Plan for the session Questions? Complete some random topics Lecture on Design of Dynamic Systems (Signal /Response Systems Recitation on hw#5? mit 人 16881
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 u Before Factor Effects on the s/N Ratio moz 16 After 12 10 Set sp2=sp3 今今寸小$$小小 Factor Effects on the s/N Ratio urises Predictions 16 unaffected 回o 12 10 人 16881
Dummy Levels and µ • B e fo re • A ft er – Set SP2=SP3 – µ rises – Predictions unaffected Factor Effects on t he S/N R atio 10 12 14 16 18 SP1 SP2 SP3 DA1 DA2 DA3 CUP1 CUP2 CUP3 PP1 PP2 PP3 S/N Ratio (dB) Factor Effects on t he S/N R atio 10 12 14 16 18 SP1 SP2 SP3 DA1 DA2 DA3 CUP1 CUP2 CUP3 PP1 PP2 PP3 S/N Ratio (dB) 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 mit 人 16881
Number of Tests • One at a time – Listed as small • Orthogonal Array – Listed as small • W h i t e B ox – Listed as medium 16.881 MIT
Linear regression Fits a linear model to data 阝0+β1X1+8 y-intercept rIse run slopeβ,=se X Y=B0+BiX 16881
Linear Regression • Fits a linear model to data Yi β 0 β 1 Xi . εi 16.881 MIT
Error terms Error should be independent Within replicates Between X values [X: .Y: on r xxX Population data points mit 人 16881
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 bo and b, that minimize the sum squared error Ssb1)-∑|y(b0+b1x) Take the derivatives, set them equal to zero and you get ∑ Xi- mean(x))(y: mean(y) 0:= mean(y)-b 1 mean(x) b mean x
Least Squares Estimators • We want to choose values of bo and b1 that minimize the sum squared error SSE b , 0 b 1 i 2 yi b 0 b 1 xi . • Take the derivatives, set them equal to zero and you get b 1 i xi mean () x yi . mean ( ) y i xi mean () x 2 b 0 mean () y b ( ) 1 .mean x MIT
Distribution of error Homoscedasticity Heteroscedasticity xxX Population data points Error terms mit 人 16881
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 What will result 2·10 if you run a linear y expo. IIod regression on these data sets? 2 0.012684 9.885085 521501 30 l3690990 a D o Scatterplot of data Estimated regression line 0012684 9.885085 mit 人 16881
Cautions Re: Regression 2 104 . • What will result 2 104 if you run a linear y expok 1 104 regression on 0 0 0 2 4 6 8 10 these data sets? 0.012684 xk 9.885085 28.521501 30 20 y quadk 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 xk 9.885085 MIT
Linear regression A ssumptions 1. The average value of the dependent variable y is a linear function ofX. 2. The only random component of the linear model is the error term 8. 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 mit 人 16881
Linear Regression Assumptions 1. The average value of the dependent variable Y 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 B You can test hypotheses Test for zero slope B1=0 Test for zero intercept 0.7 0.80.8509 You can compute ×× Scatterplot of data Estimated regression line Upper prediction band prediction intervals Lower pi rediction band Prediction band for the mean ofx mit 人 16881
If The Assumptions Hold • You can compute confidence intervals on β1 0.8 • You can test hypotheses – Test for zero slope β1=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
