[9] R Ã : (1) x 0 R A(x 0 ) = 1; (2) α [0 1] Ã α = {x A(x) α} = [A α A α ]. A(x) Ã. R R. Ã 1 m x m α x m α > 0; α A(x) = 1 x m m x m +

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Download "[9] R Ã : (1) x 0 R A(x 0 ) = 1; (2) α [0 1] Ã α = {x A(x) α} = [A α A α ]. A(x) Ã. R R. Ã 1 m x m α x m α > 0; α A(x) = 1 x m m x m +"

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1 Chinese Journal of Applied Probability and Statistics Vol.28 No.6 Dec ( ) Euclidean Lebesgue... :. : O212.2 O Zadeh [1 2]. Tanaa (1982) ; Diamond (1988) (FLS) FLS LS ; Savic Pedrycz (1991) FLS ; Chang (2001).. [7]. Euclidean Lebesgue. ( ) (BK )

2 [9] R Ã : (1) x 0 R A(x 0 ) = 1; (2) α [0 1] Ã α = {x A(x) α} = [A α A α ]. A(x) Ã. R R. Ã 1 m x m α x m α > 0; α A(x) = 1 x m m x m + β β > 0; β 0 Ã (m α β). α = β Ã (m α). : 2.1 [10] (1) Ã = ( a m a l a r ); Ã = (a m a l a r ) B = (b m b l b r ) (2) Ã + B = (a m + b m a l + b l a r + b r ); (3) Ã B = (a m b m a l + b l a r + b r ); (4) tã = (ta m t a l t a r ) t R. ( ỹ i ) (i = n) ỹ i R. ỹ = β 0 + β 1 x β0 β 1 R. (2.1) β 0 β 1 d ( ỹ i ) (i = n) min φ( β 0 β 1 ) = n d 2 (ỹ i β 0 + β 1 ). φ( β δ β0 0 β 1 ) = 0 φ( β δ β1 0 β 1 ) = 0 R.

3 : Euclidean Lebesgue. 2.2 [10] (Euclidean ) Ã B Ã = (a m a l a r ) B = (b m b l b r ) d 2 E(Ã B) = (a m b m ) 2 ω m + (a l b l ) 2 ω l + (a r b r ) 2 ω r (2.2) ω m > 0 ω l > 0 ω r > 0. Ã = (a α) B = (b β) d 2 E(Ã B) = (a b) 2 + (α β) 2. (2.3) (2.2) d 2 E (Ã B) ( R d 2 E ). 2.3 [11] (Lebesgue ) Ã B (Ã = (a (λ) a + (λ)) B = (b (λ) b + (λ)) d(ã B) ( 1 1/2 = f(λ)d 2 (A λ B λ )dλ) (2.4) 0 Ã = A λ = (a (λ) a + (λ)) B = Bλ = (b (λ) b + (λ)) Ã B λ Ã = A λ = {u µã(u) λ 0 λ 1} B = Bλ = {u µ B(u) λ 0 λ 1} µã(u) µ B(u) Ã B. d 2 (A λ B λ ) = (a (λ) b (λ)) 2 + (a + (λ) b + (λ)) 2 f(λ) [0 1] f(0) = f(λ)dλ = 1/2. (2.4) f(λ) = λ : 2.4 Ã = (a m a l a r ) B = (b m b l b r ) d(ã B) = ([a m b m + (a l b l )] 2 + [a m b m (a l b l )] 2 + (a m b m ) 2 ) 1/2 Ã B. Ã B d(ã B) = (3(a m b m ) 2 + 2(a l b l ) 2 ) 1/2 a l = a r b l = b r.

4 ỹ = β 0 + β 1 x β 0 R β 1 R 2.1 ỹ = β 0 + β 1 x ( ỹ i ) i = n ỹ i = (y i l i r i ) i = n β 0 = n n y i n n y i n n ( n ) 2 : β 1 = (β 1 β 1l β 1r ) n n y i n n y i β 1 = n n ( n ) 2 n l i n n r i n. φ(β 0 β 1 ) = n d 2 E(ỹ i β 0 + β 1 ) = n [(y i β 0 β 1 ) 2 + (l i β 1l ) 2 + (r i β 1r ) 2 ] φ = 2 β 0 n (y i β 0 β 1 ) = 0 φ = 2 n (y i β 0 β 1 ) = 0 β 1 φ = 2 n (l i β 1l ) = 0 β 1l φ = 2 n (r i β 1r ) = 0. β 1r n n x 2 i y i n n y i β 0 = n n ( n ) 2 n n y i n n n y i l i β 1 = n ( n ) 2 β1l = β1r = n n r i n. (2.5)

5 : β0 R β 1 R 2.2 ỹ = β 0 + β 1 x ( ỹ i ) i = n ỹ i = (y i l i r i ) i = n n n x 2 i y i n n n n y i l i r i n n x β 0 = i y i n n y i n n ( n ) 2 n n β1 = x 2 i n n ( n ) 2. (2.6) β 0 R β 1 R 2.1 [8] β 0 = 1 0 β 1 (λ) = ỹ = β 0 + β 1 x ( ỹ i ) i = n [ n n y i (λ) n n ] y i (λ) dλ n n ( n ) 2 n n y i + (λ) β 0 β+ n 1 (λ) = n n yi (λ) β 0 y i (λ) = [y i (λ) + y+ i (λ)]/ [8] n (2.7) ỹ = β 0 + β 1 x ( ỹ i ) i = n ỹ i = (y i l i r i ) i = n n n y i n n y i 1 n 4 (l i r i ) + n n (l i r i ) β 0 = n n ) 2 β 1 = n y i β 0 n n ( n n n l i r i n n. (2.8) ỹ i = (y i µ i ) n n y i n n n y i x i y i β 0 β 0 = n n ( n ) 2 β 1 = n n n µ i n

6 630 β1. 2. β0 R β 1 R 2.2 [8] 2.2 [8] ỹ = β 0 + β 1 x ( ỹ i ) i = n n y β 0 i (λ) β n n 1 y i + (λ) β n 1 (λ) = β+ n 0 (λ) = n n 1 (n n )(yi (λ) + y+ i (λ))dλ 0 β 1 = 2 (n n ( n ) 2 ). (2.9) ỹ = β 0 + β 1 x ( ỹ i ) i = n ỹ i = (y i l i r i ) i = n ( n y i β n n n 1 l i r i ) β 0 = n n n n (n n ) y i 1 n (n n )(l i r i ) 4 β 1 = n n ) 2 (2.10) β0. ( n ỹ i = (y i µ i ) ( n y i β n n 1 µ i ) n (n n )y i β 0 = β1 = n n n n ) 2 β0. ( n (2.5) (2.6) (2.8) (2.10) ỹ = β 0 + β 1 x ỹ = β 0 + β 1 x (2.1) i ( ỹ i )

7 : 631 i ( ỹ i ). i ỹ [i] = β 0[i] + β 1[i] x β 0[i] β 1[i] R. ỹ [i] ỹ i [8] : 3.1 ỹ [i] = β 0[i] + β 1[i] x β 0[i] R β 1[i] R (x ỹ ) i = n ỹ = (y l r ) x 2 y x x y β 0[i] = (n 1) ( ) 2 x (n 1) x y x y x l x r β 1[i] = (n 1) ( ) 2 x x 2 x 2. (3.1) 3.2 ỹ [i] = β 0[i] + β 1[i] x β 0[i] R β 1[i] R (x ỹ ) i = n ỹ = (y l r ) x 2 y x x y x l x r β 0[i] = (n 1) ( ) 2 x x 2 x 2 β 1[i] = (n 1) x y x y (n 1) ( ) 2. (3.2) x 3.3 ỹ [i] = β 0[i] + β 1[i] x β 0[i] R β 1[i] R (x ỹ ) i = n ỹ = (y l r ) β 0[i] = y x x y 1 4 (l r ) + x x (l r ) (n 1) ( ) 2 x x y β 0[i] β 1[i] = x x l x r. (3.3)

8 ỹ [i] = β 0[i] + β 1[i] x β 0[i] R β 1[i] R (x ỹ ) i = n ỹ = (y µ ) x 2 y x x y β 0[i] = (n 1) ( ) 2 x x y β 0[i] x x µ x µ β 1[i] =. (3.4) 3.4 ỹ [i] = β 0[i] + β 1[i] x β 0[i] R β 1[i] R (x ỹ ) i = n ỹ = (y l r ) ( y β 1[i] x l r ) β 0[i] = n 1 n 1 n 1 ( (n 1)x ) x y 1 ( (n 1)x x )(l r ) 4 β 1[i] = (n 1) ( ) 2. (3.5) x 3.2 ỹ [i] = β 0[i] + β 1[i] x β 0[i] R β 1[i] R (x ỹ ) i = n ỹ = (y µ ) ( y β 1[i] x µ µ ) β 0[i] = n 1 n 1 β1[i] = n ( (n 1)x x )y (n 1) ( ) 2. x (3.6) ỹ [i] ỹ ( ỹ i ) ( R ỹ i R i = n) ỹ i ỹ = β 0 + β 1 x. [ 1 n s = d 2 (ỹ i n 2 ỹ 1/2 i )] (3.7) ỹ i ỹ i...

9 : n 2 β 0 β 1. i [ 1 1/2. s = d 2 (ỹ n 3 [i] ỹ [i] )] (3.8) i ỹ [i] = β 0[i] + β 1[i] x s 2 = x x y y + x x l l 2 n 3 x 2 y + x ( x y (n 1) (n 1) ( ) 2 x n 3 x x l + r 2 ) x y. (3.9) i ỹ [i] = β 0[i] + β 1[i] x s 2 = ( y β 1[i] y n ( y ( y x n 3 y β 1[i] n 1 x n 3 y β 1[i] n 1 x β 1[i] x ) 2 n 3 β 1[i] x β 1[i] x ( l )) 2 n 1 l ( r n 1 r )) 2 2. (3.10) i ỹ [i] = β 0[i] +β 1[i] x i ỹ [i] = β 0[i] + β 1[i] x. i i ; i i.

10 ỹ i.. 1 ( [8]) ỹ i = (y i l i u i ) ỹ i = (y i l i u i ) 1987 (23021) 1992 (25721) 1988 (23632) 1993 (26232) 1989 (24123) 1994 (27633) 1990 (24612) 1995 (28123) 1991 (25222) 1996 (28612) = 1986 ỹ = β 0 + β 1 x (2.6) β 0 = ( ) β1 = ỹ = ( ) x. (3.2) (3.8) i ( 2). 2 i 0 ỹ = ( ) x ỹ = ( ) x ỹ = ( ) x ỹ = ( ) x ỹ = ( ) x ỹ = ( ) x ỹ = ( ) x ỹ = ( ) x ỹ = ( ) x ỹ = ( ) x ỹ = ( ) x

11 : : 4 4. ỹ = β 0 + β 1 x (2.8) β 0 = β 1 = ( ). ỹ = ( )x. (3.4) (3.8) i ( 3). 3 i 0 ỹ = ( )x ỹ = ( )x ỹ = ( )x ỹ = ( )x ỹ = ( )x ỹ = ( )x ỹ = ( )x ỹ = ( )x ỹ = ( )x ỹ = ( )x ỹ = ( )x :

12 636 [1] Zadeh L.A. The concept of a linguistic variable and its application to approximate reasoning I II Information Sciences 8(3)(1975) (4)(1975) [2] Zadeh L.A. Fuzzy sets as a basis for a theory of possibility Fuzzy Sets and Systems 1(1)(1978) [3] Tanaa H. Uejima S. and Asai K. Linear regression analysis with fuzzy model IEEE Transactions on Systems Man and Cybernetics 12(6)(1982) [4] Diamond P. Fuzzy least squares Information Sciences 46(3)(1988) [5] Savic D.A. and Pedrycz W. Evaluation of fuzzy linear regression models Fuzzy Sets and Systems 39(1)(1991) [6] Chang Y.-H.O. and Ayyub B.M. Fuzzy regression methods a comparative assessment Fuzzy Sets and Systems 119(2)(2001) [7] : [8] (I) ( ) 42(2)(2006) [9] Diamond P. and Körner R. Extended fuzzy linear models and least squares estimates Computers & Mathematics with Applications 33(9)(1997) [10] Arabpour A.R. and Tata M. Estimating the parameters of a fuzzy linear regression model Iranian Journal of Fuzzy Systems 5(2)(2008) [11] Xu R.N. and Li C.L. Multidimensional least-squares fitting with a fuzzy model Fuzzy Sets and Systems 119(2)(2001) [12] Coppi R. D Urso P. Giordani P. and Santoro A. Least squares estimation of a linear regression model with LR fuzzy response Computational Statistics & Data Analysis 51(1)(2006) Impact Assessment for Fuzzy Linear Regression Model Based on Case Deletion Zhang Aiwu (School of Mathematical Sciences Yancheng Teachers University Yancheng ) Least squares method based on Euclidean distance and Lebesgue distance between fuzzy data is used to study parameter estimation of fuzzy linear regression model based on case deletion respectively. And the parameter estimations on two inds of distance are compared. The input of the above model is real data and output is fuzzy data. The statistical diagnosis estimation standard error of regression equations is constructed to test highly influential point or outlier in observation data. At last through identifying highly influential point or outlier in actual data it shows that the statistic constructed in this paper is effective. Keywords: Fuzzy linear regression model case deletion standard error of estimate. AMS Subject Classification: 62G68.

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: 29 : n ( ),,. T, T +,. y ij i =, 2,, n, j =, 2,, T, y ij y ij = β + jβ 2 + α i + ɛ ij i =, 2,, n, j =, 2,, T, (.) β, β 2,. jβ 2,. β, β 2, α i i, ɛ i 2009 6 Chinese Journal of Applied Probability and Statistics Vol.25 No.3 Jun. 2009 (,, 20024;,, 54004).,,., P,. :,,. : O22... (Credibility Theory) 20 20, 80. ( []).,.,,,.,,,,.,. Buhlmann Buhlmann-Straub