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Finals-Review-2005

Finals-Review-2005 - EAD 115 Numerical Solution of...

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EAD 115 Numerical Solution of Engineering and Scientific Problems David M. Rocke Department of Applied Science
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Least Squares Regression
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Curve Fitting Given a set of n points (x i , y i ), find a fitted curve that provides a fitted value y = f(x) for each value of x in a range. The curve may interpolate the points (go through each one), either linearly or nonlinearly, or may approximate the points without going through each one, as in least-squares regression.
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Simple Linear Regression We have a set of n data points, each of which has a measured predictor x and a measured response y. We wish to develop a prediction function f(x) for y. In the simplest case, we take f(x) to be a linear function of x, as in f(x) = a 0 + a 1 x
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The least-squares criterion minimizes There are many other possible criteria. Use of the least-squares criterion does not imply any beliefs about the data Use of the linear form for f(x) assumes that this straight-line relationship is reasonable Assumptions are needed for inference about the predictions or about the relationship itself ( ) ( ) 2 2 1 1 n n i i i i i SS r y f x = = = =
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Computing the Least-Squares Solution We wish to minimize the sum of squares of deviations from the regression line by choosing the coefficients a 0 and a 1 accordingly Since this is a continuous, quadratic function of the coefficients, one can simply set the partial derivatives equal to zero
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( ) ( ) ( ) ( ) ( ) 2 2 2 0 1 0 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 2 0 1 0 1 0 1 1 1 1 1 1 0 1 1 ( , ) ( , ) 0 2 2 ( , ) 0 2 2 n n n i i i i i i i i n n n n i i i i i i i i n n n n i i i i i i i i i i i i i SS a a r y f x y a a x SS a a y a a x y a a x a SS a a y a a x x x y a x a x a na a x = = = = = = = = = = = = = = = = = − = − = = − = − + 1 2 0 1 1 1 1 n n i i n n n i i i i i i i y a x a x x y = = = = = =
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