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lecture9 - Numerical Methods in Engineering ENGR 391 Curve...

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Numerical Methods in Engineering ENGR 391 Faculty of Engineering and Computer Sciences Concordia University Curve fitting and interpolation Chapter 5
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Curve fitting 1. Obtain estimates between data points 2. Obtain a simplified version of a complicated function Least Square Regression - desire a curve (equation) to follow the general trend of the data - “best fit” Interpolation - Use with precise data - curves that intersect all of the data points Data usually available at discrete points from measurements, tables of data, etc. We may want to:
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Regression Used when there is error associated with data Desire a general trend of the data Least square regression method to determine the best fit of an equation to data simplest approximation ± Straight Line Objective y=a 0 +a 1 x+e Higher order polynomial or linearization of nonlinear relationship
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The upward velocity of a rocket is given as a function of time in Table 1. 901.67 30 602.97 22.5 517.35 20 362.78 15 227.04 10 0 0 v(t) [m/s] t [s] 0 250 500 750 1000 0 10 20 30 40 t [s] v (t) [s]
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Least Squares Linear Regression Linear Regression Fitting a straight line to a set of paired observations: (x 1 , y 1 ), (x 2 , y 2 ),…,(x n , y n ). y=a 0 +a 1 x+e a 1 - slope a 0 - intercept e- error, or residual, between the model and the observations Why Least Squares??
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Criteria for a “Best” Fit Minimize the sum of the residual errors for all available data: N = total number of points However, this is an inadequate criterion, so is the sum of the absolute values ¦ ¦ ± ± N i i o i N i i r x a a y e S 1 1 1 ) ( ¦ ¦ ± ± N i i i N i i r x a a y e S 1 1 0 1
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Chapra & Canele Error positive and negative cancel Anywhere between dashed lines gives the same amount of total error Minimizing each distance (maximum error) for all data points.
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Best strategy is to minimize the sum of the squares of the residuals between the measured y and the y calculated with the linear model: Yields a unique line for a given set of data. ¦ ¦ ¦ ± ± ± N i N i i i i i N i i r x a a y y y e S 1 1 2 1 0 2 1 2 ) ( ) model , measured , ( Criteria for a “Best” Fit Best fit
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1. Minimize S r – Differentiate S r w.r.t. each coefficient a i 2. Set each equation equal to zero 3. Solve N equation for a n unknowns ¦ ¦ ¦ ± ± ± N i N i i i i i N i i r x a a y y y e S 1 1 2 1 0 2 1 2 ) ( ) model , measured , ( Linear Least Square Regression
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Least-Squares Fit of a Straight Line > @ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ± ± ± ± ± ± ± w w ± ± ± w w 2 1 0 1 0 1 1 1 0 0 0 ) ( 2 0 ) ( 2 i i i i i i i i o i r i o i o r x a x a x y x a a y x x a a y a S x a a y a S ² ³ x a y a x x N y x y x N a i i i i i i 1 0 2 2 1 ± ± ± ¦ ¦ ¦ ¦ ¦ Mean values Solution of these equations: We minimize S r Two equations for your unknowns a 0 and a 1
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0 0 0 0 1 0 . . . na a a a a n i ´ ´ ´ ¦ ¦ ¦ ´ n i i n i i y x a na 1 1 1 0 ¦ ¦ ¦ ´ n i i i n i i n i i y x x a x a 1 1 2 1 1 0 2 1 1 2 1 1 1 1 ¸ ¹ · ¨ © § ± ± ¦ ¦ ¦ ¦ ¦ n i i n i i n i i n i i n i i i x x n y x y x n a 2 1 1 2 1 1 1 1 2 0 ¸ ¹ · ¨ © § ± ± ¦ ¦ ¦ ¦ ¦ ¦ n i i n i i n i i i n i i n i i n i i x x n y x x y x a n x x n i i ¦ 1 _ n y y n i i ¦ 1 _ _ 1 _ 0 x
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