next up previous contents
Next: Fitting Parameterized Models Up: Curve Fitting Previous: Fitting Straight Lines

Fitting General Curves

The line is a special case of the general parametric curve , where t moves x along the curve defined by parameters a. Finding the parameters, a, which best fit such a general curve through a set of 2D data points can be complicated because of the difficulty of defining and evaluating a suitable likelihood function. Simply calculating the closest approach distance of the curve to a single data point requires a minimization in itself (ie Find t which minimizes ). This leads to nasty nested minimization problems of the form

A special case worth describing is when the curve can be written in the form

where are arbitrary fixed functions of x called basis functions. The only unknown parameters are the m weights . If we wish to find the best fit of such a function to N points where the are known precisely but the have a gaussian measurement error with standard deviation of , we can use and approach called General Least Squares. We define a log-likelihood function

The parameters a which minimize this can be found as follows. Define an N x m design matrix D such that

Define an N element vector r such that

It can be shown [6] that the optimal choice of parameters is given by using a singular value decomposition to give the least squares solution to the linear equation

If D is decomposed as with V m x m and orthonormal, then the covariance of the estimate of parameters a is given by .

Bob Fisher
Fri Mar 28 14:12:50 GMT 1997