试验优化设计-第四章(2013)

第四章二次回归组合设计§1二次回归模型一、问题的提出1、线性设计单元多元纯线性纯线性非纯线性――含一级二次交互作用1°确定因素范围Z2jZ1jZ0j2°因素编码正交设计+1-10单纯形3°编制方案编码空间非整体设计整体设计自然空间非整体设计整体设计4°计算，5°检验6°方程转换…2、非线性问题高次项因素：交互项因素：不能同时为1非线性因素：指数、对数二、二次回归模型1、自然空间回归模型代数式不完全矩阵式2、自然空间的回归方程3、编码空间的回归方程--第j个自然因素的二次项回归系数—第j个编码因素的二次项的回归系数三、必要条件1、N——总试验次数；q——二次回归方程欲求的回归系数总项数；2、在回归分析和回归设计中，因素应取的最小水平数bmin应等于该因素在回归方程中的最高次数加1。要求取二次回归方程，则每个因素应取的最小水平数为：bj

为因素xj的水平数…1°传统方法为全面试验2°试验量太大§2二次组合设计一、概念在编码空间中，选择几类具有不同特点的试验点，适当组合起来形成的试验方案。二、组合设计一般由三类不同的试验点组成mc:二水平点，各因素皆取二水平+1，-1，的全面试验点mr

:分布在p个坐标轴上，他们与中心点的距离r,r≥1m0

：零点，各因素均取零水平的试验点，即中心点X1X21+11-1-1+1-1-1+r0-r00+r0-r000000三、组合设计1、满足和的试验条件2、降低试验次数传统111727457912308424472661015212823456组合qp3、点分布合理4、可进行连贯设计X1X21+11-1-1+1-1-1+r0-r00+r0-r000000四、mc的部分实施组合设计中，安排mc试验点，主要是为了求取因素的一次项和所有一级交互作用的回归系数共有

部分实施a—能安排下q′个因素与交互作用的正交表123456789101112131415ababcacbc

abcdadbd

abd

cd

acd

bcd

abcd列号列名x1因素五、分类1、二次回归正交设计2、二次回归连贯设计3、二次回归旋转设计x2x1x2x1x3x3x4x2x3x2x4x1x4x3x4x5x1x5x2x5x3x5x4x5§3二次回归正交组合设计一、概念从正交优良性出发进行组合设计寻求二次方程二、设计阵Z0000000000000001119101100

r2r2

r2r200000000r-r

r-r0011115678111111111-1-111-11-111-1-111111234X22X12X1X2X2X1X0jijIX0X1X2X1X2X12X221234111111-1-11-11-11-1-111111111156781111r-r0000r-r0000r2r20000r2r291011111000000000000000三、正交性1、考察正交性正交性条件满足条件不满足条件2、正交性条件1°2、正交性条件臂长r与因素个数p，中心点试验次数m0

有关;当组合设计正交时，r，p，m0

应具有确定的函数关系;要使组合设计正交，对于确定的p,r的大小取决于m0

的大小。2°中心化处理jiX0X1X2X1X2X12X221234111111-1-11-11-11-1-111111111156781111r-r0000r-r0000r2r20000r2r291011121111000000000000000000001.21-1.211.21-1.21x1′0.4230.4230.4230.4230.8870.887-0.577-0.577-0.577-0.577-0.577-0.577x2′0.4230.4230.4230.423-0.577-0.5770.8870.887-0.577-0.577-0.577-0.577四、正交设计的主要步骤应用二次回归设计研究离合器摩擦系数的变化规律，根据专业知识和以往经验，选定结合压力Z1，滑摩速度Z2，表面温度Z3三个试验因素。1、组合设计1、确定的范围1°上下限（,）2°零水平3°水平间隔2、因素编码1°编码公式83.488.782.26Z0j-△j66.522.220.74216.5213.223.74Z0j+△j6082Z1j150113240144Z2jZ3Z2Z1Zj因素编码表(Xj)r10-1-r4、编制方案结构阵1°因素Xj

逐步裂分

2°交互作用互乘解决

3°Xj2自乘x32x22x12x2x3x1x3x1x2x3x2x1x0jI1234567891011121314151617111111111111111111111-1-1-1-1r-r000000011-1-111-1-100r-r000001-11-11-11-10000r-r00011-1-1-1-1110000000000000000000000000001-11-1-11-111-1-111-1-1111111111r2r200000001111111100r2r200000111111110000r2r2000x1′0.3140.3140.3140.3140.3140.3140.3140.3141.1451.145-0.686-0.686-0.686-0.686-0.686-0.686-0.686x2′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.6861.1451.145-0.686-0.686-0.686-0.686-0.686x3′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.686-0.686-0.6861.1451.145-0.686-0.686-0.6861.353-1.3531.353-1.3531.353-1.3535、回归系数bj

计算同线性回归设计一样规范化和程式化6、F检验同线性回归设计一样公式化和程式化………x32x22x12x2x3x1x3x1x2x3x2x1x0jI1234567891011121314151617111111111111111111111-1-1-1-1r-r000000011-1-111-1-100r-r000001-11-11-11-10000r-r00011-1-1-1-1110000000000000000000000000001-11-1-11-111-1-111-1-1111111111r2r200000001111111100r2r200000111111110000r2r2000x1′0.3140.3140.3140.3140.3140.3140.3140.3141.1451.145-0.686-0.686-0.686-0.686-0.686-0.686-0.686x2′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.6861.1451.145-0.686-0.686-0.686-0.686-0.686x3′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.686-0.686-0.6861.1451.145-0.686-0.686-0.6861.353-1.3531.353-1.3531.353-1.353YI0.680.460.520.480.600.500.490.470.540.530.550.450.540.460.500.480.52Dj

1711.66211.662

11.662

8886.7056.7076.705Bj

8.770.0810.4150.49800.140.260.140.14

0.14

bj

0.5160.0070.0360.04300.0180.0330.0210.0020.002

x32x22x12x2x3x1x3x1x2x3x2x1x0jI1234567891011121314151617111111111111111111111-1-1-1-1r-r000000011-1-111-1-100r-r000001-11-11-11-10000r-r00011-1-1-1-1110000000000000000000000000001-11-1-11-111-1-111-1-1111111111r2r200000001111111100r2r200000111111110000r2r2000x1′0.3140.3140.3140.3140.3140.3140.3140.3141.1451.145-0.686-0.686-0.686-0.686-0.686-0.686-0.686x2′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.6861.1451.145-0.686-0.686-0.686-0.686-0.686x3′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.686-0.686-0.6861.1451.145-0.686-0.686-0.6861.353-1.3531.353-1.3531.353-1.353YI0.680.460.520.480.600.500.490.470.540.530.550.450.540.460.500.480.52Dj

1711.66211.662

11.662

8886.7056.7076.705Bj

8.770.0810.4150.49800.140.260.140.14

0.14

bj

0.5160.0070.0360.04300.0180.0330.0210.0020.002

Sj

4.5245.67×10-40.01480.021302.54×10-38.58×10-32.94×10-32.8×10-5

2.8×10-5Fj

3753.256.3021.457.35αj

0.050.050.250.050.05

Se=0.053

fe=2x32x22x12x2x3x1x3x1x2x3x2x1x0jI1234567891011121314151617111111111111111111111-1-1-1-1r-r000000011-1-111-1-100r-r000001-11-11-11-10000r-r00011-1-1-1-1110000000000000000000000000001-11-1-11-111-1-111-1-1111111111r2r200000001111111100r2r200000111111110000r2r2000x1′0.3140.3140.3140.3140.3140.3140.3140.3141.1451.145-0.686-0.686-0.686-0.686-0.686-0.686-0.686x2′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.6861.1451.145-0.686-0.686-0.686-0.686-0.686x3′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.686-0.686-0.6861.1451.145-0.686-0.686-0.6861.353-1.3531.353-1.3531.353-1.353YI0.680.460.520.480.600.500.490.470.540.530.550.450.540.460.500.480.52Dj

1711.66211.662

11.662

8886.7056.7076.705Bj

8.770.0810.4150.49800.140.260.140.14

0.14

bj

0.5160.0070.0360.04300.0180.0330.0210.0020.002

Sj

4.5245.67×10-40.01480.021302.54×10-38.58×10-32.94×10-32.8×10-5

2.8×10-5Fj

3753.256.3021.457.35αj

0.050.050.250.050.05

S回=0.0501,

F回=5Se=0.053

fe=2x32x22x12x2x3x1x3x1x2x3x2x1x0jI1234567891011121314151617111111111111111111111-1-1-1-1r-r000000011-1-111-1-100r-r000001-11-11-11-10000r-r00011-1-1-1-1110000000000000000000000000001-11-1-11-111-1-111-1-1111111111r2r200000001111111100r2r200000111111110000r2r2000x1′0.3140.3140.3140.3140.3140.3140.3140.3141.1451.145-0.686-0.686-0.686-0.686-0.686-0.686-0.686x2′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.6861.1451.145-0.686-0.686-0.686-0.686-0.686x3′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.686-0.686-0.6861.1451.145-0.686-0.686-0.6861.353-1.3531.353-1.3531.353-1.353YI0.680.460.520.480.600.500.490.470.540.530.550.450.540.460.500.480.52Dj

1711.66211.662

11.662

8886.7056.7076.705Bj

8.770.0810.4150.49800.140.260.140.14

0.14

bj

0.5160.0070.0360.04300.0180.0330.0210.0020.002

Sj

4.5245.67×10-40.01480.021302.54×10-38.58×10-32.94×10-32.8×10-3

2.8×10-3Fj

1.403753.256.3021.457.35αj

0.050.050.250.050.05

000000000000S=0.053f=16S回=0.0501,

F回=5Se=0.053

fe=2x32x22x12x2x3x1x3x1x2x3x2x1x0jI1234567891011121314151617111111111111111111111-1-1-1-1r-r000000011-1-111-1-100r-r000001-11-11-11-10000r-r00011-1-1-1-1110000000000000000000000000001-11-1-11-111-1-111-1-1111111111r2r200000001111111100r2r200000111111110000r2r2000x1′0.3140.3140.3140.3140.3140.3140.3140.3141.1451.145-0.686-0.686-0.686-0.686-0.686-0.686-0.686x2′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.6861.1451.145-0.686-0.686-0.686-0.686-0.686x3′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.686-0.686-0.6861.1451.145-0.686-0.686-0.6861.353-1.3531.353-1.3531.353-1.353YI0.680.460.520.480.600.500.490.470.540.530.550.450.540.460.500.480.52Dj

1711.66211.662

11.662

8886.7056.7076.705Bj

8.770.0810.4150.49800.140.260.140.14

0.14

bj

0.5160.0070.0360.04300.0180.0330.0210.0020.002

Sj

4.5245.67×10-40.01480.021302.54×10-38.58×10-32.94×10-32.8×10-3

2.8×10-3Fj

1.403753.256.3021.457.35αj

0.050.050.250.050.05

SR=0.0029fR=11S=0.053f=16S回=0.0501F回=5Se=0.053

fe=2x32x22x12x2x3x1x3x1x2x3x2x1x0jI1234567891011121314151617111111111111111111111-1-1-1-1r-r000000011-1-111-1-100r-r000001-11-11-11-10000r-r00011-1-1-1-1110000000000000000000000000001-11-1-11-111-1-111-1-1111111111r2r200000001111111100r2r200000111111110000r2r2000x1′0.3140.3140.3140.3140.3140.3140.3140.3141.1451.145-0.686-0.686-0.686-0.686-0.686-0.686-0.686x2′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.6861.1451.145-0.686-0.686-0.686-0.686-0.686x3′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.686-0.686-0.6861.1451.145-0.686-0.686-0.6861.353-1.3531.353-1.3531.353-1.353YI0.680.460.520.480.600.500.490.470.540.530.550.450.540.460.500.480.52Dj

1711.66211.662

11.662

8886.7056.7076.705Bj

8.770.0810.4150.49800.140.260.140.14

0.14

bj

0.5160.0070.0360.04300.0180.0330.0210.0020.002

Sj

4.5245.67×10-40.01480.021302.54×10-38.58×10-32.94×10-32.8×10-3

2.8×10-3Fj

1.403753.256.3021.457.35αj

0.050.050.250.050.05

SR=0.0029fR=11S=0.053f=17S回=0.0501F回=5Se=0.053

fe=2x32x22x12x2x3x1x3x1x2x3x2x1x0jI1234567891011121314151617111111111111111111111-1-1-1-1r-r000000011-1-111-1-100r-r000001-11-11-11-10000r-r00011-1-1-1-1110000000000000000000000000001-11-1-11-111-1-111-1-1111111111r2r200000001111111100r2r200000111111110000r2r2000x1′0.3140.3140.3140.3140.3140.3140.3140.3141.1451.145-0.686-0.686-0.686-0.686-0.686-0.686-0.686x2′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.6861.1451.145-0.686-0.686-0.686-0.686-0.686x3′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.686-0.686-0.6861.1451.145-0.686-0.686-0.6861.353-1.3531.353-1.3531.353-1.353YI0.680.460.520.480.600.500.490.470.540.530.550.450.540.460.500.480.52Dj

1711.66211.662

11.662

8886.7056.7076.705Bj

8.770.0810.4150.49800.140.260.140.14

0.14

bj

0.5160.0070.0360.04300.0180.0330.0210.0020.002

Sj

4.5245.67×10-40.01480.021302.54×10-38.58×10-32.94×10-32.8×10-5

2.8×10-5Fj

1.403753.256.3021.457.35αj

0.050.050.250.050.05

Slf=0.0021

flf=9SR=0.0029fR=11S=0.053f=16S回=0.0501F回=5Se=0.053

fe=2x32x22x12x2x3x1x3x1x2x3x2x1x0jI1234567891011121314151617111111111111111111111-1-1-1-1r-r000000011-1-111-1-100r-r000001-11-11-11-10000r-r00011-1-1-1-1110000000000000000000000000001-11-1-11-111-1-111-1-1111111111r2r200000001111111100r2r200000111111110000r2r2000x1′0.3140.3140.3140.3140.3140.3140.3140.3141.1451.145-0.686-0.686-0.686-0.686-0.686-0.686-0.686x2′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.6861.1451.145-0.686-0.686-0.686-0.686-0.686x3′0.3140.3140.3140.3140.3140.3140.3140.314-0.686-0.686-0.686-0.6861.1451.145-0.686-0.686-0.6861.353-1.3531.353-1.3531.353-1.353YI0.680.460.520.480.600.500.490.470.540.530.550.450.540.460.500.480.52Dj

1711.66211.662

11.662

8886.7056.7076.705Bj

8.770.0810.4150.49800.140.260.140.14

0.14

bj

0.5160.0070.0360.04300.0180.0330.0210.0020.002

Sj

4.5245.67×10-40.01480.021302.54×10-38.58×10-32.94×10-32.8×10-3

2.8×10-3Fj

1.403753.256.3021.457.35αj

0.050.050.250.050.05

Slf=0.0021

flf=9SR=0.0029fR=11S=0.053f=17S回=0.0501F回=5Se=0.053

fe=27、方程转换（检验合格后）§4二次回归连贯设计一、概念在一次回归正交设计的基础上，补充少量试验点即星号点，进行二次回归设计的方法，称为二次回归连贯设计。对于线性回归方程失拟的场合，欲充分利用一次回归试验信息，再继续寻求二次回归方程，以满足实际需要，连贯设计是行之有效的。条件：1°非单纯形，基础为正交设计+1+1+1-12°一次失拟3°形式①一次二次②一次纯线性一次非线性③一次纯线性一次非纯线性二次二、主要步骤1、已有基础：一次正交设计将一次回归正交设计中，水平1与-1的试验点和零水平试验点分别作为mc

与m0

。2、再补充星号点mr

编制组合设计方案，寻求二次回归方程。三、主要特点1、充分利用原来的信息2、转换式二次连贯设计与二次回归正交组合设计不同，因素编码仍通用一次回归正交设计的编码公式。3、扩大了试验范围，在一次试验时要留有余地。二次连贯设计，自然因素的实际上、下限不是在实验方案编制前预先选定而是根据编码因素的星号臂长r，利用一次回归设计编码公式计算确定。二次连贯设计将自然因素的实际试验范围扩大了，在设计时，应注意实际试验范围是否容许。4、使得mc

点是整数，mr

各点是小数5、二次回归连贯设计的基本公式同于二次回归正交组合设计组合设计连贯设计例利用二次回归正交设计寻求镓溶液导电率与因素Z1（镓的浓度）和Z2（苛性碱浓度）的二次回归方程。302083.725.8Z1′(-r)9030Z1j(-1)12050Z0j(0)15070Z2j(1)156.374.2Z2′

(r)Z2Z1Zj

(Xj

)X0X1(Z2)X2(Z2)X1X2X12(x1′)X22(x2′)YiYi211(70)1(150)11(0.423)1(0.423)5.025.0011(70)-1(90)-11(0.423)1(0.423)6.744.891-1(30)1(150)-11(0.423)1(0.423)8.572.251-1(30)-1(90)11(0.423)1(0.423)2.04.0011.21(74.2)0(120)0r2(0.887)0(-0.577)5.934.811-1.21(25.8)0(120)0r2(0.887)0(-0.577)4.924.0110(50)1.21(156.3)00(-0.577)r2(0.887)5.833.6410(50)-1.21(83.7)00(-0.577)r2(0.887)2.98.4110(50)0(120)00(-0.577)0(-0.577)2.87.8410(50)0(120)00(-0.577)0(-0.577)3.210.2410(50)0(120)00(-0.577)0(-0.577)3.411.5610(50)0(120)00(-0.577)0(-0.577)3.09.00Dj126.9286.92844.2874.287S=41.7499f=11Se=0.200Fe=3S回=40.616f回=5SR=1.133,fR=6Slf=0.933,flf=3Bj54.1002.4108.309-8.2006.7963.722bj4.5080.3481.199-2.0501.5850.868Sj243.9010.8389.96516.81010.7723.231Fj12.57169.48252.15161.5848.46aj0.050.010.010.010.011mc2345mr6789m0101112§5二次旋转组合设计一、旋转性在p维编码空间中，如果回归方程在以零水平试验点即试验中心为球心，为半径的p维球面上各点的方差皆相等，就称该方程具有旋转性。二、旋转设计从旋转性出发，设计方案求取回归方程。只要是线性（一次）正交设计三、二次旋转设计必要条件1、旋转性条件2、非退化条件…jiX0X1X2X1X2X12X221234111111-1-11-11-11-1-111111111156781111r-r0000r-r0000r2r20000r2r291011111000000000000000r2=2jiX0X1X2X1X2X12X221234111111-1-11-11-11-1-11111111115678111100000000000000000000…四、二次旋转设计1、设计方法在三个或更多变量的场合，实现旋转设计常借助于组合设计思想。

分布在三个半径不相等的球面上

个点分布在半径为的球面上；

个点分布在半径为的球面上；

个点集中在半径为的球面上。2、具体设计jiX0X1X2X1X2X12X221234111111-1-11-11-11-1-111111111156781111r-r000