I'm following up on the informative discussion here concerning choice of m (order of derivative) for MGCV splines. Using the default options in MGCV (thin plate spline, REML for optimization, k=10, etc.) I noticed that the choice of m affects the estimated degrees of freedom, which is not surprising. However, when using m=1, this often results in edf that are close to zero. Reading online (including here) I think that this might occur since when using m=2 (the default), the linear function is in the penalty null space, since the penalty is based on the curvature of the function (second derivative) and the linear function has no curvature. I'm guessing that when m is changed to 1 this no longer applies, since now the linear function would contribute to the penalty, but I admittedly don't entirely understand why. Also, going back to the previous discussion on the choice of m, does it make sense to choose m=1 in cases where the edf is shrunk towards zero? What would be the interpretation of the spline in such cases? And then finally, why is the default to penalized the second derivative compared to the first?