Regression analysis for non linear function

Question

I am trying to model a problem with damped sine wave, $f(x) = a\sin(bx)\exp(-cx)$. I want to find optimum $a,b,c$ for my data. Can anyone please shed some light on this?

Answer 1

Please, have a look at :http://en.wikipedia.org/wiki/Non-linear_least_squares. In fact, we could say the problem is very similar to linear regression if you linearize locally the problem. This is why good initial estimates are required. So basically, you can use Newton-Raphson procedure in order to solve, in your case, for three equations (the derivatives of the sum of squares with respect to parameters a, b and c) and three unknowns (namely a, b and c). Please understand that this is something which works but I made it very simple in this post. In ageneral manner, the problem is treated as a minimization problem. If there is anything you want me to clarify, jus ask.

Answer 2

Assuming that your data start "close" to zero, the slope of your function at the origin is a * b. I suppose that, for some value of x, the curve shows a maximum; this takes place at a point which is such that Tan[b * x] = b / c. The first value of x where your function is almost zero corresponds to b * x = Pi. So, you have reasonable estimates and you can start the least square procedure. I hope this helps. I shall look at your data and be back.

Answer 3

The current methods to fit a function to a given data set often require a first guess of the parameters, followed by an interative process. This is a non-linear regression problem. A different method consists in transforming the non-linear regression to a linear regression thanks to a convenient integral equation. Then, there is no need for initial guess and no need for iterative process : the fitting is directly obtained. This is explained in the paper "Régressions et équations intégrales" published on Scribd : http://www.scribd.com/JJacquelin/documents The case of the function y = a*sin(b*x)*exp(-c*x) is not among the various examples given in the paper, but could be treated on a similar manner, thanks to the general method using a convenient integral equation related to this particular function.

Answer 4

The function to be fitted is a case of damped sinusoidal function. A straightforward method (no guess to start, not itterative process) is given with a numerical example in the section "Damped sinusoidal regression", pp.64-70 : http://fr.scribd.com/doc/14674814/Regressions-et-equations-integrales