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Lecture 30

# Lecture 30 - Chapter 14 General Linear Squares and...

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Chapter 14 General Linear Squares and Nonlinear Regression

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y = - 20.5717 +3.6005x Error S r = 4201.3 Correlation r = 0.4434 x = [-2.5 3.0 1.7 -4.9 0.6 -0.5 4.0 -2.2 -4.3 -0.2]; y = [-20.1 -21.8 -6.0 -65.4 0.2 0.6 -41.3 -15.4 -56.1 0.5];
Preferable to fit a parabola Large error, poor correlation

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Polynomial Regression Quadratic Least Squares y = f ( x ) = a + a x + a x = - - - = n 1 i 2 2 i 2 i 1 0 i 2 1 0 r x a x a a y a a a S ) ( ) , , ( ( 29 ( 29 ( 29 - - - - = = - - - - = = - - - - = = = = = n 1 i 2 i 2 i 1 0 i 2 i 2 r n 1 i 2 i 2 i 1 0 i i 1 r n 1 i 2 i 2 i 1 0 i 0 r x a x a a y x 2 0 a S x a x a a y x 2 0 a S x a x a a y 2 0 a S
Quadratic Least Squares Use Cholesky decomposition to solve for the symmetric matrix or use MATLAB function z = A\r = = = = = = = = = = = = n 1 i i 2 i n 1 i i i n 1 i i 2 1 0 n 1 i 4 i n 1 i 3 i n 1 i 2 i n 1 i 3 i n 1 i 2 i n 1 i i n 1 i 2 i n 1 i i y x y x y a a a x x x x x x x x n

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Standard Error for 2 Polynomial Regression / 3 r y x S s n = - where n observations 2 nd order polynomial (3 coefficients) (start off with n degrees of freedom, use up m+1 for m th -order polynomial)

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» [x,y]=example2 » z=Quadratic_LS(x,y) x y (a0+a1*x+a2*x^2) (y-a0-a1*x-a2*x^2) -2.5000 -20.1000 -18.5529 -1.5471 3.0000 -21.8000 -22.0814 0.2814 1.7000 -6.0000 -6.3791 0.3791 -4.9000 -65.4000 -68.6439 3.2439 0.6000 0.2000 -0.2816 0.4816 -0.5000 0.6000 -0.7740 1.3740 4.0000 -41.3000 -40.4233 -0.8767 -2.2000 -15.4000 -14.4973 -0.9027 -4.3000 -56.1000 -53.1802 -2.9198 -0.2000 0.5000 0.0138 0.4862 err = 25.6043 Syx = 1.9125 r = 0.9975 z = 0.2668 0.7200 -2.7231 y = 0.2668 + 0.7200 x - 2.7231 x 2 Correlation coefficient r Standard error of the estimate function [x,y] = example2 x = [ -2.5 3.0 1.7 -4.9 0.6 -0.5 4.0 -2.2 -4.3 -0.2]; y = [-20.1 -21.8 -6.0 -65.4 0.2 0.6 -41.3 -15.4 -56.1 0.5];
Quadratic Least Square: y = 0.2668 + 0.7200 x - 2.7231 x 2 Error S r = 25.6043 Correlation r = 0.9975

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Cubic Least Squares =
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Lecture 30 - Chapter 14 General Linear Squares and...

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