What kind of functions can have non whole degrees? Thanks for the help in advance.
I am reading a technical report on a regression algorithm that reports a pair of functions as having a total degree of freedom of 5.4.  I believe that both of these functions are splines, but it isn't directly specified in the report (they are graphed but the explicit functions are not given \ they could also be polynomials).
How might a function, or a set of functions, have a total degree of freedom that is not a whole number?  What would be an example function exhibiting this behavior?
 A: If you conceive of model degrees of freedom in terms of say the trace of the hat-matrix or smoother matrix (in cases where you can write your fitted model in the form $\hat{y}=Ay$ for some square matrix $A$), or more generally in terms of $\sum_i\frac{\partial \hat{y_i}}{\partial y_i}$, then for some models this isn't necessarily a whole number, yet it still in some sense captures the same kind of thing as we conceive d.f. being in other cases (including reproducing the d.f. of simpler models).
Conceived this way, various smoothing, shrinkage and other regularization methods yield noninteger degrees of freedom.
See, for example, ESLII$^{[1]}$ (e.g. Sec5.4.1 for one example), or Ye (1998)$^{[2]}$.
[1]: Hastie, T., Tibshirani, R. and Friedman, J. (2009),
The Elements of Statistical Learning: Data Mining, Inference, and Prediction (2e),
Springer Series in Statistics, Springer, New York.
[2]: Ye, J. (1998),
"On measuring and correcting the effects of data mining and model selection",
Journal of the American Statistical Association 93(441):120-131
