I noticed that spline interpolation with a degree higher than 3 (everything beyond cubic splines) have a very high interpolation error, hence the prediction is mostly awful. I've come across various lecture notes, slides and Youtube videos that simply indicate that cubic splines (3rd degree) are optimal and that anything beyond that is a bad idea. These sources however never mention why this is the case.
Can anyone explain to me why this is the case and maybe give me a title/link to a journal/conference paper that explains this or maybe even gives a proof.
Answer
There is no such proof because it's not always true. It's a rule of thumb, because I guarantee that you could come up with a situation- an infinite number of situations actually- where higher order splines would do better than cubic splines. The optimal spline order for any given situation is the exact same order as the system you are trying to model. If the order is the same and your data points are error free (never the case, of course, except for in theoretical problems), then you should be able to model the system perfectly.
The reason they recommend not to go higher than cubic splines is because overfitting is really, really bad. Overfitting can greatly magnify errors, while "underfitting" (choosing a spline method with an order lower than the order of the system you are modelling) introduces some low pass filtering that is either not that bad or sometimes even beneficial.
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