Fitting Equation to Data with Regression Techniques
Date: 04/20/2007 at 07:36:47 From: Daniel Subject: Curvilinear or non-linear regression I have a set of values that I'm trying to find the best fit of. I've used linear regression with little success. I'll list some of the values below. x y --- ----- 7 .438 10 1.091 12 1.111 13 1.625 14 1.680 17 2.519 48 2.800 52 3.500 55 3.850 My problem is that I've only dealt with linear regression. I don't know how to go about any version of non-linear regression. Can you step me through how I need to approach that?
Date: 04/20/2007 at 08:40:24 From: Doctor George Subject: Re: Curvilinear or non-linear regression Hi Daniel, Thanks for writing to Doctor Math. Let's say you want to fit a second order polynomial to the data. The general form would be this. y = ax^2 + bx + c For each (x,y) point you get an equation in a, b and c. In matrix form it looks like the following. _ _ _ _ _ _ | y1 | |x1^2 x1 1| | a | | y2 | |x2^2 x2 1| | b | | y3 | |x3^2 x3 1| | c | | . | = | . . .| - - | . | | . . .| | . | | . . .| | yn | |xn^2 xn 1| - - - - Now you need the least squares solution to find a, b and c. Does that make sense? Write again if you need more help. - Doctor George, The Math Forum http://mathforum.org/dr.math/
Date: 04/20/2007 at 12:18:20 From: Daniel Subject: Curvilinear or non-linear regression Thanks for the reply. My last calculus class was 12 years ago. I feel like a freshman in college all over again! Does the matrix shown mean that I need to solve for each set of variables then use normal linear regression? How do I come up with a, b, and c?
Date: 04/20/2007 at 14:03:21 From: Doctor George Subject: Re: Curvilinear or non-linear regression Hi Daniel, This is actually a linear algebra problem. Substitute your (x,y) pairs into the matrices. _ _ _ _ _ _ | y1 | |x1^2 x1 1| | a | | y2 | |x2^2 x2 1| | b | | y3 | |x3^2 x3 1| | c | | . | = | . . .| - - | . | | . . .| | . | | . . .| | yn | |xn^2 xn 1| - - - - Now I will give the vectors and the matrix the names b', A and x'. With these names we can write b' = A x' or A x' = b' We want to find the vector x' that is the least squares solution to this overdetermined system of equations. Check the index of a linear algebra book and you should find more information. The vector x' gives you the a, b, and c for the best fit second order regression polynomial. Write again if you need more help. - Doctor George, The Math Forum http://mathforum.org/dr.math/
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