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add interpolator functor
People often want to interpolate histograms. Interpolating profiles is even more useful. Interpolation should be done via a functor, so that the interpolation can have state.
auto h = ... // histogram
auto interp = interpolator(h, interpolator::linear);
@HDembinski : There are tons of splines in Boost.Math. Presumably one of any of the functions in boost/math/interpolators would do?
@NAThompson Good point, I see that there were several added in the last two releases. Catmull-Rom splines and the vector-valued barycentric rational interpolation are candidates. Other interpolators only work for 1D data and are not suitable here.
The interpolators should be added in a way that at least linear interpolation can be used without requiring Boost.Math as a dependency.
On closer look, it seems like neither works. Both interpolate a multi-dimensional point as a function of the 1D variable. A histogram is the opposite. The interpolated value is 1D and the dependent variable is an ND point. So none of the interpolators in Boost.Math can be used.
Basically, interpolation for Histograms is R^n -> R instead of R -> R^n.
R^n->R interpolation is more difficult and it's been in the back of my head to add it for some time. Do you have an arbitrary pointset in R^n or will a uniform grid on compact subsets of R^n be sufficient?
Also, what are reasonable values of n for this problem?
Depending on the axis type for the histogram, it can be uniform, but in general will be non-uniform. Reasonable values of n are 1 to 6, but can be arbitrarily large in principle.
I agree, R^n -> R is way more difficult. Only linear interpolation is easy in any dimension, so that's definitely the first algorithm that will be implemented. I spend a lot of time reading up on fast interpolation a few years back for another project. I can recommend the famous Numerical Recipes http://numerical.recipes on the subject. The references in the SciPy docs are also helpful. https://docs.scipy.org/doc/scipy/reference/interpolate.html
I think that the Makima spline might be the way to go for this:
https://blogs.mathworks.com/cleve/2019/04/29/makima-piecewise-cubic-interpolation/
Thank you for this interesting blog post, I didn't know about the pchip and the makima methods, which both look very interesting.
pchip is interesting, because we often do not want the interpolation to leave the value range of the data, so example, interpolated values in a histogram should not become less than zero.
The main reason this is a good method is that it has a natural generalization to higher dimensions; see Akima's 1974 paper: https://doi.org/10.1145/360767.360779