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My problem can best be described by first looking at my plot:

enter image description here

It is very simple I would like to interpolate between the 2D splines that sit atop the coloured faces (they are only there for illustration).

Each spline represents time-series evolution of a variable. The conditions of the variable have been changed four times, which produces the four splines above. As you can also see; the process get shorter as the conditions are changed.

Now I do not quite know how to attack this.

  • I have considered e.g. Gaussian processes to take into my uncertainty of the regions inbetween.
  • Standard interpolations techniques should work too, but I am not familiar at all with them.

Hence any guidance would be most welcome.

EDIT:

I found this awesome post: https://scicomp.stackexchange.com/questions/19137/what-is-the-preferred-and-efficient-approach-for-interpolating-multidimensional

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Astrid
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1 Answers1

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As also mentioned in the comments to the post you cite, when you go to 2D the span of interpolation approaches wides dramatically, and which to take largely depends on considerations about the physics (or economics, etc.) underlying the process you are examining.
What you already noticed, that changing the conditions produces "similar" behaviour, would suggest (to my mind) the approach to interpolate the 1d spline parameters, with an additional one as time scale.
That is, in the case of a standard cubic spline, starting from the representation as $$ \left\{ \begin{gathered} q(x_{\,1} ) = y_1 \hfill \\ q(x_{\,2} ) = y_2 \hfill \\ q'(x_{\,1} ) = k_1 \hfill \\ q'(x_{\,2} ) = k_2 \hfill \\ \end{gathered} \right. $$ Once you have $y_1,\; y_2, \; k_1,\; k_2$ and $x_2$ (suppose $x_1$ is fixed) for each of the four cases ($s=1,2,3,4$), then put each one as $f(s)$ and interpolate.
But you could start instead from the resulting polynomial coefficients.
Again the physical evidence might suggest you to reduce some parameters, e.g. it might be a reasonable assumption to assume a linear behaviour for $x_2$, but it could be also that an exponential (or ..) might be a more suitable hypothesis for the process considered.

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