( ) xi, yi, yi , xi - . .
y = f(x), (1)
, .
, y = f(x) :
y = a0 + a1x + a2x2 +... + amxm,
a0,a1 ,a2,... am , .
y = f(x) = a0 + a1x. (2)
.
y ( ) . , .
(.. ) () [3-5].
, , y .
, y1, y2, , yn , f(x1), f(x2), , f(xn) - , , , :
. (3)
, .
( 3).
3 -
n () n (x1, y1); (x2, y2);....; (xn, yn). , (2).
a0 a1 , (x1, y1); (x2, y2);....; (xn, yn), XOY, y = a0 + a1x.
a0 a1, (2) :
, (4)
n - , .. y x.
|
|
F a0 a1 , ..
; .
, :
(5)
:
; ; ,
:
. (6)
. 1- a0:
.
(6):
.
:
, . (7) - (8)
. y x y=a0 + a1x ( ). ( 1).
1
xi | yi | xi2 | xi yi | |
1.0 | 1.25 | 1.00 | 1.250 | |
1.5 | 1.40 | 2.25 | 2.100 | |
3.0 | 1.50 | 9.00 | 4.500 | |
4.5 | 1.75 | 20.25 | 7.875 | |
5.0 | 2.25 | 25.00 | 11.250 | |
S | 15.0 | 8.15 | 57.50 | 26.975 |
(7)-(8):
; .
, :
y = 1.024 + 0.202x.
. . y .
:
, (9)
, - x y,
; . (10)
+1 -1 a1.
. , n=5; = 1.63; = 3.0. ( 2).
2
xi | yi | xi - | yi - | (xi - ) (yi - ) | (xi - )2 | (yi - )2 | |
1.0 | 1.25 | -2.0 | -0.38 | 0.760 | 4.00 | 0.1444 | |
1.5 | 1.40 | -1.5 | -0.23 | 0.345 | 2.25 | 0.0529 | |
3.0 | 1.50 | -0.13 | 0.0169 | ||||
4.5 | 1.75 | 1.5 | 0.12 | 0.180 | 2.25 | 0.0144 | |
5.0 | 2.25 | 0.62 | 1.240 | 4.00 | 1.240 | ||
S | 2.525 | 12.50 | 0.613 |
, ,
.. .
- 4.
3.
, , . , .
. ( ):
|
|
. (11)
:
. (12)
y' = ln y, x' = x, (13)
a0 = ln a, a1 = ln b, (14)
(2)
y' = a0 + a1x',
3
n | |
xi, yi | x y. [1..n]. |
yri | y, - . [1..n]. |
s1 | |
s2 | |
s3 | |
s4 | |
a0, a1 | |
xsr, ysr | x y |
s5 | |
s6 | |
s7 | |
r |
(7) (8). (11) :
a = exp(a0), b = exp(a1). (15)
, , :
, (16)
- , .
, s .
4 - -
.
, , .
, , .
20..30 . , .
[1].
, , . .
, . , .
, . .
y=f(x1, x2,..., xn).
:
1) , .. ;
2) , .. .
.
, , .. , .
, , .
, , -.
|
|
, , , . () .
n
n = qk, (17)
q - ;
k - .
2k, .
( ):
, (18)
y - ();
b0 - ;
k - ;
bi, bj, bij - ;
xi, xj - ;
c - k ;
i, j - .
, :
1) ;
2) - ;
3) ;
4) ;
5) ;
6) .
, .
, , ( 5).
, .
. .
5 -
, . , - .
, , - .
Xi: - Xi, - Xi, - Xi0.
:
DXi = Xi0 - Xi = Xi - Xi0. (19)
.
:
, (20)
xi - ;
Xi - ;
Xi0 - ;
DXi - .
,
.
. , ( 6), ( 6). , , . k k- k- .
6 - 22 :
|
|
- ; -
, , ( 2).
:
1) ;
2) , .
, (-1; +1) . 0 - . , .
, , 0, , - , - , - .. 4 , , , .
4 - 22 24
x0 | x1 | x2 | x3 | x4 | ||||
24 | 23 | 22 | +1 | -1 | -1 | -1 | -1 | |
+1 | +1 | -1 | -1 | -1 | ||||
+1 | -1 | +1 | -1 | -1 | ||||
+1 | +1 | +1 | -1 | -1 | ||||
+1 | -1 | -1 | +1 | -1 | ||||
+1 | +1 | -1 | +1 | -1 | ||||
+1 | -1 | +1 | +1 | -1 | ||||
+1 | +1 | +1 | +1 | -1 | ||||
+1 | -1 | -1 | -1 | +1 | ||||
+1 | +1 | -1 | -1 | +1 | ||||
+1 | -1 | +1 | -1 | +1 | ||||
+1 | +1 | +1 | -1 | +1 | ||||
+1 | -1 | -1 | +1 | +1 | ||||
+1 | +1 | -1 | +1 | +1 | ||||
+1 | -1 | +1 | +1 | +1 | ||||
+1 | +1 | +1 | +1 | +1 |
.
1. (18).
:
, i = 0, 1, , n, (21)
, i, j = 1, 2, , n, i ¹ j, (22)
n - ;
xiu - i u;
yu - u;
u - .
2. .
: . , .. , . 3-4 :
, (23)
n0 - ;
- ;
- .
3. .
, , - . .
:
, (24)
.. , n.
:
, (25)
- i - ;
- i - .
, (26)
t - , a f = n0 - 1 ( 5).
(26) , i - . - (26) , ( ) (18). 0.05.
5 a = 0.05
f | ||||||||||||||||||
¥ | ||||||||||||||||||
12.7 | 4.30 | 3.18 | 2.78 | 2.57 | 2.45 | 2.37 | 2.31 | 2.26 | 2.23 | 2.20 | 2.18 | 2.16 | 2.15 | 2.13 | 2.09 | 2.04 | 2.00 | 1.96 |
4. .
, .. .
|
|
, (27)
z - ;
- u - , .
f = n - z.
:
. (28)
,
F £ F, (29)
F - a f f ( 6).
6 - F(f; f) a = 0.05
f | f | ||||||||||||||
¥ | |||||||||||||||
161.4 | 199.5 | 215.7 | 224.6 | 230.2 | 233.9 | 236.8 | 238.9 | 240.5 | 241.8 | 248.0 | 250.1 | 252.2 | 253.3 | 254.3 | |
18.5 | 19.0 | 19.2 | 19.2 | 19.3 | 19.3 | 19.4 | 19.4 | 19.4 | 19.3 | 19.4 | 19.5 | 19.5 | 19.4 | 19.5 | |
10.1 | 9.6 | 9.3 | 9.1 | 9.0 | 8.9 | 8.9 | 8.8 | 8.8 | 8.7 | 8.7 | 8.6 | 8.6 | 8.5 | 8.5 | |
7.7 | 6.9 | 6.6 | 6.4 | 6.3 | 6.2 | 6.1 | 6.0 | 5.9 | 5.9 | 5.8 | 5.7 | 5.7 | 5.6 | 5.6 | |
6.6 | 5.8 | 5.4 | 5.2 | 5.1 | 4.95 | 4.9 | 4.8 | 4.8 | 4.7 | 4.6 | 4.5 | 4.4 | 4.4 | 4.3 | |
5.9 | 5.1 | 4.8 | 4.5 | 4.4 | 4.3 | 4.2 | 4.1 | 4.1 | 4.0 | 3.9 | 3.8 | 3.7 | 3.7 | 3.6 | |
5.6 | 4.7 | 4.3 | 4.1 | 4.0 | 3.9 | 3.8 | 3.7 | 3.7 | 3.6 | 3.4 | 3.4 | 3.3 | 3.2 | 3.2 | |
5.3 | 4.5 | 4.1 | 3.8 | 3.7 | 3.6 | 3.5 | 3.4 | 3.4 | 3.3 | 3.2 | 3.1 | 3.0 | 2.9 | 2.9 | |
5.1 | 4.3 | 3.9 | 3.6 | 3.5 | 3.4 | 3.3 | 3.2 | 3.2 | 3.1 | 2.9 | 2.8 | 2.8 | 2.7 | 2.7 | |
4.9 | 4.1 | 3.7 | 3.5 | 3.3 | 3.2 | 3.1 | 3.1 | 2.0 | 2.9 | 2.8 | 2.7 | 2.6 | 2.6 | 2.5 | |
4.5 | 3.7 | 3.3 | 3.1 | 2.9 | 2.8 | 2.7 | 2.6 | 2.6 | 2.5 | 2.3 | 2.2 | 2.16 | 2.1 | 2.0 | |
4.4 | 3.5 | 3.1 | 2.9 | 2.7 | 2.6 | 2.5 | 2.4 | 2.4 | 2.3 | 2.1 | 2.0 | 1.9 | 1.9 | 1.8 | |
4.2 | 3.3 | 2.9 | 2.7 | 2.5 | 2.4 | 2.3 | 2.3 | 2.2 | 2.2 | 1.9 | 1.8 | 1.7 | 1.7 | 1.6 | |
4.0 | 3.2 | 2.8 | 2.5 | 2.4 | 2.3 | 2.2 | 2.1 | 2.0 | 1.9 | 1.7 | 1.65 | 1.5 | 1.5 | 1.4 | |
3.9 | 3.1 | 2.7 | 2.45 | 2.3 | 2.2 | 2.1 | 2.0 | 1.9 | 1.9 | 1.7 | 1.6 | 1.4 | 1.3 | 1.25 | |
¥ | 3.8 | 3.0 | 2.6 | 2.4 | 2.2 | 2.1 | 2.0 | 1.9 | 1.8 | 1.8 | 1.6 | 1.5 | 1.3 | 1.2 | 1.0 |
.
, bi, bij y, .. xi (i=0k) - . bi, bij y. .
, . (20). () . .
. 360 1440 =4.4 / ( 7).
, , , , .. .
t , . : a w.
7
- , .
7, - 8.
7
a (X1), . | w (X2), / | |
+1,0 | 2,92 | |
-1,0 | 1,46 | |
0,0 | 2,19 | |
1,0 | 0,73 |
8 - -
x0 | x1 | x2 | a, . | w, / | t, . | |
+1 | -1 | -1 | -1,0 | 1,46 | 3,70 | |
+1 | +1 | -1 | +1,0 | 1,46 | 1,95 | |
+1 | -1 | +1 | -1,0 | 2,92 | 2,55 | |
+1 | +1 | +1 | +1,0 | 2,92 | 1,20 | |
+1 | 0,0 | 2,19 | 2,52 | |||
+1 | 0,0 | 2,19 | 2,40 | |||
+1 | 0,0 | 2,19 | 2,38 |
( ):
y = b0 + b1x1 + b2x2 + b12x1x2,
y t;
x1 a;
x2 - w.
(21)(22):
b0 = (3,7 + 1,95 + 2,55 + 1,2) / 4 = 2,35;
b1 = (-3,7 + 1,95 - 2,55 + 1,2) / 4 = - 0,775;
b2 = (-3,7 - 1,95 + 2,55 + 1,2) / 4 = - 0,475;
b12 = (3,7 - 1,95 - 2,55 + 1,2) / 4 = 0,1.
(n0=3) , 5-7:
,
:
.
:
, .
:
; ;
; .
, 5 a=0.05 f = n0 1=3-1=2:
t = 4.303.
b12 .
:
y = 2.35 0.775 x1 - 0.475 x2.
. 8:
;
;
;
.
(27):
.
.
6 a=0.05 f=4-3=1, f=2 - F=18.5.
(29) , .. .
, :
,
:
.
, , :
.
8 - 22 . 9.
-
8 - - 22
8 ()
8 ()
9 -
xij | , . - [1..7, 0..2], 7- , 2 - |
yi | , . - [1..7]. |
bj | . - [0..3]. 4 6 . |
trj | . - [0..3]. |
yri | . - [1..4]. |
ys0 | |
Sv2 | |
Sa2 | |
Sb2 | |
.9 | |
Fr | |
t | |
F | |
i | |
j | |
s | |
p | |
m | |
z |