Plan for the Session ? Questions? ? Complete some random topics ? Lecture on Design of Dynamic Systems (Signal / Response Systems) ? Recitation on HW#5? 16.881 MIT Dummy Levels and μ ?Before ?After – Set SP2=SP3 – μ rises – Predictions unaffected Factor Effects on the S/N Ratio 10 12 14 16 18 S P 1 SP 2 SP 3 DA 1 DA 2 DA 3 CU P 1 C U P2 C U P3 P P1 P P2 P P 3 S / N Ra t i o ( d B ) Factor Effects on the S/N Ratio 10 12 14 16 18 S P 1 SP 2 SP 3 DA 1 DA 2 DA 3 CU P 1 C U P2 C U P3 P P1 P P2 P P 3 S / N Ra t i o ( d B ) μ μ 16.881 MIT Number of Tests ? One at a time – Listed as small ? Orthogonal Array – Listed as small ?White Box – Listed as medium 16.881 MIT Linear Regression ? Fits a linear model to data Y i β 0 β 1 X i . ε i 16.881 MIT Error Terms ? Error should be independent – Within replicates – Between X values 6 4 2 0 0 0.5 1 1.5 2 Population regression line 2 Population data points 16.881 Error terms MIT Least Squares Estimators ? We want to choose values of b o and b 1 that minimize the sum squared error , SSE b 0 b 1 i 2 y i b 0 b 1 x i . ? Take the derivatives, set them equal to zero and you get b 1 i x i mean x() y i mean y() . i x i mean x() 2 b 0 mean y ()() b 1 . mean x MIT Distribution of Error ? Homoscedasticity ? Heteroscedasticity 4 2 0 2 4 6 0 0.5 1 1.5 2 Population regression line Population data points Error terms 16.881 MIT Cautions Re: Regression 2 10 4 . ? What will result 210 4 if you run a linear y expo k 1 10 4 regression on 0 0 0 2 4 6 8 10 these data sets? 0.012684 x k 9.885085 28.521501 30 20 y quad k 1.369099 10 0 16.881 1 2 3 0 50 Scatterplot of data Estimated regression line 0 2 4 6 8 10 0.012684 x k 9.885085 MIT Linear Regression Assumptions 1. The average value of the dependent variableY is a linear function ofX. 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 ofX, the errors are are normally distributed with a mean of zero and a constant variance. 16.881 MIT If The Assumptions Hold ? You can compute confidence intervals on β 1 0.8 ? You can test hypotheses – Test for zero slope β 1 =0 0.6 – Test for zero intercept β 0 =0 0.7 0.75 0.8 0.85 0.9 Scatterplot of data Estimated regression line Upper prediction band Lower prediction band Prediction band for the mean of x ? You can compute prediction intervals 16.881 MIT Design of Dynamic Systems (Signal / Response Systems) 16.881 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 Response Product / Process Response Signal Factor Signal 16.881 Control Factors MIT Examples of Dynamic Systems ? Calipers ? Automobile steering system ? Aircraft engine ? Printing ?Others? 16.881 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) 16.881 MIT S/N Ratios for Dynamic Problems Signals Continuous Digital Continuous Responses Digital C-C C-D D-C D-D Examples of each? 16.881 MIT Continuous - continuous S/N ? Vary the signal among Response discrete levels y β ? Induce noise, then compute m n ∑∑ y ij M i β = i=1 j=1 m n 2 ∑∑ M i i=1 j=1 m n σ e 2 = 1 ∑∑ ( y ij ?βM i ) 2 M mn ? 1 i=1 j=1 Signal Factor β 2 η = 10log 10 σ 2 e 16.881 MIT C - C S/N and Regression C-C S/N Linear Regression () y i ()x i β = m n ∑∑ y ij M i mean x . mean y ∑∑ M i 2 b 1 mean x 2 i=1 j=1 i i= 1 j=1 i m n ()x i m n σ e 2 = 1 ∑∑ ( y ij ?βM i ) 2 SSE b 0 b 1 , y i b 0 b 1 x i . 2 mn ? 1 i =1 j=1 i β 2 η = 10log 10 σ 2 e 16.881 MIT Non-zero Intercepts ? Use the same formula Response for S/N as for the zero y α{ β intercept case ? Find a second scaling factor to independently adjust β and α M Signal Factor 16.881 MIT Continuous - digital S/N Response β β y d =1 y d =0 ? Define some on y continuous response y ? The discrete output y d is off a function of y M Signal Factor 16.881 MIT Temperature Control Circuit ? Resistance of Response β β Current in R T =0 R T >0 Current in on thermistor decreases R T with increasing temperature off ? Hysteresis in the circuit lengthens life R 3 Signal Factor 16.881 MIT System Model ? Known in closed form R T_ON R 3 R 2 . E z R 4 . E o R 1 .. R 1 E z R 2 . E z R 4 . E o R 2 .. R T_OFF R 3 R 2 . R 4 . R 1 R 2 R 4 . 16.881 MIT Problem Definition Noise Factors R 3 Product / Process Response Signal Factor R T-ON R 1 , R 2 , R 4 , E o , E z What if R 3 were also a noise? What if R 3 were also a CF? Control Factors R T-OFF R 1 , R 2 , R 4 , E z 16.881 MIT R 1 Results (Graphical) R 2 R 4 E z 35 35 30 25 35 30 25 35 30 25 ,m η ON A mean η ON level ,m η OFF A mean η OFF level 30 25 20 20 20 20 1 2 3 1 2 3 1 2 3 1 2 level MIT16.881 m 20 log β ON . A, level mean 20 log β ON . m 20 log β OFF . A, level mean 20 log β OFF . level 1 2 3 0 10 20 1 2 3 0 10 20 1 2 3 0 10 20 1 2 3 0 10 20 3 Results (Interpreted) ?R T 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 R T-OFF smaller than for R T-ON ? Best choices for R T-ON tend to negatively impact R T-OFF ? Why not consider factor levels outside the chosen range? 16.881 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 16.881 MIT