Response Surface Methods Response surface methodology (RSM) is a - - PowerPoint PPT Presentation

ST 516 Experimental Statistics for Engineers II

Response Surface Methods

Response surface methodology (RSM) is a combination of statistics (regression modeling) and mathematics (optimization) used to find factor settings that optimize the average value of a response, or some function of that average. E.g. the yield of a chemical process (y) is influenced by temperature (x1) and pressure (x2): y = f (x1, x2) + ǫ, where ǫ has mean zero.

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ST 516 Experimental Statistics for Engineers II

Typically the form of f (x1, x2, . . . , xk) is unknown, so we approximate it by either a first-order model y = β0 + β1x1 + β2x2 + · · · + βkxk + ǫ

• r a second-order model

y = β0 +

k

• i=1

βixi +

k

• i=1

βi,ix2

i + i<j

βi,jxixj + ǫ. We expect these approximations to work well only for a small region in (x1, x2, . . . , xk) space.

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ST 516 Experimental Statistics for Engineers II

Steepest Ascent

Typical approach Begin with a 2k factorial design augmented with center points, centered around some initial settings; Fit the first-order model, then check for interaction and pure quadratic terms; If neither interaction nor quadratic effects are significant, use steepest ascent to improve the mean response.

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ST 516 Experimental Statistics for Engineers II

Example Improving yield as a function of reaction time and temperature Initial settings: ξ1 = 35 minutes, ξ2 = 155◦F. Experimental levels: ξ1 = 30, 40; ξ2 = 150, 160; coded as x1 = −1, 1 and x2 = −1, 1. Design: unreplicated 2 × 2 with 5 center points.

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ST 516 Experimental Statistics for Engineers II

Data:

xi1 xi2 x1 x2 y 30 150 -1 -1 39.3 30 160 -1 1 40.0 40 150 1 -1 40.9 40 160 1 1 41.5 35 155 0 40.3 35 155 0 40.5 35 155 0 40.7 35 155 0 40.2 35 155 0 40.6

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ST 516 Experimental Statistics for Engineers II

First-order model

summary(lm(y ~ x1 + x2, t11d1))

Output

Call: lm(formula = y ~ x1 + x2, data = t11d1) Residuals: Min 1Q Median 3Q Max

• 0.244444 -0.044444

0.005556 0.055556 0.255556 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 40.44444 0.05729 705.987 5.45e-16 *** x1 0.77500 0.08593 9.019 0.000104 *** x2 0.32500 0.08593 3.782 0.009158 **

• Signif. codes:

0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.1719 on 6 degrees of freedom Multiple R-Squared: 0.941, Adjusted R-squared: 0.9213 F-statistic: 47.82 on 2 and 6 DF, p-value: 0.0002057

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ST 516 Experimental Statistics for Engineers II

Check interaction and pure quadratic effects

summary(aov(y ~ x1 + x2 + I(x1 * x2) + I(x1^2) + I(x2^2), t11d1))

Output

Df Sum Sq Mean Sq F value Pr(>F) x1 1 2.4025 2.4025 55.872 0.00171 ** x2 1 0.4225 0.4225 9.826 0.03503 * I(x1 * x2) 1 0.0025 0.0025 0.058 0.82132 I(x1^2) 1 0.0027 0.0027 0.063 0.81374 Residuals 4 0.1720 0.0430

• Signif. codes:

0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Both are small ⇒ first-order model is adequate.

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ST 516 Experimental Statistics for Engineers II

How to improve yield? Mean yield increases as either x1 or x2 is increased. If we fix ∆x2

1 + ∆x2 2 = ∆2, the maximum improvement is at

∆x1 = β1 × ∆ β2

1 + β2 2

, ∆x2 = β2 × ∆ β2

1 + β2 2

. This is the direction of steepest ascent. The engineer chose ∆x1 = 1, ∆x2 = ˆ β2/ˆ β1 = 0.42.

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ST 516 Experimental Statistics for Engineers II

But note that this direction depends on the choice of experimental levels for ξ1 and ξ2. Also it’s not clear that the constraint ∆x2

1 + ∆x2 2 = ∆2 is relevant.

So other similar directions could also be searched. In this case, yield increased for 10 steps, then declined. New initial values are 85 minutes and 175 degrees F.

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ST 516 Experimental Statistics for Engineers II

Similar augmented 2 × 2 experiment at the new initial conditions has significant pure quadratic effect, so first-order model is not adequate. Engineer added 4 axial points to estimate second-order model.

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ST 516 Experimental Statistics for Engineers II

Data:

xi1 xi2 x1 x2 y 80 170

• 1
• 1

76.5 80 180

• 1

1 77.0 90 170 1

• 1

78.0 90 180 1 1 79.5 85 175 79.9 85 175 80.3 85 175 80.0 85 175 79.7 85 175 79.8 92.07 175 1.414 0 78.4 77.93 175

• 1.414 0

75.6 85 182.07 0 1.414 78.5 85 167.93 0

• 1.414 77.0

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ST 516 Experimental Statistics for Engineers II

Regression model

t11d6Lm <- lm(y ~ x1 + x2 + x1:x2 + I(x1^2) + I(x2^2), t11d6) summary(t11d6Lm)

Output

Call: lm(formula = y ~ x1 + x2 + x1:x2 + I(x1^2) + I(x2^2), data = t11d6) Residuals: Min 1Q Median 3Q Max

• 0.23995 -0.18089 -0.03995

0.17758 0.36005 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 79.93995 0.11909 671.264 < 2e-16 *** x1 0.99505 0.09415 10.568 1.48e-05 *** x2 0.51520 0.09415 5.472 0.000934 *** I(x1^2)

• 1.37645

0.10098 -13.630 2.69e-06 *** I(x2^2)

• 1.00134

0.10098

• 9.916 2.26e-05 ***

x1:x2 0.25000 0.13315 1.878 0.102519

• Signif. codes:

0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

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ST 516 Experimental Statistics for Engineers II

Output, continued

Residual standard error: 0.2663 on 7 degrees of freedom Multiple R-squared: 0.9827, Adjusted R-squared: 0.9704 F-statistic: 79.67 on 5 and 7 DF, p-value: 5.147e-06

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ST 516 Experimental Statistics for Engineers II

ANOVA for second-order model

summary(aov(y ~ x1 + x2 + x1:x2 + I(x1^2) + I(x2^2), t11d6))

Output

Df Sum Sq Mean Sq F value Pr(>F) x1 1 7.9198 7.9198 111.6873 1.484e-05 *** x2 1 2.1232 2.1232 29.9413 0.000934 *** I(x1^2) 1 10.9816 10.9816 154.8663 4.979e-06 *** I(x2^2) 1 6.9721 6.9721 98.3225 2.262e-05 *** x1:x2 1 0.2500 0.2500 3.5256 0.102519 Residuals 7 0.4964 0.0709

• Signif. codes:

0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

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ST 516 Experimental Statistics for Engineers II

To test for lack of fit in this quadratic model, we must separate the sum of squares for “Residuals” into the sum of squares for lack of fit, and the sum of squares for pure error. Pure error comes from replicated values (center points). We can find it by treating x1 and x2 as categorical variables, because then each unique combination of factor levels has its own mean.

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ST 516 Experimental Statistics for Engineers II

Add factor(x1):factor(x2) to the quadratic model:

summary(aov(y ~ x1 * x2 + I(x1^2) + I(x2^2) + factor(x1) : factor(x2), t11d6))

Output

Df Sum Sq Mean Sq F value Pr(>F) x1 1 7.9198 7.9198 149.4303 0.0002571 *** x2 1 2.1232 2.1232 40.0594 0.0031894 ** I(x1^2) 1 10.9816 10.9816 207.2009 0.0001354 *** I(x2^2) 1 6.9721 6.9721 131.5491 0.0003298 *** x1:x2 1 0.2500 0.2500 4.7170 0.0956108 . factor(x1):factor(x2) 3 0.2844 0.0948 1.7885 0.2885640 Residuals 4 0.2120 0.0530

• Signif. codes:

0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

The sum of squares for lack of fit (factor(x1):factor(x2)) is not significant, so the quadratic model seems to be adequate.

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