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!
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.