A Note on Linear Regression Model Using Non-Symmetric Triangular Fuzzy Number Coefficients

(1)

A Note on Linear Regression Model Using

Non-Symmetric Triangular Fuzzy Number Coefficients

Dug Hun Hong¹⁾ ․ Kyung Tae Kim²⁾

Abstract

Yen et al. [Fuzzy Sets and Systems 106 (1999) 167-177] calculated the fuzzy membership function for the output to find the non-symmetric triangular fuzzy number coefficients of a linear regression model for all given input-output data sets. In this note, we show that the result they obtained in their paper is invalid.

Keywords : Fuzzy regression analysis, Fuzzy triangular coefficients, Minimization of fuzziness, Non-symmetric coefficients

The following model shows the dependence of the output variable on the inputs variables,

˜ = f( x, AY ˜ ) = A˜

0+ A˜

1x₁+ …+ A˜

nx_n, (1) where ˜ is the fuzzy output, x = [xY ₁, x₂, …, x_n]^T is the real-valued input vector, and A˜ = { A˜

0, A˜

1, …, A˜

n} is a set of fuzzy numbers.

The membership function for the set ˜ is defined by Zadeh's extension Y principle as follows:

μ ˜_Y( y )=

{

⁰^max ^{( a}¹^{,… , a}ⁿ^{) = f}^{- 1}^{( y, x)}^{{ min}^j^μ ^˜^A^j^(a^j^{) }} ^otherwise.^{if f} ^{- 1}^{(y,x) ≠φ,}

Then the regression analysis problem is defined as: given a set of crisp data

1) First Author : Professor, Department of Mathematics, Myongji University, Kyunggi 449-728, Korea

E-mail : [email protected]

2) Professor, Department of Electronics and Electrical Engineering, Kyungwon University, Kyunggi 461-701, Korea

E-mail : [email protected]

(2)

points < x₁, y₁>, < x₂, y₂>, … , < x_m, y_m>, we want to find a set of fuzzy parameters ˜A

0, A˜

1, … , A˜

n for the Eq.(1) which is the best fit to the given data points, according to some criteria of goodness of fit.

In (1), ˜A

i is the fuzzy coefficient of the variable x_i in the regression model of additive form. If ˜A

is have triangular membership functions, then each fuzzy number coefficient ˜A_i can be uniquely defined by

˜A

i= {a^L_i, a^C_i, a^U_i },

where a^L_i is the lower limit, a^U_i is the upper limit, and a^C_i is the point having the property that μ ˜_A

i(a^C_i) = 1. The property of symmetry of the fuzzy coefficient ˜A

i enables us to establish the following two relations:

a^C_i = ( a^L_i + a^U_i )/2 and

a^S_i= a^C_i - a^L_i = a^U_i- a^C_i, where a^C_i is the center and a^S_i the spread of ˜A

i.

For symmetric triangular fuzzy number coefficients, the membership function μ ^˜_A

i for ˜A

i , i = 1,…,n can be described as

μ ˜_A

i(a_i) =

{

^{1 - ( a}^{1 - ( a}⁰ ^Cⁱⁱ^{- a}^{- a}^Cⁱⁱ^)/a^)/a^Sⁱ^Sⁱ^{, a}^{, a} otherwise.^C^Cⁱⁱ^≤a^{- a}ⁱ^Sⁱ^≤a^≤a^Cⁱⁱ^{+ a}^≤a^Cⁱ^Sⁱ^,^,

Having established the membership function for each fuzzy coefficient ˜A

i, the fuzzy output from the linear model f( x, A˜ ) in (1) can be expressed according to the principle of extension and fuzzy arithmetic on fuzzy numbers [1] as

˜ = f( x, AY ˜ ) = (f^C( x), f^S( x)),

where f^C(x) is the center of the fuzzy linear model f( x, A˜ ) and has the form f^C( x) = a^C₀+ a^C₁x₁+ … + a^C_nx_n

and f^S( x) is the spread of f( x, A˜ ) and defined as f^S( x) = a^S₀+ a^S₁|x₁| + … + a^S_n|x_n|.

(3)

where [f( x_j)]_h= [ A˜₀_]_h_{+ [ A}^˜₁_]_h_x_j1_{+ … + [ A}^˜_n_]_h_x _jn such that [∙]_h represents the h-level set of a fuzzy number.

In regression , the goal is to find the fuzzy coefficients that minimize the above-mentioned spread of fuzzy output for all the data sets. The cost function, Z , to be minimized can be written as

Z = a^S₀+ ∑ⁿ

i = 1

[

^a^Sⁱ ^{j = 1}^∑^m ^|x ^ji^|

]

which can also be expressed as

Minimize Z =f^S( x₁) + f^S( x_{2 )}+ … + f^S( x_m) subject to the set of constraints

y_j∈[ f( x_j)]_h.

However, if these triangles are not symmetric , we need minimum three parameters to uniquely describe each. For example, ˜A

i can be described by the triplet {a^L_i, a^P_i, a^U_i } or by {s^L_i, a^P_i, s^R_i}, where a^P_i is the point at which

μ ˜_A

i(a^P_i) = 1, s^L_i is the left-side spread from the peak point a^P_i, and s^R_i represents the right-side spread. The membership function for each ˜A

i has the form

μ ˜_A

i(a_i) =

ꀊ

ꀖ ꀈ

︳︳

1 - a_i- a^P_i

s^R_i , a^P_i≤a_i≤a^P_i+ s^R_i, 1 - a^P_i- a_i

s^L_i , a^P_i- s^L_i≤a_i≤a^P_i,

0 otherwise.

Following the principle of extension, Yen et al.[2] used the fuzzy membership function for the output by

μ ˜_Y( y )=

ꀊ

ꀖ ꀈ

︳︳

︳

︳︳

︳︳ 1 -

y -∑

i a^P_ix_i- a^P₀ s^R₀+∑

i s^R_i|x_i| , a^P₀+∑^a^Pix_i≤y≤a^P₀+∑

i a^P_ix_i+

(

^s^R⁰⁺^∑i s^R_i|x_i|

)

^,

1 -

a^P₀+∑

i a^P_ix_i- y s^L₀+∑

i s^L_i|x_i| , a^P₀+∑

i a^P_ix_i-

(

^s^L⁰⁺^∑i s^L_i|x_i|

)

^≤y≤a^P⁰⁺^∑i a^P_ix_i,

0 otherwise.

(2) i.e, ˜ = {SY ^L₀+∑

i S^L_i|x_i|, a^P₀+∑

i a^P_ix_i, S^R₀+∑

i S^R_i|x_i| }.

(4)

But this result is wrong. Let A˜

i= {S^L_i, a^P_i, S^R_i} , i = 1, 2 and λ∈R be a real number. By the extension principle, the following rules for addition and real multiplication can be represented as

λ A˜ =

{

^{λS^{|λ|S^L^{, λa}^R^{, λa}^P^{, λS}^P^{, |λ|S}^R^}^L^} ^{if λ≥0,}if λ < 0,

˜A

1+ A˜

2= {S^L₁+ S^L₂, a^P₁+ a^P₂, S^R₁+ S^R₂}.

This result can be generalized to linear combinations of fuzzy numbers as follows:

˜ = AY ˜

0+ A˜

1x₁+ … + A˜

nx_n

= {S^L₀+ ∑

xi≥0|x_i|S^L_i+ ∑

xi< 0|x_i|S^R_i, a^P₀+∑

i a^P_ix_i, S^R₀+ ∑

xi≥0|x_i|S^R_i+ ∑

xi< 0|x_i|S^L_i}, i.e.

μ ˜_Y( y )=

ꀊ

ꀖ ꀈ

︳︳

︳

︳︳

︳ 1 -

y -∑

i a^P_ix_i- a^P₀ S^R₀+ ∑

xi≥0|x_i|S^R_i+ ∑

xi< 0|x_i|S^L_i , if a^P₀+∑^a^Pix_i≤y≤a^P₀+∑

i a^P_ix_i+

(

^S^R⁰⁺ ^x^∑ⁱ^≥0^|xⁱ^|S^Rⁱ⁺ ^∑^xⁱ^{< 0}^|xⁱ^|S^Lⁱ

)

^,

1 -

a^{P +}₀ ∑

i a^P_ix_i- y S^L₀+ ∑

xi≥0|x_i|S^L_i + ∑

xi< 0|x_i|S^R_i , if a^P₀+∑

i a^P_ix_i-

(

^S^L⁰⁺ ^x^∑ⁱ^≥0^|xⁱ^|S^Lⁱ ⁺ ^x^∑ⁱ^{< 0}^|xⁱ^|S^Rⁱ

)

^≤y≤a^P⁰⁺^∑ⁱ ^a^Pⁱ^xⁱ^,

0, otherwise.

(3) So μ˜_A( y ) in (2) is not true but μ ˜_A( y ) in (3) is true.

We consider the following example.

Example. Let ˜A

0= {S^L₀= 1, a^P₀= 2, S^R₀= 3 },

˜A

1= {S^L₁= 1, a^P₁= 4, S^R₁= 6 }, and x₁=- 2. Then by (3),

˜ = AY ˜

0+ ( - 2) A˜

1= { left - side spread = 1 + ( | - 2|)6 = 13, peak point

= 2 + ( - 2)4 =- 6, right-side spread = 3 + ( | - 2|)1 = 5 } = {13, - 6, 5 }. But by (2), we have ˜ = {3, - 6, 15 } which is not true.Y

From the expression of (3), we get the constraints of the regression as

(5)

1 -

y -∑

i a^P_ix_i- a^P₀ S^L₀+ ∑

xi≥0|x_i|s^L_i+ ∑

xi< 0|x_i|s^R_i ≥h and

1 -

a^P₀+∑

i a^P_ix_i- y S^R₀+ ∑

xi≥0|x_i|s^R_i+ ∑

xi< 0|x_i|s^L_i ≥h if we have taken membership function value at h-cut.

References

1. Tanaka, H. Vejima, S. and Asia ,K. (1982). Linear regression analysis with fuzzy model, IEEE Trans. Systems Man Cybernet. 12(6), 903-907.

2. Yen, K. K. Ghoshray, S. and Roig, G. (1999). A Linear regression model using triangular fuzzy number coefficients, Fuzzy Sets and Systems 106, 167-177.

[ received date : Dec. 2004, accepted date Apr. 2005 ]