# Regressing/interpolating between quasi-periodic sinusoids

This is a cross-post (on recommendation) from CV.

My problem is very simple.

I currently intend on using Kriging (Gaussian process regression) to perform regression between the trajectories marked with red dots on the below plot, where a simple linear-regression surface has been superimposed as well.

As you can see there is a strong correlation in a particular direction of the four time-series. Through long and very interestind discussion with @DeltaIV, we started consdering other methods such as harmonic regression. The latter would have been very elegant except the example time-series in the above plot have different periods (fundamental frequencies) thus HR will not work.

Hence I would like to ask you excellent people what you would recommend for either regression or interpolation between quasi-periodic time-series such as the above? I should not that the above plot shows only one period, that approximately repeats itself but not quite (hence 'quasi').

Alas currently (in order of preference):

• Multivariate splines (recommendations here welcome too)
• Gaussian process regression (with complex kernel choice)
• Some magical way of making harmonic regression work with time-series with different periodicity

UPDATE:

Here is another image, showing the periodicity of ONE of the signals annotated by the red dots above. Thus, this time-series shows the periodicity of the time-series over a longer period of recorded time. Again, this is just ONE of the above four time-series, recorded at a certain regime.

• So, as far as I understood, you know that each time series is highly periodic, right? But: you can't compare things at fixed time offsets, because the period is different for every realization. Would estimating that period and then normalizing to a phase (i.e. $\frac{\text{sample time}}{\text{Period}}$) be an option? – Marcus Müller Dec 16 '16 at 21:56
• @Marcus could you give an example? Your description is correct regarding the problem. Though I do not quite understand your proposal. – Astrid Dec 18 '16 at 15:09
• Assume that you were able to measure the frequency of the periodic signal. Since we can assume sufficient bandwidth limitation (you've sampled the signal, after all), it's possible to resample the signals in a manner that brings them all to the same frequency (measured in $f_\text{sample}$) and thus, making the resulting samples comparable – Marcus Müller Dec 18 '16 at 17:15
• I think I understand what you mean; thus using a first sampled signal as a basis function almost, we can measure the rest of the sampled signals in terms of this base? – Astrid Dec 19 '16 at 15:52
• First, i should confirm that the thousands axis are time, and the decimal axis is space? Though this could be viable, i fail to see a periodical behaviour on any section, hence i cannot suggest any periodic strategy on that sense. Unless you clarify us which is the physical meaning of the measurements?. Also, using dynamical system techniques for the time axis is a way better approach than kernels. If and only if we can plug in causality on the estimation...... Can you paste another clearer image of what we should achieve? Or the physical meaning of all this? At least, the meaning of the axes – Brethlosze Dec 21 '16 at 22:04