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 .

Fri Mar 28 14:12:50 GMT 1997