, .
(. . 5).
1. ? , ?
2. ? . .
3. , ? ?
4. ?
5. ?
6. ?
7. ?
8. , ?
9. ?
10. ?
11. ?
12. ?
13. ?
14. , ?
5
: , .
: 1. ;
2. , , .
, . , (). - 23 23-1 . 5.1.
. 5.1. : ; ,
23 ( ), 23-1 , (. 5.1 ). , , 23-1 (. 5.1 ).
|
|
, , .
, , () :
y = b0 + b 1 x 1 + b2 x 2 ++b x + b12 x 1 x 2 + b13 x 1 x 3+ + b 1, x 1 x
. , , .
: ½; ¼; 1/8; 1/16 .. , . 24 5.1.
5.1. 24
Yj | |||||||||
0 | 1 | 2 | 3 | 4 | |||||
+ | - | - | - | - | 1 | 1/8 | 1/4 | 1/2 | |
+ | + | - | - | - | 2 | ||||
+ | - | + | - | - | 3 | 1/8 | |||
+ | + | + | - | - | 4 | ||||
+ | - | - | + | - | 5 | 1/8 | 1/4 | ||
+ | + | - | + | - | 6 | ||||
+ | - | + | + | - | 7 | 1/8 | |||
+ | + | + | + | - | 8 | ||||
+ | - | - | - | + | 9 | 1/8 | 1/4 | 1/2 | |
+ | + | - | - | + | 10 | ||||
+ | - | + | - | + | 11 | 1/8 | |||
+ | + | + | - | + | 12 | ||||
+ | - | - | + | + | 13 | 1/8 | 1/4 | ||
+ | + | - | + | + | 14 | ||||
+ | - | + | + | + | 15 | 1/8 | |||
+ | + | + | + | + | 16 |
24-1 24, 25. . , p , 2 k-p, , , k-p .
, =1 , , 23. 23-1. =2 -. , 25 32 , -, 25-2. =3 26-3, 64 8. 5.2.
5.2
1/2 23 | 231 | |||
1/2 24 | 241 | |||
1/4 25 | 252 | |||
1/8 26 | 263 | |||
1/16 27 | 274 | |||
1/2 25 | 251 | |||
1/4 26 | 262 | |||
1/8 27 | 273 | |||
1/16 28 | 284 | |||
1/32- 29 | 295 | |||
1/64- 210 | 2106 | |||
1/128- 211 | 2117 | |||
1/256- 212 | 2128 | |||
1/512- 213 | 2139 | |||
1/1024- 214 | 21410 | |||
1/2048- 215 | 21511 |
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|
, (2 k-p) (k+1), .. :
k+1 ≤ 2 k-p (5.1)
. , , , . .
() . 2 k-p .
23-1 23, k =3 ( ), =1 ( , ), N=4 ( ). 23-1 (. 5.3).
5.3. 23-1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
|
x 1 x 2 , 22, x 3 = x 1 x 2
:
y = b 0 + b 1x1 + b 2x2 + b 3x3 + b 12x1x2 + b 13x1x3+ b 23x2x3+ b 123 x1x2 x3
. , : 1 8; 2 7; 3 6; 4 5. , , :
(5.2)
b1 + b23, b2 + b13, b3 + b12 . (8) (4). , , , .. .
, b 123 =0,
(5.3)
, ,
(5.4)
, , , , .
23-1 : 3 (+ x 1 x 2) ( x 1 x 2). 23-1 (. 5.4).
5.4. 23-1 22
x 1 | x 2 | x 3 | x 1 x 2 x 3 | |
+ | + | + | + | |
+ | + | |||
+ | + | |||
+ | + |
x 1 | x 2 | x 3 | x 1 x 2 x 3 | |
+ | + | |||
+ | + | |||
+ | + |
|
|
: +1= x 1 x 2 x 3, : -1= x 1 x 2 x 3.
, (+1) (1), .
. , , , , .
, +1= x 1 x 2 x 3, x 1 x 1= x 12 x 2 x 3= x 2 x 3, x 12 = 1. x 2 x 2= x 1 x 22 x 3= x 1 x 3, x 3 x 3= x 1 x 2 x 32= x 1 x 2.
,
b1=β1 + β 23, b2= β 2 + β 13, b3= β 3 + β 12.
, , , .
, , III ( ). : .
, , . 24-1 x 4= x 1 x 2 x 3 ( 1= x 1 x 2 x 3 x 4), , 4; x 4= x 1 x 2 1 = x 1 x 2 x 4, 3.
.
, , ( ). , , .
24-1 x 4:
1. x 4= x 1 x 2, 2. x 4= - x 1 x 2, 3. x 4= x 2 x 3, 4. x 4= - x 2 x 3,
5. x 4= x 1 x 3, 6. x 4= - x 1 x 3, 7. x 4= x 1 x 2 x 3, 8. x 4= - x 1 x 2 x 3.
. , 16 , 78 . 7 8 .
. , .
, , . , .
, , , , IV ( ). . , .
, 24-1 x 4= x 1 x 2 x 3. : 1= x 1 x 2 x 3 x 4.
:
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|
x 0 = x 1 x 2 x 3 x 4 b0= β 0 + β 1234
x 1 = x 2 x 3 x 4 b1= β 1 + β 234
x 2 = x 1 x 3 x 4 b2= β 2 + β 134
x 3 = x 1 x 2 x 4 b3= β 3 + β 124
x 4 = x 1 x 2 x 3 b4= β 4 + β 123
x 1 x 2 = x 3 x 4 b12= β 12 + β 34
x 1 x 3 = x 2 x 4 b13= β 13 + β 24
x 1 x 4 = x 3 x 2 b14= β 14 + β 32
, - , . b1,b2, b3, b4, β 234, β 134, β 124, β 123 .
:
y = b0 + b 1x1 + b2x2 + b3x3 + b4x4 + b12x1x2+ b13x1x3+ b14 x1 x4,
, .
25-1 :
1. 5 . III. , ;
2. 5 . IV, ;
3. x 5 = x 1 x 2 x 3 x 4 x 5 =- x 1 x 2 x 3 x 4.
+1= x 1 x 2 x 3 x 4 x 5 -1= x 1 x 2 x 3 x 4 x 5. V .
26-1 . 26-1 32 , 26-2 26-3 16 8 . .
, .
. , . : , . , , .
.
1/4- 25-2. 32 , 8, . . 25-2. , 4 , 5 .
x 4= x 1 x 2= x 1 x 3= x 1 x 4= x 1 x 5= x 2 x 3= x 2 x 4= x 2 x 5= x 3 x 4= x 3 x 5= x 4 x 5;
x 5= x 1 x 2 x 3= x 2 x 3 x 4= x 3 x 4 x 5= x 1 x 3 x 4= x 1 x 4 x 5= x 1 x 3 x 5= x 2 x 4 x 5 ..
, x 4= x 1 x 3 x 5= x 1 x 2 x 3. 1= x 1 x 3 x 4 1= x 1 x 2 x 3 x 5.
, , 1= x 2 x 4 x 5.
, :
1= x 1 x 3 x 4= x 1 x 2 x 3 x 5= x 2 x 4 x 5
1, 2, 3, 4, 5, 1 2, 1 5.
x 1 = x 3 x 4= x 2 x 3 x 5= x 1 x 2 x 4 x 5 b1= β 1 + β 34 + β 235 + β 1245
x 2 = x 1 x 2 x 3 x 4= x 1 x 3 x 5= x 4 x 5 b2= β 2 + β 1234 + β 135 + β 45
x 3 = x 1 x 4= x 1 x 2 x 5= x 2 x 3 x 4 x 5 b3= β 3 + β 14 + β 125 + β 2345
x 4 = x 1 x 3= x 1 x 2 x 3 x 4 x 5= x 2 x 5 b4= β 4 + β 13 + β 12345 + β 25
x 5 = x 1 x 3 x 4 x 5= x 1 x 2 x 3= x 2 x 4 b5= β 5 + β 1345 + β 123 + β 24
x 1 x 2 = x 2 x 3 x 4= x 3 x 5= x 1 x 4 x 5 b12= β 12 + β 234 + β 35 + β 145
x 1 x 5= x 3 x 4 x 5= x 2 x 3= x 1 x 2 x 4 b15= β 15 + β 345 + β 23 + β 124
, , . , , :
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b12 = β 12 + β 234 + β 35 + β 145 b15= β 15 + β 345 + β 23 + β 124,
, , , . , 1/4-.
. , - .
, -. . -, , .
. 1/8- 26-3 - , , :
1. x 4 = x 1 x 2, x 5 = x 1 x 3, x 6 = x 2 x 3;
2. x 4 = x 1 x 3, x 5 = x 2 x 3, x 6 = x 1 x 2 x 3;
3. x 4 = x 1 x 2, x 5 = x 2 x 3, x 6 = x 1 x 2 x 3;
4. x 4 = x 1 x 2, x 5 = x 1 x 3, x 6 = x 1 x 2 x 3;
. 24 1/8-. , .
, . , 1 2, , 1 3 , 2 3 .
, , , 4, 5, 6 4. 4 :
x 4 = x 1 x 2 x 3, x 5 = x 1 x 2, x 6 = x 1 x 3.
:
1= x 1 x 2 x 3 x 4, 1= x 1 x 2 x 5, 1= x 1 x 3 x 6.
(, x i2 =1), : 1= x 3 x 4 x 5, 1= x 2 x 4 x 6, 1= x 2 x 3 x 5 x 6.
1= x 1 x 4 x 5 x 6.
1/8-, :
1= x 1 x 2 x 3 x 4= x 1 x 2 x 5= x 1 x 3 x 6= x 3 x 4 x 5= x 2 x 4 x 6= x 2 x 3 x 5 x 6= x 1 x 4 x 5 x 6.
( ):
x 1 = x 2 x 5= x 3 x 6= x 2 x 3 x 4= x 4 x 5 x 6 b1= β 1 + β 25 + β 36 + β 234+ β 456;
x 2 = x 1 x 5= x 4 x 6= x 1 x 3 x 4= x 3 x 5 x 6 b2= β 2 + β 15 + β 46 + β 134+ β 356;
x 3 = x 1 x 6= x 4 x 5= x 1 x 2 x 4= x 2 x 5 x 6 b3= β 3 + β 16 + β 45 + β 124+ β 256;
x 4 = x 3 x 5= x 2 x 6= x 1 x 2 x 3= x 1 x 5 x 6 b4= β 4 + β 35 + β 26 + β 123+ β 156;
x 5 = x 1 x 2= x 3 x 4= x 2 x 3 x 6= x 1 x 4 x 6 b5= β 5 + β 12 + β 34 + β 236+ β 146;
x 6 = x 1 x 3= x 2 x 4= x 2 x 3 x 5= x 1 x 4 x 5 b6= β 6 + β 13 + β 24 + β 235+ β 145.
:
y = b0 + b 1x1 + b2x2 + b3x3 + b4x4 + b5x5+ b6x6,
.
1/16- 27-4. 1/16 27-4 8 128.
:
x 4 = x 1 x 2, x 5 = x 1 x 3, x 6 = x 2 x 3, x 7 = x 1 x 2 x 3.
:
1= x 1 x 2 x 4, 1= x 1 x 3 x 5, 1 = x 2 x 3 x 6, 1= x 1 x 2 x 3 x 7.
, :
1= x 1 x 2 x 4= x 1 x 3 x 5 = x 2 x 3 x 6= x 1 x 2 x 3 x 7= x 2 x 3 x 4 x 5= x 1 x 3 x 4 x 6 = x 3 x 4 x 7= x 1 x 2 x 5 x 6= =x 2 x 5 x 7= x 1 x 6 x 7 = x 4 x 5 x 6= x 1 x 4 x 5 x 7 =x 2 x 4 x 6 x 7= x 3 x 5 x 6 x 7 = x 1 x 2 x 3 x 4 x 5 x 6 x 7.
, , , :
x 1 = x 2 x 4= x 3 x 5= x 6 x 7 b1= β 1 + β 24 + β 35 + β 67;
x 2 = x 1 x 4= x 3 x 6= x 5 x 7 b2= β 2 + β 14 + β 36 + β 57;
x 3 = x 1 x 5= x 2 x 6= x 4 x 7 b3= β 3 + β 15 + β 26 + β 47;
x 4 = x 1 x 2= x 5 x 6= x 3 x 7 b4= β 4 + β 12 + β 56 + β 37;
x 5 = x 1 x 3= x 4 x 6= x 2 x 7 b5= β 5 + β 13 + β 46 + β 27;
x 6 = x 2 x 3= x 4 x 5= x 1 x 7 b6= β 6 + β 23 + β 45 + β 17;
x 7 = x 3 x 4= x 2 x 5= x 1 x 6 b7= β 7 + β 34 + β 25 + β 16.
, . , . , . :
x 4 = - x 1 x 2, x 5 =- x 1 x 3, x 6 = - x 2 x 3, x 7 =- x 1 x 2 x 3.
, , , . , . , , , .
. . :
y = b 0 + b 1x1 + b 2x2 + b 3x3 + b 4x4 + b 5x5+ b 6x6+ b 7 x7,
.
9 15 16 . 16 31 32 , 32 63 64 . . (. 5.2). . . .
, , , , , . , , .
1. , , 4, , .
2. .
3. . , .
4. .
5. .
6. .
7. ( ).
. 4 . , x1x2 x1x3 . : - , 25-2.
. 5.5. x1, x2, x3 , 4.4 (. . 4), x4 x5 x4 = x1x2 x5 = x1x3 (. 5.6).
i = (xi - xoi) / dxi, i = 1, 2, , n; dxi i. i . xi max xi min, . dxi.
5.5.
1 = , , | 2 = τ, , | 3=, , 10-2 | 4=, , % | 5=, , /2 | |
10 | |||||
dxi | |||||
xi max | |||||
xi min |
, i min (1), Xi max (+1).
: =(13030) , τ= (3020) , =(52) 10-2 , = (4030) %, =(2010) /2.
1= x1x2x4 1= x1x3x5, 1= x2x3x4x5, 1 = x1x2x4 = x1x3x5 = x2x3x4x5.
:
x 1 = x 2 x 4= x 3 x 5= x 1 x 2 x 3 x 4 x 5 b1= β 1 + β 24 + β 35 + β 12345= β 1 + β 24 + β 35;
x 2 = x 1 x 4= x1x2x 3 x 5= x 3 x 4 x 5 b2= β 2 + β 14 + β 1235 + β 345= β 2 + β 14;
x 3 = x1x2 x3x4 = x1x5 = x2x4x5 b3= β 3 + β 1234 + β 15 + β 245= β 3 + β 15;
x 4 = x1x2 = x1x3 x4x5 = x2x3x5 b4= β 4 + β 12 + β 1345 + β 235= β 4 + β 12;
x 5 = x1x2x4 x5 = x1x3 = x2x3x4 b5= β 5 + β 1245 + β 13 + β 234= β 5 + β 13;
x 2 x3 = x1x3x4 = x1x2x5 = x4x5 b23= β 23 + β 134 + β 125 + β 45= β 23 + β 45;
x1x2x3 = x3x4 = x2x5 = x1x4x5 b123= β 123 + β 34 + β 25 + β 145= β 34 + β 25.
, β 12345; β 1235; β 345; β1234; β245; β 1345; β 1245; β 235; β 234; β 134; β 125; β 123; β 145, . b1,b2, b3, b4.
:
y = b0 + b 1x1 + b2x2 + b3x3 + b4x4 + b5x5+ b23x2x3+ b123x1x2x3,(5.5)
, .
32 8. 25-2, 1/4 23, 5.6.
5.6. 25-2.
( ), N | 1 | 2 | 3 | 4 (12) | 5 (13) | 23 | 123 | 11 | 12 | 13 | ỹ | -, S2 | G | ΣS2 |
1, (4), (2) | - | - | - | + | + | + | - | 4,5 | 4,9 | 4,2 | 4,53 | 0,123 | 0,256 | 1,454 |
2, (2), (4) | + | - | - | - | - | + | + | 5,9 | 6,3 | 6,5 | 6,23 | 0,09 | ||
3, (1), (6) | - | + | - | - | + | - | + | 5,2 | 5,7 | 5,9 | 5,6 | 0,13 | ||
4, (7), (8) | + | + | - | + | - | - | - | 12,4 | 12,1 | 12,5 | 0,21 | |||
5, (8), (1) | - | - | + | + | - | - | + | 6,2 | 6,6 | 6,9 | 6,56 | 0,123 | ||
6, (3), (7) | + | - | + | - | + | - | - | 10,4 | 10,7 | 9,8 | 10,63 | 0,373 | ||
7, (5), (3) | - | + | + | - | - | + | - | 8,1 | 8,9 | 9,0 | 8,66 | 0,232 | ||
8, (6), (5) | + | + | + | + | + | + | + | 5,6 | 5,8 | 6,4 | 5,93 | 0,173 |
: .
, :
. . ,
N- , i, j, m ();
. , , :
. , i -
.
. a=0,05 ( 4.7) ( 4.8) . 5.6.
(5.5) ( 4.15) α =0,05. (4.17) , α =0,05.
G , a=0,05 ( ) f1 f2 ( ).
f1=m 1=3-1=2; f2 =N =8, , N - , (m 1) , , .. .
G=0,816; G>Gp, .. . , , s2 - .
4.11.
:
y = 7,33 + 1,24x1 + 0,59x2 + 0,37x3 - 0,2x4 1,4x5 1,24x2x3- 1,5 x1x2 x3
S2 :
S2 =ΣS2/ 8 = 1,454/8 = 0,182.
S b:
S b 2 = S2 / N, N - ;
S b 2 =0,182/8=0,022, S b =0,15.
t. , |b|≥ S b t. t = |bi| / S b : b0 =7,33; b1 =1,24; b2 =0,59; b3 = 0,37; b4 = -0,2; b5 = -1,4; b23 =-1,24; b123 =-1,5.
=0,95, f=N(-1) = 8(3-1)=16, t=2,1190 ( 2). t0 = |b0| / S b = 7,33/0,15=48,86>2,119;
t1 = |b1| / S b = 1,24/0,15=8,26>2,119;
t2 = |b2| / S b = 0,59/0,15=3,93>2,119;
t3 = |b3| / S b = 0,37/0,15=2,46>2,119;
t4 = |b4| / S b = 0,2/0,15=1,33<2,119;
t5 = |b5| / S b = 1,4/0,15=9,33>2,119;
t23 = |b23| / S b = 1,24/0,15=8,26>2,119;
t123 = |b123| / S b = 1,5/0,15=10>2,119;
, b4 .
:
: y=7,33+1,24x1 + 0,59x2 + 0,37x3 1,4x5 (5.6)
: y=7,33+1,24x1+0,59x2+0,37x31,4x5 1,24x2x3-1,5x1x2 x3 (5.7)
. Fp = S2 / S2
:
= Σ(ỹi - ỹi )2 / (N ), N ; .
ỹi .
y1 = 7,33 + 1,24(-1) + 0,59(-1) +0,37(-1)-1,4(+1)-1,24 (+1)-1,5(-1) = 3,99
y2 = 7,33 + 1,24(+1) + 0,59(-1) +0,37(-1)-1,4(-1)-1,24 (+1)-1,5(+1) = 6,27
y3 = 7,33 + 1,24(-1) + 0,59(+1) +0,37(-1)-1,4(+1)-1,24 (-1)-1,5(+1) = 4,65
y4 = 7,33 + 1,24(+1) + 0,59(+1) +0,37(-1)-1,4(-1)-1,24 (-1)-1,5(-1) = 12,93
y5 = 7,33 + 1,24(-1) + 0,59(-1) +0,37(+1)-1,4(-1)-1,24 (-1)-1,5(+1) = 7,01
y6 = 7,33 + 1,24(+1) + 0,59(-1) +0,37(+1)-1,4(+1)-1,24 (-1)-1,5(-1) = 9,69
y7 = 7,33 + 1,24(-1) + 0,59(+1) +0,37(+1)-1,4(-1)-1,24 (+1)-1,5(-1) = 8,71
y8