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I am embarking on a project for which I'd like to use functional Data Analysis (FDA). I have several thousand discrete curves objects on which I'd like to fit continuous time curves. These discrete curve objects each contain about 6 points on which to fit each continuous time curve.

I am concerned that 6 is too few a point to fit a curve through reliably. What would be the minimum number of points to apply FDA as a rule of thumb? The examples I have seen fit curves on data with 2 orders of magnitude more points per discrete curve on which we fit the continuous curve.

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In a sense it depends if your observations points for each curve are random or not, it also depends on the task you're trying to accomplish.

I'm not very familiar with what's done in practice, but from what I know of the literature on discretized functional data, the number of observations available for each curve will influence the statistical performance you want to achieve in different ways depending on :

  • the statistical task you are trying to perform with your data.
  • the number of curves observed (in your case its several thousands).
  • the assumed smoothness of your curves (this depends on what you are trying to model/your dataset).
  • the dimension of the curves observed.

For example, if you're looking to estimate the mean function "hidden" behind your observations by referring to Cai and Yuan :

  • If your curves are all observed on the same grid of size m=6 and you only suppose your curves to be continuous, you will have an error in $L_2$-estimation of the mean function proportional to $\frac{1}{36}$.
  • If your curves are observed at random evaluation points and again suppose that your function are only continuous, you will have an error proportional to $\left(\frac{1}{6n}\right)^{2/3}$ (where $n$ is the number of curves in your dataset).

In the end, in my opinion, if your curve evaluation points are random, you have lots of curves and you assume that they are very smooth, you may be able to get away with a result that's not too distressing. On the other hand, if the evaluation points are the same for all the curves, you'd better find another approach.

If you're interested in this subject, the key word "phase transition" is often used to refer to the phenomenon of change in the behavior of the error of statistical procedures on functional data, in view of the relationship between the number of curves observed, the number of evaluation points for each curve and the assumed smoothness of the data.

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