I am working on a problem where I have to model a system and its temporal evolution. The data provided is sampled with a temporal resolution of 1 minute. The physical description of the system does not include any temporal dependency which will result in independent models at each timestamp. In order to avoid an unnatural temporal evolution I have introduced B-splines. I found the resulting smoothing to have a big effect, but the model parameters temporal evolution were at times too fast. For this reason I introduced an additional regularization term of the second derivative (from the beginning I have regularized the model 2-norm).
When trying to determine the optimal value of the regularization parameters I test a wide range of value, calculate the GCV-score and found the minimum. Without the regularization term of the second derivative this works very well. With the regularization term of the second derivative I find the behavior of the GCV-score a bit odd.
In the attached image I have tested a wide range of parameter values which result in the surface. The optimal value of alpha is very clear (regularization parameter of the model 2-norm). The beta value (regularization parameter of the second derivative) on the other hand is continuously decreasing in the interval I tested. As a result I fixed the alpha parameter and expanded the search of the beta parameter illustrated by the red line. In the second image I show this line independently. Initially, the shape of the curve is as expected from a GCV-score curve, but at the end it decrease rapidly.
My question: How do I select a regularization parameter in this case? Is the GCV-score method insufficient when regularizing the second derivative? Is this behavior expected?
Thanks in advance!
