(2.3). , ei = yi , , i = 1,, n. (2.3) e = (e 1,, en). , b 0 ( 0), (, ) 0,
å i = 0 (3.1)
S / b 0 = 0 b 0 = b 0, b 1 = b 1 (. . 9). , , , .
3.1. . , , i = 1, 2, , n y 1,, yn. ,
= , i = 1, 2, , n; =( ++ )/ n,
= . (3.2)
ij, (2.10)
= / n = ( ( + b 1(xi )))/ n = + b 1 b 1 = .
(3.2) 0 e:
S ei = S (yi ) = n n = 0.
(, , S 1 n).
3.2. .
S(yi )2 = S(yi )2 + S( )2. (3.3)
. ( .) (3.3) .
S(yi )2 = S(yi + )2 = S(yi )2 + S( )2 +
+ 2S( ) ( ).
, 0. (2.10),
= b 1(xi ), yi = yi b 1(xi ).
S b 1(xi )((yi ) b 1(xi )) = b 1(Sx y b 1 Sx x) = 0
( (2.9)). г (3.3) .
,
S( )2 = S(b 1(xi ))2 = b 12 Sx x = b 1 Sx y. (3.4)
, (3.3) . ; ; , .
3.3. .
R 2 = . (3.5)
(3.3)
R 2 £ 1. (3.6)
, R 2 , . , 100. R 2 . R 2 . R 2 1, (2.9) .
3.2.1. R 2 R x y .
, x, h
|
|
rxh = Cov(x, h) / (Dx × Dh)1 / 2,
Cov(x, h) = xh x h, D .
( ) x h
Rxh = , (3.7)
(xi, hi), i = 1,..., n (x, h) n , , 1 n.
Rxy , , x y y ŷ .
= sign (b 1) × Rxy (3.8)
sign x =
R 2 = (Rxy)2, (3.9)
R 2 = ()2 (3.10)
ij, (3.4)
= sign (b 1)× Rxy (3.11)
,
R 2 = = (Rxy)2.
(3.11) (3.8) (3.10).
IJ 4.
.
4.1. , .
, , . . ε - ( ε = yi (β 0 + β 1 xi)), = 1,..., n ε = (ε 1,..., εn) , 0:
Mε = 0, = 1,..., n, (4.1)
σ2:
Dε = σ 2, = 1,..., n. (4.2)
4.1.1. . (4.1) (4.2) ( ):
M () = β 0 + β 1 , (4.3)
D () = σ 2, (4.4)
D = σ 2 ∕ n (4.5)
4.1.2. . :
1) Mb 0 = β 0, Mb 1 = β 1; (4.6)
2) Db 0 = σ 2, Db 1 = . (4.7)
3) Cov (, b 1) = 0, (4.8)
4) Cov (b 0, b 1) = . (4.9)
5) 0 . ŷ 0 ŷ 0.
D ŷ 0 = . (4.10)
, (4.6) , b 0 b 1 , , β0 β1. г (4.7) (2.3). г (4.8) b 1. г (4.9) b 0, b 1, (4.10) . , ŷ 0 , 0 .