Why Does Successive Residual Polynomial Fits Work?

  To see why the successive residual polynomial fit works so well we can compute the polynomials for the sequence of fits and measure the distances between the fits and the original curve represented by the data points. A simple definition of the distance between the curves represented by two sets of points, f and g, is the square root of the sum of the squares of the distances for each set of points.

The calculation of the successive fits proceeds as follows.

A plot of the distances shows that the successive fits steadily approach the original data points.

The root mean square error is another measure of the distance between two sets of points. One needs to be careful about the density of points and that could be taken into account by including a weight function in the sum of the squares. There may also be some error correction of the coefficients for the fit polynomial taking place as one gets closer to the desired function. Blunders may be eliminated by removing unusually large deviations from the fit.