Nonlinear Least Squares Fitting

Given a function of a variable tabulated at values , ..., , assume the function is of known analytic form depending on parameters , and consider the overdetermined set of equations

We desire to solve these equations to obtain the values , ..., which best satisfy this system of equations. Pick an initial guess for the and then define

Now obtain a linearized estimate for the changes needed to reduce to 0,

for , ..., , where . This can be written in component form as

where is the matrix

In more concise matrix form,

where is an -vector and is an -vector.

Applying the transpose of to both sides gives

Defining

in terms of the known quantities and then gives the matrix equation

which can be solved for using standard matrix techniques such as Gaussian elimination. This offset is then applied to and a new is calculated. By iteratively applying this procedure until the elements of become smaller than some prescribed limit, a solution is obtained. Note that the procedure may not converge very well for some functions and also that convergence is often greatly improved by picking initial values close to the best-fit value. The sum of square residuals is given by after the final iteration.

An example of a nonlinear least squares fit to a noisy Gaussian function

is shown above, where the thin solid curve is the initial guess, the dotted curves are intermediate iterations, and the heavy solid curve is the fit to which the solution converges. The actual parameters are , the initial guess was (0.8, 15, 4), and the converged values are (1.03105, 20.1369, 4.86022), with . The partial derivatives used to construct the matrix are

The technique could obviously be generalized to multiple Gaussians, to include slopes, etc., although the convergence properties generally worsen as the number of free parameters is increased.

An analogous technique can be used to solve an overdetermined set of equations. This problem might, for example, arise when solving for the best-fit Euler angles corresponding to a noisy rotation matrix, in which case there are three unknown angles, but nine correlated matrix elements. In such a case, write the different functions as for , ..., , call their actual values , and define

and

where are the numerical values obtained after the th iteration. Again, set up the equations as

and proceed exactly as before.