ChE253K Spring09 Lecture07.Rev06

# ChE253K Spring09 Lecture07.Rev06 - Class Business HW01 Back...

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1 UT ChE 253K Lecture 07 Class Business HW01 Back HW02 Due HW03 Posted

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2 UT ChE 253K Lecture 07 How To Fit Data Using Linear Regression Lecture 07 -- Descriptive Statistics
3 UT ChE 253K Lecture 07 What To Learn And Why How to calculate a linear regression line

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4 UT ChE 253K Lecture 07 Objective Of Linear Regression A reasonable line Ŷ = a + bX Ŷ(i) = a + b(C(i)) that minimizes the deviations or residuals e(i) = Y(i) - Ŷ(i) between measured Y(i) and the line’s Ŷ(i) Sources: Miller & Freund text
5 UT ChE 253K Lecture 07 Least Squares Linear Regression Sources: www-micro.lsb.le.ac.uk

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6 UT ChE 253K Lecture 07 Linear Reg’n Optimization Prob’m Objective function to minimize Sum squared residuals (SSE) Least-squares linear reg’n Sum signed residuals Canceling Unreasonable Sum absolute residuals “Robust regression” 1 ( ) n i i i Y a bX = - + 1 [ ( )] n i i i Y a bX = - + 2 1 [ ( )] n i i i Y a bX = - +
7 UT ChE 253K Lecture 07 Minimization Procedure & Result Find a & b to minimize SSE find the zero’s of the partial derivatives After some algebra (given in the Text) ( 29 2 1 [ ( )] 0 n i i i SSE Y a bX a a = = - + = ( 29 b SSE =0 a Y bX = - xy xx b S S = ˆ ˆ ( ) ( ) Y Y bX bx Y Y b X X = - + - = -

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8 UT ChE 253K Lecture 07 S xy S yy S xx 5.4 2 Mean 17.00 29.20 10 0 0 27 10 Sum 7.20 12.96 4 3.6 2 9 4 1.60 2.56 1 1.6 1 7 3 0.00 0.16 0 -0.4 0 5 2 1.40 1.96 1 -1.4 -1 4 1 6.80 11.56 4 -3.4 -2 2 0 Tools For Regression Calculations Sources: Means X = ____ Y = ____ N = ____ Sums of Squares Sxx = ____ Syy = ____ Sxy = ____ Slope, Intercept, CofD a = ____ b = ____ r 2 = ____ StdError, α , ν , t s e = ____ α = ____ ν = ___ t = ____ Confidence Intervals a ± ____ b ± ____ 0 2 4 6 8 10 0 1 2 3 4 5 Additive, qty Life, hrs 1 1 1 1 1 ( )( ) n n n n xy i i i i i i i i i i S X X Y Y X Y X Y n = = = =  - - = -  
9 UT ChE 253K Lecture 07 Data Scatter Plot. Linear? Additive Life, hrs 0 2 1 4 2 5 3 7 4 9 0 2 4 6 8 10 0 1 2 3 4 5 Additive, qty Life, hrs Sources: Text Exercise 11.2

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10 UT ChE 253K Lecture 07 Data Scatter Plot. Linear! Additive Life, hrs 0 2 1 4 2 5 3 7 4 9 0 2 4 6 8 10 0 1 2 3 4 5 Additive, qty Life, hrs Sources: Text Exercise 11.2
11 UT ChE 253K Lecture 07 Sums of Squares: S xx , S yy , S xy Deviation of X from its mean Deviation of Y from its mean Cross-product of X & Y deviations 2 2 2 1 1 1 1 ( ) n n n xx i i i i i i S X X X X n = = = - = - 1 1 1 1 1 ( )( ) n n n n xy i i i i i i i i i i S X X Y Y X Y X Y n = = = =  - - = -   2 2 2 1 1 1 1 ( ) n n n yy i i i i i i S Y Y Y Y n = = = - = -

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12 UT ChE 253K Lecture 07 Sums of Squares Co/Variance Deviation of X from its mean
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