# Advantages of Penalized B-Splines over Regular B-Splines

My understanding: choosing knots for a B-spline can be an arcane task filled with guessing and eye-balling. Penalized B-splines attempt to do away with the choice of knot picking, fitting a spline by:

1) Using many basis B-splines (yes, that sounds oxymoronic since bases over a vector space have the same number of elements -- but the idea is that we consider a linear combination of many B-splines)

2) Penalizing the coefficients of these basis B-splines using some penalty while fitting them to some data set.

Assuming that my understanding above is correct: is there any other point to using penalized B-splines?

I think that another important point is related to the interpolation and extrapolation properties of Penalized B-splines (aka P-splines).

When interpolating, the B-spline coefficients form a sequence of degree $$2d − 1$$, when extrapolating, of degree $$d−1$$. Thus, when $$d = 2$$, we get cubic interpolation and linear extrapolation
where $$d$$ is the order of the difference penalty.
• you don't even have to cross-validate, modern packages for estimation of penalized splines can also directly optimize ML or REML criterion (e.g. mgcv in R) – adibender May 21 at 16:14