I am looking for a method for extending a curve, like the ones shown in the figure below.
The known part of the curve is defined only for integer arguments, which however span a range from one to millions. This curve is rather easy to evaluate.
The way I do extrapolation now is by choosing a set of points in the known part of the curve, fitting a polynomial, and then evaluating this polynomial for values beyond the end of the known curve. Sometimes it looks reasonable, but sometimes it explodes (not unexpectedly for polynomial fitting).
The question is thus two-fold:
- If using the polynomial fitting: how to choose the points in the most optimal way (number and placement), and how to evaluate the performance?
- What are the alternative methods? Could we use functions other than polynomials? - there are probably established methods that I am not aware of, so tips/references would be greatly appreciated.
Background
- I know that the derivative of the curve is decreasing, but I cannot say with certainty that the curve saturates.
- I have some background knowledge about the curve and there are methods of extrapolation, based on its domain-specific content. However, I am looking for a non-parametric alternative.