On constructing an approximation and measuring the badness of your approximation, of some arbitrary function, for use in computation.
I mean computable things here here, rather than abstract tools such as truncated functional Taylor expansions, which are used in theory not practice.
Choosing the best approximation
In what sense? Most compact? Most easy to code?
This is not in itself a statistical thing; you might want a simpler or more convenient approximation, or a more compact one, as in MP3 audio or JPEG images.
We can use cross-validation, especially so-called “generalized” cross validation, to choose smoothing parameter this efficiently, in some sense.
Or you might have noisy data, in which case you now have a function approximation and inference problem, with error due to both approximation and sampling complexity. Compressive sensing has some finite-sample guarantees.
To discuss: loss functions.
An interesting sub-problem here is how you align the curves that are your objects of study; That is a problem of warping.
Polynomial spline smoothing of observations
The classic, and not just for functional data, but filed here because that’s where the action is now.
Special superpowers: Easy to differentiate and integrate.
Special weakness: many free parameters, not easy to do in high dimension.
Radial basis function approximation
I actually care about this mostly for densities, so see mixture models, for what information I do have.
Rational approximation (Padé)
Really handy for computation. Trivial to implement once calculated.
Easy to differentiate. OK to integrate if you cheat using a computation mathematics package.
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