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Unformatted text preview: 4/15/2011 http://numericalmethods.eng.usf.edu 1 Nonlinear Regression Major: All Engineering Majors Authors: Autar Kaw, Luke Snyder http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates Nonlinear Regression http://numericalmethods.eng.usf.edu Nonlinear Regression ) ( bx ae y = ) ( b ax y = + = x b ax y Some popular nonlinear regression models: 1. Exponential model: 2. Power model: 3. Saturation growth model: 4. Polynomial model: ) ( 1 m m x a ... x a a y + + + = http://numericalmethods.eng.usf.edu 3 Nonlinear Regression Given n data points ) , ( , ... ), , ( ), , ( 2 2 1 1 n n y x y x y x best fit ) ( x f y = to the data, where ) ( x f is a nonlinear function of x . Figure. Nonlinear regression model for discrete y vs. x data ) ( x f y = ) , ( n n y x ) , ( 1 1 y x ) , ( 2 2 y x ) , ( i i y x ) ( i i x f y − http://numericalmethods.eng.usf.edu 4 Regression Exponential Model http://numericalmethods.eng.usf.edu 5 Exponential Model ) , ( , ... ), , ( ), , ( 2 2 1 1 n n y x y x y x Given best fit bx ae y = to the data. Figure. Exponential model of nonlinear regression for y vs. x data bx ae y = ) , ( n n y x ) , ( 1 1 y x ) , ( 2 2 y x ) , ( i i y x ) ( i i x f y − http://numericalmethods.eng.usf.edu 6 Finding Constants of Exponential Model ( ) ∑ = − = n i bx i r i ae y S 1 2 The sum of the square of the residuals is defined as Differentiate with respect to a and b ( )( ) 2 1 = − − = ∂ ∂ ∑ = i i bx n i bx i r e ae y a S ( ) ( ) 2 1 = − − = ∂ ∂ ∑ = i i bx i n i bx i r e ax ae y b S http://numericalmethods.eng.usf.edu 7 Finding Constants of Exponential Model Rewriting the equations, we obtain 1 2 1 = ∑ + ∑ − = = n i bx n i bx i i i e a e y 1 2 1 = ∑ − ∑ = = n i bx i n i bx i i i i e x a e x y http://numericalmethods.eng.usf.edu 8 Finding constants of Exponential Model Substituting a back into the previous equation 1 2 1 2 1 1 = ∑ ∑ ∑ − ∑ = = = = n i bx i n i bx bx n i i bx i n i i i i i i e x e e y e x y The constant b can be found through numerical methods such as bisection method. ∑ ∑ = = = n i bx n i bx i i i e e y a 1 2 1 Solving the first equation for a yields http://numericalmethods.eng.usf.edu 9 Example 1Exponential Model t(hrs) 1 3 5 7 9 1.000 0.891 0.708 0.562 0.447 0.355 Many patients get concerned when a test involves injection of a radioactive material. For example for scanning a gallbladder, a few drops of Technetium99m isotope is used. Half of the techritium99m would be gone in about 6 hours. It, however, takes about 24 hours for the radiation levels to reach what we are exposed to in daytoday activities. Below is given the relative intensity of radiation as a function of time....
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida  Tampa.
 Spring '08
 Kaw,A

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