Recently I just learnt Basis Expansion and Regularization. There are several interesting techniques including: cubic spline, natural spline, b-spline and smoothing spline. The question is, what is the Pros and Cons(if there is any) of smoothing spline compared to the "typical" cubic and natural spline where users have to select the knots ?

Apr 12, 2018 · What is a smoothing spline? The Wikipedia article on smoothing splines does a good job in explaining that. To recap, given a set of data points, $\{ (x_i, y_i)_{i=1}^n \}$, a smoothing spline is a solution to the interpolation problem:

Smoothing splines try to minimize the function $$\sum_{i=1}^n(y_i-g(x_i))^2+\lambda\int g''(t)^2dt$$ Smoothing splines shrink the parameters. Moreover, Smoothing Splines are basically natural cubic splines, and thus they're smooth too. So, how can these differences( and properties) save the Smoothing Spline from overfitting?

Jun 7, 2018 · This is because the smoothing spline is a direct basis expansion of the original data; if you used 100 knots to make it that means you created ~100 new variables from the original variable. Loess instead just estimates the response at all the values experienced (or a stratified subset for large data). In general, there are established ...

Mar 6, 2018 · A thin plate spline would smooth through/across the boundary between the two arms of the rotated U; the low values of the lower arm of the U would leak across the boundary and vice versa, producing biased estimated values of the function as it approaches the boundary. Question 2: P splines are useful in several non-standard settings. For example

Sep 6, 2022 · I am trying to fit some time series data to a smoothing spline in R. However, it seems like the spline is fitting the data too perfectly, meaning overfitting. I was trying to figure out what settings to change to try and adjust the level of smoothing.

Jan 12, 2016 · 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(.)).

Jan 18, 2017 · A smoothing spline is basically a natural cubic spline at every unique level of x(i). The tuning parameter λ controls the roughness of the smoothing spline, and hence the effective degrees of freedom which I have to add in the gam(), if I recall correctly.

What effect would it have on a smoothing spline to use the third (or fourth) derivative for the penalty term? Specifically, what would be the effect on the RSS if the tuning parameter were to be varied from 0 to infinity? $$ RSS=∑(y_i−f(x_i))^2 + λ∫((f(t)′′)^2dt $$

Oct 31, 2016 · The default setting in gam is the thin plate spline. So I was wondering what it basically does. From my understanding so far, the advantage in that method is, that you don't have to specify the number of knots k. You start with the maximal number of knots, then gam chooses via GCV which k suits best for the function.