lecture15 - http:/ocw.mit.edu _ MIT OpenCourseWare Spring...

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MIT OpenCourseWare ____________ http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: ________________ http://ocw.mit.edu/terms .
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2.830J/6.780J/ESD.63J 1 M anufacturing Control of Manufacturing Processes Subject 2.830/6.780/ESD.63 Spring 2008 Lecture #15 Response Surface Modeling and Process Optimization April 8, 2008
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2.830J/6.780J/ESD.63J 2 M anufacturing Outline •L a s t T i m e – Fractional Factorial Designs – Aliasing Patterns – Implications for Model Construction • Today – Response Surface Modeling (RSM) • Regression analysis, confidence intervals – Process Optimization using DOE and RSM Reading: May & Spanos, Ch. 8.1 – 8.3
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2.830J/6.780J/ESD.63J 3 M anufacturing Regression Fundamentals • Use least square error as measure of goodness to estimate coefficients in a model • One parameter model: – Model form – Squared error – Estimation using normal equations – Estimate of experimental error – Precision of estimate: variance in b – Confidence interval for β – Analysis of variance: significance of b – Lack of fit vs. pure error • Polynomial regression
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2.830J/6.780J/ESD.63J 4 M anufacturing Measures of Model Goodness – R 2 Goodness of fit – R 2 – Question considered: how much better does the model do than just using the grand average? – Think of this as the fraction of squared deviations (from the grand average) in the data which is captured by the model Adjusted R 2 – For “fair” comparison between models with different numbers of coefficients, an alternative is often used – Think of this as (1 – variance remaining in the residual). Recall ν R = ν D - ν T
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2.830J/6.780J/ESD.63J 5 M anufacturing Least Squares Regression We use least-squares to estimate coefficients in typical regression models One-Parameter Model: • Goal is to estimate β with “best” b • How define “best”? – That b which minimizes sum of squared error between prediction and data – The residual sum of squares (for the best estimate) is
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2.830J/6.780J/ESD.63J 6 M anufacturing Least Squares Regression, cont. Least squares estimation via normal equations – For linear problems, we need not calculate SS( β ); rather, direct solution for b is possible – Recognize that vector of residuals will be normal to vector of x values at the least squares estimate Estimate of experimental error – Assuming model structure is adequate, estimate s 2 of σ 2 can be obtained:
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2.830J/6.780J/ESD.63J 7 M anufacturing Precision of Estimate: Variance in b We can calculate the variance in our estimate of the slope, b : Why?
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2.830J/6.780J/ESD.63J 8 M anufacturing Confidence Interval for β • Once we have the standard error in b , we can calculate confidence intervals to some desired (1- α )100% level of confidence • Analysis of variance – Test hypothesis: – If confidence interval for β includes 0, then β not significant – Degrees of freedom (need in order to use t distribution) p = # parameters estimated by least squares
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2.830J/6.780J/ESD.63J 9 M anufacturing
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This note was uploaded on 09/24/2010 for the course MECHE 2.830J taught by Professor Davidhardt during the Spring '08 term at MIT.

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lecture15 - http:/ocw.mit.edu _ MIT OpenCourseWare Spring...

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