{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture 6

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

This preview shows pages 1–20. Sign up to view the full content.

Part Five Curve Fitting

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Fig PT5.6
Chapter 17 Least-Squares Regression

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Least Square Regression Curve Fitting Statistics Review Linear Least Square Regression Linearization of Nonlinear Relationships MATLAB Functions
Wind Tunnel Experiment Measure air resistance as a function of velocity Curve Fitting

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
( a ) Least-squares regression ( b ) Linear interpolation ( c ) Curvilinear interpolation Regression and Interpolation Curve fitting
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Interpolation and Regression
Regression and Residual

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
Linear Regression Fitting a straight line to observations Small residual errors Large residual errors

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
Least Squares Approximation Minimizing Residuals (Errors) minimum average error (cancellation) minimum absolute error minimax error (minimizing the maximum error) least squares (linear, quadratic, ….)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
) 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
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 ) ( ) (

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 )
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 ) ( ) , (    

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 148

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

This preview shows document pages 1 - 20. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online