+bt) +εB. We replace the parametersα,β_A,β_B,γ_A,γ_Bwhich allow us to transform these equations into a regression model, namely,

mt=Fα, at=βAmt+γA1, bt=βBmt+γB1, where Fis a matrix with rows f_i^T = [f₁(x_i), . . . , f_p(x_i)] (see again [2]).

The theoretical values of the output variables are yei∗

= ((mt)_i,(at)_i,(bt)_i)_LR, i= 1, n.

Next, we will generalize the distanced²and the theoretical results from [2].

4. WEIGHTED LR CASE

Definition 4.1. Forw= (w1, w2, w3), where w1, w2, w3∈R^∗₊, define d²(w;y,eye^∗) =d²(w; (m,a,b)_LR,(m_t,a_t,b_t)_LR) =

=w₁km−m_tk²+w₂k(m−λa)−(m_t−λa_t)k²+w₃k(m+ρb)−(m_t+ρb_t)k², where λ=R1

0 L⁻¹(r) dr and ρ=R1

0 R⁻¹(r) dr.

Theorem 4.1. The relation

d²(w;y,eey^∗) = (w1+w2+w3)

(m−mt)^T(m−mt)

−

−2w₂λ(m−m_t)^T(a−a_t) +w₂λ²(a−a_t)^T(a−a_t) + +2w3ρ(m−mt)^T(b−bt) +w3ρ²(b−bt)^T(b−bt) holds.

Proof. We have

d²(w;y,eye^∗) =w1km−mtk²+w2k(m−λa)−(mt−λat)k²+ +w₃k(m+ρb)−(m_t+ρb_t)k² =w₁(m−m_t)^T(m−m_t) +

+ h

w2(m−mt)^T(m−mt)−2w2λ(m−mt)^T(a−at) + +w2λ²(a−at)^T(a−at)

i + +

h

w3(m−mt)^T(m−mt) + 2w3ρ(m−mt)^T(b−bt) + +w3ρ²(b−bt)^T(b−bt)

i

=

= (w₁+w₂+w₃)h

(m−m_t)^T(m−m_t)i

−2w₂λ(m−m_t)^T(a−a_t) + +w₂λ²(a−a_t)^T(a−a_t) + 2w₃ρ(m−m_t)^T(b−b_t) +

+w₃ρ²(b−b_t)^T(b−b_t).

We will discuss how the model given in [2] is affected after the introduc- tion of the weighted distance, d²(w;y,eye^∗). For w₁ =w₂ =w₃ = 1 we obtain the results from [2].

A procedure of finding α, β_A, β_B, γ_A, γ_B is to minimize the weighted distance d²(w;·,·) between the experimental measurements of the response variable ye_i,i= 1, n and the theoretical valuesye_i^∗. In other words, we have to solve the problem

α,βA,βminB,γA,γB

d²(w;ey,ey^∗) = min

α,βA,βB,γA,γB

D²_LR(w;α, βA, βB, γA, γB).

Theorem 4.2. The problem min

α,βA,βB,γA,γB

D²_LR(w;α, βA, βB, γA, γB)ad- mits local solutions which can be improved using an iterative estimation algo- rithm as follows:

α=

"

X³

i=1

w_i

−w₂λβ_A(2−λβ_A) +w₃ρβ_B(2 +ρβ_B)

#−1

×

× F^TF−1

F^T[(w1+w2+w3)m−w2λ(mβ_A+a−1γ_A) + +w₂λ²(aβ_A−1β_Aγ_A) +w₃ρ(mβ_B+b−1γ_B) +w₃ρ²(bβ_B−1β_Bγ_B)

, βA=λ⁻¹ α^TF^TFα−1

λ α^TF^Ta−α^TF^T1γA

− α^TF^Tm−α^TF^TFα , βB=ρ⁻¹ α^TF^TFα−1

ρ α^TF^Tb−α^TF^T1γB

+ α^TF^Tm−α^TF^TFα , γA= (nλ)⁻¹

λ1^T(a−FαβA)−1^T(m−Fα) , γB = (nρ)⁻¹

ρ1^T(b−FαβB) +1^T(m−Fα) . Proof. We have

D_LR² (w;α, βA, βB, γA, γB) = (w1+w2+w3)

(m−Fα)^T(m−Fα)

−

−2w₂λ(m−Fα)^T(a−FαβA−1γA) + +w2λ²(a−FαβA−1γA)^T(a−FαβA−1γA) +

+2w3ρ(m−Fα)^T(b−FαβB−1γB) + +w₃ρ²(b−Fαβ_B−1γ_B)^T(b−Fαβ_B−1γ_B) =

= (w1+w2+w3) m^Tm−2m^TFα+α^TF^TFα

−

−2w₂λ m^Ta−m^TFαβ_A−m^T1γ_A−α^TF^Ta+α^TF^TFαβ_A+α^TF^T1γ_A + +w₂λ² a^Ta−2a^TFαβ_A−2a^T1γ_A+α^TF^TFαβ²_A+2α^TF^T1β_Aγ_A+nγ_A²

+ +2w3ρ m^Tb−m^TFαβB−m^T1γB−α^TF^Tb+α^TF^TFαβB+α^TF^T1γB

+ +w₃ρ² b^Tb−2b^TFαβ_B−2b^T1γ_B+α^TF^TFαβ_B² + 2α^TF^T1β_Bγ_B+nγ_B²

. Equaling to zero the partial derivatives of D_LR² (w;α, βA, βB, γA, γB):

and obtain the equations F^TFα^T

(w1+w2+w3)−2w2λβA+w2λ²β_A² + 2w3ρβB+w3ρ²β_B²

=

= (w1+w2+w3)F^Tm−w2λ F^TmβA+F^Ta−F^T1γA

+ +w2λ² F^TaβA−F^T1βAγA

+w3ρ F^TmβB+F^Tb−F^T1γB

+ +w₃ρ² F^Tbβ_B−F^T1β_Bγ_B

, w₂

α^TF^Tm−α^TF^TFα−λ α^TF^Ta−α^TF^TFαβ_A−α^TF^T1γ_A

= 0, w3

−α^TF^Tm+α^TF^TFα−ρ α^TF^Tb−α^TF^TFαβB−α^TF^T1γB

= 0, w2

α^TF^T1−m^T1+λ a^T1−α^TF^T1βA−nγA

= 0,

w3

α^TF^T1−m^T1−ρ b^T1−α^TF^T1βB−nγB

= 0.

An iterative solution is given by the relations

α= [(w₁+w₂+w₃)−w₂λβ_A(2−λβ_A) +w₃ρβ_B(2 +ρβ_B)]⁻¹×

× F^TF−1

F^T[(w1+w2+w3)m−w2λ(mβA+a−1γA) + +w2λ²(aβA−1βAγA) +w3ρ(mβB+b−1γB) +w3ρ²(bβB−1βBγB)

, β_A=λ⁻¹ α^TF^TFα⁻¹

λ α^TF^Ta−α^TF^T1γ_A

− α^TF^Tm−α^TF^TFα , βB=ρ⁻¹ α^TF^TFα−1

ρ α^TF^Tb−α^TF^T1γB

+ α^TF^Tm−α^TF^TFα , γA= (nλ)⁻¹

λ1^T(a−FαβA)−1^T(m−Fα) , γB = (nρ)⁻¹

ρ1^T(b−FαβB) +1^T(m−Fα) , and the proof is complete.

5. THE WEIGHTED LL SYMMETRICAL CASE

In this case L = R, a = b. Consequently, we have Ye = (m, a)_LL. Moreover, ρ = λ, β_A = β_B and γ_A = γ_B (see [2]). We have the following results.

Theorem5.1. In the special LL symmetrical case the distance becomes d²(w;y,eey^∗) = (w₁+w₂+w₃)

(m−m_t)^T(m−m_t) +

+2 (−w₂+w3)λ(m−mt)^T(a−at) + (w2+w3)λ²(a−at)^T(a−at). Proof. We have

d²(w;y,eye^∗) =w1km−mtk²+

+w₂k(m−λa)−(m_t−λa_t)k²+w₃k(m+ρb)−(m_t+ρb_t)k²=

=w₁km−m_tk²+w₂k(m−λa)−(m_t−λa_t)k²+ +w3k(m+λa)−(mt+λat)k² =

=w₁km−m_tk²+w₂k(m−m_t)−λ(a−a_t)k²+ +w3k(m−mt) +λ(a−at)k² =w1(m−mt)^T(m−mt) + +

h

w2(m−mt)^T(m−mt)−2w2λ(m−mt)^T(a−at)+w2λ²(a−at)^T(a−at) i

+ +

h

w3(m−mt)^T(m−mt)+2w3λ(m−mt)^T(a−at)+w3λ²(a−at)^T(a−at) i

=

= (w₁+w₂+w₃)h

(m−m_t)^T(m−m_t)i +

+2 (−w₂+w3)λ(m−mt)^T(a−at) + (w2+w3)λ²(a−at)^T(a−at).

Thus we have the problem

α,βminA,γA

d²(w;y,eye^∗) = min

α,βA,γA

D²_LL(w;α, β_A, γ_A). Theorem5.2. The problem min

α,βA,γA

D_LL² (w;α, β_A, γ_A)admits local solu- tions which can be improved using an iterative estimation algorithm as follows:

α=

(w₁+w₂+w₃) + 2 (−w₂+w₃)λβ_A+ (w₂+w₃)λ²β²_A−1

×

× F^TF⁻¹

F^T[(w₁+w₂+w₃)m+ (−w₂+w₃)λ(mβ_A+a−1γ_A) + + (w₂+w₃)λ²(aβ_A−1β_Aγ_A)

, β_A= α^TF^TFα⁻¹

α^TF^T(a−1γ_A)+λ⁻¹(−w₂+w₃)

w₂+w₃ α^TF^Tm−α^TF^TFα

, γA= 1

n

1^T(a−FαβA) + λ⁻¹(−w₂+w3) w2+w3

−α^TF^T1+m^T1

. Proof. We have

D_LL² (w;α, βA, γA) = (w1+w2+w3) h

(m−Fα)^T(m−Fα) i

+ +2 (−w₂+w3)λ(m−Fα)^T(a−FαβA−1γA) + + (w₂+w₃)λ²(a−Fαβ_A−1γ_A)^T(a−Fαβ_A−1γ_A) =

= (w₁+w₂+w₃) m^Tm−2m^TFα+α^TF^TFα +

+2(−w₂+w₃)λ m^Ta−m^TFαβ_A−m^T1γ_A−α^TF^Ta+α^TF^TFαβ_A+α^TF^T1γ_A + (w2+w3)λ² a^Ta−2a^TFαβA−2a^T1γA+α^TF^TFαβ_A²+2α^TF^T1βAγA+nγ_A²

. Similarly to Theorem 4.2, we obtain the equations

F^TFα^T

(w₁+w₂+w₃) + 2 (−w₂+w₃)λβ_A+ (w₂+w₃)λ²β_A²

=

= (w1+w2+w3)F^Tm+ (−w₂+w3)λ F^TmβA+F^Ta−F^T1γA

+ + (w2+w3)λ² F^TaβA−F^T1βAγA

, (−w₂+w3) −α^TF^Tm+α^TF^TFα

+ + (w2+w3)λ −α^TF^Ta+α^TF^TFαβA+α^TF^T1γA

= 0, (−w₂+w3) α^TF^T1−m^T1

+ (w2+w3)λ −a^T1+α^TF^T1βA+nγA

= 0.

Thus,

α= [(w1+w2+w3) + +2 (−w₂+w₃)λβ_A+ (w₂+w₃)λ²β_A²−1

F^TF−1

F^T[(w₁+w₂+w₃)m+

+ (−w₂+w3)λ(mβA+a−1γA) + (w2+w3)λ²(aβA−1βAγA) ,

β_A= α^TF^TFα−1

α^TF^T(a−1γ_A) + λ⁻¹(−w₂+w₃) w2+w3

α^TF^Tm−α^TF^TFα

, γ_A= 1

n

1^T(a−Fαβ_A) +λ⁻¹(−w₂+w3)

w₂+w₃ −α^TF^T1+m^T1

, and the proof is complete.

Forw1=w2 =w3 = 1 we obtain the solutions from [2].

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Received 15 July 2007 Academy of Economic Studies

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

010374 Bucharest, Romania cippx@yahoo.com