Plan for the Session

1. Plan for the Session • Questions? • Complete some random topics • Lecture on Design of Dynamic Systems (Signal / Response Systems) • Recitation on HW#5? MIT

2. Dummy Levels and μ • Before • After • Set SP2=SP3 • μrises • Predictions unaffected MIT

3. Number of Tests • One at a time • Listed as small • Orthogonal Array • Listed as small •White Box • Listed as medium MIT

4. Linear Regression • Fits a linear model to data MIT

5. Error Terms • Error should be independent • Within replicates • Between X values MIT

6. Least Squares Estimators • We want to choose values of bo and b1 that minimize the sum squared error • Take the derivatives, set them equal to zero and you get MIT

7. Distribution of Error • Homoscedasticity • Heteroscedasticity MIT

8. Cautions Re: Regression • What will result if you run a linear regression on these data sets? MIT

9. Linear Regression Assumptions • The average value of the dependent variable Y is a linear function of X. • The only random component of the linear model is the error term ε. The values of X are assumed to be fixed. • The errors between observations are uncorrelated. In addition, for any given value of X, the errors are normally distributed with a mean of zero and a constant variance. MIT

10. If The Assumptions Hold • You can compute confidence intervals on β1 • You can test hypotheses • Test for zero slope β1=0 • Test for zero intercept β0=0 • You can compute prediction intervals MIT

11. Design of Dynamic Systems (Signal / Response Systems) MIT

12. Dynamic Systems Defined “Those systems in which we want the system response to follow the levels of the signal factor in a prescribed manner” – Phadke, pg. 213 Noise Factors Response Product / Process Signal Factors Response Signal Control Factors MIT

13. Examples of Dynamic Systems • Calipers • Automobile steering system • Aircraft engine • Printing • Others? MIT

14. Static Vary CF settings For each row, induce noise Compute S/N for each row (single sums) Dynamic Vary CF settings Vary signal (M) Induce noise Compute S/N for each row (double sums) Static versus Dynamic MIT

15. S/N Ratios for Dynamic Problems Signals Continuous Digital Continuous Responses Digital Examples of each? MIT

16. Continuous - continuous S/N • Vary the signal among discrete levels • Induce noise, then compute MIT

17. C - C S/N and Regression MIT

18. Non-zero Intercepts • Use the same formula for S/N as for the zero intercept case • Find a second scaling factor to independently adjust β and α Response Signal Factor MIT

19. Continuous - digital S/N • Define some continuous response y • The discrete output yd is a function of y MIT

20. Temperature Control Circuit • Resistance of thermistor decreases with increasing temperature • Hysteresis in the circuit lengthens life MIT

21. System Model • Known in closed form MIT

22. Problem Definition What if R3 were also a noise? What if R3 were also a CF? Noise Factors R1, R2, R4, Eo, Ez Product / Process Signal Factor Response R3 RT-ON RT-OFF Control Factors R1, R2, R4, Ez MIT

23. Results (Graphical) MIT

24. Results (Interpreted) • RT has little effect on either S/N ratio • Scaling factor for both βs • What if I needed to independently set βs? • Effects of CFs on RT-OFF smaller than for RT-ON • Best choices for RT-ON tend to negatively impact RT-OFF • Why not consider factor levels outside the chosen range? MIT

25. Next Steps • Homework #8 due 7 July • Next session Monday 6 July 4:10-6:00 • Read Phadke Ch. 9 --“Design of Dynamic Systems” • No quiz tomorrow • 6 July --Quiz on Dynamic Systems MIT