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Constrained Non-Linear Rationals

There are several drawbacks to standard rational fitting:

  • With noisy data, poles may appear within the X range of the data. The poles are undefined regions associated with the zeros of the denominator.
  • To fit a rational model linearly, the Y used in the design matrix must be that of the raw data rather than the computed model's Y. With noisy data, the true non-linear least-squares minimization is not achieved.
  • While SVD can sometimes deal with ill-conditioning, often the zeroing of one or more sets of basis functions produces results that are altogether useless.

To combat all three of these problems, TableCurve 2D includes 36 non-linear rational models which in form, are exactly equivalent to the more commonly used linearly-fitted rationals.

For these models, the fit begins with the coefficients for the equivalent linear fit. If a pole is detected in the X range of the data, the coefficients are adjusted in an effort to produce a model without such poles. Then, a true non-linear rational fit is made with a constraint set to eliminate, if possible, the formation of any pole within the X data range. In a non-linear fit, if the zeroing of singular values produces nonsense, that iteration's adjustments are ignored. As such, it is generally possible to keep the coefficients within a useful range for the SVD algorithm. This results in the most stable set of coefficients possible.

In general, because of 1) the constraints, 2) the SVD, and 3) the non-linear nature of the rational model, the results from these fits will differ from that of the linearly fitted rationals of the same form.

As in all non-linear fitting, constraint-based fitting is hardly foolproof and this is all the more true when the coefficients must be resolved through SVD. As such, these models are not always successful. In some cases, it is simply not possible to avoid the formation of a pole in the X range for even a modest goodness-of-fit. When these equations are successful, however, the fit will generally be quite superior to the linear procedures.

The Constrained (No-Singularity) Non-Linearly Fit SVD Rational Equations are as follows:

  • *Standard Rationals: 1/1 through 5/5, Equations 7901-7909*
  • *Log X Rationals: 1/1 through 5/5, Equations 7911-7919*
  • *Even Order Rationals: 2/2 through 10/10, Equations 7921-7929*
  • *Half Order Rationals: 0.5/0.5 through 2.5/2.5 order, Equations 7931-7939*