Question

You have a sample of data, (X1​,Y1​),…,(X100​,Y100​). You fit a natural cubic smoothing spline to the data using R's smooth.spline function. It places one knot at each data point, and uses generalized cross-validation to pick the best λ. It chooses λ>0. The effective degrees of freedom (EDF) of a regression model is the number of parameters estimated, minus the number of constraints, minus any loss in flexibility caused by penalization or regularization. (We will define EDF formally for linear smoothers in the next few lectures) What is a plausible effective degrees of freedom for our smoothing spline fit, after R selected λ>0 ?

a. 75
b. 100
c. 125

Answer 1

The plausible effective degrees of freedom for the smoothing spline fit after R selected λ > 0 is 100.

Step-by-step explanation:

The plausible effective degrees of freedom for the smoothing spline fit after R selected λ > 0 can be determined using the formula:

EDF = number of parameters estimated - number of constraints - any loss in flexibility caused by penalization or regularization

Since R's smooth.spline function places one knot at each data point, there will be 100 parameters estimated. As the smoothing spline uses generalized cross-validation to select the best λ, the number of constraints in this case will be zero. Thus, the plausible effective degrees of freedom would be:

EDF = 100 - 0 - (any loss in flexibility caused by penalization or regularization)

Therefore, the plausible effective degrees of freedom for the smoothing spline fit would be 100.