Design of experiment methodology
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Transcript of Design of experiment methodology
Design of Experimentmethodology
ANOVA、 FFD、CCD、Mixture Design
Response surface design
Kung, [email protected] engineering department in NCKU
• Help you better understand and optimize your response.
• Used to refine models after you have determined important factors using factorial designs
Advantages of Response surface design
Factorial Points : Estimated main factor & interaction
Axial Points : Estimated pure quadratic form
Center Points : Estimated pure Error
→ Building a quadratic response surface→ Resolves both main effects and interactions
Central composite design (CCD)
8
Level Temperature () Annealing time (mins)120 5.0
1 115 6.50 100 10.0-1 85 13.5- 80 15.0
Run Temp. Time Temp. Time
1 -1 -1 85 13.5
2 -1 1 85 6.5
3 1 -1 115 13.5
4 1 1 115 6.5
5 0 0 100 10
6 0 0 100 10
7 0 0 100 10
8 0 - 100 15
9 0 100 5
10 - 0 80 10
11 0 120 10
Design matrix
Reference: Michael Grätzel, Advanced Functional Materials, 24, 3250(2014)
Effect of Annealing Temperature on Film Morphology of Organic–Inorganic Hybrid Pervoskite Solid-State Solar Cells
120
5 mins
15 mins
8100
80
run Temp. Time Voc (V) Jsc (mA/cm2) FF PCE (%)
1 80 10 0.77 7.07 0.72 3.89
2 85 13.5 0.72 10.30 0.59 4.44
3 85 6.5 0.25 12.98 0.37 1.23
4 100 15 0.78 13.86 0.72 7.77
5 100 10 0.78 11.99 0.73 6.78
6 100 5 0.81 6.63 0.73 3.92
7 115 13.5 0.71 12.80 0.66 5.99
8 115 6.5 0.72 13.18 0.71 6.72
9 120 10 0.71 11.42 0.68 5.50
Origin data-CCD
SAS-ANOVA
Source:11DF: 10
變異數分析來源 DF 和平方 平均值平方 F 值 Pr > F
模型 5 31.35310
6.27062 6.47 0.0306
誤差 5 4.84396
0.96879
已校正的總計
10 36.19705
根 MSE 0.98427 R 平方 0.8662
應變平均值 5.43636 調整 R 平方
0.7324
變異係數 18.10534 Set regression equation(model y1=x1 x2 t1 t2 t3 /noint selection=forward;)Commands :
proc : procedurereg : regressionanova : calculate ANOVA
Y = a + bX1+c x2 + dx12+ ex2
2+ fx1x2
Parameters:11
參數估計值
變數 DF 參數估計
標準誤差
t 值 Pr > |t|
Intercept 1 6.78014 0.56827 11.93 <.0001
x1 1 1.16474 0.34802 3.35 0.0204
x2 1 -0.99064 0.34802 -2.85 0.0360
t1 1 -1.21157 0.41428 -2.92 0.0328
t2 1 -0.63640 0.41428 -1.54 0.1851
t3 1 0.98500 0.49214 2.00 0.1017
z=6.78+1.16*x-0.99*y-1.21*x2-0.63*y2+0.98.*x*y
Regression of PCE (%)
[x,y] = meshgrid(-2:0.01:2);z=6.78014+(1.16474.*x)-(0.99064.*y)-(1.21157.*(x.^2))-(0.63640.*(y.^2))+(0.98500.*x.*y);[C,h] = contour(x,y,z, [1,2,3,4,5,5.5,6,6.5,7,7.2]);axis([-1.5,1.5,-1.5,1.5])clabel(C,h);
12
Temperatuer(degree)
Ann
ealin
g tim
e(m
ins)
85 100 115
6.5
10
13.5
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Parameter Coefficient
Xtemp1.16
Ytime-0.99
XtempXtemp-1.21
YtimeYtime-0.63
XtempYtime0.99
Main factor: Interaction: Temp.-Time
90
110
z=6.78+1.16*x-0.99*y-1.21*x2-0.63*y2+0.98.*x*y
¿
Regression of PCE (%)
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Model is fixed Algebra equation Time-consuming
Model reduction SAS regression Interaction effect Save time
0.00 0.25 0.50 0.75 1.00
0.00
0.25
0.50
0.75
1.000.00
0.25
0.50
0.75
1.00
IBr
ClBinary Design (A) Ternary Design (B)
Modified mixture design methods
Advantages of mixture design
• Designs for these experiments are useful because many product design and development activities in industrial situations involve formulations or mixtures.
17
Mixture design methodology
RegressionExperimental data Contour plot1. 2. 3.
SAS 9.3 MATLAB R2013a
0.00 0.25 0.50 0.75 1.00
0.00
0.25
0.50
0.75
1.000.00
0.25
0.50
0.75
1.00
IBr
Cl
Origin data
Ratio MACl MABr MAI Voc(V) Jsc (mA/cm2) FF (%) PCE (%)
1 1 0 0 0.76 10.31 69% 5.422 0 1 0 0.97 5.54 71% 3.813 0 0 1 0.78 9.76 70% 5.384 0.33 0.333 0.333 0.86 11.54 66% 6.545 0.5 0.5 0 0.97 4.76 70% 3.236 0 0.5 0.5 0.92 5.97 66% 3.587 0.5 0 0.5 0.71 12.10 72% 6.128 0.67 0.17 0.17 0.73 12.37 71% 6.389 0.17 0.67 0.17 0.97 10.66 64% 6.54
10 0.17 0.17 0.67 0.85 11.46 77% 7.5111 0.75 0.25 0 0.90 7.43 64% 4.3212 0.25 0.75 0 0.96 7.68 57% 4.1513 0.25 0 0.75 0.78 13.06 68% 6.8914 0.75 0 0.25 0.80 9.93 72% 5.6815 0 0.75 0.25 0.99 10.81 72% 7.7116 0 0.25 0.75 0.82 10.05 73% 6.01
Forward selection
SAS Regression
run;
proc reg;model y1=t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 /noint selection=forward;proc anova;
前進選擇 : 步驟 1 R 平方 = 0.6006 和 C(p) = 57.8286
前進選擇 : 步驟 2R 平方 = 0.8367 和 C(p) = 17.3612
前進選擇 : 步驟 7 R 平方 = 0.9792 和 C(p) = 1.7387
Click run
Model reduction
Parameter Coefficient
X1 t1
X2 t2
X3 t3
X1X2 t4
X1X3 t5
X2X3 t6
X1X2X3 t7
X1X2(X1-X2) t8
X1X3(X1-X3) t9
X2X3(X2-X3) t10
X1X1X2X3 t11
X1X2X2X3 t12
X1X2X3X3 t13
SAS Regression-13 parameters
變異數分析來源 DF R2 平均值平方 F 值 Pr > F
模型 6 515.7
85.9 73.60
<.0001
誤差 10 11.8 1.2
未校正的總計 16 527.4
變數
參數估計
標準誤差
第二型 SS
F 值 Pr > F
t1 5.57799 0.77588 60.35377
51.69
<.0001
t2 4.53861 0.83794 34.25800
29.34
0.0003
t3 6.53751 0.73561 92.22951
78.98
<.0001
t4 -5.83005 4.15712 2.29666 1.97 0.1911
t7 64.07944 25.58609
7.32434 6.27 0.0312
t10 11.75880 7.90392 2.58452 2.21 0.1677
R 平方 = 0.9779 和 C(p) = -0.0183
Parameter Coefficient
X1 t1
X2 t2
X3 t3
X1X2 t4
X1X3 t5
X2X3 t6
X1X2X3 t7
X1X1X2X3 t11
X1X2X2X3 t12
X1X2X3X3 t13
SAS Regression-10 parameters
變異數分析來源 DF 和
平方平均值平方
F 值 Pr > F
模型 6 513.8
85.6 63.28
<.0001
誤差 10 13.53
1.35
未校正的總計 16 527.4
變數
參數估計
標準誤差
第二型 SS
F 值 Pr > F
t1 5.26373 1.01318 36.52686
26.99
0.0004
t2 5.09614 0.83751 50.10738
37.03
0.0001
t3 5.81178 0.83751 65.16845
48.15
<.0001
t4 -6.08694
4.54785 2.42432 1.79 0.2104
t5 3.33659 4.54785 0.72845 0.54 0.4800
t7 59.03570
29.04160
5.59231 4.13 0.0695
R 平方 = 0.9743 和 C(p) = 2.7857
0
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1 0
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Cl
Br I0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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R 平方 = 0.9779 和 C(p) = -0.0183 R 平方 = 0.9743 和 C(p) = 2.7857
Different parameters compared
13 parameters 10 parameters
Contour plot- e.g. PCEA=tril(meshgrid(0:0.001:1));B=tril(meshgrid(1:-0.001:0)');C=tril(1-A-B);x=tril(0.5.*(1+C-B));y=tril((3^0.5)*0.5.*A);
z=5.57799.*A +4.53861.*B+6.53751.*C -5.83005.*A.*B +64.07944.*A.*B.*C+11.75880.*B.*C.*(B-C);
[C,h] = contourf(x,y,z, [1,2,3,4,5,6,6.5,7,7.2,7.4,7.6],'LineWidth',1);
axis([0,1,0,1]);
clabel(C,h,'manual','fontsize',15);hold onplot([0.375,0.625],[0.6495,0.6495],'k:');hold onplot([0.25,0.75],[0.433,0.433],'k:');hold onplot([0.375,0.75],[0.6495,0],'k:');hold onplot([0.25,0.5],[0.433,0],'k:');hold onplot([0.125,0.25],[0.2165,0],'k:');hold onplot([0.125,0.875],[0.2165,0.2165],'k:');hold onplot([0.25,0.625],[0,0.6495],'k:');hold onplot([0.50,0.75],[0,0.433],'k:');hold onplot([0.75,0.875],[0,0.2165],'k:');
先建立矩陣數列,從 0~1,間格0.001
利用 SAS迴歸得到的 Eq.
A=tril(meshgrid(0:0.001:1)) B=tril(meshgrid(1:-0.001:0)');
x=tril(0.5.*(1+C-B)); y=tril((3^0.5)*0.5.*A);
C=tril(1-A-B);
z=5.58.*A +4.54.*B+6.54.*C -5.83.*A.*B+64.08.*A.*B.*C+11.76.*B.*C.*(B-C);
e.g. 0.5*(1+0.001-0.998) e.g. *0.5*0.001
Function
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Contour plot- e.g. PCE[C,h] = contourf(x,y,z, [1,2,3,4,5,6,6.5,7,7.2,7.4,7.6],'LineWidth',1);
axis([0,1,0,1]);
78 10
9
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12 12.5
12.7
1211
108
1110
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Br I0 0.2 0.4 0.6 0.8 1
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Br I0 0.2 0.4 0.6 0.8 1
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Br I0 0.2 0.4 0.6 0.8 1
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0.65 0.68
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0.80.85
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IBr
Cl
5 6 6.5 7 7.2
7.27
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6.5
0 0.2 0.4 0.6 0.8 10
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JscVoc FF PCE
PCE=5.6*A +4.5*B+6.5*C -5.8*A*B+64.1*A*B*C+11.8*B*C*(B-C);
SAS Regression
Jsc=9.44 X1+6.58 X2+10.75 X3-6.69 X1X2+8.07 X1X3+91.79 X1X2X3+17.10X2X3(X2-X3)
Voc=0.76 X1+0.97 X2+0.77 X3+0.38 X1X2+0.20 X2X3+101.67X1X2X3+0.23X1X2(X1-X2)+0.38 X2X3(X2-X3)-112.78 X12X2X3-99.91 X1X2
2X3-95.62 X1X2X32
FF=0.70 X1+0.70 X2+0.71 X3-0.26 X1X2+0.36 X1X2(X1-X2)-3.92 X1X22X3+4.90X1X2X3
2
27
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0 0.25 0.5 0.75 1 0.0 0.2 0.4 0.6 0.8 1.0-16
-14
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0
MAPbCl0.30
Br0.15
I0.55
MAPbCl0.30
Br0.35
I0.30
MAPbCl0.30
Br0.55
I0.15
Curr
ent den
sity
(m
A/c
m2)
Voltage (V)
PCE Contour plotrun Cl Br I Voc Jsc FF PCE
0.30 0.55 0.15 0.97 9.94 0.70 6.77
0.30 0.35 0.35 0.92 14.07 0.73 9.47
0.30 0.15 0.55 0.75 12.67 0.68 6.52
Cl
IBr
Verification by J-V curve
Factors and levels for the 23 Full Factorial Design
factorslevels
+ -A, P3HT:PCBM concentration
(wt%) 2.5 1.5
B, rpm 600 1000
C, time (s) 60 40
Full Factorial Design
1.5wt% 600rpm 40s 600rpm 60s 1000rpm 40s 1000rpm 60sVOC (V) 0.04 0.60 0.70 0.72
JSC (mA/cm2) 2.94 7.22 1.21 1.03FF 0.29 0.32 0.32 0.30
PCE (%) 0.03 2.83 0.27 0.22
R.P (Ω ・ cm2)104 0.00083 278.8 71.68 99.72
R.S (Ω ・ cm2) 1.88 2.23 7.36 1.15
Run A B C Voc (V) Jsc (mA/cm2) FF PCE (%)
1 - + - 0.04 2.94 0.29 0.03
2 - + + 0.60 7.22 0.65 2.83
3 - - - 0.70 1.21 0.32 0.27
4 - - + 0.72 1.03 0.30 0.22
factors + -A, concentration (wt%) 2.5 1.5B, revolution (rpm) 600 1000C , time (s) 60 40
2.5wt% 600rpm 40s 600rpm 60s 1000rpm 40s 1000rpm 60sVOC (V) 0.60 0.58 0.60 0.70
JSC (mA/cm2) 7.54 9.11 9.40 1.90FF 0.54 0.60 0.63 0.40
PCE (%) 2.46 3.18 3.53 0.53
R.P (Ω ・ cm2)104 0.88 40.91 143.42 168.52
R.S (Ω ・ cm2) 2.05 2.80 2.27 7.39
5 + + - 0.60 7.54 0.54 2.466 + + + 0.58 9.11 0.60 3.187 + - - 0.60 9.40 0.63 3.538 + - + 0.70 1.90 0.40 0.53
factors + -A, concentration (wt%) 2.5 1.5B, revolution (rpm) 600 1000C , time (s) 60 40
Run A B C Voc (V) Jsc (mA/cm2) FF PCE (%)
1 - + - 0.04 2.94 0.29 0.032 - + + 0.60 7.22 0.65 2.833 - - - 0.70 1.21 0.32 0.274 - - + 0.72 1.03 0.30 0.225 + + - 0.60 7.54 0.54 2.466 + + + 0.58 9.11 0.60 3.187 + - - 0.60 9.40 0.63 3.538 + - + 0.70 1.90 0.40 0.53
factors + -A, concentration (wt%) 2.5 1.5B, revolution (rpm) 600 1000C , time (s) 60 40
Design matrix
Run A B C AB BC AC ABC PCE
1 - + - - - + + 0.03
2 - + + - + - - 2.83
3 - - - + + + - 0.27
4 - - + + - - + 0.22
5 + + - + - - - 2.46
6 + + + + + + + 3.18
7 + - - - + - + 3.53
8 + - + - - + - 0.53
effect estimateA 1.59
B 0.99
C 0.12
AB -0.20
BC 1.64
AC -1.26
ABC 0.22
The estimate of A = ( ) - ( = 1.59
Calculate the estimate