I have read many documents, and I am confused about the difference between smoothing splines and penalised splines.
Are those two the same?
Can someone please suggest any good document which can explain these concepts clearly to me?
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2$\begingroup$ Take a look at pub.uni-bielefeld.de/download/2301835/2301838 "Theoretical and Practical Aspects of Penalized Spline Smoothing". Per the abstract "... penalized splines (P-splines), which have become a very powerful and applicable smoothing technique over the last decade. This nonparametric method can be viewed as a generalization of smoothing splines with a more flexible choice of bases and penalties." $\endgroup$– Mark L. StoneJan 11, 2016 at 15:38
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2 Answers
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There are sometimes some variations in how people use the terminology but usually a smoothing spline has a knot at every x-point while a penalized spline does not.
Penalized splines use a reduced knot set -- not necessarily at data points, somewhat akin to regression splines in that aspect.
Penalized splines and smoothing splines are otherwise similar in that they both include a smoothing term (roughness penalty) and a fit term (lack of fit penalty).
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There are basically two kinds of splines. Regression splines and Smoothing splines. Both can become P-splines by adding penalties to balance model fit to data and smoothness of the curve (and without needing to choose number of knots and position of knots as the penalty handles that). The difference between Regression spline and Smoothing spline is that that former includes both spline (non-linear function f(.)) and regression coefficients of linear variables while the later only has spline (non-linear function f(.)).
