Delta-Fuzzy Measures and its Application - wseas.us

Proceedings of the 8th WSEAS International Conference on Applied Computer and Applied Computational Science

Delta-Fuzzy Measures and its Application HSIANG-CHUAN LIU Department of Bioinformatics Asia University No. 500, Lioufeng Rd., Wufeng, Taichung County, 41354 TAIWAN [email protected] / [email protected] DER-BANG WU Graduate Institute of Educational Measurement Department of Mathematics Education Taichung University No. 140, Ming-Sheng Rd., Taichung, 40306 TAIWAN [email protected] / [email protected] YU-DU JHENG Graduate Institute of Educational Measurement Taichung University No. 140, Ming-Sheng Rd., Taichung, 40306 TAIWAN [email protected] TIAN-WEI SHEU Graduate Institute of Educational Measurement Taichung University No. 140, Ming-Sheng Rd., Taichung, 40306 TAIWAN [email protected] Abstract: - The well known fuzzy measures, Lambda-measure and P-measure, have only one formulaic solution, the former is not a closed form, and the latter is not sensitive enough. In this paper, a novel fuzzy measure, called Delta-measure, is proposed. This new measure proves to be a multivalent fuzzy measure which provides infinitely many solutions to closed form, and it can be considered as an extension of the above two measures. In other words, the above two fuzzy measures can be treated as the special cases of Delta-measure. For evaluating the Choquet integral regression models with our proposed fuzzy measure and other different ones, a real data experiment by using a 5-fold cross-validation root mean square error (MSE) is conducted. The performances of Choquet integral regression models with fuzzy measure based on Delta-measure, Lambdameasure and P-measure, respectively, a ridge regression model and a multiple linear regression model are compared. Experimental result shows that the Choquet integral regression models with respect to Deltameasure based on Gamma-support outperforms other forecasting models. Key-Words: - Lambda-measure, P-measure,

Delta-measure, Gamma-support, Choquet integral

regression model

1 Introduction

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When there are interactions among independent variables, traditional multiple linear regression models do not perform well enough. The traditional

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axioms:including a subsection you must use, for its heading, small letters, 12pt, left justified, bold, Times New Roman as here.

improved methods exploited ridge regression models [1]. In this paper, we suggest using the Choquet integral regression models [5],[6],[7],[8] based on some single or compounded fuzzy measures [2],[3],[4] to improve this situation. The well-known fuzzy measures, λ-measure [2],[3] and P-measure,[4] have only one formulaic solution of fuzzy measure, the former is not a closed form, and the latter is not sensitive enough. In this paper, we proposed a new fuzzy measure, δ-measure, which offers infinitely many solutions to a fuzzy measure with closed form and without changing the given singleton measure, and thereby, we can obtain an improved Choquet integral regression model with respect to this new fuzzy measure. This paper is organized as follows: The multiple linear regression and ridge regression [1] are introduced in section II; two well known fuzzy measure, λ-measure [2] and P-measure [4], are introduced in section III; our new measure, δ measure, is introduced in section IV; the fuzzy support, γ-support [7] is described in section V; the Choquet integral regression model [6],[7],[8] based on fuzzy measures are described in section VI; experiment and result are described in section VII; and final section is for conclusions and future works.

1) μ (φ ) = 0 , μ ( X ) = 1 (boundary conditions) (2) 2) A ⊆ B ⇒ μ ( A ) ≤ μ ( B ) (monotonicity) (3) 3.2 Singleton Measures [2, 6, 7]

A singleton measure of a fuzzy measure μ on a finite set X is a function s : X → [ 0,1] satisfying: s ( x ) = μ ({ x} ) , x ∈ X

(4)

s ( x ) is called the fuzzy density of singleton x .

3.1 Subsection For given singleton measures s, a λ-measure, gλ , is a fuzzy measure on a finite set X, satisfying: A, B ∈ 2 X , A ∩ B = φ , A ∪ B ≠ X

⇒ gλ ( A ∪ B )

(5)

= gλ ( A) + gλ ( B ) + λ gλ ( A) gλ ( B ) n

∏ ⎡⎣1 + λ s ( x )⎤⎦ = λ + 1 > 0, s ( x ) = gλ ({ x }) 2 The Multiple Linear Regression, Ridge Regression

model, βˆ = ( X ′X ) X ′Y be the estimated regression −1

coefficient vector, and βˆk = ( X ′X + kI n ) X ′Y be the estimated ridge regression coefficient vector, Kenard and Baldwin [1] suggested −1

x∈ X

if

i

(6)

∑ s ( x) = 1

x∈ A

then λ-measure is just the additive

x∈ X

measure

(1)

3.4 P-measure [4] For given singleton measures s, a P-measure, g P , is a fuzzy measure on a finite set X, satisfying:

3 Fuzzy Measures The two well known fuzzy measures, the λ-measure proposed by Sugeno in 1974, and P-measure proposed by Zadah in 1978, are concise introduced as follows.

∀

A∈ 2X

{

}

⇒ g P ( A ) = max {s ( x )} = max g P ({ x} ) x∈ A

x∈ A

(7)

Note that for any subset of X, A, P-measure considers only the maximum value and will lead to insensitivity.

3.1 Fuzzy Measures [2, 3, 4] A fuzzy measure μ on a finite set X is a set function μ : 2 X → [ 0,1] satisfying the following

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i

Note that once the singleton measure is known, we can obtain the values of λ uniquely by using the previous polynomial equation. In other words, λmeasure has a unique solution without closed form. Moreover, for given singleton measures s, If ∑ s ( x) = 1 then g λ ( A ) = ∑ s ( x) , in other word,

Let Y = X β + ε , ε ~ N ( 0,σ 2 I n ) be a multiple linear

nσˆ 2 kˆ = βˆ ′βˆ

i

i =1

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4.2.4 Theorem 4 If s ( x ) = 1 and δ = 0 then δ -measure is just the

4 Delta-Measures

∑

4.1 δ-measure

x∈ X

λ -measure.

For given singleton measure s, a δ-measure, gδ , is a fuzzy measure on a finite set X, X = n , satisfying:

4.2.5 Theorem 5 P -measure, additive measure and λ -measure are the special cases of δ -measure

1) δ ∈ [ −1,1] , ∑ s( x) = 1 x∈ X

2) gδ (φ ) = 0, gδ ( X ) = 1 3) ∀A ⊂ X , A ≠ X ⇒

5 γ- support [7]

∑ ∑

(1 + δ ) s( x) x∈ A ⎡ ⎤ gδ ( A ) = 1 + δ max s ( x ) − δ max s ( x ) x∈ A x∈ A ⎣⎢ ⎦⎥ 1 + δ s ( x)

For given singleton measure s of a fuzzy measure μ on a finite set X, if ∑ s ( x ) = 1 , then s is called a x∈ X

fuzzy support measure of μ, or a fuzzy support of μ, or a support of μ. Two kinds of fuzzy supports are introduced as below. Let μ be a fuzzy measure on a finite set X = { x1 , x2 ,..., xn } , yi be global response of subject

x∈ A

(8) 4.2 Important Properties of δ-measure

( )

To prove that δ-measure is a fuzzy measure, we need to prove the following theorem 1 firstly.

i and f i x j

singleton x j , satisfying:

4.2.1 Theorem 1 For given singleton measure s, If A ⊆ B ⊆ X then

( )

0 < fi x j < 1, i = 1, 2,..., N , j = 1, 2,..., n

∑ s ( x ) − ∑ s ( x ) ≥ max {s ( x )} − max {s ( x )} ≥ 0 x∈B

x∈ A

x∈B

be the evaluation of subject i for

γ (xj ) =

x∈ A

(9)

( ( ))

1+ r f xj n

∑ k =1

4.2.2 Theorem 2

( ( )) = S S

For given singleton measure s, ∀δ ∈ [ −1,1] , δ measure is a fuzzy measure.

y

S y2 =

1 N

(19)

S y, x j

where r f x j

4.2.3 Theorem 3

, j = 1, 2,..., n

⎡1 + r ( f ( xk ) ) ⎤ ⎣ ⎦

⎛ 1 ⎜ yi − N i =1 ⎝ n

∑

(20)

xj

N

∑ i =1

⎞ yi ⎟ ⎠

2

(21)

(i) δ-measure is increasing function on δ S x2j

(ii) if δ = −1 then δ -measure is just the Pmeasure (iii) if δ = 0 then δ -measure is just the additive measure

S y, x j =

(iv) if −1 < δ < 0 then δ -measure is a subadditive measure

1 N

⎡ 1 ⎢ fi x j − N i =1 ⎣ n

∑ ( )

⎛ 1 ⎜ yi − N i=1 ⎝ n

∑

∑ ( ) i =1

⎞⎡ 1 yi ⎟⎢ fi xj − N i =1 ⎠⎣ N

∑

( )

291

⎤ fi x j ⎥ ⎦

N

( )

satisfying 0 ≤ γ x j ≤ 1 and

(v) if 0 < δ < 1 then δ -measure is a supperadditive measure

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1 = N

2

(22)

⎤

N

∑ f ( x )⎦⎥ i

j

(23)

i=1

n

∑γ ( x ) = 1 j

(24)

j =1

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then

the

γ : X → [ 0,1]

function

satisfying

αˆ =

μ ({ x} ) = γ ( x ) , ∀x ∈ X is a fuzzy support of μ,

1 N

N

∑ i =1

N

1 yi − βˆ N

∑ ∫ f dg μ

(31)

i

i =1

called γ-support of μ. ⎡ 1 ⎢ yi − N ⎢ i =1 ⎣ N

6 Choquet Models

Integral

Regression

S yf =

∑ N

Let μ be a fuzzy measure on a finite set X. The Choquet integral of fi : X → R+ with respect to μ for individual i is denoted by

∫

S ff =

i =1

∫

⎤

N

∑ ∫ f dgμ ⎥⎥ k

k =1

∗

⎦

i

i

j

j =1

i

j −1

j

( )

where fi x( 0 ) = 0 , fi x( j ) indicates that the indices have been permuted so that

( ) ( ) ( ) A( ) = { x( ) , x( ) ,..., x( ) }

0 ≤ fi x(1) ≤ fi x( 2 ) ≤ ... ≤ fi x( n ) j +1

j

(26)

6.2 Choquet Integral Regression Models [6, 7, 8, 9, 10, 11, 12] Let y1 , y2 ,..., y N be global evaluations of N objects

( ) ( )

⎣

∗

−

1 N

⎤

∑ ∫ f dgμ ⎥⎥ k

k =1

∗

⎦

N −1

(33)

The data of all variables listed in Table 2 is applied to evaluate the performances of four Choquet integral regression models with P-measure, λ-measure and δ -measure based on γ-support respectively, a ridge regression model, and a multiple linear regression model by using 5-fold cross validation method to compute the root mean square error (MSE) of the dependent variable. The formula of MSE is

(27)

n

i =1

i

2

The total scores of 80 students from a junior high school in Taiwan are used for this research. The examinations of four courses, physics and chemistry, biology, geoscience and mathematics, are used as independent variables, the score of the Basic Competence Test of junior high school is used as a dependent variable.

(25)

( )

⎡

∑ ⎢⎢ ∫ f dgμ

N

7 Experiment and Result

∑⎡⎣⎢ f ( x( ) ) − f ( x( ) )⎤⎦⎥ μ ( A( ) ) ,i =1,2,..., N n

j

∑

⎤⎡ 1 yi ⎥ ⎢ fi dg μ ∗ − N ⎦⎥ ⎣⎢ N −1

(32)

6.1 Choquet Integral [3, 5, 9, 10]

C fi d μ =

N

( )

and f1 x j , f 2 x j ,..., f N x j , j = 1, 2,..., n , be their evaluations of x j , where fi : X → R+ , i = 1, 2,..., N .

Let μ be a fuzzy measure, α , β ∈ R ,

yi = α + β

∫

C fi dgμ

(

+ ei , ei ~ N 0,σ

2

)

1 MSE = N

, i =1,2,..., N

N

α ,β

i =1

i

) ⎤⎥⎦ 2

C

f i dg μ

(29)

∫

2

i

(34)

i =1

For any fuzzy measure, μ-measures, once the fuzzy support of the μ-measure is given, all event measures of μ can be found, and then, the Choquet integral based on μ and the Choquet integral regression equation based on μ can also be found by using above corresponding formulae.

Choquet integral regression equation of μ, where

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i

{0.2488, 0.2525, 0.2439, 0.2547}

then yˆi = αˆ + βˆ f i dg μ , i = 1, 2,..., N is called the

βˆ = S yf / S ff

∑ ( y − yˆ )

The singleton measures, γ-support of the Pmeasure, λ-measure and δ -measure are listed as follows which can be obtained by using the formula (20).

(28)

(αˆ , βˆ ) = arg min ⎡⎢⎣∑ ( y − α − β ∫

N

(30)

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The experimental results of five forecasting models are listed in Table I. We find that the Choquet integral regression model with δ-measure based on γ-support outperforms other forecasting regression models.

Communications in Statistics, vol. 4, No. 2, pp. 105-123, 1975. [2] Z. Wang, and G. J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992. [3] M. Sugeno, Theory of Fuzzy Integrals and its Applications, unpublished doctoral dissertation, Tokyo Institute of Technology, Tokyo, Japan, 1974. [4] L. A. Zadeh, Fuzzy Sets and Systems, vol. 1, pp. 3, 1978. [5] G. Choquet, Theory of Capacities, Annales de l’Institut Fourier, vol. 5, pp. 131-295, 1953. [6] H.-C. Liu, C.-C. Chen, D.-B. Wu, and Y.-D. Jheng, A New Weighting Method for Detecting Outliers in IPA Based on Choquet Integral, IEEE International conference on Industrial Engineering and Engineering Management 2007, Singapore, December 2007. [7] H.-C. Liu, Y.-C. Tu, C.-C. Chen, and W.-S. Weng, The Choquet Integral with Respect to λMeasure Based on γ-support, 2008 International Conferences on Machine Learning and Cybernetics, Kunming, China, July 2008. [8] J.-I Shieh, H.-H. Wu, H.-C. Liu, Applying Complexity-based Choquet Integral to Evaluate Students’ Performance, Expert Systems with Applications, 36, pp. 5100-5106, 2009. [9] H.-C. Liu, D.-B. Wu, and H.-L. Ma, Fuzzy Clustering with New Separable Criterion. WSEAS Transactions on Biology and Biomedicine, vol. 4, no. 7, pp. 99-102, July 2007. [10] H.-C. Liu, C.-C. Chen, G.-S. Chen, & Y.-D. Jheng, (2007). Power-transformed-measure and its Choquet Integral Regression Model, Proc. of 7th WSEAS International Conference on Applied Computer Science (ACS’07), pp.101-104, 2007. [11] H. C. Liu, D. B. Wu, J. M. Yih, & S. W. Liu, Fuzzy Possibility C-Mean Based on Mahalanobis Distance and Separable Criterion. WSEAS Transactions on Biology and Biomedicine, vol. 4, no. 7, pp. 93-98, July 2007. [12] G.-S. Chen, Y.-D. Jheng, H.-C. Yao, and H.-C. Liu, Stroke Order Computer-based Assessment with Fuzzy Measure Scoring. WSEAS Transactions Information Science & Applications, 2(5), 62-68, 2008.

TABLE 1 MSE OF REGRESSION MODELS Regression model measure

Choquet Integral Regression model

δ

λ p

Ridge regression Multiple linear regression

5-fold CV MSE 48.7672 49.1832 53.9582 34.3777 35.5151

8 Conclusion In this paper, multivalent fuzzy measure, δ measure, is proposed. This new measure is proved that it is of closed form with infinitely many solutions, and it can be considered as an extension of the two well known fuzzy measures, λ-measure and P-measure. By using 5-fold cross-validation MSE, an experiment is conducted for comparing the performances of a multiple linear regression model, a ridge regression model, and the Choquet integral regression model with respect to P-measure, λmeasure, and our proposedδ-measure based on γsupport respectively. The result shows that the Choquet integral regression models with respect to the proposed δ -measure based on γ-support outperforms other forecasting models. In the future, we will apply the proposed Choquet integral regression model with fuzzy measure based on γ-support to develop multiple classifier system.

9 Acknowledgment This paper is partially supported by the National Science Council grant (NSC 97-2410-H-468-014) References: [1] A. E. Hoerl, R. W. Kenard, and K. F. Baldwin, Ridge Regression: Some Simulation,

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