) 041 (CIP ),2003.9 ISBN 5037 3561 9 /,.:...-.F10 CIP (2003) 032658 / / 75 / 100826 / 6 / ( 010) 63459084 63266600-22500( ) / / / 787960mm 1/ 16 / 301 / 16.75 / 1-5000 / 2003 9 1 / 2003 9 1 / ISBN 7-5037-3561-9/ F1261 / 26.80,,,,
,,,,,,,,, 20,,,, :,,,,, ;,,,, SAS SPSS T SP, Excel ;,,,,, : ( ), (), ( ), ( ), ( ),,,,,!,,,,! 2003 9
( 1) ( 1) ( 4) (10) (13) (Delphi poll) (13) (17) (20) (24) (25) (40) (53) (74) (83) (83) (87) (92) ( 111) ( 115) ( 115) ( 116) ( 126) ( 128) ( 128) ( 133) ( 136)
2 ( 144) ( 147) ( 150) ( 152) ( 153) ( 155) ( 159) ( 163) X - 11 ( 166) ( 170) ( 170) ( 175) ( 182) ( 188) ( 188) ( 193) ( 197) AR MA ( 202) ( 210) ( 219) ( 223) ( 232) ( 232) ( 237) 1 ( 241) 2 X 2 ( 243) 3 X 2 () ( 245) 4 F ( 247) 5 D.W ( 255) 6 ( r z ) ( 259) ( 260)
( forecasting ),,,,,,,,,,,,,,, 20 30 40,,,,,,,,,,,,,
2 ), (economic forecasting),, 4,,,,,,,,, (),,,,,,,,,,,,,, :,,
3 ),,,,,, :,,,,,,, ( ),,,,,,,, ( ),,,,,,,,,,,,, (),, 5 ; 1 5, 1,
4,,,,,,,,,,,, :, 1.1 Box - Jenkins 1.1, : () :
5 ;,,, ( ) :, ;, (),,,,,, (),, ;, (),,,, ;,,, ( ), : = + = -
6 ( ) : 1.,, 2.,, (),,,,,,,,,,,,, ;,, (),,,,
7 ),,,,,,, TSP, E-views, SPSS, SAS, Excel, () 1.,, :,,,,,,,,,,,,,,,,,,,,,,, 2.,,
8,,,,,,,,,,, 3.,,, 4.,,,,,,, 5.,,,,,, ( ) 1., 2. 3. () (),,,,,,
9,, ( ),, ;,,,, 1.2 : 1.2 (),, :,,,,,,,,,,,,,,,,,,,,
10,,,,, ;, : 1.,,, ( + ),,, 2., 3. : ;, ;,,,,,,, : (1 ) : (2 ) :,,,,, :
11,,,,,,,,,,,,,,,,,,,,,,,,,, ( ),,,,,,,,,,,,,,,,,
12 ),,,,,,, (),,,,,
,,,, ;,, :,,,,,, :,,, ( ), :, ; ; :,,,,,,,,,,, (Delphi Poll),,
14,,,,,,,,, 20 60 O. N., T.J., 1964,,,, :,,, 50,,,,,,,, 1964, T.J.O. (1015 ), : ; ; ; ; ;, 50%,,,,,,,, () ( ),,, : ( ),,
( Delphi Poll) 15 20,, ( ),,,,,,,,, (),, (),,,,,,,, (),,,,,,, (),,,,,, 2.1,, B,
16,, ( ),, 26, 60,46 2.1 A 25 60 85 25 70 80 25 75 80 B 35 50 75 35 50 75 35 50 75 C 50 60 70 40 50 60 50 70 75 A 5 15 37 9 22 47 9 24 47 B 30 55 85 35 50 70 25 68 75 A 40 55 80 35 45 70 25 35 60 B 10 25 55 22 35 60 20 35 60 A 19 22 31 22 28 34 22 28 34 B 20 30 45 22 34 44 22 34 44 A 16 22 31 12 25 31 28 37 62 B 20 35 50 20 35 50 25 45 50 20 35 55 20 35 50 25 45 50 23 39 58 24 40 57 26 46 60 37 57 77 33 57 72 37 65 77 18 34 61 17 36 59 17 46 60 22 42 65 25 41 63 22 38 60 20 26 38 22 31 39 22 31 39 18 29 41 16 30 41 27 41 56 20 35 55 20 35 50 25 45 50,,,,,,,,, ;
17 ;,,,,, : ; ;, : (1 ),, ; (2 ),,, : (1 ) ; (2 ), ; (3 ), (Subjective Probability),,,, : P( Ai ) Ai (1 )0P( Ai )1 (2 ) P(= 1), : (3 )Ai Aj, ij, j = 1, 2, Ai, Aj, : P i = 1 Ai = P ( i = 1 Ai ),,, 1/ 2,
18,,,,,, :,, : = :, 2.2 ( ) * 1000 0.3 30 800 0.5 400 600 0.2 300 820 1200 0.2 240 1000 0.6 600 800 0.2 160 1000 900 0.2 180 700 0.5 350 500 0.3 150 680,, 1/ 3, (820 + 1000 + 680)/ 3 = 833.4( ) 950, 750, 50%,
19 950 + 750)/ 2 = 850( ) 60%, 40%, : 833.460 + 850 + 40 100 = 840.04,,, 8 : 2.3 1 2 3 4 5 6 7 8 0.98 1.03 1.02 0.86 0.97 1.01 0.93 1.04 0.98 98 % - 1 = - 2 % 840.04 98 % = 823.6( ),,, :, 205, 6 2.4 ( % ) 1.0 12.5 25 37.5 50 62.5 75 87.5 99 ( ) 1%,, 99%,,
20.5 1 12.5 25 37.5 50 62.5 75 87.5 99 1 2 3 4 5 6 7 8 9 10 190 178 184 194 198 168 194 180 188 200 192 190 190 195 199 179 198 185 189 201 194 192 192 196 200 180 200 186 190 202 198 194 193 196 202 184 206 189 191 205 200 198 202 193 205 190 208 192 192 207 202 200 204 199 208 192 212 195 193 209 204 204 206 200 210 194 216 198 194 212 205 205 208 201 212 196 219 200 195 213 208 225 220 202 216 198 224 205 196 220 187.4 191.8 193.2 195.9 199.2 201.4 203.8 205.4 211.4,,, 199.2, 50%,, (199.2-6, 199.2 + 6 )( 193.2, 205.2),, 62.5% (87.5% 25% ),,,,,,,,,,,, ( Leading Indicators ),,,,
21 10 M1 18 ( Coincident Indicators ),,,, : M2 10 ( Lagging Indica tors),,,, : 6,,,,, : :, yt xt tth y y th yt = f ( xt - ), yth = f ( xt + h - ), = g(x th - ),,, 2.1
22,, t, t + 2.1 2.6 ( ) Money supply H ousing Permits Stock market Mortgage debt Residential Investmen t Busines s Loans Capital goods orders Inven tories Busines s invent ory Cr edit outstanding Industr ial construction Unemployed Rate Automobile sales Free r eserves 28.9 14.8 14.5 13.7 11.9 11.5 10.4 9.3 8.5 8.5 6.6 4.2 3.9 2.9
23,,,,,,?, (Diffusion Index ), 1, 0.5, 0, : ( DI t ) = + 0.5 100 DIt t, 1 6 9,,,, :, 0 < DIt < 50 %, 50%,,,,, 50% < DIt < 100 %,,,, DIt 100 %,, 100% > DIt > 50 %,,,,,,, 50% > DIt > 0,,,,,,, ;,
,,,, ( ), :,,, : 3.1 : (1 ); (2), ; (3 ),, ; (4 )
25 ) Y, X, 1. ( ) Y = +X +, Y X ;,,, X,, : 1.Y1 =+X1 + 1 2.Y2 =+X2 + 2 3.Y3 =+ X3 + 3, : N.Y N =+ X N + N Y i = +Xi + i, E( i ) = 0 Cov( i, j ) = 2 i = j 0 ij, 50,, : Y = + X +,,, Y =+ X,
26.,, n,, : y x 1 y1 x1 : 2 y2 x2 n yn xn a,, b, Y X : y = a + bx + e(, ), a b,, y = a + bx e,, y^ = a + bx y ei = yi - y^i ( i = 1, 2n) : y^1 = a + bx e1 = y1 - y^1 y^2 = a + bx e2 = y2 - y^2 y^3 = a + bx e3 = y3 - y^3 y^ n = a + bx en = yn - y^ n,,, (),, :, a b; a b
27,,, y x 1 y1 x1 2 y2 x2 n y n x n y^ = a + bx y, x, a b, yi y^i, n n ( y i = 1 i - y^i ) 2 = i = 1 e2 i n = ( i = 1 yi - a - bx) 2, a b : : b= n n i = 1 xi y i n n i = 1 x2 i a= y - b x b = n i = 1 a = y - b x n n - x i = 1 i i = 1 yi n - ( i = 1 xi ) 2 ( xi - x) ( yi - y) n ( x i = 1 i - x) 2,, a b, : 1.,,, b,, b, 2., : (1 ); (2 ) ; (3 )
28 a b, a b,, a b,, a b, a b y, y, a b,, : 2 bn [, n ] ( x i = 1 i - x) 2 an [, ( 1 n + x 2 n ) 2 ] ( x i = 1 i - x) 2 cov( a, b) = - n ( i = 1 xi x 2 - x) 2 : a, b, 2, S y 2 = n i = 1 e2 i / ( n - 2 ), 2, yi y^i, ( ) n b - 2 S y 2 ( xi - x) i = 1 a - ( 1 n + x 2 2 n ) Sy - x) 2 i = 1 ( xi t( n - 2 ) t ( n - 2), a b,: 1. ; 2., E( ei, ej ) = 0 ( ij) ; 3., i N( 0, 2 )
29,, ( ) 1., X Y 0, X Y b - ( Sb b ) t, t Sb, H0 : = 0, H1 0 tb = b Sb = b ( x - x) 2 Sy S y 2 = ( y - y^ ) 2 / ( n - 2 ) t tc ( n - 2) t > tc ( n - 2), H0, H1 : 0, 2., y^i = a + bxi y i = a + bx i + ei,,,,, y x,, b 0,,,,,, : H0 : b= 0, F,,, = +, : n n n ( i = 1 yi - y) 2 = ( i = 1 y^i - y) 2 + ( i = 1 yi - y^i ) 2 : n - 1 1 n - 2 : ST = SR + SE,
30 ST 2 2 ( n - 1) SR/ 2 2 ( 1) SE/ 2 2 ( n - 2) SR S E, b= 0, F = SR S E/ ( n - 2) = ( y^ - y) 2 / 1 F( 1, n - 2 ) ( y - y^)/ ( n - 2 ),, F F(1, n - 2 ), F > F( 1, n - 2 ),,,, F, 3.1 F SR SE 1 n - 2 S T n - 1 SR SE/ ( n - 2) S R SE/ ( n - 2 ) F : (1 ), ; (2 ) Y X ; (3 ) Y X 3. D.W, : Cov( i, j ) = 0,,,,, t F, D. W, ij : H0 := 0,,
31 d n i = 2 ( ei - ei - 1 ) 2 n i = 1 ei 2, n p, Durbin - Wa tson D.W, dl du, d d L du, 3.2 : 3.2 D.W du < d < 4 - du,,, d 2,, 0 < d < dl,, 4 - dl < d < 4,, dl ddu 4 - du d4 - dl, : (1 ); (2 ),, ; (3 ) ; (4 ) ; (5 ) 4. ( S.E. of regression ) S y
32 Sy n i = 1 ( yi - y^) 2 n - 2 Sy,, ; Sy 0,,,, ;, Sy, Sy/ y S y/ y 15 %, 5.,, = +,,, R 2, : R 2 =,0R 2 1R 2 n i = 1 ( y^i - y) 2 n = 1 - ( i = 1 yi - y) 2 R 2,, R 2, R 2 n ( y i = 1 i - y^i ) 2 n ( y i = 1 i - y) 2 = 1,, = 0, 1, 0.8,,,,, ( ) 1. ( Mean Error) n ME = ei/ n i = 1 2. ( Mean Absolute Error) n MAE = ei / n i = 1
33. ( Sum of Squared Error ) SSE = 4. ( Mean Squared Error) n 2 i = 1 ei 2 MSE = / n n i = 1 ei 5. ( Standard devia tion of Error) SDE = n i = 1 ei 2 / ( n - 1 ),, ;,, ;,,,,,,,,,, ( ), : 1. ( Percentage Error) P Ei = ei/ y100 % 2. ( Mean Percentage Error ) n MP E = P Ei/ n i = 1 3. ( Mean Absolute Percentage Error ) n MA PE = PEi / n i = 1,,, MAP E < 10, x, y (), x,
34 y a N (, 1 n + x2 Lx x b N, 2 cov( a, b) = -, Lx x Lx x x L x x 2 ) 2 n 2 = ( xi - x) i = 1, x0,, y^0 = a + bx 0 y^0 N + x0, 1 n + ( x0 - x) 2 Lx x 2 y^0 E( y0 ) = + x0, y^0 y^0,, y0 - y^0 ( ) x0 y^0, 1 - ( ), x0, y0, P( y0 - y^0 ) = 1 - P( y^0 - y0 y^0 + ) = 1 - y^0 -, y^0 + y0 1 -?, x = x0, y0 y^0, y0, y^0 x0, y0 y^0, N (0, 2 ), y0 - y^0, E( y0 - y^0 ) = 0 var ( y0 - y^0 ) = var ( y0 ) + var ( y^0 ) = 1 + 1 n + ( x0 - x) 2 Lx x 2
35 y - y^0 N 0, 1 + 1 2 ( x0 - x) + n Lx x 2 Sy/ 2 2 ( n - 2 ), S y y0 - y^0, t y0 - y^0 1 + 1 n + ( x0 - x) 2 Lx x S y ( n - 2) 2 y0 - y^0 = ^1 + 1 n + ( x0 - x) 2 Lx x t( n - 2 ) ^2 = S y ( n - 2 ) ^ = ^ 2, P y0 - y^0 = P y0 ^1 + 1 n - y^0 t + ( x0 - x) 2 Lx x 1 + 1 n ^1 + 1 2 ( x0 - x) + n Lx x ^ + ( x0 - x) 2 Lx x = t1 - a 2 ( n - 2) = 1 - : = t1 - a ( n - 2) 2 ^ 1 + 1 n + ( x0 - x) 2 Lx x y^0 - t1 - a ( n - 2) 2 ^ 1 + 1 n + ( x 0 - x) 2 Lx x, y^0 - t1 - a ( n - 2) 2 ^ 1 + 1 n + ( x0 - x) 2 Lx x ^2 = Sy ( n - 2) = y^0 ts y 1 + 1 n + ( x0 - x) 2 ( x - x) 2
36 Lx x, x0 Lx x x0 x x, Lx x, x = x, x0, : x 3.3 n, 1 + 1 n + ( X0 - x) 2 Lx x 1, :, Sy = y^tsy,, Excel, : :, 3.2 :,, 3.4, :,,, 3.5
37.2 3.4 3.5
38, y x,, 3.6, 3.6 : 3.7
39 : (1 ) y^ = 21. 22187 + 0. 086229 (2 ) R 2 = 0. 822409, (3 ) t ta = 4.058403, tb = 9.38015 5 %, 21-2 = 19, t, t0. 02 5 = 2. 093,,, 10.27719 < a < 32.16654 0.066989 < b< 0.10547 (4 ) F ( ) 3.3 df SS MS F SignificanceF 1 20792.16 20792.16 87.98721 1.46E - 08 19 4489.87 236.3089 20 25282.03 = 5 %, ( 1, 19 ), F, F0. 0 5 (1. 19) = 4. 38, F, (5 ) S y = 15.37234, S y y = 59.49% > 15%, (6 )D.W d = 0.316878, = 1%, n = 21 p = 1, D.W, dl = 0.97 du = 1.16,0 < d < dl,,,,, 19862000,, 3.4:
40 3.4 Dependent Variable : Y Met hod: Least Squares Sample : 1986 2000 Included obse rvations : 15 Variable Coefficient Std. Error t - Statistic P rob. C X 42.89438 0.059689 3.508056 0.005232 12.22739 11.40890 0.0000 0.0000 R - squared Adjusted R - squared S.E. of regression Sum squared resid Log likelihood Durbin - Watson stat 0.909194 0.902209 6.996619 636.3848-49.39223 1.179151 Mean dependen t var S.D. dependent var Akaike info c riterion Sch warz criterion F - statistic Prob( F - statistic) 77.20264 22.37379 6.852297 6.946703 130.1630 0.000000, F d = 1.179, = 1%, n = 15 p = 1, D.W, d1 = 0.81 du = 1.07, du < d < 4 - du, : 2001 x0 = 1100, y^ 0 = 108.55228 95 % y^0 tsy 1 + 1 n 2 ( x0 - x) + ( x - x) 2 = 108.551.77096.99661 + 1 15 = 108.5513.70886 + ( 1100-436.3606 ) 2 2796352 2001 1100, 95 %, 94.84114122.25886,
41,,,,,, ( ) Y, X1, X2, X p,, Y = 0 + 1 X1 + 2 X2 + + p X p + j, X Y ; 0, j ( j = 1, 2p),, X,, : 1. Y1 = 0 + 1 X11 + 2 X12 + + p X 1 p 2.Y2 = 0 + 1 X21 + 2 X22 + + p X 2 p N. Y N = 0 + 1 X N 1 + 2 X N2 + + i X N p, : Y i = 0 + 1 Xi1 + 2 Xi2 + + p X ip, : + i + 1 + 2 + N (1 )X1, X2, X p,, (2 ) Y, (3 )0, E( i ) = 0 Cov( i, j ) = 2 i = j 0 ij : E( Y ) = 0 + 1 X1 + 2 X2 + + p X p va r( Y ) = 2 I
42 YN 0 + 1 X1 + 2 X2 + + p X, 2 I) Y = 0 + 1 X1 + 2 X2 + + p X p ( ),, n,, : y x1 x2 xp 1 y1 x11 x12 x1 p : 2 y2 x21 x22 x2 p n y n x n1 xn2 xnp j, j, b0 : ) bj ( j = 1p), Y X y = b0 + b1 x1 + b2 x2 + + bp x p + e(,, b0 bj 0 j,, y^ = b0 + b1 x1 + b2 x2 + + bp x p e,, y^ = b0 + b1 x1 + b2 x2 + + bp x p y ei = y i - y^i ( i = 1, 2n) (),,, n i = 1 e2 1 = ( yi - b0 - b1 x1 - b2 x2 - bp x p ) 2, 0, : yi = nb0 + b1 xi1 + b2 x i2 + + bp xip x i1 y i = b0 xi1 + b1 x 2 i2 + b2 x i1 xi2 + + bp xi1 xi p x ip y i = b0 xip + b1 x ip x i1 + b2 xip x i2 + + bp x 2 ip
43 b0, b1 bp Y = Xb+ e y1 b1 e1 : Y = y2 b= b2 e = e2 yn bp en 1 x11 x1 2 x1 p X = 1 x21 x2 2 x2 p 1 xn1 xn2 xnp A = X X Ab = B, ( X X ) b= ( X Y ) b= A - 1 B = ( X X ) - 1 B = X Y X Y, b : (1 )b0, b1, bp y i ( ) yi, b0, b1, bp, bi, bj N [ j, 2 ( X X ) - 1 ] (2 ), b0, b1, bp 0, 1, p, 2 ( X X ) - 1 2, ^2, bj t, n - p - 1, : t = bj - j Sb, Sb bj (3 ) b 2 A - 1, Cov( bj, bk ) = 2 Cjk ( j, k = 1, 2p) C= A - 1, A, : 1.,,
44 bj xi, bj, 2., : (1 ) ; (2 ); (3 ) ( ),, bj, 0,,,,, bj 0,,,, t H0 : j = 0, H1 : j 0 bj, t = bj Sb bj, t Cj j = S ( n - p - 1) ( j = 1, 2p), Cj j C = A - 1 j S, t tc ( n - p - 1) t > tc ( n - p - 1 ), H0, H1 : 0,, t F, F, : H0 F = ( bj - j ) 2 Cjj ( S ( n - p - 1) ) F(1, n - p - 1 ), F = b 2 j Cjj S ( n - p - 1 )
45 : ; ;, ( ),,, : H0 b1 = b2 = b3 = = bp = 0 F F( p, n - p - 1 ) = ( y^i - y) 2 / p ( yi - y^i ) 2 / n - p - 1 F( p, n - p - 1 ) > F F( p, n - p - 1 ) F,,,, : (1 ); (2 ) (), : R 2, R 2 R 2 = ( y^i - y) 2 = 1 - ( yi - y^i ) 2 ( yi - y) 2 ( yi - y) 2 1,,, ( y^i - y) 2, ( y^i - y) 2,, R 2 R 2,, : R 2 = 1 - ( yi - y^i ) 2 / n - p - 1 ( yi - y) 2 / n - 1,,,,, R 2 R 2,, R 2 R 2
46 R 2 R = 1 - (1 - R 2 ) n - 1 n - p - 1,, F R 2 : F( p, n - p - 1) = R 2 n - p - 1 1 - R 2 p y F y x j ( j = 1, 2p), ( ) (D.W ),D.W, Sy = n ( y - y^) 2 i = 1 n - p - 1,,,,,,,,,, () 1.,,, y x 1 b1 ( 1 ), y x2 b2 ( 1 ), y x 1, x2 b1 ( 2 ), b2 ( 2 ), x1 x2,, b1 ( 1 ) b1 ( 2 ), b2 ( 1 ) b2 ( 2 ), x1 x2, x2 x1,, x1 x2, x1 x2, r12 0, b1 ry1 r1 2, x2 x1,, x1 x2,,,
47 2., R 2, R 2 y, ( yi - y) 2 y,, ( y^i - y) 2, ( yi - y^) 2, R 2, S( x1 ) y x1, S( x2 ) y x2, S( x1 x2 ) y x1, x2, S( x1 x2 ) x2, x1, S( x1 x2 ) = S( x2 ) - S( x1 ) x1 x2,, S( x1 x2 ) = S( x1 ) S( x1 ) y x1, x1 x1 x2, x1 x2, x1 S( x2 ), S( x1 x2 ) < S( x1 ),,, y ( ), (1 ) F, t ; (2 ), ; (3 ); (4 ),, r > 0.7, (),, ; ;,,,,
48,, : ( ),,,,, () AIC AIC ( An information criterion ), 1973 ( Akaike) ARMA, AR, MA AIC AIC = nlog ^2 + 2 p ^2 2, S p, n, p ^2 = 1 ^ ^ ( Y - X ) ( Y - X ) n ^ = ( X X ) - 1 X Y AIC,, AIC, AIC () Cp ( Mallows), Cp Cp Cp = S^2 + 2 p ^2 = S ( n - p - 1 ), p, p ( S p - 1SS ) Cp,, Cp, Cp,,,,
49,, : 3.5 :2000 Excel :, : 3.8
50 3.9, 3.6 3.6 M ultiple R R Square Adjusted R Square 0.983782 0.967828 0.957104 873.306 17 3.7 df SS MS F Significance F 4 2.75E + 08 68828824 90.24798 7.55E - 09 12 9151960 762663.3 16 2.84E + 08 3.8 Coefficients ts t a t P - v alu e Lo wer95 % Upper 95 % 95.0 % 95.0% In t erce pt - 47482.9 13258.96-3.58119 0.003774-76371.7-18594.1-76371.7-18594.1 X Variabl e1 0.71964 0.116885 6.156799 4.9 E - 05 0.464968 0.974311 0.464968 0.974311 X Variabl e2 4.558464 0.308989 14.75284 4.7 E - 09 3.885235 5.231693 3.885235 5.231693 X Variabl e3-0.08521 0.059508-1.43192 0.177695-0.21487 0.044446-0.21487 0.044446 X Variabl e4 0.072091 0.249412 0.289045 0.77748-0.47133 0.615514-0.47133 0.615514
51, (1 ) y^ = - 47482.9 + 0.71964 x1 + 4.558464 x2-0.08521 x3 + 0.07209 x4 (2 ) R 2 = 0.967828 R 2 = 0.957104, (3 ) t tb 0 = - 3.58119, tb 1 = 6.156799, tb 2 = 14.75284, tb 3 = - 1.43192, tb 4 = 0.289045 5 %, 17-4 - 1 = 12, t, t0.025 = 2.1788,, b3, b4 (4 ) F ( ), = 5%, ( 1, 15 ), F, F0.05 (1, 15) = 4.54, F, (5 ) S y = 873.306, Sy y = 1.953% < 15%, (6 )D.W d = 1.179151, = 1%, n = 17 p = 4, D.W, dl = 0.68 du = 1. 77, dl < d < du,,,,, F,,, 3.8 3.8 x 1 x1 x2 x3 x4 1 x 2-0.01885 1 x 3-0.28563 0.49574 1 x4 0.221964 0.487609 0.397576 1,,,,, X4,,, 3.9
52.9 Dependent Variable : Y Met hod: Least Squares Sample : 1985 2001 Included obse rvations : 17 Variable Coefficient Std. Error t - Statistic P rob. C - 46841.71 12602.95-3.716726 0.0026 X1 0.731662 0.105311 6.947652 0.0000 X2 4.588584 0.280450 16.36152 0.0000 X3-0.079758 0.054413-1.465795 0.1665 R - squared 0.967604 Mean dependent var 44716.60 Adjusted R - squared 0.960128 S.D. dependent var 4216.539 S.E. of regression 841.9587 Akaike info criter ion 16.51166 Sum squared resid 9215628. Schwarz criterion 16.70771 Log likelihood - 136.3491 F - statistic 129.4277 Durbin - Watson stat 1.162342 Prob( F - statistic) 0.000000 b3, D.W x3,, 3.10 3.10 Dependent Variable : Y Met hod: Least Squares Sample : 1985 2001 Included obse rvations : 17 Variable Coefficient Std. Error t - Statistic Prob. C - 54840.94 12706.85-4.315855 0.0008 X1 0.785732 0.111642 7.037864 0.0000 X2 4.403592 0.300906 14.63444 0.0000 X4-0.041124 0.245919-0.167226 0.8698 R - squared 0.962331 Mean dependent var 44716.60 Adjusted R - squared 0.953638 S.D. dependent var 4216.539 S.E. of regression 907.9002 Akaike info criterion 16.66247 Sum squared resid 10715676 Schwarz criterion 16.85852 Log likelihood - 137.6310 F - statistic 110.7029 Durbin - Watson stat 1.603753 Prob( F - statistic) 0.000000
53 D W, x4 x3, x4,, : 3.11 Dependent Variable : Y Met hod: Least Squares Sample : 1985 2001 Included obse rvations : 17 Variable Coefficient Std. Error t - Statistic Prob. C - 55546.06 1156.24-4.803678 0.0003 X1 0.780779 0.103852 7.518179 0.0000 X2 4.378207 0.250630 17.46878 0.0000 R - squared 0.962250 Mean dependent var 44716.60 Adjusted R - squared 0.956857 S.D. dependent var 4216.539 S.E. of regression 875.8150 Akaike info c riterion 16.54697 Sum squared resid 10738727 Schwarz criterion 16.69401 Log likelihood - 137.6493 F - statistic 178.4289 Durbin - Watson stat 1.612149 P rob( F - statistic) 0.000000,,,, y^ = - 55546.06 + 0.780779 x1 + 4.378207 x2, x1, x2,,,, ( ),,,, y, y
54 1. :, ( 2 n 2. - 1 ),,,,,,,,,,, ;, 3.,,,,,,,,,,,,,, ;, 4. (2 ) ( 3 ),,,, ( ),,,,,,,,,,,
55,,, :??,,,, F ;,,,,,,, : l, bj l S= bj L jy j = 1 = S- S, L jy l x j, : : V j : V j = b2 j C jj S= S- S = S- S = S- S Cj j ( A) Cij j : V ( l) j = ( l) b( j ) 2 C ( j j l) (1 ),, k, V ( l) k V ( l) l) k = min j V( j, F= ( n - l - 1) V k ( l) S ( l) FFa 2,, (2 ),,
56 ( l + 1 ) V k k, l + 1 ) = max j V( j ( l + 1 ) V k, F= ( n - ( l + 1 ) - 1 ) V k ( l + 1 ) S ( l + 1 ) = ( n - l - 2) V k ( l + 1 ) ( S ( l) - V k ( l + 1 ) ) FF 1,, ( ) y^ = b0 + b1 x1 + b2 x2 + + bp x p y^ - y = b1 ( x1 - x1 ) + b2 ( x2 - x2 ) + + bp ( x p - xp ) : x i = xi - xi S x i Sx i = n 1 n - 1 ( x i = 1 i - xi ) 2 y^ - y S y = b1 x1 - x1 S y + b2 x2 - x2 S y + + bp x p - xp S y = b1 x1 - x1 Sx 1 S x 1 Sy + b2 x2 - x2 S x 2 S x 2 Sy + + bp x p S x p - x p S x p S y b j = bj S x j S y = bj L j j L0 0, Lj j = ( x j - x j ) 2, L0 0 = ( yi - y) 2 : y^ - y S y x 1 = b1 - x1 Sx 1 x2 + b 2 - x2 Sx 2 + + b p x p - xp S x p y^ = b 1 x 1 + b 2 x 2 + + b p x p n ( x i1 i = 1 n n - x 1 ) 2 b 1 + + i1 i = 1 - x 1 ) ( x ip - x p ) b p = i1 i = 1 - x 1 ) ( y i - y) ( x ip - x p ) ( x i1 - x 1 ) b 1 + + ( x ip - x p ) 2 = ( x ip - x p ) ( y i - y)
57 L 1 L00 L 11 b 1 + L 12 L00 L00 b L 2 + + L 1 p b 22 L p = L 10 pp Lp1 L 00 b 1 + Lp2 L 11 L 00 b 2 + + Lpp L 22 L 00 b p = Lp0 L pp k ( k = 1,2, p) L00, Lj j ( j = 1,2p) L 11 L 11 b 1 + L 11 L 11 L 12 b 2 + + L 22 L 11 L 1 p b p = L pp L 10 L 00 L 11 Lpp Lp1 L11 b 1 + Lpp L p2 L22 b 2 + + L pp Lpp = r10 r11 b 1 + r1 2 b 2 + + Vip b p L pp b p = r21 b 1 + r2 2 b 2 + + r2 pb p = r2 0 rp1 b 1 + rp2 b 2 + + rpp b p = rp0 RX = r1 1 r1 2 r1 p r2 1 r2 2 r2 p rp1 rp1 rpp b= Rx b= Ry b 1 b 2 b p b= R - 1 x R y : RY = r10 r20 L00 L p0 rp0 Lpp b j = bj L jj L00 bj = b j Sij Cij : Cij = r ( - 1 ) i j : L ii L j j L00 L jj, S= L0 0, S= L0 0 S S ( ), V = L00 V, R = R, S = L00 S t, ti = t i, Ry x i = R y x i,,
58 ), r11 r12 r1 p r1 y r21 r22 r2 p r2 y R ( 0 ) =, rp1 rp2 rpp rpy ry1 ry2 ryp ry y,, R m = r ( m) i j m, m + 1 k, m + 1, : r ( m + 1 ) i j = ( m) rkj ( m) rkk i = k, jk rij ( m) - rik ( m) rkj ( m) rkk ( m) ik, jk 1 rkk ( m) i = k, j = k - rik ( m ) rkk ( m) ik, j = k m : ( m ) r11 ( m) r12 ( m) r1 p ( m) r1 y r21 ( m ) r22 ( m) r2 p ( m) r2 y ( m) R ( m) = ryi ( m) : rp1 ( m) rp2 ( m) rpp ( m ) rpy ( m) ry1 ( m) ry2 ( m ) ryp ( m) ry y ( m ) x1, x2 x1, ( l < p), : (1 ) ( m) ( m) (2 )xi, ryi = - riy, = riy ( m), ( m) (3 ) ry y (4) R ( m) k1, k2 k1 k1, k2 k1 ( ),R 0, Cij = r ( ij - 1) Sii Sjj (5 ) : m + 1, xk m) V k ( m + 1 ) = r( k y r ( kk m) 2
59 y( / ) : x1 3CaOAl2 O3 ( % ) ; x2 3CaO SiO2 ( % ) ; x3 4CaO Al2 O3 Fe2 O3 ( % ) ;x2 4CaOSiO2 ( % )3.12 3.12 1 2 3 4 5 6 7 8 9 10 11 12 13 x 1 x 2 x 3 x 4 y 7 26 6 60 78.5 1 29 15 52 74.3 11 56 8 20 104.3 1 31 8 47 87.6 7 52 6 33 95.9 11 55 9 22 109.2 3 71 17 6 102.7 1 31 22 44 72.5 2 54 18 22 93.1 21 47 4 26 115.9 1 40 23 34 83.8 11 66 9 12 113.3 10 68 8 12 109.4 y x1, x2, x3, x4 ( ) 1. 0 r11 ( 0 ) r12 ( 0 ) r1 p ( 0 ) r1 y ( 0 ) R ( 0 ) = ( 0 ) r21 ( 0 ) r22 ( 0 ) r2 p ( 0 ) r2 y rp1 ( 0 ) rp2 ( 0 ) rpp ( 0 ) rpy ( 0 ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) ry1 ry2 ryp ryy 1 0.228579-0.824134-0.245446 0.73071 0.228579 1-0.139242-0.97295 0.816253 = - 0.824134-0.1392 1 0.029537-0.534671-0.245445-0.9729 0.029537 1-0.821305 0.73071 0.81625-0.53471-0.82105 1 2., F
60 F,, F ( ) ; F,,,,, n m, ( n - m - 1), = 0.10, 4 2-3, f1 f2 = 10, F F( 1, 10 ) = 3.28 ( ) : = 1 ( 1 ), (1 )( ), (, ) : V j ( 1 ) = ( rjy ( 0 ) ) 2 rjj ( 0 ) = U = blx y, F, ( 1 ) V1 = ( r1 y ) 2 r11 V2 ( 1 ) = ( r2 y ) 2 ( 1 ) V3 x4, : x4 ( 1 ) F4 r22 = 0.285873; V4 = ( 1 ) Fj = (0.730717) 2 = 0.533947 = (0816253 ) 2 = 0.666269 ( 1 ) = ( n - 2 ) V j 1 - V j 110.674542 1-0.674542 = 0.67542 * ( 1 ) ( 1 ) = 22.7986 > 3.28 (2 ), 1 R ( 1 ) R ( 1 ) = = ( rij ( 1 ) ) 0.9377-0.0102-0.8168 0.2454 0.5291-0.0102 0.0533-0.1105 0.9729 0.0171-0.8186-0.1105 0.9991-0.0295-0.5104-0.2454-0.9729 0.0295 1.0000-0.8213 0.5291 0.0171-0.5104 0.8213 0.3254
61 x : R ( 0 ) x4, x4-0.8213 0.3254 4. : 3.13 F 1 11 10.3254 = 0.6745 0.3254 12 1 0.6745/ 1 = 22.7986 0.3254/ 11 : (1 ), 1 R ( 1 ), V1 Fj ( 2 ) = = ( 2 ) ( 2 ) V j ( 2 ) V 1 ( 2 ) V2 = ( rjy = ( 1 ) ) 2 ( 1 ) rjj (0.5291) 2 0.9397 = 0.0055 V3, F, ( n - ( l + 1) - 1) Vj (13-2 - 1) V 1 S ( l + 1 ) ( 2 ) ( 2 ) = S- V1 ( l + 1 ) = 0.2979 * ( 2 ) (13-3 )0.2979 0.3254-0.2979 x1, x1 = 0.2607 = 108.22 > 3.28 (2 ) x1 2 : R ( 2) = (3 ) R ( 2 ) : 1.064105-0.010884-0.86925 0.261179 0.563052 0.010833 0.053248-0.119395 0.975626 0.022919 0.869250-0.119395 0.289051 0.183816-0.050463 0.261179-0.975626-0.183816 1.064105-0.683107-0.563052 0.022919-0.050463 0.683107 0.027528
62 x, x4 : b4 ( 2 ) ( 2 ) b1 = - 0.681307 = 0.563052 0.027528 : 3.14 F 2 0.972471 176.6265 10 0.027528 12 1 x1, x4 x1, x4 F, F,, V4 F4 ( 2 ) x4 : F (2 ) = ( r4 y ( 2 ) ) 2 ( 2 ) = ( - 0.683107) 2 r44 1.064105 = ( n - l - 1) V k ( l) ( l) = S = 159.29 > 3.28 = 0.4385 ( 13-2 - 1 )0.4385 0.027528 (1 ), 2 R ( 2 ), V2 ( 3 ) ( 3 ) V j V2 ( 3 ) = ( 3 ) V3, F, Fj ( 3 ) = = = ( rj y ( 2 ) ) 2 ( 2 ) rj j (0.0229119 ) 2 0.053248 = 0.008810 ( n - ( l - 1) - 1) Vj S ( l + 1 ) ( l + 1 ) (13-3 - 1)0.009865 0.027528-0.009865 = 0.009865 * = 5.026 > 3.28
x, x2 (2 ) x2 3 : R ( 3 ) = 1.06633 0.204391-0.8936 0.660589 0.567737 0.204391 18.780350-2.242271 18.3226 0.430415 0.893654 0.242271 0.021336 2.371435 3.000926 0.460589 18.322604-2.371435 18.940119-0.263182-0.567737-0.430415 0.000926 0.263182 0.017664 (3 ) R ( 3 ) : x1, x4, x2 : b ( 3 ) 4 = - 0.263182 ( 3 ) b1 = 0.567737 b2 ( 3 ) = 0.430415 0.017664 : 63 3.15 F 3 0.982336 166.8340 9 0.017664 12 1 x1, x4, x2 x2 F, F,,, x1, x4 ( 3 ) V4 ( 3 ) F4 ) 2 = ( ( 2 ) r4 y ( 2 ) = 0.003657 r4 4 ( 3 ) ( 3 ) = S = ( n - 3-1 ) V4 = 1.863 < 3.28 F ( 13-3 - 1 )0.003657 0.017664 x2, x4, x4 : (1 ) F x4 (2 )x4,,
64 1.05512-0.24118-0.835985 0.024318 0.574137-0.24118 1.055129 0.051847-0.967396 0.688507 R 4 ) = 0.855985-0.051857 0.318256-0.125207 0.033878 0.024318 0.967396-0.125207 0.052798-0.013896-0.574137-0.68501 0.033878-0.013896 0.021322 (3 )R ( 4 ) : x1, x4 : b1 ( 4 ) ( 4 ) b2 = 0.574137 = 0.685017 0.021322 : 3.16 F 2 0.978678 229.5211 10 0.021322 12 1 x1, x2 F, F1 ( 4 ) = 146.52 > 3.28 F2 ( 4 ) = 208.578 > 3.28 :, ( 5 ) V 3 ) 2 = ( ( 4 ) r3 y ( 4 ) = 0.00360628 r33 V 4 ( 5 ) = 0.00363731 x4, x4,, ( ) 1. 4 x3, x4 3 4 3 4, R ( 4 ) = 1.055129-0.241181 0.574137-0.241181 1.055129 0.685017-0.574137-0.685017 0.021322
65 x, x2 x1, x2, 1.055129-0.241181 C ( 2 ) = - 0.241181 1.055129 x1, x2 2. bj = b j L00 L jj L0 0 = 2715.7635, L11 = 415.2308, L22 = 2905.6923 b 1 = 0.574137 b 2 = 0.685017 b1 = 1.4683 b2 = 0.6623 b0 = y - b1 x1 - b2 x2 = 52.5742 : y = 52.5742 + 1.4683 x1 + 0.6623 x2 3.17 F 2 2657.8889 1328.9290 10 5.7906 5.7906 229.2511 12 2715.7635 s = R 2 = 0.9893 n ( y i = 1 i - y^i ) 2 13-2 - 1 = 2.4064 x1, x2, y x 3, x4 : r3 y 12 2 = ( r3 y ( 4 ) 2 r4 y 12 ) 2 r33 ( 4 ) ry y ( 4 ) = 0.1691 = ( r4 y ( 4 ) r44 ( 4 ) ry y ) 2 ( 4 ) = 0.1715 SPSS : : SPSS, :
66.10 : (1 ) Analyze Regression Linear (Dependent), ( Independent), 3.11 3.11
67 2 ) Statistics, 3.12 (3 ) Continue Plots, 3.13 (4 ) Continue Save,
68.14 (5 ) Continue Option s 3.15
69, 3.16 3.16 : 3.18 ( Descriptive Statistics) Mean Std. Deviation N Y 95.4231 15.0437 13 X1 7.4615 5.8824 13 X2 48.1538 15.5609 13 X3 11.7692 6.4051 13 X4 30.0000 16.7382 13 3.19 ( Correlations ) Y X1 X2 X3 X4 Pea rson Corr elation Y 1.000.731.816 -.535 -.821 X1.731 1.000.229 -.824 -.245 X2.816.229 1.000 -.139 -.973 X3 -.535 -.824 -.139 1.000.030 X4 -.821 -.245 -.973.030 1.000 Sig. ( 1 - tailed) Y..002.000.030.000 X1.002..226.000.209 X2.000.226..325.000 X3.030.000.325..462 X4.000.209.000.462.
70 3.20 ( Variables Entered/ Removed) Varia bles Va riables Model E ntered Removed M e thod 1 X4 Stepwise (Criteria: F-to-enter > = 3.840, F-to-remove < = 2.710). 2 X1 Stepwise (Criteria: F-to-enter > = 3.840, F-to-remove < = 2.710). 3 X2 Stepwise (Criteria: F-to-enter > = 3.840, F-to-remove < = 2.710). 4 X4 Stepwise (Criteria: F-to-enter > = 3.840, F-to-remove < = 2.710). a Dependen t Variable: Y 3.21 ( Model Summary) R Adju st ed Std. E rror of Mod el R R Square Squ are R Square the Es tima te Cha nge Ch a ng e St atistics F df1 df2 Ch a ng e Dur bin Sig. F -Wa tson Ch ang e 1.821.675.645 8.9639.675 22.799 1 11.001 2.986.972.967 2.7343.298 108.224 1 10.000 3.991.982.976 2.3087.010 5.026 1 9.052 4.989.979.974 2.4063 -.004 1.863 1 11.205 1.922 a b c d e Predictors : ( Constant), X4 Predictors : ( Constant), X4, X1 Predictors : ( Constant), X4, X1, X2 Predictors : ( Constant), X1, X2 Dependent Variable : Y 3.22 ( ANOVA ) Model Sum of Squares df Mean Squa re F Sig. 1 Regression 1831.896 1 1831.896 22.799.001 R esidual 883.867 11 80.352 To tal 2715.763 12 2 Regression 2641.001 2 1320.500 176.627.000 R esidual 74.762 10 7.476 To tal 2715.763 12 3 Regression 2667.790 3 889.263 166.832.000 R esidual 47.973 9 5.330 To tal 2715.763 12 4 Regression 2657.859 2 1328.929 229.504.000 R esidual 57.904 10 5.790 To tal 2715.763 12 a b c d e Predictors : ( Constant), X4 Predictors : ( Constant), X4, X1 Predictors : ( Constant), X4, X1, X2 Predictors : ( Constant), X1, X2 Dependent Variable : Y
71.23 (Coefficients ) Model Unstanda rdized Coefficie nts Std. B E rror Standardized Coefficie nts Be ta t Sig. 95 % Confide nce Int erval for B Lower U pper Bound Bound Correla tion s Zero- order Partial Part 1 ( C) 117.568 5.262 22.342.000 105.986 129.150 X4 -. 738.155 -. 821-4.775.001-1.078 -. 398 -. 821 -. 821-. 821 2 ( C) 103.097 2.124 48.540.000 98.365 107.830 X4 -. 614.049 -. 683-12.621.000 -. 722 -. 506 -. 821 -. 970-. 662 X1 1.440.138.563 10.403.000 1.132 1.748.731.957.546 3 ( C) 71.648 14.142 5.066.001 39.656 103.641 X4 -. 237.173 -. 263-1.365.205 -. 629.155 -. 821 -. 414-. 060 X1 1.452.117.568 12.410.000 1.187 1.717.731.972.550 X2.416.186.430 2.242.052 -. 004.836.816.599.099 4 ( C) 52.577 2.286 22.998.000 47.483 57.671 X1 1.468.121.574 12.105.000 1.198 1.739.731.968.559 X2.662.046.685 14.442.000.560.764.816.977.667 a Depe ndent Variable : Y 3.24 ( Ex clude d Varia bles ) Model Beta In t Sig. Partial Correlation Collinearity Statistics Tolera nce 1 X1.563 10.403.000.957.940 X2.322.415.687.130 5.336 E - 02 X3 -. 511-6.348.000 -. 895.999 2 X2.430 2.242.052.599 5.325 E - 02 X3 -. 175-2.058.070 -. 566.289 3 X3.043.135.896.048 2.134 E - 02 4 X3.106 1.354.209.411.318 X4 -. 263-1.365.205 -. 414 5.280E - 02 a b c d e P redictors in the Model: (Constant), X4 P redictors in the Model: (Constant), X4, X1 Predictors in the Model : (Constant), X4, X1, X2 Predictors in the Model : (Constant), X1, X2 Dependent Va ria ble : Y
72 3.25 Coefficient Correlations Model X4 X1 X2 a 1 Correlations X4 1.000 Covariances X4 2.390E - 02 2 Correlations X4 1.000.245 X1.245 1.000 Correlations X4 2.366E - 03 1.653E - 03 X1 1.653E - 03 1.916E - 02 X4 1.000.102.972 Correlations X1.102 1.000.046 3 X2.972.046 1.000 X4 3.003E - 02 2.078E - 03 3.125E - 02 Correlations X1 2.078E - 03 1.369E - 02 9.918E - 04 X2 3.125E - 02 9.918E - 04 3.445E - 02 4 Correlations X1 1.000 -.229 X2 -.229 1.000 Correlations X1 1.471E - 02-1.271E - 03 X2-1.271E - 03 2.103E - 03 Dependent Va riable : Y 3.26 ( Residuals Sta tistics ) Minim um Maximu m Mean Std. Deviation N P redicted Value 73.2509 114.5375 95.4231 14.8825 13 Std. P redicted Value -1.490 1.284.000 1.000 13 Standard Error of Predicted Value.6958 1.7846 1.1238.2819 13 Adjusted Predicted Value 72.8787 113.0821 95.3974 14.7715 13 Residual -2.8934 4.0475-1.2025E-14 2.1967 13 St d. Residual -1.202 1.682.000.913 13 Stud. R esidual -1.356 1.788.002 1.024 13 Deleted Residual -3.6821 4.5741 2.566 E - 02 2.7969 13 Stud. Deleted Residual -1.425 2.057.029 1.078 13 Mahal. Dista nce.080 5.677 1.846 1.447 13 Cook s Dista nce.011.290.093.081 13 Cen ter ed Lever age Value.007.473.154.121 13 a Dependent Va riable : Y
73.17 3.18 3.19
74.20 3.27,,, ( ),
75,,,,,, :, ( ), : 1. y = b0 + b1 x + b2 x 2 + + bk x k + e y = b0 + b1 x1 + b2 x2 2 + + bk x k k + e 3.21 3. 22 2. y = ab x e y = ae bx e 3.23
76. y = ax b e 3.24 4. y = a + b x + e 1 y = a + b x + e 3.25 5. y = a + bln x + e 3.26
77. y = a + bsin x + e ( ), : 1. y = b0 + b1 [ e x p( b2 x) ] + e y = L + ab x y = L + ae bx 3.27 y = L 3.27 2., ^ y = Le - ae - bx y = La bx 3.28,y = L/ e, x = lna/ b, 3.28
78., ^ y = L 1 + ae - bx, y = L/ 2, x = ln a/ b, 3.29 S, S, ( ),,, : y = b0 + b1 x + b2 x 2 + + bk x k + e X1 = x, X2 = x 2, Xk = x k y = b0 + b1 X1 + b2 X2 + + bk X k + e y = a+ b x + e X = 1 x, y = a + b X + e
79 y a + bln x + e X = ln x, y = a + b X + e y = a + bsin x + e X = sin x, y = a + b X + e,,,, ( ), y = ae bx e ln y = ln a + bx + lne Y = ln y, A = ln a, B = b, E = ln e Y = A + Bx + E,, y = ax b e ln y = ln a + bln x + ln e Y = ln y, A = ln a, B = b, X = ln x, E = lne Y = A + B X + E,, - ae - bx y = Le : y L = e- ae - bx ln y L = - aebx ln L y = aebx ln( ln L y ) = ln a - bx
80 L y L y > 1, ln L y > 0, Y = ln( ln L y ), A = ln a, Y = A - bx, L,,, y = L 1 + ae - bx : L y - 1 = ae- bx ln( L y - 1) = ln a - bx Y = ln( L y - 1), A = ln a, Y = A - bx, L,,,,,, y,,,,,,,,,,, ( x) ( y),
81.28 :,, y^ = a+ b x + e, X = 1 x y = a + b X + e 3.30 :, Excel,,, 3.31
82.31 a= 2.225407, b= 7.621271, y^ = 2.225407 + 7.621271 x 28.5, 29 30,, :2.49282 % 2.488209% 2.479449%
( ) Y t, {Y t, t = t0, t1 }Y t, t = 1, 2,, 20 60 70,,, Warren Persons (),,,,,,,,,,, ( T) ( S) ( C) ( I)
84. (T ),,,,,, 2. (S ), 5, 3. (C ),,,, 12 4 1 1, 4. (I ),,,,, ( ),, : 1.,,,,,,,,
85.,,,,,, 12 4 7, 3.,, 4., ( ),,, 5.,,,,,,,, t, t, t,, ( ),
86,,,, : Y t = f( T1, S1, C1, I1 ), : : Y t = T1 St Ct It : Y t = Tt + St + Ct + It Y t Y t = Tt St + Ct It,, ;, ;,,,, ( ),,,,,,, Box - Jenkins 4.1, Box - Jenkins 4.1
87,,,,, : n : y1, y2, y3 yn - 1, yn t, t + 1 : Ft + 1 = y = i = 1 t, y t, Ft + 1 t + 1 : t yi et + 1 = yt + 1 - Ft + 1 t + 2, : Ft + 2 t + 1 yi = y = i = 1 t + 1 et + 2 = yt + 2 - Ft + 2, : ;, Ft + 1 = y = i = 1 t t yi
88 t + 1 yi Ft 2 = y = i = 1 t + 1 = y1 + y2 + + yi + y i + 1 t + 1 = t Ft + 1 + yt + 1 t + 1 Ft + 3 = t + 1 t + 2 Ft + 2 + 1 t + 2 yt + 2 = t t + 1 Ft + 1 + 1 t + 1 yt + 1,,,,,,,,,, t,, : t + 1 t t + 2 Ft + 1 = Ft + 2 = y1 + y2 + + yt t y2 + y3 + + yt + 1 t, = 1 t t i = 1 yi = 1 t + 1 t i = 2 yi, Excel, : : : C5 = AVERAGE( B2: B4 ), : C5, C14,, 4.2
89 4.1 4.2,,,,,, :, t, 12, 4,
90 t + 1 Ft 2 = 1 t i = 2 yi = 1 t ( y1 + y2 + yt + yt + 1 - y1 ) = Ft + 1 + 1 t ( yt + 1 - y1 ),, ( ),,,,,,, : Ft + 1 =, i, wi Ft + 1 t i = 1 i y i t i i = 1 t = i = 1 wi y i t, w1 w2 wt, wi = 1 i = 1 Excel,, : :, 0 = 1.5, 1 = 1, 2 = 0.5 : D5 : = ( B41.5 + B31 + B20.5)/ 3, : D5, D14, : 4.2
91.3,,,,,,, : ( t ), ;, ;,,,,,, 4.3
92 4.3,,,,,,,,,, Ft + 1 = Ft + 1 n ( y t - yt - n ) n, t Ft y t - n, Ft + 1 = Ft + 1 ( yt - Ft ) n Ft + 1 = 1 n y t + 1-1 n Ft = 1, 0 < < 1, n Ft + 1 =yt + (1 - ) Ft, Ft t Ft + 1 t + 1, t + 1 : St + 1 =yt + (1 - ) St
93, : St = y t + ( 1 - a) St - 1, 0.2, 0.5, 0.7, 4.3 4.3 ( ) y t = 0.2 = 0.5 = 0.7 1 371.5 2 267.4 371.5 371.5 371.5 3 372.4 350.68 319.45 298.63 4 368.2 355.02 345.93 350.27 5 349.4 357.66 356.06 362.82 6 362.8 356.01 353.23 353.43 7 420.9 357.37 358.02 359.99 8 380.4 370.07 389.46 402.63 9 385.6 372.14 384.93 387.07 10 335.0 374.83 285.27 386.04 11 338.5 366.86 360.13 350.31 12 306.6 361.19 349.32 342.04 1 350.27 327.96 317.23 : = 0.2 S1 = 0.2371.5 + 0.8371.5 = 371.5 S2 = 0.2267.4 + 0.8371.5 = 350.68 : 4.4
94 0.2,,= 0.7,,, (1 - ) n S t - n + 1 St = yt - 1 St =yt + (1 - ) St - 1 =yt + ( 1 - ) St - 1 St - 1 =yt - 2 + (1 - ) St - 2 + (1 - ) yt - 1 + ( 1 - ) 2 yt - 2 + + ( 1 - ) n y t - n + 1 +,, t,, 4.4 = 0.3 = 0.2 = 0.1 1 0.012106 0.026844 0.038742 2 0.01729 0.033554 0.043047 3 0.02471 0.041943 0.047830 4 0.03529 0.052429 0.053144 5 0.05042 0.065536 0.059049 6 0.07203 0.08192 0.06561 7 0.1029 0.1024 0.0729 8 0.147 0.128 0.081 9 0.21 0.16 0.09 10 0.3 0.2 0.1,,,, ;,, : ( ),
95 0.100.30, 0.30,,,,,,,, SSE MSE MAE ( ), n, (1 - ) n,,, ;,, ;,,, :,,,, 8 3,,,,,, ( ),,, (), : St + 1 = St + ( yt - St ) St + 1 = St + ( et ) = +,,,, ( 4.5 ),
96, : S ( 1 ) t = Yt + (1 - ) S ( 1 ) t - 1 S ( 2 ) t = S ( 2 ) t + ( 1 - ) S ( 2 ) t - 1 4.5 1 2 1.00 0.500 2 4 2.50 0.375 3 6 4.25 2.313 4 8 6.13 4.219 5 10 8.06 6.141 6 12 10.03 8.086 7 14 12.02 10.053 8 16 14.01 12.031 9 18 16.00 14.016,,,,,,, : : at bt Ft + m = at + bt m t ; t ; m at bt, () ( Brown ) 1. (1 ), : S ( t 1 ) = Yt + (1 - ) S ( t 1 - ) 1 S ( t 2 ) = S ( t 1 ) + ( 1 - ) S ( t 2 - ) 1,,,, 1 - bt, bt t
97,, (2 ), : t bt Ft + m t t m,, 2. = t + bt m at = S ( 1 ) t + S ( 1 ) t - S ( 2 ) t = 2 S ( 1 ) t - S ( 2 ) t bt = 1 - ( S( t 1 ) - S ( 2 ) t ) 4.6 4.6 ( ) 1 1 143 2 2 152 3 3 161 4 4 139 5 5 137 6 6 174 7 7 142 8 8 141 9 9 162 10 10 180 11 11 164 12 12 171 1 13 206 2 14 193 3 15 207 4 16 218 5 17 229 6 18 225 7 19 204 8 20 227 9 21 223 10 22 242 11 23 239 12 24 266
98 Excel : : S ( 1 ) 0 = S ( 2 ) 0 = 143,= 0.2 0, D2, E2 143 D3 : = 0.2 * C3 + 0.8 * D2, D2 6, : E3 E26, : F3 F26, : G3 G2 6, b : = 0.2 * D3 + 0.8 * E2, : = 2 * D3 - E3, : = 0.2 * (D3 - E3 )/ 0.8, : 1 ( m = 1 ), H4 H27, 225 : = F3 + G3, 2, Ft + m = at + bt m m = 2, F26 = a24 + b24 2 = 252.246 + 5.5142 = 263.27 ( ) m = 3, F27 = a24 + b24 3 = 252.246 + 5.5143 = 268.79 ( ), : 4.7
99 4.5 3. ( ) ( H OLT ) 1., ( ) : St =yt + (1 - ) ( St - 1 + bt - 1 ) bt - 1 St - 1,, St, yt ( ) bt = ( St - St - 1 ) + ( 1 - ) bt - 1, St,,, : : St bt Ft + m = St + bt m t ; t ; m - St - 1,, 2., :
100 b0 b0 = (152-143) + (161-152) + (139-161) 3 S0 = 143, b0 = - 1.33, = - 1.33 = 0.2,= 0.3 D2 143, E2-1.33 D3 : = 0.2 * C3 + 0.8 * (D2 + E2), : E3 : = 0.3 * (D3 - D2) + 0.7 * E2, : D3, E3, E26, : 1 ( m = 1 ), F3 : = D2 + E2, H27, 225 2, Ft + m = at + bt m : m = 2, F26 = a24 + b24 2 = 254.6857 + 6.0035472 = 266.693( ) m = 3, F27 = a24 + b24 3 = 254.6857 + 6.0035473 = 272.696( ), 4.8 4.5: 4.8
101 3..6 ( H OLT ),,,,, bt 0,,,,, h, b h = 1 - (1 - b ) 2,= 1 - b,, 2(1 - b ), 2 - b, ( H OLT ) 4.64.7 4.7
102.8 : ( ) 1.,, S ( 1 ) t = Yt + (1 - ) S ( 1 ) t - 1 S ( t 2 ) = S ( t 1 ) S ( 3 ) t = S ( 2 ) t + ( 1 - ) S ( t 2 - ) 1 + ( 1 - ) S ( 3 ) t - 1,, Ft + m = at + bt m + 1 2 ct m2, at = 3 S ( t 1 ) - 3 S ( t 2 ) + S ( t 3 ) bt = ct = 2 (1 - ) S ( 1 ) t ) 2 [ ( 6-5 2 ( S ( 1 ) (1 - ) 2 t - 2 S ( 2 ) t + S ( 3 ) t ) - (10-8 ) S ( 2 ) t + (4-3 ) S ( 3 ) t ],,,
103. : 4.9 ( ) 1 1 3.4 2 2 3 3 3 3.4 4 4 3.7 5 5 3.3 6 6 4.6 7 7 3.2 8 8 3.8 9 9 4.2 10 10 4 11 11 6.1 12 12 7.3 1 13 5.7 2 14 4.5 3 15 6 4 16 7 5 17 7.6 6 18 9.3 7 19 11.8 8 20 19.9 9 21 15.5 10 22 20.1 11 23 16.1 12 24 18.4 Excel, : : S ( 1 ) 0 = S ( 2 ) 0 = S ( 3 ) 0 = 3.4,= 0.250, D2, E2, F2 3.4D3 : = 0.25 * C3 + 0.75 * D2, D26, : E3 E26, : F3 F26, : G3 G2 6, a : H3 : = 0.25 * D3 + 0.75 * E2, : = 0.25 * E3 + 0.75 * F2, : = 3 * D3-3 * E3 + F3, :
104 0.25 * ( ( 6-5 * 0.25 ) * D3-2 * (5-4 * 0.25 ) * E3 + ( 4-3 * 0.25) * F3 )/ (2 * 0.75 * 0.75 ), H26, b : I3 : = 0.25 * 0.25 * ( D3-2 * E3 + F3 )/ (2 * 0.75 * 0.75 ), I2 6, c : 1 ( m = 1 ), J4 : = G3 + H3 + I3, J27, 225 2, Ft + m = at + bt m + 1 2 ct m2, : m = 2, F2 6 = a2 4 + b2 4 2 + 1 2 c24 22 = 19.2130 + 1.3362 + 1 2 0.01858422 = 21.922168 ( ), : 4.10 :
105.9 3. ( ) ( ) 1., St = y t b1 It - L + ( 1 - ) ( St - 1 + bt - 1 ) = ( St - St - 1 ) + ( 1 - ) bt - 1 It = y t S t + ( 1 - ) It - L L, I,, Ft + m = ( St + bt m) It - L + m : ( St ), ( bt ) ( It ) 2. : 4.11 ( ) 1996 1997 1998 1999 2000 2001 2002 115 109.9 110.1 136.6 151.6 212.2 218 69.9 68.2 90.3 93 94.6 97.6 103 64.5 84.2 117.3 51.7 123.6 99.6 105.6 92.7 89.4 90.9 74 144.8 156.6 135.2
106 4.9,, :,,,,,,, : :, S0 = 77.254, b0 = 2.335, : I - 3 = 1.37, I - 2 = 0.81, I - 1 = 0.851, I0 = 0.964 = 0.2,= 0.2,= 0.3, : D6 : = 0.2 * ( C6/ F2) + 0.8 * ( D5 + E5), E6 : = 0.2 * (D6 - D5) + 0.8 * E5, F6 : = 0.3 * ( C6/ D6) + 0.7 * F2, D7 : = 0.2 * ( C7/ F3) + 0.8 * ( D6 + E6), E7 : = 0.2 * (D7 - D6) + 0.8 * E6, F7 : = 0.3 * ( C7/ D7) + 0.7 * F3, D8 : = 0.2 * ( C8/ F4) + 0.8 * ( D7 + E7), E8 : = 0.2 * (D8 - D7) + 0.8 * E7, F8 : = 0.3 * ( C8/ D8) + 0.7 * F4, D9 : = 0.2 * ( C9/ F5) + 0.8 * ( D8 + E8), E9 : = 0.2 * (D9 - D8) + 0.8 * E8, F9 : = 0.3 * ( C9/ D9) + 0.7 * F5, : D6, E6, F6, D9, E9, F9, F33, : 1 ( m = 1), G6 : = ( D5 + E5) * F2, G7 : = ( D6 + E6) * F3, G8 : = ( D7 + E7) * F4, G9 : = ( D8 + E8) * F5, G6G9 G34, 229 2, Ft + m = ( St + bt m) It - L + m : m = 2, F30 = ( S28 + 2b28 )I26 = (146.8142 + 21.8930 )0.78
107 117.4681 ( ) m = 3, F31 = ( S28 + 3b28 )I27 = (146.8142 + 31.8930 )0.80 = 121.9946 ( ), : 4.12 ( ) 4.10 3.
108 1. ) ( ), St = ( yt - It - L ) + ( 1 - ) ( St - 1 + bt - 1 ) bt = ( St - St - 1 ) + ( 1 - ) bt - 1 It = ( yt - St ) + (1 - ) It - L L, I,, 2. Ft + m = ( St + bt m ) + It - L + m, : :, S0 = 77.254, b0 = 2.335, : I - 3 = 39.76786, I - 2 = - 22.6321, I- 1 = - 18.3607, I0 = 1.225 = 0.2,= 0.2,= 0.3, : D6 : = 0.2 * ( C6 - F2 ) + 0.8 * (D5 + E5 ), E6 : = 0.2 * (D6 - D5) + 0.8 * E5, F6 : = 0.3 * ( C6 - D6 ) + 0.7 * F2, D7 : = 0.2 * ( C7 - F3 ) + 0.8 * (D6 + E6 ), E7 : = 0.2 * (D7 - D6) + 0.8 * E6, F7 : = 0.3 * ( C7 - D7 ) + 0.7 * F3, D8 : = 0.2 * ( C8 - F4 ) + 0.8 * (D7 + E7 ), E8 : = 0.2 * (D8 - D7) + 0.8 * E7, F8 : = 0.3 * ( C8 - D8 ) + 0.7 * F4, D9 : = 0.2 * ( C9 - F5 ) + 0.8 * (D8 + E8 ), E9 : = 0.2 * (D9 - D8) + 0.8 * E8, F9 : = 0.3 * ( C9 - D9 ) + 0.7 * F5, : D6, E6, F6, D9, E9, F9, F33, X : 1 ( m = 1),
109 G : = ( D5 + E5) + F2, G7 : = ( D6 + E6) + F3, G8 : = ( D7 + E7) + F4, G9 : = ( D8 + E8) + F5, G6G9 G34, 229 2, Ft + m = ( St + bt m) + It - L + m : m = 2, F30 = ( S28 + 2b28 + I2 6 = ( 146.8355 + 21.908335) + 53.08 = 203.7322 ( ) m = 3 F31 = ( S2 8 + 3 b28 ) + I27 = ( 146.8355 + 3 1.908335 ) - 29.32 = 123.2405 ( ), 4.13, 4.11 4.13 ( ) 4.12, 4.13
110.11 4.12 4.13,,
111.,,,,,,,,,, :,,,,,,,,, ( ),,,,,,,,,,,,,,,
112,,,,, (, ) ; : 4.14 1 2 3 A ( ) B ( ) C( ) A1 B1 C1 A2 B2 C2 A3 B3 C3 : 4.15 A - 1 A - 2 A - 3 B - 1 B - 2 B - 3 :, 1996 1
113 C 1 C - 2 C - 3 : 4.16 A1 A2 A3 B1 B1 S t + 1 = y t + ( 1 - ) S t F t + 1 = S t + 1 S t = ( y t - I t - L ) + ( I - ) S t - 1 F t + m = S t + I t - m + L I t = ( y t - S t ) + ( 1 - ) I t - L S t =( y t / I t - L ) + ( 1 - ) S t - 1 F t + m = S t I t - m+ L I t =( y t / S t ) + ( 1 - ) I t - L S (1 ) t =Y t + ( 1 - ) S ( t 1 - ) 1 S ( t 2 ) =S ( t 1 ) + ( 1 - ) S ( t 2 - ) 1 S t =y t + (1 - ) ( S t - 1 + b t - 1 ) a t = S (1) t + ( S ( t 1) - S (2) t ) = 2 S (1 t ) - S ( t 2) b t = 1 - ( S( t 1 ) - S ( t 2 ) F t + m = a t + b t m F t + m = S t + b t m b t = ( S t - S t- 1 ) + (1 - ) b t - 1 C1 S (1 ) t =Y t + ( 1 - ) S ( t 1 - ) 1 S ( t 2 ) =S ( t 1 ) S ( t 3 ) =S ( t 2 ) + ( 1 - ) S ( t 2 - ) 1 + ( 1 - ) S ( t 3 - ) 1 a t = 3 S ( t 1) - 3 S ( t 2 ) + S ( t 3) b t = 2( 1 - ) 2 [(6-5 ) S (1) t - (10-8 )S (2) t + (4-3 ) S (3) t ] c t = 2 (1 - ) 2 ( S( t 1 ) - 2 S ( t 2 ) + S ( t 3 ) ) F t + m = a t + b t m + 1 2 c t m 2
114 B B3 S t = (y t - I t- L ) + (1 - )(S t - 1 + b t- 1 ) b t = ( S t - S t- 1 ) + (1 - ) b t - 1 F t + m = ( S t + b t m ) + I t - L + m I t = ( y t - S t ) + ( 1 - ) I t - L S t = y t + (1 - ) ( S I t - 1 + b t - 1 ) t- L b t = ( S t - S t- 1 ) + (1 - ) b t - 1 F t + m = ( S t + b t m ) I t - L + m I t = y t S t + ( 1 - ) I t - L
( self - adaptive filtering) -,,,,,,, y1, y2 yt, y^t + 1 = w1 yt + w2 yt - 1 + + wn y t - n + 1 y^t + 1 n = i = 1 wi y t - i + 1 t + 1, W i, n yt - i + 1 t - i + 1,,,,, yt + 1 t + 1, yt + 1 = y^t + 1 + et + 1 = w1 yt + w2 yt - 1 + + wn y t - n + 1 + et + 1 et + 1 et + 1 = yt + 1 - y^ t + 1,,,,
116 et + 1 ( ),, : w i = wi - ke 2 t + 1, W i, W i, e 2 t + 1 e 2 t + 1 ; k,,,, B.Widrow k : k n 1 i = 1 y2 i n, n ( ) ma x wi et + 1 = y t + 1 - y^t + 1 = y t + 1 - w1 yt - w2 yt - 1 - - wn y t - n - 1 e 2 t + 1 = ( yt + 1 - w1 yt - w2 y t - 1 - - wn y t - n - 1 ) 2 : e 2 t + 1 = e2 t + 1 wi w i = wi et + 1 = 2 et + 1 wi = 2 et + 1 ( - yt - i + 1 ) = - 2 et + 1 yt - i + 1 w i = wi + 2 ket + 1 yt - i + 1 - ke 2 t + 1 ( i = 1, 2n), n yt - i + 1 t - i + 1 ( ),, : 5.1 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20
117 :, n 4 : k k n 1 i = 1 y2 i max = 1 14 2 + 16 2 + 18 2 + 20 2 = 0.00085 :, W i = 1 n = 1 4 = 0.25, n = 4, t t = 4 y^t + 1 = y^5 = w1 y4 + w2 y3 + w3 y2 + w4 y1 = 0.258 + 0.256 + 0.254 + 0.253 = 5 et + 1 = e5 = y5 - y^ 5 = 10-5 = 5 W i = wt, + 2 ket + 1 y t - i + 1 : w 1 = 0.25 + 20.0008558 = 0.314 w 2 = 0.25 + 20.0008556 = 0.296 w 3 = 0.25 + 20.0008554 = 0.282 w 4 = 0.25 + 20.0008552 = 0.266 :,,,, : t = 5, y^t + 1 = y^6 = w1 y5 + w2 y4 + w3 y3 + w4 y2 = 0.31410 + 0.2988 + 0.2826 + 0.2664 = 8.28 et + 1 = e6 = y6 - y^6 = 12-8.28 = 3.72 w i = wi + 2 ket + 1 yt - i + 1 : w 1 = 0.314 + 20.000853.7210 = 0.374 w 2 = 0.298 + 20.000853.728 = 0.349 w 3 = 0.282 + 20.000853.726 = 0.320 w 4 = 0.266 + 20.000853.724 = 0.291, t = 6, t = 7, t = 8,, t = 10, y1 1, e11 wi,,, t = 4, ( ),
118, : w1, w2, w3, w4 11, 12 : 11 y^1 1 = w1 y1 0 + w2 y9 + w3 y8 + w4 y7 12 y^1 2 = w1 y^1 1 + w2 y10 + w3 y9 + w4 y8, Excel, : :, 4,, 5.2 5.2 : K6 : = G6 * C6 + H6 * D6 + I6 * E6 + J6 * F6, : L6 : = B6 - K6, : G7 : = G6 + 2 * 0.0008 * L6 * B5, w1 H7 : = H6 + 2 * 0.0008 * L6 * B4, w2 I7 : = I6 + 2 * 0.0008 * L6 * B3, w3 J7 : = J6 + 2 * 0.0008 * L6 * B2, w4 K6, L6, L7,
119 5.3 : G7, H7, I7, J7, K7, L7, L11, : 5.4 0.384368 0.353706 0.323044 0.292383 word,, G6 J6 (, ),, 5.5,, :
120 5.6 5.7,,,,, : ( ) ( n),, 4 8, 12 n,,, n 2 6 n, n (),,, wi = 1 n r1, r2 Yule - Walker
121 n, w1 = ( ) k r1 ( 1 - r2 ) 1 - r 2 1, w2 = r2 - r2 1 1 - r 2 1, k, n,,,, k, k 1 n : 5.8 ( ) 1 1 3.4 2 2 3 3 3 3.4 4 4 3.7 5 5 3.3 6 6 4.6 7 7 3.2 8 8 3.8 9 9 4.2 10 10 4 11 11 6.1 12 12 7.3 1 13 5.7 2 14 4.5 3 15 6 4 16 7 5 17 7.6 6 18 9.3 7 19 11.8 8 20 19.9 9 21 15.5 10 22 20.1 11 23 16.1 12 24 18.4 Excel, : :, 1 3, wi = 1/ 3
122 k 1 15.5 2 + 20.1 2 + 16.1 2 = 0.0000095271 : J5 : = G5 * D5 + H5 * E5 + I5 * F5, : K5 : = C5 - J5, L5 : = K5 * K5, : G6 : = G5 + 2 * 0.0000095271 * K5 * C5, w1 H6 : = H5 + 2 * 0.0000095271 * K5 * C4, w2 I6 : = I5 + 2 * 0.0000095271 * K5 * C3, w3 J5, K5, L5, L6, : G6, H6, I6, J6, K6, L6, L25, L26 176.159/ 21 = 8.388526 : 5.9
123.3408 0.3382 0.33769 word,, G5 I5 (, ),, 5.10: 5.10 21, :
124.11 8.388526 8.040016 7.753617 7.520465 7.330499 7.175578 7.051652 6.952486 6.874098 6.813025 6.765576 6.729515 6.703358 6.68437 6.671673 6.664001 6.660141 6.659421 6.661035 6.664456 6.66923,,, 18, :
125 5.11 18 : : w1 = 0.4178, w2 = 0.3733, w3 = 0.37066 18 5.1 : y25 = 0.417818.4 + 0.373316.1 + 0.370620.1 = 20.33( ) y26 = 0.417820.33 + 0.373318.4 + 0.370616.1 = 21.33( )
126 (1 ),, (2 ),,,,,,,,,,,,,,,,, n wi = 1,, i = 1,,,,, 1,,, ( Weight ed Sum) ( ),,,,,,,,,
127, ;, 1, ( ) ARMA AR MA : y^t + 1 = 1 yt + 2 yt - 1 + + n y t - n + 1 n : y^t + 1 = w1 yt + w2 yt - 1 + + wn y t - n + 1,, ARMA,, 1973,,,,,,
,,,,, t, y, y^ = F( t), t, (trend projection ) t,,, : y^ = F( t), n n Q = ( y - y^) 2 i = 1 n 2 = [ y - F( t) ] i = 1 (),, : Q = 0 ( i = 1,2p) i p
129 :,,, t, y, : ( ) yt ^ = a+ bt 6.1 () yt ^ = b0 + b1 t + b2 t 2 6.2
130 ) ^ yt = at b 6.3 ( 1) ( b > 0 ) 6.4 (2 ) ( a < 0) () yt ^ = b0 + b1 t + b2 t 2 + b3 t 3 6.5 () yt ^ = a+ blnt
131.6 ( ) yt ^ = a+ b 1 t 1 ^ = a + b1 t yt 6.7 () yt ^ = ae bt 6.8
132 ) ^ yt = L + ae bt ( a< 0, b< 0 ) yt ^ = L + ab t ( a< 0, 0 < b < 1) 6.9 () yt ^ = La bt - ae - bt yt ^ = Le 6.10 () yt ^ = 1 L + ab t ( )
133 yt = L 1 + ae - bt 6.11, t,,, 1991 2002 : 6.1 19911992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 704 846 ( ) 1093 1444 1812 2299 2971 4123 5553 7621 9575 13131 t, :
134.12,,,,,,,,,,, : 6.2,, 6.2 ( t) yt = a + bt ( yt - yt - 1 ) 1 2 3 4 a + b a + 2 b a + 3 b a + 4 b - b b b t - 1 t a + ( t - 1) b a + tb b b 6.3,,
135 6.3 yt b0 b1 b2 ( yt - yt - 1 ) ( t) = + t + t 2 1 b 0 + b 1 + b 2-2 b 0 + 2 b 1 + 4b 2 b 1 + 3 b 2 3 b 0 + 3 b 1 + 9b 2 b 1 + 5 b 2 4 b0 + 4 b1 + 16b2 b1 + 7 b2 t - 1 b 0 + ( t - 1 ) b 1 + ( t - 1 ) 2 b 2 b 1 + ( 2t - 3) b 2 t b0 + b1 t + b2 t 2 b 1 + ( 2t - 1) b 2 [ ( y t - y t - 1 ) - ( yt - 1 - yt - 2 ) ] - - 2 b 2 2 b2 2 b 2 2 b2 6.4,, 6.4 ( t) y t = b 0 + b 1 t + b 2 t 2 + b 3 t 3 1 b 0 + b 1 + b 2 + b 3 - - 2 b 0 + 2 b 1 + 4 b 2 + 8b 3 - - 3 b 0 + 3 b 1 + 9 b 2 + 27b 3 2 b 2 + 12b 3-4 b 0 + 4b 1 + 16 b 2 + 64 b 3 2 b 2 + 18b 3 6 b3 t - 1 b0 + ( t - 1) b1 + ( t - 1) 2 b2 + ( t - 1) 3 b3 2 b 2 + 6( t - 1 ) b 3 6 b 3 t b0 + b1 t+ b2 t 2 + b3 t 3 2 b 2 + 6tb 3 6 b3 6.5,, 6.5 ( t) y t = ae bt ( y 1 / y t - 1 ) 1 2 3 4 t - 1 t ae b ae 2b ae 3b ae 4b ae ( t - 1) b ae tb - e b e b e b e b e b
136.6,, 6.6 ( t) yt = a + bc t ( y t - y t - 1 ) y t - y t - 1 1 2 3 4 t - 1 t y t = a + bc yt = a + bc 2 y t = a + bc 3 y t = a + bc 4 yt = a + bc ( t - 1) y t = a + bc t - bc( c - 1 ) bc 2 ( c - 1) bc 3 ( c - 1) bc t - 2 ( c - 1) bc t - 1 ( c - 1) - - c c c c yt - 1 - yt - 2 6.7,, 6.7 ( t) y t = La bt lg y t = lg L + b t lg a lg y t - lg y t - 1 lg yt - lg yt - 1 lg y t - 1 - lg y t - 2 1 2 3 4 t - 1 t La b La b2 La b3 La b4 La bt - 1 La bt lg L + blga lg L + b 2 lga lg L + b 3 lga lg L + b 4 lga lg L + b t - 1 lg a lg L + b t lg a - b( b - 1) lga b 2 ( b - 1) lga b 3 ( b - 1) lga b t - 1 ( b - 1)lg a b t ( b - 1) lga - - b b b b, : ;,
137,,,, t 6.1, y^t = ae bt ln y^ t = ln a + bt, ln y, t ln yt, ln a b Excel : : ln yt D2 : = LN ( C2 ), D13 6.8 : t ln y t, : 6.13
138 :.14 6.9 M ultiple R R Square Adjusted R Square 0.998627 0.997256 0.996981 0.053461 12 6.10 df SS MS F Significance F 1 10.38541 10.38541 3633.75 3.84E - 14 10 0.02858 0.002858 11 10.41399 6.11 Coefficie nts t Sta t P - value Lower 95 % Upper 95 % 95.0 % 95.0 % Int ercept 6.193556 0.0329031 88.2382 4.4 E - 19 6.120245 6.266868 6.120245 6.266868 XVariable1 0.269491 0.0044716 0.18059 3.84 E - 14 0.25953 0.279452 0.25953 0.279452 : ln a = 6.193556, b= 0.269491 :
139 y t = 489.5842 0.2 694 91t e 0.2 6949 1t y^ t = 489.58422.718282 :, ln yt E, F2 : = EXP ( E2 ),, F13, yt F14 : = 489.5842 * ( 2.718282 )^(0.269491 * 13 ), 2003 y^2 003 = 16267.75 : 6.12 6.15,,, :
140 6.13 y = a + bx y = a + b/ x y = L + ab x y = L + ax b y = 1/ (1/ L + ab x ) y = a + bx y = a + b( 1/ x) lg( y - L) = lg a + lgbx lg( y - L) = lg a + blg x lg (1/ y - 1/ L) = lg a + lgbx ( ),, (), () 1. y^t = L + ae bt ( a< 0, b< 0 ),,,, y0, y1, yn - 1,, y3 ( n - 1 ) 3 n,, yi, yn - 1 y3 n - 1 y0 = L + ae 0 y1 = L + ae b = L + ae ( n - 1 ) b = L + ae ( 3 n - 1 ) b, n,, n - 1 1 y = yi = nl + a( e0 + i = 0 e b + + e ( n - 1 ) b ) = nl + a enb - 1 e b - 1
2 n - 1 y = y i = nl + aenb i = n e nb - 1 ( e 0 + e b + e ( n - 1 ) b ) = nl + ae nb e b - 1 3 n - 1 3 y = y i = 2 n i = nl + nb ae2 ( e 0 + e b + e ( n - 1 ) b 2 nb enb - 1 ) = nl + ae e b - 1 nb ( e nb - 1) 2 ae 3 y - 2 y 2 y - 1 y = ( e b - 1 ) a ( ebn - 1) 2 ( e b - 1 ) 2 y - 1 y = a ( enb - 1 ) 2 ( e b - 1 ) a = (2 y - 1 y) ( eb - 1 ) ( e nb - 1 ) 2 b= ln n 3 y - 2 y 2 y - 1 y L = 1-1 (1 y - aenb n e b - 1 ) = 1 n [ 1 y3 y - (2 y) 2 1 y + 3 y - 22 y ], 3,,,,,, y^t = L + ab t ( a< 0, 0 < b < 1) 2. a = ( 2 y - 1 y) b - 1 ( b n - 1) 2 b= n 3 y - 2 y 2 y - 1 y L = 1 n [ 1 y - a( bn - 1 b - 1 ) ] y^t = La b t 141
142 ln yt = ln L + ( ln a) b t, 3. ln a = (2 ln y - 1 ln y) b - 1 ( b n - 1) 2 b= n 3 ln y - 2 ln y 2 ln y - 1 ln y ln L = 1 n [1 ln y - ln a( bn - 1 b - 1 ) ] ^ y t = 1 L + ab t 1 ^ = L + abt yt, a = (2 b= n 3 2 4. L ^ yt = 1 + ae - bt 1 ^ = 1 L + a L e - bt yt : 1 b c = e- ( 1 - e - nb ) 1 - e - b 1 y - 1 1 y ) b - 1 ( b n - 1 ) 2 1 y - 1 2 y 1 y - 1 1 y L = 1 n [1 1 y - a( bn - 1 b - 1 ) ] 1 y, 1 2 y, 1 3 y
143 1 y = 1 ( n + ac) ( t = 1, t = 0 ) L 2 3 1 2 1 y = 1 L ( n + ace- nb ), 1 y = 1 L ( n + ace- 2 nb ) 1 y - 2 1 y = ac L ( 1 - e- nb ) 1 y - 3 1 y = ac L e - nb (1 - e - nb ) 2 e - nb = 1 1 y - 1 3 y 1 y - 1 2 y b = 1 n [ln(1 1 y - 2 1 y ) - ln (2 1 y - 3 1 y ) ] 1 a = L (1 c y - 1 2 y ) 2 1 1 y + 1 3 y - 1 22 y L = n 1 1 1 y + 1 3 y - 1 22 y 1 1 y 3 y - 1 (2 y ) 2 :, 6.14 : : I10 = P OWE R ( ( E14 - E8 )/ ( E8 - E2), 1/ 6 ), b; I9 = ( E8 - E2 ) * ( ( I10-1)/ ( I10^6-1)^2 ), a; I9 = 1/ 6 * ( E2 - I9 * ( ( I10^6 ) - 1 )/ (I10-1) ), L : y^t = 1 0.0000937061 + 0.0001165570.965171532 t : F 2 : 1/ ( 0.0 0 0 0 9 3 7 0 6 1 + 0.0 0 0 1 1 6 5 5 7 * (0.965171532 )^ B2 ), F21, 2003 2004, 6.14
144 6.14,,, ;,,,,,, ( ) 1.,,, 6.15,,,
145 A B C yt = ae bt yt = b0 + b1 t + b2 t 2 y^ t = b0 + b1 t + b2 t 2 + b3 t 3, y^t 0.12 077 8t = 93.88849e y^t = 182.9483-25.1507 t + 3.556453 t 2 y^t = 160.016-15.1493 t + 2.57644 t 2 + 0.026134 t 3 6.15 6.15,, B C,
146.16 2.,, : 6.16 A B C 4.96 1.11318E - 13-6.15804E - 14 55.69197 59.611699 57.51389 7.780241 14.40331 12.7175 7208.743 5552.718 5507.231 86.73042 76.11925 75.80683, C,,, ( ),,, : ;, :
147,, ;,,,,, ;,,,,,,,,,,,,, 6.17, 6.17,,, A B C,, y^t 0.13 628 5t = 173416.6e yt = 327697.2-51132.6 t + 8505.003 t 2 y^t = 269234.1-25635.4 t + 6006.579 t 2 + 66.62464 t 3
148.17 ( ) ( y) ( ) ( y) 1978 207222 1990 961640 1979 226592 1991 1061667 1980 264779 1992 1282631 1981 291566 1993 1561087 1982 311925 1994 1876878 1983 344704 1995 2299106 1984 402855 1996 2595144 1985 518104 1997 2785205 1986 619019 1998 3037155 1987 721103 1999 3315515 1988 927708 2000 3627022 1989 1006533 2001 3954305 6.17
149,,,, 6.18,,,,,,, 6.18 6.18 A B C 7.0888 8.912377 6.685767 32784099801 9135759662 8840122771
150, 6.19: 6.19 ( ) ( ) 2002 25 4365009 4423470.975 2003 26 4747632 4834155.777 2004 27 5147264 5267247.18 2005 28 5563907 5723144.933,,,,,,,,, :
151 6.20 F t + m = a t + b t m ^t, t y = a + bt, t, m F t + m = a t + b t m + 1 2 c t m 2 ^t y = b0 + b1 t + b2 t 2, t, t, m,,,,,,,,
,,,,,,,, (),, 12, 4, ;, ;,,,, :, ( S ) ; (T ) ;,, (C ),, ( I ) : Y = TCI S, ; X - 11 X - 11
153 y^t = yf i y ; f i, 35 ( ) 1991 2001, 7.1 : (), : 7.1
154 ( ) :,, B13 : = AVERAGE( B2: B12),, M13 N2 : = A VE RAGE ( A2: M2 ),, N13, 112 (),, 12 1200%,,, 1200, : f i = ( ) ( ) B14 : = B13/ 92.40909 100,, M14 B15 : = B14 * 1200/ 1200.002,, M15 7.2
155.2 : 7.2 ( ) 2001 135.583, : y^i = ( 135.583) f i B16 : = 135.583 * B15/ 100,, M162002 7.3 26.143 33.479 86.165 138.851 285.4381 381.207 335.99 166.995 80.029 38.1473 26.676 27.8769 y^t = ( a + bt) f i : ( a + bt) ; f i,
156 997 1 2001 12 : 7.4 ( ) 7.3,,,, 7.3 : (), (1 ), Ft = a + bt,,, 12 ; EXCEL :
157.5 Coefficie nts t St a t P - value Lower 95 % U pper 95 % 95.0 % 95.0 % Intercept 190.1555361 5.223376 36.40472 1.2E -41 179.6998 200.6113 179.6998 200.6113 XVariable1 3.577468782 0.148926 24.02185 7.88 E-32 3.279362 3.875576 3.279362 3.875576 7.6 M ultiple R R Square Adjusted R Square 0.95324 0.90867 0.90709 19.9777 60 7.7 df SS MS F Significance F 1 230305.1 230305.1 577.05 7.87805E - 32 58 23148.27 399.108 59 253453.4 : Ft = 190.1555 + 3.5774 t, 7.8 7.8 (2 ), ; E2 : = C2/ D2,, E61