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I am reading up on splines and as a beginner I have a basic question -

Does it make sense to say - "I will fit a cubic b-spline to the data". As b-spline is just a representation of spline in terms of its bases. I think, a more accurate statement will be - "I will fit a natural/not-a-knot/clamped/etc. cubic spline to the data and present results in terms of its basis."

My question arises out of my limited understanding of relevant concepts. Could someone please confirm this. Many thanks!

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    $\begingroup$ People talk about Newton interpolating polynomial and Lagrange interpolating polynomial, even though they are the same. When the focus is on algorithm, a different representation of the same mathematical object can make enough difference to be emphasized like that. $\endgroup$ – user147263 Dec 29 '15 at 20:49
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You are correct in thinking that the statement "fit a cubic b-spline" is pretty vague.

A spline is just a piecewise polynomial. As you said, "b-spline" is just a way of representing a spline, using a particular basis. In fact every polynomial spline is a b-spline, because b-spline basis functions can be used to construct a basis for any spline space (i.e. any space of piecewise polynomials).

Instead of just saying "fit a b-spline" it would be more informative to say what type of spline will be used, what properties it has, and what type of algorithm will be used to construct it.

The answers to this question have a bit more information on this topic, and it's all explained very clearly in the book "A Practical Guide to Splines" by Carl deBoor.

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  • $\begingroup$ Thank you. I have read a lot of your answers on related topics on this website, you are obviously an expert. Thanks for sharing the knowledge. $\endgroup$ – Innocent Dec 30 '15 at 20:10
  • $\begingroup$ I am not sure I agree with the statement that "every spline is a b-spline". A spline is not necessarily polynomial either. Try to goggle "exponential spline" and you will find that there are other splines that are simply non-polynomial. $\endgroup$ – fang Dec 31 '15 at 2:08
  • $\begingroup$ @fang: My guess was that the OP was only considering polynomial splines, since these are the ones used 99% of the time, and cubic splines are certainly polynomial. But, I added the word "polynomial" to further clarify. $\endgroup$ – bubba Dec 31 '15 at 2:26

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