) 041 (CIP ), ISBN /,.:...-.F10 CIP (2003) / / 75 / / 6 / ( 010) ( ) / / / mm 1/ 16 / 301

Transcription

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2 ) 041 (CIP ), ISBN /,.:...-.F10 CIP (2003) / / 75 / / 6 / ( 010) ( ) / / / mm 1/ 16 / 301 / / / / / ISBN / F1261 / 26.80,,,,

3 ,,,,,,,,, 20,,,, :,,,,, ;,,,, SAS SPSS T SP, Excel ;,,,,, : ( ), (), ( ), ( ), ( ),,,,,!,,,,!

4 ( 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)

5 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)

6 ( forecasting ),,,,,,,,,,,,,,, ,,,,,,,,,,,,,

7 2 ), (economic forecasting),, 4,,,,,,,,, (),,,,,,,,,,,,,, :,,

8 3 ),,,,,, :,,,,,,, ( ),,,,,,,, ( ),,,,,,,,,,,,, (),, 5 ; 1 5, 1,

9 4,,,,,,,,,,,, :, 1.1 Box - Jenkins 1.1, : () :

10 5 ;,,, ( ) :, ;, (),,,,,, (),, ;, (),,,, ;,,, ( ), : = + = -

11 6 ( ) : 1.,, 2.,, (),,,,,,,,,,,,, ;,, (),,,,

12 7 ),,,,,,, TSP, E-views, SPSS, SAS, Excel, () 1.,, :,,,,,,,,,,,,,,,,,,,,,,, 2.,,

13 8,,,,,,,,,,, 3.,,, 4.,,,,,,, 5.,,,,,, ( ) 1., () (),,,,,,

14 9,, ( ),, ;,,,, 1.2 : 1.2 (),, :,,,,,,,,,,,,,,,,,,,,

15 10,,,,, ;, : 1.,,, ( + ),,, 2., 3. : ;, ;,,,,,,, : (1 ) : (2 ) :,,,,, :

16 11,,,,,,,,,,,,,,,,,,,,,,,,,, ( ),,,,,,,,,,,,,,,,,

17 12 ),,,,,,, (),,,,,

18 ,,,, ;,, :,,,,,, :,,, ( ), :, ; ; :,,,,,,,,,,, (Delphi Poll),,

19 14,,,,,,,,, O. N., T.J., 1964,,,, :,,, 50,,,,,,,, 1964, T.J.O. (1015 ), : ; ; ; ; ;, 50%,,,,,,,, () ( ),,, : ( ),,

20 ( Delphi Poll) 15 20,, ( ),,,,,,,,, (),, (),,,,,,,, (),,,,,,, (),,,,,, 2.1,, B,

21 16,, ( ),, 26, 60, A B C A B A B A B A B ,,,,,,,,, ;

22 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,

23 18,,,,,, :,, : = :, 2.2 ( ) * ,, 1/ 3, ( )/ 3 = 833.4( ) 950, 750, 50%,

24 )/ 2 = 850( ) 60%, 40%, : = ,,, 8 : % - 1 = - 2 % % = 823.6( ),,, :, 205, ( % ) ( ) 1%,, 99%,,

25 ,,, 199.2, 50%,, ( , )( 193.2, 205.2),, 62.5% (87.5% 25% ),,,,,,,,,,,, ( Leading Indicators ),,,,

26 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

27 22,, t, t ( ) 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 23,,,,,,?, (Diffusion Index ), 1, 0.5, 0, : ( DI t ) = DIt t, 1 6 9,,,, :, 0 < DIt < 50 %, 50%,,,,, 50% < DIt < 100 %,,,, DIt 100 %,, 100% > DIt > 50 %,,,,,,, 50% > DIt > 0,,,,,,, ;,

29 ,,,, ( ), :,,, : 3.1 : (1 ); (2), ; (3 ),, ; (4 )

30 25 ) Y, X, 1. ( ) Y = +X +, Y X ;,,, X,, : 1.Y1 =+X Y2 =+X 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,

31 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

32 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 )

33 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 )

34 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 n - 2 : ST = SR + SE,

35 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,,

36 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

37 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

38 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,

39 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 ) = n + ( x0 - x) 2 Lx x 2

40 35 y - y^0 N 0, ( x0 - x) + n Lx x 2 Sy/ 2 2 ( n - 2 ), S y y0 - y^0, t y0 - y^ 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 n ^ ( x0 - x) + n Lx x ^ + ( x0 - x) 2 Lx x = t1 - a 2 ( n - 2) = 1 - : = t1 - a ( n - 2) 2 ^ n + ( x0 - x) 2 Lx x y^0 - t1 - a ( n - 2) 2 ^ n + ( x 0 - x) 2 Lx x, y^0 - t1 - a ( n - 2) 2 ^ n + ( x0 - x) 2 Lx x ^2 = Sy ( n - 2) = y^0 ts y n + ( x0 - x) 2 ( x - x) 2

41 36 Lx x, x0 Lx x x0 x x, Lx x, x = x, x0, : x 3.3 n, n + ( X0 - x) 2 Lx x 1, :, Sy = y^tsy,, Excel, : :, 3.2 :,, 3.4, :,,, 3.5

42

43 38, y x,, 3.6, 3.6 : 3.7

44 39 : (1 ) y^ = (2 ) R 2 = , (3 ) t ta = , tb = %, 21-2 = 19, t, t = ,,, < a < < b< (4 ) F ( ) 3.3 df SS MS F SignificanceF E = 5 %, ( 1, 19 ), F, F (1. 19) = 4. 38, F, (5 ) S y = , S y y = 59.49% > 15%, (6 )D.W d = , = 1%, n = 21 p = 1, D.W, dl = 0.97 du = 1.16,0 < d < dl,,,,, ,, 3.4:

45 Dependent Variable : Y Met hod: Least Squares Sample : Included obse rvations : 15 Variable Coefficient Std. Error t - Statistic P rob. C X R - squared Adjusted R - squared S.E. of regression Sum squared resid Log likelihood Durbin - Watson stat Mean dependen t var S.D. dependent var Akaike info c riterion Sch warz criterion F - statistic Prob( F - statistic) , 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 = % y^0 tsy n 2 ( x0 - x) + ( x - x) 2 = = ( ) , 95 %, ,

46 41,,,,,, ( ) Y, X1, X2, X p,, Y = X1 + 2 X2 + + p X p + j, X Y ; 0, j ( j = 1, 2p),, X,, : 1. Y1 = X X p X 1 p 2.Y2 = X X p X 2 p N. Y N = X N X N2 + + i X N p, : Y i = Xi1 + 2 Xi2 + + p X ip, : + i N (1 )X1, X2, X p,, (2 ) Y, (3 )0, E( i ) = 0 Cov( i, j ) = 2 i = j 0 ij : E( Y ) = X1 + 2 X2 + + p X p va r( Y ) = 2 I

47 42 YN X1 + 2 X2 + + p X, 2 I) Y = 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

48 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.,,

49 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 )

50 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

51 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 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,,,

52 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, (),, ; ;,,,,

53 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,,,,

54 49,, : 3.5 :2000 Excel :, : 3.8

55 50 3.9, M ultiple R R Square Adjusted R Square df SS MS F Significance F E E E Coefficients ts t a t P - v alu e Lo wer95 % Upper 95 % 95.0 % 95.0% In t erce pt X Variabl e E X Variabl e E X Variabl e X Variabl e

56 51, (1 ) y^ = x x x x4 (2 ) R 2 = R 2 = , (3 ) t tb 0 = , tb 1 = , tb 2 = , tb 3 = , tb 4 = %, = 12, t, t0.025 = ,, b3, b4 (4 ) F ( ), = 5%, ( 1, 15 ), F, F0.05 (1, 15) = 4.54, F, (5 ) S y = , Sy y = 1.953% < 15%, (6 )D.W d = , = 1%, n = 17 p = 4, D.W, dl = 0.68 du = 1. 77, dl < d < du,,,,, F,,, x 1 x1 x2 x3 x4 1 x x x ,,,,, X4,,, 3.9

57 52.9 Dependent Variable : Y Met hod: Least Squares Sample : Included obse rvations : 17 Variable Coefficient Std. Error t - Statistic P rob. C X X X R - squared Mean dependent var Adjusted R - squared S.D. dependent var S.E. of regression Akaike info criter ion Sum squared resid Schwarz criterion Log likelihood F - statistic Durbin - Watson stat Prob( F - statistic) b3, D.W x3,, Dependent Variable : Y Met hod: Least Squares Sample : Included obse rvations : 17 Variable Coefficient Std. Error t - Statistic Prob. C X X X R - squared Mean dependent var Adjusted R - squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F - statistic Durbin - Watson stat Prob( F - statistic)

58 53 D W, x4 x3, x4,, : 3.11 Dependent Variable : Y Met hod: Least Squares Sample : Included obse rvations : 17 Variable Coefficient Std. Error t - Statistic Prob. C X X R - squared Mean dependent var Adjusted R - squared S.D. dependent var S.E. of regression Akaike info c riterion Sum squared resid Schwarz criterion Log likelihood F - statistic Durbin - Watson stat P rob( F - statistic) ,,,, y^ = x x2, x1, x2,,,, ( ),,,, y, y

59 54 1. :, ( 2 n ),,,,,,,,,,, ;, 3.,,,,,,,,,,,,,, ;, 4. (2 ) ( 3 ),,,, ( ),,,,,,,,,,,

60 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 ),,

61 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 b p x p - xp S x p y^ = b 1 x 1 + b 2 x b p x p n ( x i1 i = 1 n n - x 1 ) 2 b 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 ( x ip - x p ) 2 = ( x ip - x p ) ( y i - y)

62 57 L 1 L00 L 11 b 1 + L 12 L00 L00 b L L 1 p b 22 L p = L 10 pp Lp1 L 00 b 1 + Lp2 L 11 L 00 b 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 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 L pp Lpp = r10 r11 b 1 + r1 2 b Vip b p L pp b p = r21 b 1 + r2 2 b r2 pb p = r2 0 rp1 b 1 + rp2 b 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,,

63 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

64 59 y( / ) : x1 3CaOAl2 O3 ( % ) ; x2 3CaO SiO2 ( % ) ; x3 4CaO Al2 O3 Fe2 O3 ( % ) ;x2 4CaOSiO2 ( % ) x 1 x 2 x 3 x 4 y 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 = , F

65 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 = ; V4 = ( 1 ) Fj = ( ) 2 = = ( ) 2 = ( 1 ) = ( n - 2 ) V j 1 - V j = * ( 1 ) ( 1 ) = > 3.28 (2 ), 1 R ( 1 ) R ( 1 ) = = ( rij ( 1 ) )

66 61 x : R ( 0 ) x4, x : 3.13 F = / 1 = / 11 : (1 ), 1 R ( 1 ), V1 Fj ( 2 ) = = ( 2 ) ( 2 ) V j ( 2 ) V 1 ( 2 ) V2 = ( rjy = ( 1 ) ) 2 ( 1 ) rjj (0.5291) = V3, F, ( n - ( l + 1) - 1) Vj ( ) V 1 S ( l + 1 ) ( 2 ) ( 2 ) = S- V1 ( l + 1 ) = * ( 2 ) (13-3 ) x1, x1 = = > 3.28 (2 ) x1 2 : R ( 2) = (3 ) R ( 2 ) :

67 62 x, x4 : b4 ( 2 ) ( 2 ) b1 = = : 3.14 F x1, x4 x1, x4 F, F,, V4 F4 ( 2 ) x4 : F (2 ) = ( r4 y ( 2 ) ) 2 ( 2 ) = ( ) 2 r = ( n - l - 1) V k ( l) ( l) = S = > 3.28 = ( ) (1 ), 2 R ( 2 ), V2 ( 3 ) ( 3 ) V j V2 ( 3 ) = ( 3 ) V3, F, Fj ( 3 ) = = = ( rj y ( 2 ) ) 2 ( 2 ) rj j ( ) = ( n - ( l - 1) - 1) Vj S ( l + 1 ) ( l + 1 ) ( ) = * = > 3.28

68 x, x2 (2 ) x2 3 : R ( 3 ) = (3 ) R ( 3 ) : x1, x4, x2 : b ( 3 ) 4 = ( 3 ) b1 = b2 ( 3 ) = : F x1, x4, x2 x2 F, F,,, x1, x4 ( 3 ) V4 ( 3 ) F4 ) 2 = ( ( 2 ) r4 y ( 2 ) = r4 4 ( 3 ) ( 3 ) = S = ( n ) V4 = < 3.28 F ( ) x2, x4, x4 : (1 ) F x4 (2 )x4,,

69 R 4 ) = (3 )R ( 4 ) : x1, x4 : b1 ( 4 ) ( 4 ) b2 = = : 3.16 F x1, x2 F, F1 ( 4 ) = > 3.28 F2 ( 4 ) = > 3.28 :, ( 5 ) V 3 ) 2 = ( ( 4 ) r3 y ( 4 ) = r33 V 4 ( 5 ) = x4, x4,, ( ) 1. 4 x3, x , R ( 4 ) =

70 65 x, x2 x1, x2, C ( 2 ) = x1, x2 2. bj = b j L00 L jj L0 0 = , L11 = , L22 = b 1 = b 2 = b1 = b2 = b0 = y - b1 x1 - b2 x2 = : y = x x F s = R 2 = n ( y i = 1 i - y^i ) = x1, x2, y x 3, x4 : r3 y 12 2 = ( r3 y ( 4 ) 2 r4 y 12 ) 2 r33 ( 4 ) ry y ( 4 ) = = ( r4 y ( 4 ) r44 ( 4 ) ry y ) 2 ( 4 ) = SPSS : : SPSS, :

71 66.10 : (1 ) Analyze Regression Linear (Dependent), ( Independent),

72 67 2 ) Statistics, 3.12 (3 ) Continue Plots, 3.13 (4 ) Continue Save,

73 68.14 (5 ) Continue Option s 3.15

74 69, : 3.18 ( Descriptive Statistics) Mean Std. Deviation N Y X X X X ( Correlations ) Y X1 X2 X3 X4 Pea rson Corr elation Y X X X X Sig. ( 1 - tailed) Y X X X X

75 ( 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 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 R esidual To tal Regression R esidual To tal Regression R esidual To tal Regression R esidual To tal a b c d e Predictors : ( Constant), X4 Predictors : ( Constant), X4, X1 Predictors : ( Constant), X4, X1, X2 Predictors : ( Constant), X1, X2 Dependent Variable : Y

76 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) X ( C) X X ( C) X X X ( C) X X a Depe ndent Variable : Y 3.24 ( Ex clude d Varia bles ) Model Beta In t Sig. Partial Correlation Collinearity Statistics Tolera nce 1 X X E - 02 X X E - 02 X X E X X E - 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

77 Coefficient Correlations Model X4 X1 X2 a 1 Correlations X Covariances X E Correlations X X Correlations X E E - 03 X E E - 02 X Correlations X X X E E E - 02 Correlations X E E E - 04 X E E E Correlations X X Correlations X E E - 03 X E E - 03 Dependent Va riable : Y 3.26 ( Residuals Sta tistics ) Minim um Maximu m Mean Std. Deviation N P redicted Value Std. P redicted Value Standard Error of Predicted Value Adjusted Predicted Value Residual E St d. Residual Stud. R esidual Deleted Residual E Stud. Deleted Residual Mahal. Dista nce Cook s Dista nce Cen ter ed Lever age Value a Dependent Va riable : Y

78

79 ,,, ( ),

80 75,,,,,, :, ( ), : 1. y = b0 + b1 x + b2 x bk x k + e y = b0 + b1 x1 + b2 x bk x k k + e y = ab x e y = ae bx e 3.23

81 76. y = ax b e y = a + b x + e 1 y = a + b x + e y = a + bln x + e 3.26

82 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 , ^ y = Le - ae - bx y = La bx 3.28,y = L/ e, x = lna/ b, 3.28

83 78., ^ y = L 1 + ae - bx, y = L/ 2, x = ln a/ b, 3.29 S, S, ( ),,, : y = b0 + b1 x + b2 x 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

84 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

85 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),

86 81.28 :,, y^ = a+ b x + e, X = 1 x y = a + b X + e 3.30 :, Excel,,, 3.31

87 82.31 a= , b= , y^ = x 28.5, 29 30,, : % % %

88 ( ) Y t, {Y t, t = t0, t1 }Y t, t = 1, 2,, ,,, Warren Persons (),,,,,,,,,,, ( T) ( S) ( C) ( I)

89 84. (T ),,,,,, 2. (S ), 5, 3. (C ),,,, , 4. (I ),,,,, ( ),, : 1.,,,,,,,,

90 85.,,,,,, , 3.,, 4., ( ),,, 5.,,,,,,,, t, t, t,, ( ),

91 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

92 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 Ft + 1 t + 2, : Ft + 2 t + 1 yi = y = i = 1 t + 1 et + 2 = yt Ft + 2, : ;, Ft + 1 = y = i = 1 t t yi

93 88 t + 1 yi Ft 2 = y = i = 1 t + 1 = y1 + y2 + + yi + y i + 1 t + 1 = t Ft yt + 1 t + 1 Ft + 3 = t + 1 t + 2 Ft t + 2 yt + 2 = t t + 1 Ft 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

94 ,,,,,, :, t, 12, 4,

95 90 t + 1 Ft 2 = 1 t i = 2 yi = 1 t ( y1 + y2 + yt + yt y1 ) = Ft t ( yt 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 : = ( B B31 + B20.5)/ 3, : D5, D14, : 4.2

96 91.3,,,,,,, : ( t ), ;, ;,,,,,, 4.3

97 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 n Ft = 1, 0 < < 1, n Ft + 1 =yt + (1 - ) Ft, Ft t Ft + 1 t + 1, t + 1 : St + 1 =yt + (1 - ) St

98 93, : St = y t + ( 1 - a) St - 1, 0.2, 0.5, 0.7, ( ) y t = 0.2 = 0.5 = : = 0.2 S1 = = S2 = = : 4.4

99 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 (1 - ) St (1 - ) yt ( 1 - ) 2 yt ( 1 - ) n y t - n + 1 +,, t,, 4.4 = 0.3 = 0.2 = ,,,, ;,, : ( ),

100 , 0.30,,,,,,,, SSE MSE MAE ( ), n, (1 - ) n,,, ;,, ;,,, :,,,, 8 3,,,,,, ( ),,, (), : St + 1 = St + ( yt - St ) St + 1 = St + ( et ) = +,,,, ( 4.5 ),

101 96, : S ( 1 ) t = Yt + (1 - ) S ( 1 ) t - 1 S ( 2 ) t = S ( 2 ) t + ( 1 - ) S ( 2 ) t ,,,,,,, : : 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

102 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 ) ( )

103 98 Excel : : S ( 1 ) 0 = S ( 2 ) 0 = 143,= 0.2 0, D2, E2 143 D3 : = 0.2 * C * D2, D2 6, : E3 E26, : F3 F26, : G3 G2 6, b : = 0.2 * D * 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 = = ( ) m = 3, F27 = a24 + b24 3 = = ( ), : 4.7

104 ( ) ( H OLT ) 1., ( ) : St =yt + (1 - ) ( St 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., :

105 100 b0 b0 = ( ) + ( ) + ( ) 3 S0 = 143, b0 = , = = 0.2,= 0.3 D2 143, E D3 : = 0.2 * C * (D2 + E2), : E3 : = 0.3 * (D3 - D2) * E2, : D3, E3, E26, : 1 ( m = 1 ), F3 : = D2 + E2, H27, 225 2, Ft + m = at + bt m : m = 2, F26 = a24 + b24 2 = = ( ) m = 3, F27 = a24 + b24 3 = = ( ), : 4.8

106 ( H OLT ),,,,, bt 0,,,,, h, b h = 1 - (1 - b ) 2,= 1 - b,, 2(1 - b ), 2 - b, ( H OLT )

107 102.8 : ( ) 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 ct m2, at = 3 S ( t 1 ) - 3 S ( t 2 ) + S ( t 3 ) bt = ct = 2 (1 - ) S ( 1 ) t ) 2 [ ( ( S ( 1 ) (1 - ) 2 t - 2 S ( 2 ) t + S ( 3 ) t ) - (10-8 ) S ( 2 ) t + (4-3 ) S ( 3 ) t ],,,

108 103. : 4.9 ( ) Excel, : : S ( 1 ) 0 = S ( 2 ) 0 = S ( 3 ) 0 = 3.4,= 0.250, D2, E2, F2 3.4D3 : = 0.25 * C * D2, D26, : E3 E26, : F3 F26, : G3 G2 6, a : H3 : = 0.25 * D * E2, : = 0.25 * E * F2, : = 3 * D3-3 * E3 + F3, :

109 * ( ( 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 ct m2, : m = 2, F2 6 = a2 4 + b c24 22 = = ( ), : 4.10 :

110 ( ) ( ) 1., St = y t b1 It - L + ( 1 - ) ( St 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 ( )

111 ,, :,,,,,,, : :, S0 = , b0 = 2.335, : I - 3 = 1.37, I - 2 = 0.81, I - 1 = 0.851, I0 = = 0.2,= 0.2,= 0.3, : D6 : = 0.2 * ( C6/ F2) * ( D5 + E5), E6 : = 0.2 * (D6 - D5) * E5, F6 : = 0.3 * ( C6/ D6) * F2, D7 : = 0.2 * ( C7/ F3) * ( D6 + E6), E7 : = 0.2 * (D7 - D6) * E6, F7 : = 0.3 * ( C7/ D7) * F3, D8 : = 0.2 * ( C8/ F4) * ( D7 + E7), E8 : = 0.2 * (D8 - D7) * E7, F8 : = 0.3 * ( C8/ D8) * F4, D9 : = 0.2 * ( C9/ F5) * ( D8 + E8), E9 : = 0.2 * (D9 - D8) * E8, F9 : = 0.3 * ( C9/ D9) * 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 = ( )0.78

112 ( ) m = 3, F31 = ( S28 + 3b28 )I27 = ( )0.80 = ( ), : 4.12 ( )

113 ) ( ), St = ( yt - It - L ) + ( 1 - ) ( St 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 = , b0 = 2.335, : I - 3 = , I - 2 = , I- 1 = , I0 = = 0.2,= 0.2,= 0.3, : D6 : = 0.2 * ( C6 - F2 ) * (D5 + E5 ), E6 : = 0.2 * (D6 - D5) * E5, F6 : = 0.3 * ( C6 - D6 ) * F2, D7 : = 0.2 * ( C7 - F3 ) * (D6 + E6 ), E7 : = 0.2 * (D7 - D6) * E6, F7 : = 0.3 * ( C7 - D7 ) * F3, D8 : = 0.2 * ( C8 - F4 ) * (D7 + E7 ), E8 : = 0.2 * (D8 - D7) * E7, F8 : = 0.3 * ( C8 - D8 ) * F4, D9 : = 0.2 * ( C9 - F5 ) * (D8 + E8 ), E9 : = 0.2 * (D9 - D8) * E8, F9 : = 0.3 * ( C9 - D9 ) * F5, : D6, E6, F6, D9, E9, F9, F33, X : 1 ( m = 1),

114 109 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 = ( ) = ( ) m = 3 F31 = ( S b28 ) + I27 = ( ) = ( ), 4.13, ( ) 4.12, 4.13

115 ,,

116 111.,,,,,,,,,, :,,,,,,,,, ( ),,,,,,,,,,,,,,,

117 112,,,,, (, ) ; : 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 :,

118 113 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 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 c t m 2

119 114 B B3 S t = (y t - I t- L ) + (1 - )(S t 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 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

120 ( self - adaptive filtering) -,,,,,,, y1, y2 yt, y^t + 1 = w1 yt + w2 yt 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 et + 1 = w1 yt + w2 yt wn y t - n et + 1 et + 1 et + 1 = yt y^ t + 1,,,,

121 116 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 y^t + 1 = y t w1 yt - w2 yt wn y t - n - 1 e 2 t + 1 = ( yt w1 yt - w2 y t 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 ke 2 t + 1 ( i = 1, 2n), n yt - i + 1 t - i + 1 ( ),, :

122 117 :, n 4 : k k n 1 i = 1 y2 i max = = :, 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 = = 5 et + 1 = e5 = y5 - y^ 5 = 10-5 = 5 W i = wt, + 2 ket + 1 y t - i + 1 : w 1 = = w 2 = = w 3 = = w 4 = = :,,,, : t = 5, y^t + 1 = y^6 = w1 y5 + w2 y4 + w3 y3 + w4 y2 = = 8.28 et + 1 = e6 = y6 - y^6 = = 3.72 w i = wi + 2 ket + 1 yt - i + 1 : w 1 = = w 2 = = w 3 = = w 4 = = 0.291, t = 6, t = 7, t = 8,, t = 10, y1 1, e11 wi,,, t = 4, ( ),

123 118, : 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,, : K6 : = G6 * C6 + H6 * D6 + I6 * E6 + J6 * F6, : L6 : = B6 - K6, : G7 : = G6 + 2 * * L6 * B5, w1 H7 : = H6 + 2 * * L6 * B4, w2 I7 : = I6 + 2 * * L6 * B3, w3 J7 : = J6 + 2 * * L6 * B2, w4 K6, L6, L7,

124 : G7, H7, I7, J7, K7, L7, L11, : word,, G6 J6 (, ),, 5.5,, :

125 ,,,,, : ( ) ( n),, 4 8, 12 n,,, n 2 6 n, n (),,, wi = 1 n r1, r2 Yule - Walker

126 121 n, w1 = ( ) k r1 ( 1 - r2 ) 1 - r 2 1, w2 = r2 - r r 2 1, k, n,,,, k, k 1 n : 5.8 ( ) Excel, : :, 1 3, wi = 1/ 3

127 122 k = : J5 : = G5 * D5 + H5 * E5 + I5 * F5, : K5 : = C5 - J5, L5 : = K5 * K5, : G6 : = G5 + 2 * * K5 * C5, w1 H6 : = H5 + 2 * * K5 * C4, w2 I6 : = I5 + 2 * * K5 * C3, w3 J5, K5, L5, L6, : G6, H6, I6, J6, K6, L6, L25, L / 21 = : 5.9

128 word,, G5 I5 (, ),, 5.10: , :

129 ,,, 18, :

130 : : w1 = , w2 = , w3 = : y25 = = 20.33( ) y26 = = 21.33( )

131 126 (1 ),, (2 ),,,,,,,,,,,,,,,,, n wi = 1,, i = 1,,,,, 1,,, ( Weight ed Sum) ( ),,,,,,,,,

132 127, ;, 1, ( ) ARMA AR MA : y^t + 1 = 1 yt + 2 yt n y t - n + 1 n : y^t + 1 = w1 yt + w2 yt wn y t - n + 1,, ARMA,, 1973,,,,,,

133 ,,,,, 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

134 129 :,,, t, y, : ( ) yt ^ = a+ bt 6.1 () yt ^ = b0 + b1 t + b2 t 2 6.2

135 130 ) ^ yt = at b 6.3 ( 1) ( b > 0 ) 6.4 (2 ) ( a < 0) () yt ^ = b0 + b1 t + b2 t 2 + b3 t () yt ^ = a+ blnt

136 131.6 ( ) yt ^ = a+ b 1 t 1 ^ = a + b1 t yt 6.7 () yt ^ = ae bt 6.8

137 132 ) ^ 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 ( )

138 133 yt = L 1 + ae - bt 6.11, t,,, : ( ) t, :

139 134.12,,,,,,,,,,, : 6.2,, 6.2 ( t) yt = a + bt ( yt - yt - 1 ) 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,,

140 yt b0 b1 b2 ( yt - yt - 1 ) ( t) = + t + t 2 1 b 0 + b 1 + b 2-2 b b 1 + 4b 2 b b 2 3 b b 1 + 9b 2 b 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 yt - 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 b b b 2 + 8b b b b b 3 2 b b 3-4 b 0 + 4b b b 3 2 b b 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 ) 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

141 136.6,, 6.6 ( t) yt = a + bc t ( y t - y t - 1 ) y t - y t 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 yt ,, 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 lg y t 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, : ;,

142 137,,,, 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 ), D : t ln y t, : 6.13

143 138 : M ultiple R R Square Adjusted R Square df SS MS F Significance F E Coefficie nts t Sta t P - value Lower 95 % Upper 95 % 95.0 % 95.0 % Int ercept E XVariable E : ln a = , b= :

144 139 y t = t e t y^ t = :, ln yt E, F2 : = EXP ( E2 ),, F13, yt F14 : = * ( )^( * 13 ), 2003 y^2 003 = : ,,, :

145 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 y = yi = nl + a( e0 + i = 0 e b + + e ( n - 1 ) b ) = nl + a enb - 1 e b - 1

146 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 n 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

147 142 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 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 y ) b - 1 ( b n - 1 ) 2 1 y y 1 y y L = 1 n [1 1 y - a( bn - 1 b - 1 ) ] 1 y, 1 2 y, 1 3 y

148 143 1 y = 1 ( n + ac) ( t = 1, t = 0 ) L y = 1 L ( n + ace- nb ), 1 y = 1 L ( n + ace- 2 nb ) 1 y y = ac L ( 1 - e- nb ) 1 y y = ac L e - nb (1 - e - nb ) 2 e - nb = 1 1 y y 1 y y b = 1 n [ln(1 1 y y ) - ln (2 1 y y ) ] 1 a = L (1 c y y ) y y y L = n y y 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 = t : F 2 : 1/ ( * ( )^ B2 ), F21, , 6.14

149 ,,, ;,,,,,, ( ) 1.,,, 6.15,,,

150 145 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 t = e y^t = t t 2 y^t = t t t ,, B C,

151 ,, : 6.16 A B C E E , C,,, ( ),,, : ;, :

152 147,, ;,,,,, ;,,,,,,,,,,,,, 6.17, 6.17,,, A B C,, y^t t = e yt = t t 2 y^t = t t t 3

153 ( ) ( y) ( ) ( y)

154 149,,,, 6.18,,,,,,, A B C

155 150, 6.19: 6.19 ( ) ( ) ,,,,,,,,, :

156 F t + m = a t + b t m ^t, t y = a + bt, t, m F t + m = a t + b t m c t m 2 ^t y = b0 + b1 t + b2 t 2, t, t, m,,,,,,,,

157 ,,,,,,,, (),, 12, 4, ;, ;,,,, :, ( S ) ; (T ) ;,, (C ),, ( I ) : Y = TCI S, ; X - 11 X - 11

158 153 y^t = yf i y ; f i, 35 ( ) , 7.1 : (), : 7.1

159 154 ( ) :,, B13 : = AVERAGE( B2: B12),, M13 N2 : = A VE RAGE ( A2: M2 ),, N13, 112 (),, %,,, 1200, : f i = ( ) ( ) B14 : = B13/ ,, M14 B15 : = B14 * 1200/ ,, M15 7.2

160 155.2 : 7.2 ( ) , : y^i = ( ) f i B16 : = * B15/ 100,, M y^t = ( a + bt) f i : ( a + bt) ; f i,

161 : 7.4 ( ) 7.3,,,, 7.3 : (), (1 ), Ft = a + bt,,, 12 ; EXCEL :

162 157.5 Coefficie nts t St a t P - value Lower 95 % U pper 95 % 95.0 % 95.0 % Intercept E XVariable E M ultiple R R Square Adjusted R Square df SS MS F Significance F E : Ft = t, (2 ), ; E2 : = C2/ D2,, E61

163 (3 ),, 1200 %,, f i = Si = yi F i, Si + Si + T + Si + 2 T + + Si + ( m - 1 ) T m m, T,,, : 7.10 () : y^t = ( t)f i

164 159,, C2 : = * B2,, C13,, E2 : = C2 * D2,, E : 7.11 yt + m = ( at + bt m)f t + m at, bt f t + m (1) y1 (2 ): yj b0 = yj - y1 T - 12 T (3 ) t = 0, a0 : (4 ) t a0 = y1-6.5 b0 at = a0 + b0 t

165 160 5 ) f t f t = yt (6 ) at ( t = 1, 2,, T) f = ^1 f = ^2 1 J ( f 1 + f f T ) 1 J ( f 2 + f f T ) f = ^12 1 J ( f12 + f f T ) (7 ),,, at = yt bt ^t f + (1 - ) ( at bt - 1 ) = ( at - at - 1 ) + ( 1 - ) bt - 1 f t + 12 = yt at + (1 - ) ^f (1 ) y1 - y1 - = , yj (2 )b0 = = , J = 5 - yj - y1 T (3 ) t = 0, a0 - yj = a0 = y1-6.5 b0 = = = ;

166 161 4 ) t at = a0 + b0 t = t, D3 : = * B3,, D62 (5 ) f t E3 : = C3/ D3,, E62 (6 ) F3 : = ( E3 + E15 + E27 + E39 + E51)/ 5,, F14 (7 ) = 0.1,= 0.2,= 0.3 at, bt a0, b0 ; G3 : = 0.1 * (C3/ F3) * ( G2 + H2),, H3 : = 0.2 * (G3 - G2) * H2, G14, H14 G15 : = 0.1 * ( C15/ F3) * (G14 + H14),, G26, H26 G27 : = 0.1 * ( C27/ F3 ) * ( G26 + H26 ),, G38, H38 G39 : = 0.1 * ( C39/ F3 ) * ( G38 + H38 ),, G50, H50 G51 : = 0.1 * ( C51/ F3 ) * ( G50 + H50 ),, G62, H62 f t + 12 I51 : = 0.3 * ( C51/ G51 ) * F3,, I6212, 7.13

167 t y a at bt f ( t + 12 ) f ( t + 12 )

168 163 t y a at bt f t + 12 ) f ( t + 12 ) : y6 0 = ( m) f6 0 + m m = 1,212, y^t = a + bt + di : ( a + bt) ; di (),

169 164 ( ), : y1 yj ; - y1 : b= yj t - 12 : t : a = y1-6.5b : Ft = a + bt Ft () di dt dt = yt - Ft () di di = : i = 1, 2, 12 1, 2,, 4; T ; di + di + T + + di + ( m - 1 ) T m m di,, :

170 165 ),, : Ft = t, () di E2 : = C2 - D2,, E61 di F2 : = ( E2 + E14 + E26 + E38 + E50 )/ 5,, F : Ft = t + di () 7.19,,, C2 : = * B2,, C13, E2 : = C2 + D2,

171 166 E13 : 7.19 X - 11 X ( Bureau of Census Department of Commerce) (NBE R ) ( The Ratio - Moving Average Method),,,, X 1960, X - 3, X - 3, 1961, X X - 11,, X - 11,,,,,,,, X - 11,, X - 11, ( IM F ),

172 X X - 11, : ( TC) ; ( S ) ; ( I) ; (D),,, : Dp: ; Dr: ; P : ; E : ; I: : T = T CSID T = T CSID T = T CSID T = T CSI : D= Dp Dr, D= Dr, I= PEI, I= EI : T = T C+ S + I+ D T = T C+ S + I+ D T = T C+ S + I+ D T = T C+ S + I : I = P + E + I, I= E + I : Y = T CSI, ( I= EI) : Y = T C+ S + I, ( I= E + I)

173 168 X 11 ( ) X - 11 TC S, 1. : { xt, t = 1, 2, n}, y L = 1 L t L xt + i t = 1, 2,, N - L + 1 i = 0 L, M AL, yt x t y L - 1 = M AL ( xt ) + t 2, x t, xt, xt + 1,, x t + L - 1 L, y L t t, t + 1,, t + L - 1, 2.,, ( 12, 4 ), M A2 m y 2 m t, y S t 2 t = 1, 2,, N - S1 + 1 { xt }, t = 1, 2, n M As1 : Z S S t 2 = M As 2 y S1-1 + t 2, = M As 2 ( M As 1 ( xt ) ) t = 1, 2,, N - S1 - S , M As s , M A, M As 2 s 1, Z S2-1 2, S1, S2 + S t M A M A2 m X - 11 M A M A212, TC, TCI

174 X M A 3, St, St 4.H enderson ( MA H ) TCI, TC H enderson 9,13, 23,, X - 11, TCI, MAH9, MA H13 MAH23, : (1 ) T C, MA H13 (2 ) I, TCI/ T C (3 ) I/ - TC -, I, - TC - I TC ( ),, MAH 7.20 MAH I/ T - C - MA H MAH9 MA H13 MA H23 ( ), S S2, y s S 2 N - S 2 + t N, ( ),,,

175 ( ), Y ( t) t ti, Y ( ti ) Y ( t) t = ti,,, Y, Y () tk, t > tk tk, tk,,,,,,,, ( ),,, () t, () Y ( t) t = ti Y ( ti )

176 171 t ti,,, ;,,,, : ;,,,,,, ( ),,,,, () AB, P( B A) A, B A, P( B A) A, B A B, AB =, P( B A) A B () Pij n,, n ( ),,,, A B n,, n, Pi1, Pi2, Pii, Pin () n, nn, P1 1 P12 P1 n P = P2 1 P22 P2 n Pn1 Pn2 Pnn

177 172 () n Pij Pij = 1 j = 1 1, 2,, n n P, k P ( k ), P ( k) = P ( k - 1 ) P = PPP ( k ), 1000 (,,), ,2 500, 50,50 ; 400,20,80 ; 100,10,10, : 8.1, P11 = = 0.8 ; P12 = = 0.1,,, : 8.2, P ( 3 ) : = PPP

178 173 C 5: E17, C15 =, MMULT,, 8.1, : 8.2 Ctrl + Shift + Enter,, 8.3: 8.3

179 174 C 0: E22, C20 =, MMULT,, : 8.3 Ctrl + Shift + Enter,, 8.4: : 8.5

180 175, 55%, 21 %, 24 % ;,, 48%, 14%, 38% ;,, 58%, 21 %, 21% ( ) Y t t = k + 1 Y t t = k, t = k,,, Y t t = k + 1 Y t t = k t = k - 1, t = k - 1, (),,,, () 1.,,,,, 2., Ei ( i = 1, 2,, n),, Ei M i, Ei : Pi Fi = Mi N N = Mi n i = 1

181 176 :. Ei E j M ij, Pij = P( Ei Ej ) Fi j = Mij Mi : p11 p1 2 p1 n P = p21 p2 2 p2 n pn1 pn2 pnn,, 4.,,,, ;,, ( ) 20, ,,

182 177,,, ,,,,,,,,, 8.4 : 3.,,, < 60 ; ; > ,, 8.7:

183 178 Mij., , M11 = 3, M12 = 4, M13 = 0, M21 = 1, M22 = 1, M23 = 3, M31 = 2, M32 = 0, M33 = 5, P = , : P31 = 2 7, P3 2 = 0, P33 = 5 7 P33 P3 1 P32, ( ) : S ( k+ 1 ) = S k P k+ 1 P11 P12 P1 n P21 P22 P2 n = S ( 0 ) Pn1 Pn2 Pnn, S ( k) t = k; P ; S ( k+1 ) t = k + 1, S ( 0 ), (),,

184 179 ( ) (), 8 9, 1. : , B17 : = B14/ 1839,, E17 : B21 : = B10/ 545,, E21; B22 : = B11/ 495,, E22; B23 : = B12/ 417,, E23; B24 : = B13/ 382,, E :

185 : S ( k+ 1 ) = S ( k) P 10 B27: E27, B27 =, MMULT,, : 8.5 Ctrl + Shift + Enter,, B28: E28, C28 =, MMULT,, :

186 181.6 Ctrl + Shift + Enter,, B29: E29, C29 =, MMULT,, : 8.7 Ctrl + Shift + Enter,, :

187 ,,,,,,,,,,,,, ( ),,, n + 1, : S ( n+ 1 ) = S ( n),,,,,

188 183 ),,,, (),,,,,, (), S ( k+ 1 ) = S ( k), S ( k + 1 ) = S ( k) P = S ( k) S ( k) = ( x1, x2, xn ) x i : n i = 1 = 1, k, p11 p12 p1 n p = p21 p22 p2 n pn1 pn2 pnn S ( k+ 1 ) = S ( k) p = S ( k) : p11 x1 + p21 x2 + + pn1 xn = x1 p12 x1 + p22 x2 + + pn2 xn = x2 p1 n x1 + p2 n x2 + + pnn x n = xn x1 + x2 + + xn = 1 n, ( n + 1 ), n,, ( p11-1 ) x1 + p21 x2 + + pn1 xn = 0 p1 2 x1 + ( p22-1 ) x2 + + pn2 xn = 0 p1 ( n - 1 ) x1 + p2 ( n - 1 ) x2 + + ( pn( n - 1 ) - 1 ) xn = 0 x1 + x2 + + xn = 1

189 184 x1 x2 = xn p1 1-1 p21 pn1 p1 ( n - 1 ) p2 ( n - 1 ) pn( n - 1 ) ,, Excel : k S ( k) = ( x1, x2, xn ) (1 ) (2 ), 8.13 B9: E12, B9 =, T RANSPOSE, 8.8, 8.8, :

190 185.9 Ctrl + Shift + Enter,, (3 ), 1, 8.13 (4 ) 8.14 B21: E24, B21 =, MIN VE RSE8.10

191 186.10, : 8.11 Ctrl + Shift + Enter,, (5 ), 8.14:

192 , 27.24%, 33.30%, 21.83%, 17.63% :,,,,,,,,

193 - -, B - J,, : ( Autoregressive), A R ; ( Moving Average), MA ; ( Autoregressive Moving Average), AR MA -, (G.U.Yule) ( H.Wold) ( G.U.Yule) 1926 ( AR ), ( Walker) 1934, 1937 ( Slutzky ) ( MA ), 1938 ( H.Wold) ARMA A RMA,, ( ARMA), ARMA,,, ( G.E.P.Box) ( G.M.Jenkins ),1970,,,, B - J ( ) - -, ARMA

194 { x( t), t T},, T, t, x( t), t T,, t T = {- 2, - 1,0,1,2, }, x( t),, 2.,,, { xt } : Ft 1, t 2 t n ( x1, x2, xn ) = F t1 + r, t2 + r t n+ r ( x1, x2, xn ), n, r, t1, tn, { x1 }, ( xt 1, xt n ) ( xt xt 1+ r n+ r ), 3.,,,,,,,, ( x t 1, xt n ) ( xt xt 1+ r n+ r ),,,, m : n, r, t1, t2, tn, ( xt 1, x t n ) m, E[ x m t 1 1 xt m 2 2 x t m n n ] = E[ xt m r x m t r x t m n n+ r ], m1, mn, m1 + m2 + + mn m, { x1 } m { x1 }, : t,, Ex t = ; E[ x 2 t ] = 2, 2 t, 2 < + ; t, s, E[ x t, xs ] t - s :,, t, s

195 189 t, s,,,,,,,, ( ),, ( ) - - :,,,,,,,,, B - J (),,, : y = b0 + b1 x1 + b2 x2 + + bk x k + e, y, x1 xk, b1 bk, b0, e,,, yt yt- 1, yt- 2, yt- k,, yt = b0 + b1 y t- 1 + b2 yt bk y t- k + et,,,,,,,ar, yt = 1 yt yt p y t - p + et

196 190 - yt 1 yt yt p y t - p + et, et, et, (et et - 1, et- 2 ), et y k ( k < t) ( et yt - 1, yt - p ), yt - 1, y t- p, 1 yt yt p y t- p ; yt - 1, yt - p, et,, et, AR, et, et, et- 1, et- 2 et- q, - 1 et- 1-2 et qet- q,, et,, et e t,, : e t = et - 1 et- 1-2 et q et- q e t ( ), : yt = 1 yt yt p y t - p = 1 y t y t p y t- p yt = 1 yt yt p y t- p + e t - 1 et et q et - q - 1 et- 1-2 et q et- q, p q, ARMA( p, q) p, q 1, 2 q, 1, 2 p : et, q = 0,, AR ( p), AR MA( p, 0) ; AR ( p) y1 = 1 yt yt p y t - p p = 0,, MA( q), A RMA( 0, q) MA( q) + et y t = et - 1 et- 1-2 et q et- q ARMA( p, q) Y t AR ( p) MA( q) ( ) + et + et A RMA( p, q),, ARMA( p, q) B k k, By t = yt- 1

197 191 B y t = yt- 2 B k y t = yt- k B k e t = et- k B k C C( C ) ( B) = 1-1 B - 2 B p B P ( B) = 1-1 B - 2 B q B q AR M A ( p, q) B - J ( B) Y t = ( B) et 9.1 B - J

198 ,,,, yt y t - 1, yt - 2,,, rk n t = k+ 1 = ( yt - y) ( yt - k - y) n ( yt - y) 2 t = 1 n, k, y,,,, - 11, - 1 rk 1, rk 1,, r1, r2, r3 9.1 t yt : :

199 ( t) y t y t- 1 y t- 2 y t , y = t = 1 10 t = 2 r1 r t = 1 y i = ( yt - y) 2 = t = 3 10 t = 4 ( yt - y) ( yt- 1 - y) ( y t t = 1 - y) 2 10 = ( yt - y) ( yt- 2 - y) ( y t t = 1 - y) 2 10 = ( yt - y) ( yt- 3 - y) r3 10 = ( y t - y) 2 t = 1, ; ( ),,, k,, k

200 194 -, 0, = 1 n ( ), F( t), ( - t, t ),, ( - t, + t) 95% ( , ), yt, yt- 1, yt- 2 yt- k+ 1, yt k yt- k yt- 1, yt- 2 yt- k+ 1, yt yt- k kk,, - 1 kk 1 : kk = r1 k = 1, r k kk k j = 1 k - 1, j rk - j k k - 1, j rj j = 1 k, j = k - 1, j k = 2, 3,, - kk k - 1, k - j k = 2, 3, j = 1, 2, k - 1, r1 = 0.148, r2 = , r3 = , : 11 = r1 = = r2 - ( 11 r1 ) 1 - ( 1 1 r1 ) = = = = r3 - ( 21 r2 ) - ( 2 2 r1 ) 1 - ( 1 1 r1 ) - ( 22 r2 ) 31 = = = = = ,,,,

201 195,, E - Views : 1. ( ) View Correlogram, : ,,OK,, 9.3

202 AC, PAC,, -,,, ( ),,, (),

203 197 ;, 9.4, 9.4 ( ) B - J B - J ( ), (),,, (), k = 2(k = 3 ),, 0, ;, 9.5, k = 1,,, 0, 9.6,

204 ( ) B - J B - J, ARMA, B - J (),, ARMA, AR MA, B - J

205 199 ARMA,, yt yt = yt - yt- 1 ( t > 1 ) (y t ) = 2 yt = ( yt - yt- 1 ) = yt - yt- 1 2 yt = ( yt - yt- 1 ) - ( yt- 1 - yt- 2 ) = yt - 2 yt- 1 + yt- 2 ( t > 2), Y t, d, Zt, : zt = d y t ( t < d),, 9.3 ;, 9.6;, 9.5 k = 1,,,, ( ) ( ), (),, k = 12, 24, 36, 48 0,,, 0,,, 9.7

206 , k = 12, 24 0, ( ),,, 12, yt 1 2 yt = y t - yt- 12 ( t > 12 ) ( 1 2 y t ) = 12 y t = ( yt - yt- 12 ) = y t - yt- 12 yt = ( yt - yt- 12 ) - ( yt yt- 24 ) = yt - 2 yt yt- 24 ( t > 24), Y t, D, W t, : wt = s D y t ( t > Ds ), s,,

207 ARMA ARMA,,, -,,, ARMA ARMA ( ) MA( q) MA( q) yt = et - 1 et et qe t- q Y t = ( B) et MA( q) rk = - k + 1 k q - k q q 1 k q 0 k > q, MA( q) y t, yt y t- k, k > q

208 202 -, rq 0, -,, MA( q) : 9.9 MA(1) MA( q),, k,, 0 MA( q) MA (2)

209 ARMA MA(q) 9.12 MA( q) ( ) AR ( p) AR ( p) AR ( p) rk yt = 1 yt yt p y t - p ( B) yt = et : B ( rk ) = 0 k > 0, AR( p),, + et

210 AR ( p) 9.14 AR ( p) AR ( p) : kj = j 1 j p 0 p + 1 j k AR ( p) kk p, 0 k p k k = 0 k > p,, A R( p), kj j, AR ( p), kj k j, k k 9.15, 9.16 AR ( p)

211 ARMA AR(1) 9.16 A R(2 ) A R( p),, -,, ( ) ARM A( p, q) ARMA( p, q), AR MA( p, q) AR MA, ARMA(1,1 )

212 A RMA( 1,1 ), ARMA ( p, q), ARMA( p, q), 9.4 ARMA ( p, q) AR( p) MA ( q) ARMA( p, q) ( B) y t = e t y t = ( B) e t ( B) y t = ( B) e t () ( p) ( q) ( ) ( ) ( ) ARMA( p, q),, AR MA( p, q),,, A RMA( p, q),,

213 ARMA 207 ) ARIMA( p, d, q) ARIMA( p, d, q),, d, ARMA( p, q), ARMA( p, q), ARIMA ( p, d, q) yt, d zt zt = d y t t > d zt ARMA( p, q), yt A RMA d, ARIMA( p, d, q),, d, p q z t B, yt = yt - yt- 1 = yt - By t = (1 - B) yt 2 yt = y t - 2 yt- 1 + yt- 2 = yt - 2 By t + B 2 yt = ( 1 - B) 2 yt ARMA( p, q) zt = d y t = ( 1 - B) d y t ( B) yt = ( B) et ARIMA ( p, d, q) ( B) (1 - B) d y t = ( B) et ( B) d y t = ( B) et, ARIMA( 0, 1, 0) (1 - B) yt = et ARIMA(0, d, 0) ( 1 - B) d y t = et ARIMA(1,1,1 ) ( B) (1 - B) yt = ( B) et (1 - B) ( 1-1 B) yt = ( 1-1 B) et AR( 1) MA ( 1) yt () ARIMA( p, d, q) ( P, D, Q) s = ( ) y t- 1-1 y t- 2 + et - 1 et- 1, ARMA,,, ARMA, ARIMA( p, d, q) ( P, D, Q) s ( P, D, Q) s, s, P D Q (1,1,2) 12

214 208-2, 12 12, 12 ARIMA( p, d, q) ( P, D, Q) s,4,, Y t 4 y t = yt - yt - 4 = ( 1 - B 4 ) yt 12, 1 2 yt = yt - yt - 12 = ( 1 - B 12 ) yt s D W t, wt = s D y t w t = ( 1 - B s ) D y t t > Ds 4, ARIMA( p, d, q) ( P, D, Q) s ARIMA(1,1,1 ) ( 1, 1, 1) 4 (1-1 B) ( 1-1 B) 4 ( 1 - B) (1 - B 4 ) y t = (1-1 B) ( 1-1 B 4 ) et AR( 1) AR (1 ) MA (1 ) MA( 1) : yt = ( ) yt ( ) y t ( ( ) yt y t ( yt et - 1 et et ) yt et- 5 1 ) yt - s, 1, 1, 1, 1,, ARIMA( p, d, q) ( P, D, Q) s p ( B) p ( B s ) ( 1 - B) d ( 1 - B s ) D y t = q ( B) Q ( B s ) et p ( B) p ( B s ) d D s y t = q ( B) Q ( B s ) et p ( B s ) = 1-1 B s - 2 B 2 s - - p B p s, P Q ( B s ) = 1-1 B s - 2 B 2 s - - p B Qs, Q p ( B), q ( B), p ( B s ), Q ( B s ) ARMA, ARMA,,, ARMA,

215 209 ARIMA( p, d, q) ( P, D, Q) s,, -, AR, MA, ARMA, ARIMA : :,, d, D, p, q, P, Q : 1, 2, p 1, 2 q 1, 2, p 1, 2 Q : et ), ( ) (, k = 2(k = 3),, 0, ;,, k = 12, 24, 36,48 0,,, 0, (),,, d, D d, d D,, D, d, D 0, 1 2, d,, D () ( ) p, q, p,, p q

216 210 -,, q,,, 0, 0, 0,,,, ARMA,,,,, : p,, ;,, p ;, p q,, 0 ;, k = q0, q = q0 P, Q ( ), p, q, k = 12(4),24 (8),, P,, Q0,, P, Q,, P, Q 0 12,, () ( )

217 211.AR AR ( p) yt yt- k, k = 1,2,, p yt yt- k = 1 yt yt p y t- p + e1 = 1 y t- 1 yt- k + 2 yt- 2 yt - k + + p y t - p yt - k + e1 yt - k E( yt yt- k ) = 1 E( yt- 1 y t- k ) + 2 E( yt- 2 y t- k ) + + p E ( yt - py t- k ) + E( e1 yt - k ) et, yt- k, E( yt- ket ) 0, k yt, : k = 1 k k p k - p 0, rk = 1 rk rk p rk - p k = 1, 2,, p, r1 r2 rp = r1 + + p rp - 1 = 1 r p rp - 2 = 1 rp rp p Yule - Walker, 1 2 = p, AR( 1) 1 = r1 AR (2 ), 1 r1 rp - 1 r1 1 rp - 2 rp - 1 rp r1 r1 1 1 = 2.MA( q) MA( q) - 1 r1 r2 r1 (1 - r2 ) 1 - r1 2 = r2 - r r r1 r2 rp

218 212 - y t et - 1 et- 1-2 et q et- q t - k, y t- k = et- k - 1 et- k et- k q et - k - q, yt yt- k = ( et - 1 et- 1-2 et q et- q ) ( et- k - 1 et - k et- k q et - k - q ) E( y t yt - k ) = E ( et - 1 et et q et - q ) ( et - k - 1 et- k et - k q et- k - q ) k = 0,,, k = k, rk 0 = = - k E( et- k et - k ) + 1 k+ 1 E( et- k - 1 et- k - 1 ) + 2 k+ 2 E( et - k - 2 et- k - 2 ) + + q - k q E( et- q et - q ) = ( - k + 1 k k q - k q ) 2 e 2 q 2 e, MA( 1), q = 1, rk - k + 1 k k q - k q q r1 = r r1 = 0, 1 = r2 1 2 r1 1 < 1, 1, MA( 1) MA( 2), q = 2, MA( 3), q = 3, r1 = r2 =

219 r = r2 = r3 = MA,,,, (50),,, 0 3.AR MA AR MR,, ARM R,,, ( ), ARMA, AR,,, MA ARMA,,,,,,,,,,,, :, () E - Views ARMA,,,,,,, d = 2 213

220 ,,, p = 1, q = 1, AR MA( 1, 2, 1) 9.18 (1-1 B) 2 y t = (1-1 ) Bet, Genr,, : dy1 = y - y( - 1), dy2 = dy1 - dy1( - 1) dy1 dy QuickEstimate Equation,

221 : 9.5 ARMA Dependen t Variable :DY2 Method: Least Squares Sample(adjusted ) : Included obse rvations : 46 afte r adjusting endpoints Convergence achieved after 7 iterations Backcast : 1954 Va riab le Coefficient Std. Error t - Statistic Prob. C AR (1) MA( 1) R - squared Mean dependent var Adjusted R - squared S.D. dependent var S.E. of regression A kaike info criterion Sum squared resid Sch warz criterion Log likelihood F - statistic Durbin - Watson stat P rob( F - statistic) Inverted AR Roots -.52 Inverted MA Roots -.17 : 1 = , 1 =

222 B) 2 y t = ( ) Be1, et (),,, ( ) et ( ),, (),, et,,, : 0,, ;, 0,, 9.21 ViewResidual testscorrelogram - Q - Statistics, 9.22,,,

223 ( ) 1970, m N (0, 1 ) m 2 n, n - k t - 1 rk ( e) = et et+ k e2 t nrk ( e) N (0,1 ) K = 1, 2, m, 2 2 m Q = nr k ( e) k = 1 m = nr 2 k ( e) k = 1 n ( ) ; m 1 -, 2 m 2 x 2 a ( m), Q 2 a ( m) Q 2 a ( m),,, ; Q > 2 a ( m),,, 2

224 218 - m Q = nr 2 k ( e) = = k = 1 = 0.05, , Q < 2 a ( m),,,,, AR Y t, AR, : y^t (1 ) = ^1 yt + ^2 yt ^p y t - p + 1 y^t (2 ) = ^1 y^t (1 ) + ^2 yt + + ^ p y t- p + 2 ^ p - 1 y^t ( p) = ^1 y^t ( p - 1) + ^2 y^t ( p - 2 ) + + yt ( 1) + p y t y^t ( L) = ^1 y^ t ( L - 1 ) + ^2 y^t ( L - 2 ) + + ^ p - 1 yt ( L - p + 1) + p y t ( L - p) ( L > p), L MA y, MA, : y^t + 1 ( 1) y^t+ 1 (2 ) y^t+ 1 ( q) = ^ y^t (1 ) ^ y^t (2 ) - ^q y^t ( q) ^q ^1 ^2 y t+ 1 ^q,t y^t ( q) = y^t (1) y (2) ^t yt+ 1, t + 1 y ^ ( q) t+ 1 = ^y t+ 1 ( 1 ) ^y t+ MA( 1) y (q) ^t T ^ t+ T 1 (2 ) y 1 ( q)

225 219 ^t ^ ^1 yt yt+ 1 ( 1) = ^1 y (1 ) - ^ t+ MA( 2) t+ 1 y^ ( 1) = y^ t+ 1 ( 2) ^t ^y t ^1 yt + 1 y 1 ( 1) = (1 ) - ^1 1 ^2 0 y + 1 (1 ) = ^1 y^ t ( 1) + y ^1 yt + 1 y^ t ( 1) y^ t ( 2) - ^ t + 1 (2 ) = ^2 t y^ ( 1) - ARMA ^1 yt + 1 ^2 (2 ) - ^2 y t+ 1 ^1 yt+ 1 ARMA MA,,,, d, wt, wt d, d = 1, yt L, y^ k ( L) = yk + w^ k (1 ) + w^ k ( 2) + + w^ k ( L) U L = y^ t ( L) + T V( L ) Se LL = y^t ( L) - T V ( L) Se, U L, LL, T k, Se, Se 2 = 1 = = = t = 1 L - 1 V( L ) = j 4 = j = 1 k - p - q V ( L) 1 e^2 t

226 220 - j 1 j j p j - p, - j,, 9.23,,, : 9.24

227 221,, 9.6, d = 2, y^ k ( L) = yk + w^ k (1 ) + w^ k ( 2) + + w^ k ( L), dy1 (2001) = dy1( 2000 ) + d y2^(2001) = dy1 (2002) = dy1 (2000) + dy2^( 2001 ) + dy2^(2002) = , : 9.7 dy2^ dy1^ ^y

228 , : : :, : , y,,

229 223.26, 9.27,, r = 12, 24, 26 0,,

230 , (12 ), (9.29) ( 9.30 ) : 9.29

231 225.30,, 0, k = 1,,,, d = 1, D = 1,,,, 0,,, p = 1, q = 1, k = 12,24, 36,,, ( 1), (12) (24), P = 1, Q = 1 ARIMA (1, 1, 1 ) ( 1, 1, 1) 12, (1-1 B) ( 1-1 B 12 ) 12 yt = (1-1 B) (1-1 B 12 ) et :

232 226 - Genr, : dy1 = y - y( - 1), dy2 = dy1 - dy1( - 12 )dy1, dy2 9.31: 9.31 QuickEstimate Equation, ARMA Dep enden t Variable: DY2, ncl uded obse rvations: 97 af ter adj usting endpoints Sample(adjusted) : 1995 : :03, Method: Least Squares Failure to improve SSR after 23 iterations, Backcast : 1994: :02 Variable Coefficient Std. Error t - Statistic P rob. C A R(1 ) SAR(12) MA( 1) SMA( 12 )

233 227 Va riab le Coefficien t Std Error t - Statistic P rob. R - squared Adjusted R - squared S.E. of regression Sum squared resid Log likelihood Durbin - Watson stat Mean dependent var S.D. dependent var Akaike info c riterion Sch warz criterion F - statistic Prob( F - statistic) Inverted AR Roots i i i i i i i i i i Inverted MA Roots i i i i i i i i i = : 1 = , 1 = , 1 = , 1 ( B) ( B 12 ) 12 y t = ( B) ( B 12 ) et 1. : 9.33

234 228 - ViewResidual testscorrelogram Q - Statistics, 9.14,,, m Q = nr 2 k ( e) = = k = 1 = 0.05, , 95, Q < ( 2 m),,, :, (9.35), 9.35

235 :,, 9.36,,, ( 9.10) 9.10, d = 1, D = 1, y^ k ( L) = yk + w^ k (1 ) + w^ k ( 2) + + w^ k ( L), dy1 (03.04 ) = dy1 (02.04 ) + d y2^(03.04 ) =

236 230 - dy (03.05 ) = dy1 (02.05 ) + d y2^(03.05 ) = y( 03.04) = y(03.03 ) + d y1^(03.04 ) = y( 03.05) = y(03.03 ) + d y1^(03.04 ) + dy1^ (03.05 ) = , 9.11 d y2^ d y1^ y^ ,, 9.37

237 , ;,,,,,,,, ( ),,,,,,,,, : (1 ) ; (2 ),,,, ; (3 ); (4 ), : (1 ),,, ;

238 232 2 ),,,,,, ; (3 ),,,,,,,,,, : (1 ),,, ; (2 ); (3 ), ;,,,,,,,,,, ( ),,,,,,,,,,,,,,, 73%,

239 233 57%,,,,,,,,,,,,,,,, ( ),,, : 1.,,,,,,,,,,,,,,, 2.,,,,,,,,

240 234.,,,, 4., :,,,,,,, 5.,,,, 6.,,,,,,, ( ) 1.,,,,,,,,,,,,,

241 235.,,,, 3.,,,,,,,,,,,, 4.,,, ;,,,,,,,,,,,,,, 5. : ;,,,,,,

242 236 ) :,, ( ), d1, ( ), d2 ; ( ), d2, ( ), s2, m2, r,, : = d1 + s1 = d2 + s2 = r + m + s2 : = d1 + s1 i + d2 j + s2 + ( r + m) : i ; j,,,,,,, ( AR ),,,,,,

243 237, AR,, B - J f ( t),,,, B - J,,, D.W, B - J,, B - J : y = b0 + b1 x1 + + bm x m + et e1 = 1 et p et - p - 1 ut q ut - q,,,,,,,,,,,, + ut,,,,,,,,,,,,,, :

244 238 k yi = j = 1 wj y^j i ^ wj = n 1 i = 1 e 2 j i k n 1 j = 1 i = 1 e 2 ji, k k, w j j, ^yj i j, e 2 ji j, wj, y^ ai y^bi y i, w y^ ai, ( 1 - w)y^ bi,,, y^ i = w y^ ai + ( 1 + w) y^ bi w, y^i y^ ai y^bi eai = y i - y^ ai, ebi = yi - y^ bi,, E( eai ) = 0, E( ebi ) = 0, ei = y i - y^i = yi - [ w y^ai + ( 1 - w) y^bi ] = w( yi - y^ ai ) + (1 - w) ( yi - y^ bi ) = weai + (1 - w) ebi, E( ei ) = 0, Var ( ei ) = E( e 2 i ) - [ E( ei ) ] 2 = E( e 2 i ) = MS E, MS E Var ( ei ), Var ( ei ) = Var [ weai + (1 - w) ebi ] = w 2 V ar( eai ) + (1 - w) 2 Var ( ebi ) + 2 w(1 - w) Cov( eai, ebi ) Cov( eai, ebi )eai ebi w, 0, dvar( ei ) d w w = = 2 wvar( eai ) - 2(1 - w) Var( ebi ) + 2(1-2 w) Cov( eai, ebi ) = 0 : V ar( ebi ) - Cov( eai, ebi ) Var ( eai ) + V ar( ebi ) - 2 Cov( eai, ebi ) eai ebi,

245 w Var ( ebi ) Var ( eai ) + V ar( ebi ), w Var ( ei ), Var ( ei ) Var ( eai )Va r( ebi ),, Var ( eai ) = E( e 2 ai ) - [ E( eai ) ] 2 = E( e 2 ai ) = Var ( ebi ) = E( e 2 bi ) - [ E( ebi ) ] 2 = E( e 2 bi ) = w w = n i = 1 e2 ai n i = 1 e2 ai n + i = 1 e2 bi n i = 1 e2 ai n n i = 1 e2 bi,,,, 239 n

246 240 1 t P{ t( n) > ta ( n) } = a n a =

247

248 P{ 2 ( n) > 2 ( n)} = a n a =

249

250 () n a =

251

252 246 4 F n2 P{ F( n1, n2 ) > Fa ( n1, n2 )} = a 1 %, 5%

253

254

255

256

257

258

259

260 254 5 D.W 5 % n 15 k = 2 k = 3 k = 4 k = 5 k = 6 d L d V d L d V d L d V d L d V d L d V :n ; k,

261 5 255 n k = 2 k = 3 k = 4 k = 5 k = 6 d L d V d L d V d L d V d L d V d L d V :n ; k,

262 % n k = 2 dl dv k = 3 dl dv k = 4 dl dv k = 5 dl dv k = 6 dl dv

263 5 257 n k = 2 k = 3 k = 4 k = 5 k = 6 d L d V d L d V d L d V d L d V d L d V

264 258 6 ( r z ) r : r,, z, r = 0.34 z, r = 0.3 r = 0.04 z

265 1 ] 1 :,1990 [2 ] 1 :,1986 [3 ]1 :,1994 [4 ] 1 :, 1991 [5 ] 1 :, 1989 [6 ] 1 :, 2001 [7 ] Excel 1 :,2001 [8 ] 1 :, 1998 [9 ] George E.P.Box, Gwilym M. Jenkins Time Series Analysis: Forecasting and Control Rev. Ed.Holden - Day. ( T hird Edition ) [10] S.Makridakis, S.C.W heelwright, V.E.Mcqee Forecasting: Methods and Applications, John Wily and Sons, 1983