lecture 6

lecture 6 - Part Five Curve Fitting Fig PT5.6 Chapter 17...

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Part Five Curve Fitting
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Fig PT5.6
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Chapter 17 Least-Squares Regression
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Least Square Regression Curve Fitting Statistics Review Linear Least Square Regression Linearization of Nonlinear Relationships MATLAB Functions
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Wind Tunnel Experiment Measure air resistance as a function of velocity Curve Fitting
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( a ) Least-squares regression ( b ) Linear interpolation ( c ) Curvilinear interpolation Regression and Interpolation Curve fitting
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Least-squares fit of a straight line 1450 830 1220 610 550 380 70 25 80 60 50 40 30 20 10 v , m/s F , N
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Simple Statistics Measurement of the coefficient of thermal expansion of structural steel [ 10 6 in/(in  F)] 535 6 667 6 670 6 592 6 633 6 627 6 598 6 499 6 621 6 445 6 451 6 403 6 703 6 542 6 659 6 624 6 733 6 662 6 721 6 396 6 564 6 435 6 625 6 555 6 655 6 478 6 621 6 543 6 399 6 552 6 667 6 325 6 495 6 775 6 554 6 485 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean, standard deviation, variance, etc.
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Statistics Review Arithmetic mean Standard deviation about the mean Variance (spread) Coefficient of variation (c.v.) n y y i   2 i t t y y y S 1 n S s ;     1 n n y y 1 n y y s 2 i 2 i 2 i 2 y / % . . 100 y s v c y
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Interpolation and Regression
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Regression and Residual
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Advantage of Least Squares Positive differences do not cancel negative differences Differentiation is straightforward weighted differences Small differences become smaller and large differences are magnified
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Linear Regression Fitting a straight line to observations Small residual errors Large residual errors
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Equation for straight line Difference between observation and line e i is the residual or error x a a y 1 0   i i 1 0 i e x a a y Linear Regression
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Least Squares Approximation Minimizing Residuals (Errors) minimum average error (cancellation) minimum absolute error minimax error (minimizing the maximum error) least squares (linear, quadratic, ….)
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) n 1 i i 1 0 i n 1 i i x a a (y e n 1 i i 1 0 i n 1 i i x a a y e Minimize the Maximum Error Minimize Sum of Errors Minimize Sum of Absolute Errors
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Linear Least Squares Minimize total square-error • Straight line approximation • Impossible to pass all points if n > 2 • n equations, but only two unknowns ) , ( , , ) , ( , ) , ( , ) , ( 3 3 2 2 1 1 n n y x y x y x y x i 1 0 i i 1 0 x a a x f y x a a x f ) ( ) (
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Linear Least Squares Total square-error function: sum of the squares of the residuals Minimizing square-error S r ( a 0 ,a 1 ) ) , ( , , ) , ( , ) , ( , ) , ( 3 3 2 2 1 1 n n y x y x y x y x n 1 i 2 i 1 0 i n 1 i 2 i r x a a y e S ) ( 0 a S 0 a S 1 r 0 r Solve for ( a 0 ,a 1 )
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Linear Least Squares Minimize Normal equation y = a 0 + a 1 x n 1 i 2 i 1 0 i 1 0 r x a a y a a S ) ( ) , (    
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This note was uploaded on 10/10/2011 for the course MAE 107 taught by Professor Rottman during the Spring '08 term at UCSD.

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lecture 6 - Part Five Curve Fitting Fig PT5.6 Chapter 17...

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