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求助:响应曲面设计中最速上升法求法

各位:
以下是小第做的一道题,如下,请帮助计算响应曲面设计中最速上升法求法是如何做的?

请分析以下实验,实验第一轮结果如下:
X1:压力 X2:温度 X3:催化剂用量 Y:强度
300 160 20 68.3
400 160 20 50.3
300 170 20 89.1
400 170 20 58.3
300 160 24 52.8
400 160 24 50.3
300 170 24 87.5
400 170 24 61.6
350 165 22 70.1
350 165 22 62.8
350 165 22 68.4
350 165 22 61.9

做完第一轮实验后,请根据缩减后的方程式,寻找最速上升路线,已知在步骤1,2,3,4,5,6,7的结果分别为:84, 88, 92, 98, 103, 110, 89.
(假定以X1 为参考,请将您的最速上升路径使用表格显示出来)
然后在山峰最高点设计22+3个中心点的实验,(假定X1以中心点+-50,X2以中心点+-5来设计)实验中Y的数据如下:
110,118,120,135,110,109,109(实验没有随机化),分析中心点是否显著,如显著,修改实验,增加的实验实施后,其数据结果如下, 110,120,123,132.继续分析实验,判断Block是否显著?继续缩减模型,最后使用反应优化器优化模型,顾客要求是:最少120, 目标140,越大越好.请找出最佳设置.
本题要求详细写出每步的关键步骤和关键结论。

答:
把以上数据输入到Minitab ,实行3因子2水平,中心点重复4次实验:

Factorial Fit: Y:强度 versus X1:压力, X2:温度, X3:催化剂用量

Estimated Effects and Coefficients for Y:强度 (coded units)

Term Effect Coef SE Coef T P
Constant 64.775 1.436 45.12 0.000
X1:压力 -19.300 -9.650 1.436 -6.72 0.007
X2:温度 18.700 9.350 1.436 6.51 0.007
X3:催化剂用量 -3.450 -1.725 1.436 -1.20 0.316
X1:压力*X2:温度 -9.050 -4.525 1.436 -3.15 0.051
X1:压力*X3:催化剂用量 5.100 2.550 1.436 1.78 0.174
X2:温度*X3:催化剂用量 4.300 2.150 1.436 1.50 0.231
X1:压力X2:温度X3:催化剂用量 -2.650 -1.325 1.436 -0.92 0.424
Ct Pt 1.025 2.486 0.41 0.708


S = 4.06038 PRESS = *
R-Sq = 97.23% R-Sq(pred) = *% R-Sq(adj) = 89.85%


Analysis of Variance for Y:强度 (coded units)

Source DF Seq SS Adj SS Adj MS F P
Main Effects 3 1468.16 1468.16 489.388 29.68 0.010
2-Way Interactions 3 252.80 252.80 84.268 5.11 0.107
3-Way Interactions 1 14.05 14.05 14.045 0.85 0.424
Curvature 1 2.80 2.80 2.802 0.17 0.708
Residual Error 3 49.46 49.46 16.487
Pure Error 3 49.46 49.46 16.487
Total 11 1787.28


Unusual Observations for Y:强度

St
Obs StdOrder Y:强度 Fit SE Fit Residual Resid
1 1 68.3000 68.3000 4.0604 0.0000 * X
2 2 50.3000 50.3000 4.0604 -0.0000 * X
3 3 89.1000 89.1000 4.0604 0.0000 * X
4 4 58.3000 58.3000 4.0604 0.0000 * X
5 5 52.8000 52.8000 4.0604 0.0000 * X
6 6 50.3000 50.3000 4.0604 0.0000 * X
7 7 87.5000 87.5000 4.0604 0.0000 * X
8 8 61.6000 61.6000 4.0604 0.0000 * X

X denotes an observation whose X value gives it large leverage.


Estimated Coefficients for Y:强度 using data in uncoded units

Term Coef
Constant 3141.10
X1:压力 -7.3870
X2:温度 -16.9300
X3:催化剂用量 -198.300
X1:压力*X2:温度 0.0402000
X1:压力*X3:催化剂用量 0.462750
X2:温度*X3:催化剂用量 1.14250
X1:压力X2:温度X3:催化剂用量 -0.00265000
Ct Pt 1.02500


Effects Pareto for Y:强度


Alias Structure
I
X1:压力
X2:温度
X3:催化剂用量
X1:压力*X2:温度
X1:压力*X3:催化剂用量
X2:温度*X3:催化剂用量
X1:压力X2:温度X3:催化剂用量

从以上中心点Ct Pt P值 = 0.708>0.05, 表明中心点不显著,删除中心点,及删除不显著因子缩减模型重新运行。

Factorial Fit: Y:强度 versus X1:压力, X2:温度

Estimated Effects and Coefficients for Y:强度 (coded units)

Term Effect Coef SE Coef T P
Constant 65.117 1.366 47.67 0.000
X1:压力 -19.300 -9.650 1.673 -5.77 0.000
X2:温度 18.700 9.350 1.673 5.59 0.001
X1:压力*X2:温度 -9.050 -4.525 1.673 -2.70 0.027


S = 4.73170 PRESS = 496.608
R-Sq = 89.98% R-Sq(pred) = 72.21% R-Sq(adj) = 86.22%


Analysis of Variance for Y:强度 (coded units)

Source DF Seq SS Adj SS Adj MS F P
Main Effects 2 1444.36 1444.36 722.180 32.26 0.000
2-Way Interactions 1 163.80 163.80 163.805 7.32 0.027
Residual Error 8 179.11 179.11 22.389
Curvature 1 2.80 2.80 2.802 0.11 0.749
Pure Error 7 176.31 176.31 25.187
Total 11 1787.28


Unusual Observations for Y:强度

Obs StdOrder Y:强度 Fit SE Fit Residual St Resid
1 1 68.3000 60.8917 3.2034 7.4083 2.13R
5 5 52.8000 60.8917 3.2034 -8.0917 -2.32R

R denotes an observation with a large standardized residual.


Estimated Coefficients for Y:强度 using data in uncoded units

Term Coef
Constant -1221.16
X1:压力 2.79350
X2:温度 8.20500
X1:压力*X2:温度 -0.0181000


Effects Pareto for Y:强度


Alias Structure
I
X1:压力
X2:温度
X1:压力*X2:温度

方程式如下:(此为编码值)
Y:强度=65.117 – 9.65X1:压力 + 9.35X2:温度 – 4.525X1:压力*X2:温度

残差图如下:

等值线图及曲面图:


设定急剧上升轨迹如下:
线的斜率为:0.93/(-9.65) = -0.97, 以X1:压力为1,其它变量步幅大小如下表。
Steps X1:压力 X2:温度 X1:压力 X2:温度 Y:强度
origin 0 0 350 165  
A 1 -0.97 50 4.9  
origin+A 1 -0.97 300 169.9 84
origin+2A 2 -1.94 250 174.8 88
origin+3A 3 -2.91 200 179.7 92
origin+4A 4 -3.88 150 184.6 98
origin+5A 5 -4.84 100 189.5 103
origin+6A 6 -5.81 50 194.4 110
origin+7A 7 -6.78 0 199.3 89

从上表可以看出,当在第6步时为最高点,第七步强度则下降,出现拐点,现在以最高点X1:压力=50, X2:温度=194为基础,为下个实验选择中心点: X1:压力=50, X2:温度=194
在最高点设计22+3个中心点的实验, 数据如下:
StdOrder RunOrder CenterPt Blocks X1:压力 X2:温度 Y:强度
1 1 1 1 0 189 110
2 2 1 1 100 189 118
3 3 1 1 0 199 120
4 4 1 1 100 199 135
5 5 0 1 50 194 110
6 6 0 1 50 194 109
7 7 0 1 50 194 109

分析结果如下:
Factorial Fit: Y:强度 versus X1:压力, X2:温度

Estimated Effects and Coefficients for Y:强度 (coded units)

Term Effect Coef SE Coef T P
Constant 120.75 0.2887 418.29 0.000
X1:压力 11.50 5.75 0.2887 19.92 0.003
X2:温度 13.50 6.75 0.2887 23.38 0.002
X1:压力*X2:温度 3.50 1.75 0.2887 6.06 0.026
Ct Pt -11.42 0.4410 -25.89 0.001


S = 0.577350 PRESS = *
R-Sq = 99.88% R-Sq(pred) = *% R-Sq(adj) = 99.64%


Analysis of Variance for Y:强度 (coded units)

Source DF Seq SS Adj SS Adj MS F P
Main Effects 2 314.500 314.500 157.250 471.75 0.002
2-Way Interactions 1 12.250 12.250 12.250 36.75 0.026
Curvature 1 223.440 223.440 223.440 670.32 0.001
Residual Error 2 0.667 0.667 0.333
Pure Error 2 0.667 0.667 0.333
Total 6 550.857


Unusual Observations for Y:强度

St
Obs StdOrder Y:强度 Fit SE Fit Residual Resid
1 1 110.000 110.000 0.577 0.000 * X
2 2 118.000 118.000 0.577 0.000 * X
3 3 120.000 120.000 0.577 0.000 * X
4 4 135.000 135.000 0.577 0.000 * X

X denotes an observation whose X value gives it large leverage.


Estimated Coefficients for Y:强度 using data in uncoded units

Term Coef
Constant -79.0000
X1:压力 -1.24300
X2:温度 1.00000
X1:压力*X2:温度 0.00700000
Ct Pt -11.4167


Effects Pareto for Y:强度


Alias Structure
I
X1:压力
X2:温度
X1:压力*X2:温度



从以上分析结果来看,中心点P值=0.001<0.05, 此为显著。需要增加实验点。如下表:
StdOrder RunOrder CenterPt Blocks X1:压力 X2:温度 Y:强度
1 1 1 1 0 189 110
2 2 1 1 100 189 118
3 3 1 1 0 199 120
4 4 1 1 100 199 135
5 5 0 1 50 194 110
6 6 0 1 50 194 109
7 7 0 1 50 194 109
8 8 -1 2 -20.7107 194 110
9 9 -1 2 120.7107 194 120
10 10 -1 2 50 186.9289 123
11 11 -1 2 50 201.0711 132

对于以上数据分析如下:
Response Surface Regression: Y:强度 versus Block, X1:压力, X2:温度

The analysis was done using coded units.

Estimated Regression Coefficients for Y:强度

Term Coef SE Coef T P
Constant 109.583 2.029 54.003 0.000
Block -0.250 1.060 -0.236 0.825
X1:压力 4.643 1.060 4.381 0.012
X2:温度 4.966 1.060 4.686 0.009
X1:压力*X1:压力 2.583 1.368 1.888 0.132
X2:温度*X2:温度 8.833 1.368 6.457 0.003
X1:压力*X2:温度 1.750 1.499 1.168 0.308


S = 2.99733 PRESS = 565.808
R-Sq = 95.88% R-Sq(pred) = 35.09% R-Sq(adj) = 89.69%


Analysis of Variance for Y:强度

Source DF Seq SS Adj SS Adj MS F P
Blocks 1 74.029 0.500 0.500 0.06 0.825
Regression 5 761.671 761.671 152.334 16.96 0.008
Linear 2 369.731 369.731 184.865 20.58 0.008
Square 2 379.690 379.690 189.845 21.13 0.007
Interaction 1 12.250 12.250 12.250 1.36 0.308
Residual Error 4 35.936 35.936 8.984
Lack-of-Fit 2 35.269 35.269 17.635 52.90 0.019
Pure Error 2 0.667 0.667 0.333
Total 10 871.636




Estimated Regression Coefficients for Y:强度 using data in uncoded units

Term Coef
Constant 14473.9
Block -0.250000
X1:压力 -2.60848
X2:温度 -140.650
X1:压力*X1:压力 0.00103333
X2:温度*X2:温度 0.353333
X1:压力*X2:温度 0.00700000

从以上可知区组及X1:压力X1:压力,X1:压力X2:温度的P值是>0.05, 应删减模型,重新分析。

Response Surface Regression: Y:强度 versus X1:压力, X2:温度

The analysis was done using coded units.

Estimated Regression Coefficients for Y:强度

Term Coef SE Coef T P
Constant 111.882 1.507 74.232 0.000
X1:压力 4.643 1.268 3.660 0.008
X2:温度 4.966 1.268 3.915 0.006
X2:温度*X2:温度 8.162 1.443 5.656 0.001


S = 3.58783 PRESS = 266.940
R-Sq = 89.66% R-Sq(pred) = 69.37% R-Sq(adj) = 85.23%


Analysis of Variance for Y:强度

Source DF Seq SS Adj SS Adj MS F P
Regression 3 781.529 781.529 260.510 20.24 0.001
Linear 2 369.731 369.731 184.865 14.36 0.003
Square 1 411.798 411.798 411.798 31.99 0.001
Residual Error 7 90.107 90.107 12.872
Lack-of-Fit 5 89.441 89.441 17.888 53.66 0.018
Pure Error 2 0.667 0.667 0.333
Total 10 871.636

Estimated Regression Coefficients for Y:强度 using data in uncoded units

Term Coef
Constant 12201.6
X1:压力 0.0928553
X2:温度 -125.677
X2:温度*X2:温度 0.326471

残差图分析:


Contour与表面Plot制做:


优化响应器如下:


最佳设置:X1:压力取120.7107, X2:温度取201.1 达到最大化Y:强度=141.8
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laoccai47 (威望:1) (上海 闵行区) 生物医药

赞同来自:

问的是怎么求y对x2的偏导吗?
把x1看成常数项,对x2求导就行了~
=9.35-4.525x1

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