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Ch 9 - Curve Fitting z z z Process of finding simple...

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Curve Fitting Process of finding simple analytical function to approximate a set of data Often from experimental measurements Contains parameters that are adjusted to agree with data Usually two types of “fitting” functions 1. Derived from fundamentals/physics 2. Empirical equation that just matches data well 1 Linear Regression Straight-line fit is simplest example Mathematical expression: “e” is error (or residual) between model and observations Error is discrepancy between the true value of “y” and the approximate value: a 0 +a 1 x 0 1 y a a x e 0 1 e y a a x 2
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Criteria for a “Best” Fit Options 1. Line through the data that minimizes the sum of the residual errors? 2. Minimize the sum of the absolute values of the discrepancies? 3. Minimize the maximum distance a point falls from a line 0 1 1 1 n n i i i i i e y a a x 0 1 1 1 n n i i i i i e y a a x Has the effect of cancelling errors Does not always yield a unique fit Gives undue influence to an outlier 3 Better option -- minimize sum of squares of residuals between the measured “y” and calculated “y” with linear model 2 2 , ,mod 0 1 1 1 1 n n n r i i measured i el i i i i i S e y y y a a x • Overcomes shortcomings of previous approaches • Unique line for given set of data 4
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