I have a model given by cubic splines within a certain number of intervals defining a valid range in total and some end conditions (second derivatives are zero). The whole model can be shifted in x and scaled in y with an amplitude and an offset.

The fit of the model to the given data is a least squares minimization giving the final shift, amplitude and offset. Some kind of gradient method will be used.

I do not know if the model completely overlaps with the data, it might not be the case. Additionally, the initial estimate of the model shift may be so that not all of the data is covered by the spline region. The data may be very noisy.

The question is: How to treat data values being located outside of the valid splines interval during the fit? My initial approach was to disregard any data points not currently within the model region. My fear is that the fitting will then be biased towards large shifts and using less data points (small overlap).

Possible solutions:

  • Extrapolating the spline outside its original region, so that the model is defined everywhere. However, since I do not know the model outside, that might introduce severe errors of it's own. I would like to avoid that.
  • Normalizing the least-squares term by the number of current data points used (by the overlap of data and model). However, this may still not be enough, because less data points might still be fitted better than the true number of data points. And it introduces non-continuities in the minimization term.

Are there better approaches?


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