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

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MIT Dummy Levels and μ Before After –Set SP2=SP3 –μrises –Predictions unaffected

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

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MIT Linear Regression Fits a linear model to data

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MIT Error Terms Error should be independent –Within replicates –Between X values

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MIT Least Squares Estimators We want to choose values of b o and b 1 that minimize the sum squared error Take the derivatives, set them equal to zero and you get

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MIT Distribution of Error Homoscedasticity Heteroscedasticity

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

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MIT Linear Regression Assumptions 1.The average value of the dependent variable Y is a linear function of X. 2.The only random component of the linear model is the error term ε. The values of X are assumed to be fixed. 3.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.

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MIT 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

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MIT Design of Dynamic Systems (Signal / Response Systems)

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MIT 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 Product / Process Control Factors Signal Factors Response Signal

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

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MIT Static versus Dynamic 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)

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

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

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MIT C - C S/N and Regression

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MIT 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

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

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

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MIT System Model Known in closed form

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MIT Problem Definition What if R 3 were also a noise? What if R 3 were also a CF? Noise Factors R1, R2, R4, Eo, Ez Response Product / Process Control Factors Signal Factor R 1, R 2, R 4, E z R3 RT-ON RT-OFF

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MIT Results (Graphical)

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MIT 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?

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MIT 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

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