Non-Linear Curve Fits

INTRODUCTION

Frequently we have a table of values as a function of a single variable which may have measurement noise or other inaccuracies.   These would be of the form What we want to do then is to find the “m” parameters “λ” of some function f(x; λ) which is “non-linear” in the parameters and provides the best fit to the tabulated data.

NON-LINEAR FITS

Given some function, then at each data point we can write for the “m” parameters and where “ ” represents the uncertainty or measurement error  as a consequence of an imperfect initial estimate of the parameters.   We can then approximate the errors as a linear combination of corrections to our parameter set as where which could be used to iteratively update the original set as Writing out the entire matrix we get or more concisely But this is an over determined set of equations and not in form allowing for a unique determination of the parameter corrections, .   So we multiply each side by the transpose of the matrix “A” which switches rows and columns as So that where and for each element And also by recalling then or finally which allows for a unique solution for values.

GAUSSIAN FUNCTION

The well known Gaussian distribution (also called the “Normal” distribution or Bell curve) has the general functional form where the three parameters are given by In this parameter set, A is area under the curve of G(x).  The arithmetic average or mean is .   And the last parameter σ is the standard distribution.   These are further described in the Appendix: Gaussian Parameters.

We can write the derivatives of the Gaussian with respect to its three parameters as follows   The iterative corrections to the initial guess for the three parameters can be expressed in the equation or in detail, this expands to the following For the measurements of then the individual matrix coefficients are      And defining then the matrix elements are   We could solve the matrix equation using Gauss-Jordan but for only three parameters it is easier to write the solutions directly as     And this allows us to iteratively update the initial guesses for the parameters APPENDIX: GAUSSIAN PARAMETERS

For a random variable, X, perhaps with a Gaussian probability distribution, the results of “n” independent measurements can be represented as The arithmetic average or mean of this series of “n” values is simply given by and the variance (or square of the standard distribution) for a large number of “n” trials can be written These equations are exact and work well for a small number of samples.  But as the number of samples increase, it is easier to create a histogram.

Basically the data is grouped into a set of “m” bins where “yj” represents the number of values which lie within some small range, “Δx”, around a midpoint of “xj”.  Each small range represents a separate bin and can be drawn as As before the expression for the mean is  and the variance can be written as As the number of discrete bins increases and the width “Δx” of each gets smaller, then the bins come to resemble a continuous curve as follows: For a continuous curve the area under the Gaussian is given by The mean value can then be written as And the variance is given by APPENDIX: INITIAL ESTIMATE

For a discrete set of data points then if we first define then the arithmetic average or mean value of the distribution is given by and the variance or square of the standard deviation is given by But since we are linear in A, we can write where then the total error associated with an initial guess for A is and we can minimize A in a least squares sense by setting the derivative to zero or 