Here are 2 questions. Your expertise on those issues would be highly appreciated.
In the GAM approach, it makes sense to start with a highly flexible approach and then apply penalties to achieve the smoothness required for a plausible shape. While fitting GAMs, I always use P-spline (=penalized B-splines). However, S. Wood recommend to use the penalized thin-plate spline as it tends to give the best MSE performance.
1) There are plenty of different types of smoothing splines. Why should we use other smoothing splines than the penalized thin-plate spline as it is the one that gives best MSE performance?
2) S. Wood writes:
"Broadly speaking the default penalized thin plate regression splines tend to give the best MSE performance, but they are slower to set up than the other bases. The knot based penalized cubic regression splines (with derivative based penalties) usually come next in MSE performance, with the P-splines doing just a little worse. However the P-splines are useful in non-standard situations".
For P-splines, what does "non-standard situations" mean? When P-splines can be preferred than penalized thin-plate spline?
Wood, S. N. (2003). Thin plate regression splines. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 65(1):95–114.