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 
.