Математическое моделирование при планировании экспериментов на трех, четырех, пяти уровнях фактора и при неодинаковом количестве уровней первого и второго фактора. Черный А.А. - 23 стр.

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23
a
1s
= f
′
o
+ f
2n
⋅
x
2n
+ f
2r
⋅
x
2r
+ f
2s
⋅
x
2s
+ f
2w
⋅
x
2w
.
После подстановки, перемножений и замены коэффициентов полу-
чаются следующие полиномы.
Для плана 3
⋅ 4 (табл. 4);
y = b
′
o
⋅
x
o
+ b
1n
⋅
x
1n
+ b
2n
⋅
x
2n
+ b
1n,2n
⋅
x
1n
⋅
x
2n
+ b
1r
⋅
x
1r
+ b
2r
⋅
x
2r
+
+ b
1n,2r
⋅
x
1n
⋅
x
2r
+ b
2n,1r
⋅
x
2n
⋅
x
1r
+ b
2r,1r
⋅
x
1r
⋅
x
2r
+ b
2s
⋅
x
2s
+
+ b
2s,1n
⋅
x
1n
⋅
x
2s
+ b
2s,1r
⋅
x
1r
⋅ x
2s
(38)
Для плана 3
⋅ 5 (см. табл. 5);
y = b
′
o
⋅
x
o
+ b
1n
⋅
x
1n
+ b
2n
⋅
x
2n
+ b
1n,2n
⋅
x
1n
⋅
x
2n
+ b
1r
⋅
x
1r
+ b
2r
⋅
x
2r
+
+ b
1n,2r
⋅
x
1n
⋅
x
2r
+ b
2n,1r
⋅
x
2n
⋅
x
1r
+ b
2r,1r
⋅
x
1r
⋅
x
2r
+ b
2s
⋅
x
2s
+
+ b
2s,1n
⋅
x
1n
⋅
x
2s
+ b
2s,1r
⋅
x
1r
⋅
x
2s
+ b
2w
⋅
x
2w
+ b
2w,1n
⋅
x
1n
⋅
x
2w
+
+ b
2w,1r
⋅
x
1r
⋅
x
2w
(39)
Для плана 4 ⋅ 5 (см. табл. 6);
y = b
′
o
⋅
x
o
+ b
1n
⋅
x
1n
+ b
2n
⋅
x
2n
+ b
1n,2n
⋅
x
1n
⋅
x
2n
+ b
1r
⋅
x
1r
+ b
2r
⋅
x
2r
+
+ b
1n,2r
⋅
x
1n
⋅
x
2r
+ b
2n,1r
⋅
x
2n
⋅
x
1r
+ b
2r,1r
⋅
x
1r
⋅
x
2r
+ b
1s
⋅
x
1s
+
+ b
2s,1n
⋅
x
1n
⋅
x
2s
+ b
1s,2n
⋅
x
2n
⋅
x
1s
+ b
1r,2s
⋅
x
1r
⋅
x
2s
+ b
2r,1s
⋅
x
2r
⋅
x
1s
+
+ b
2s1s
⋅
x
2s
⋅
x
1s
+ b
2w
⋅
x
2w
+ b
2w,1n
⋅
x
1n
⋅
x
2w
+ b
2w,1r
⋅
x
1r
⋅
x
2w
+
+ b
2w,1s
⋅
x
1s
⋅
x
2w
(40)
В уравнениях регрессии (38) - (40)
y - показатель (параметр) процесса;
x
o
= + 1; x
1n
=x
n
1
+ v
1
;
x
1r
= x
r
1
+ a
1
⋅
x
n
1
+ c
1
; x
1s
= x
s
1
+ d
1
⋅
x
r
1
+ e
1
⋅
x
n
1
+ f
1
;
x
2n
=x
n
2
+ v
2
;
x
2r
= x
r
2
+ a
2
⋅
x
n
2
+ c
2
; x
2s
= x
s
2
+ d
2
⋅
x
r
2
+ e
2
⋅
x
n
2
+ f
2
;
x
2w
= x
w
2
+ g
2
⋅
x
s
2
+ h
2
⋅
x
r
2
+ k
2
⋅
x
n
2
+ l
2
;
x
1
, x
2
-1, 2-й факторы (независимые переменные); n, r, s, w -
изменяемые числа показателей степени факторов;
v
1
, a
1
, c
1
, d
1
, e
1
, f
1
, - ко-
эффициенты ортогонации, определяемые
при четырех уровнях 1-го фактора,
m = 1, N = 4 по формулам (12) -
(17);
при трех уровнях 1-го фактора,
m = 1, N = 3 по формулам (2)-(4);
v
2
, a
2
, c
2
, d
2
, e
2
, f
2
, g
2
, h
2
, k
2
, l
2
-коэффициенты ортогонализации, определяе-
мые при пяти уровнях 2-го фактора,
m = 2, N = 5 по формулам (23)-(32);
при четырех уровнях 2-го фактора,
m = 2, N = 4 по формулам (12)-
(17);
при трех уровнях 2-го фактора,
m = 2, N = 3 по формулам (2)-(4);
b
0
′
, b
1n
, b
2n
, b
1n,2n
, b
1r
, b
2r
, b
1n,2r
, b
2n,1r
, b
1r,2r
, b
1s
, b
2s
, b
1n,2s
, b
2n,1s
, b
1r,2s
, b
2r,1s
, b
1s,2s
,
b
1w
, b
2w
, b
1n,2w
, b
2n,1w
, b
1r,2w
, b
22r,1w
, b
1s,2w
, b
2s,1w
b
1w,2w
- коэффициенты регресии. 
          a1s = f′o + f2n ⋅ x2n + f2r ⋅ x2r + f2s ⋅ x2s + f2w ⋅ x2w .
     После подстановки, перемножений и замены коэффициентов полу-
чаются следующие полиномы.
        Для плана 3 ⋅ 4 (табл. 4);
      ′
y = b o ⋅ xo + b1n ⋅ x1n + b2n ⋅ x2n + b1n,2n ⋅ x1n ⋅ x2n + b1r ⋅ x1r + b2r ⋅ x2r +
+ b1n,2r ⋅ x1n ⋅ x2r + b2n,1r ⋅ x2n ⋅ x1r + b2r,1r ⋅ x1r ⋅ x2r + b2s ⋅ x2s +
+ b2s,1n ⋅ x1n ⋅ x2s + b2s,1r ⋅ x1r ⋅ x2s                                             (38)
        Для плана 3 ⋅ 5 (см. табл. 5);
y = b′o ⋅ xo + b1n ⋅ x1n + b2n ⋅ x2n + b1n,2n ⋅ x1n ⋅ x2n + b1r ⋅ x1r + b2r ⋅ x2r +
+ b1n,2r ⋅ x1n ⋅ x2r + b2n,1r ⋅ x2n ⋅ x1r + b2r,1r ⋅ x1r ⋅ x2r + b2s ⋅ x2s +
+ b2s,1n ⋅ x1n ⋅ x2s + b2s,1r ⋅ x1r ⋅ x2s + b2w ⋅ x2w + b2w,1n ⋅ x1n ⋅ x2w +
+ b2w,1r ⋅ x1r ⋅ x2w                                                                  (39)
        Для плана 4 ⋅ 5 (см. табл. 6);
      ′
y = b o ⋅ xo + b1n ⋅ x1n + b2n ⋅ x2n + b1n,2n ⋅ x1n ⋅ x2n + b1r ⋅ x1r + b2r ⋅ x2r +
+ b1n,2r ⋅ x1n ⋅ x2r + b2n,1r ⋅ x2n ⋅ x1r + b2r,1r ⋅ x1r ⋅ x2r + b1s ⋅ x1s +
+ b2s,1n ⋅ x1n ⋅ x2s + b1s,2n ⋅ x2n ⋅ x1s + b1r,2s ⋅ x1r ⋅ x2s + b2r,1s ⋅ x2r ⋅ x1s +
+ b2s1s ⋅ x2s ⋅ x1s + b2w ⋅ x2w + b2w,1n ⋅ x1n ⋅ x2w + b2w,1r ⋅ x1r ⋅ x2w +
+ b2w,1s ⋅ x1s ⋅ x2w                                                                  (40)
В уравнениях регрессии (38) - (40) y - показатель (параметр) процесса;
            xo = + 1; x1n =xn1 + v1 ;
            x1r = xr1 + a1⋅ xn1 + c1; x1s = xs1 + d1⋅ xr1 + e1⋅ xn1 + f1;
            x2n =xn2 + v2 ;
            x2r = xr2 + a2⋅ xn2 + c2; x2s = xs2 + d2⋅ xr2 + e2⋅ xn2 + f2;
            x2w = xw2 + g2 ⋅ xs2 + h2 ⋅ xr2 + k2 ⋅ xn2 + l2;
        x1, x2 -1, 2-й факторы (независимые переменные); n, r, s, w -
изменяемые числа показателей степени факторов; v1, a1, c1, d1, e1, f1, - ко-
эффициенты ортогонации, определяемые
        при четырех уровнях 1-го фактора, m = 1, N = 4 по формулам (12) -
(17);
        при трех уровнях 1-го фактора, m = 1, N = 3 по формулам (2)-(4);
v2, a2, c2, d2, e2, f2, g2, h2, k2, l2 -коэффициенты ортогонализации, определяе-
мые при пяти уровнях 2-го фактора, m = 2, N = 5 по формулам (23)-(32);
        при четырех уровнях 2-го фактора, m = 2, N = 4 по формулам (12)-
(17);
        при трех уровнях 2-го фактора, m = 2, N = 3 по формулам (2)-(4);
  ′
b0 , b1n, b2n, b1n,2n, b1r, b2r, b1n,2r, b2n,1r, b1r,2r, b1s, b2s, b1n,2s, b2n,1s, b1r,2s, b2r,1s, b1s,2s,
b1w, b2w, b1n,2w, b2n,1w, b1r,2w, b22r,1w, b1s,2w, b2s,1w b1w,2w - коэффициенты регресии.




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