I'm following up on the informative disussion here concerning choice of m (order of derivative) for MGCV splines. Using the defualt 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 suprising. However, when using m=1, this often results in edf that are close to zero. Reading online (include 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 (the second derivative) and the linear function has no curvature. I'm guessing that when m is changed to 1 this no longer applies, but I 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 interpretaion of the spline in such cases?