[R] How to get the penalized log likelihood from smooth.spline()?

[Chong Gu]

   Date: Wed, 20 Feb 2002 17:49:46 +0100 (MET)
   From: Henrik Bengtsson <hb at maths.lth.se>
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   I use smooth.spline(x, y) in package modreg and I would like to get
   value of penalized log likelihood and preferable also its two parts. To
   make clear what I am asking for (and make sure that I am asking for the
   right thing) I clarify my problem trying to use the same notation as in
   help(smooth.spline):

   I want to find the natural cubic spline f(x) such that

      L(f) = \sum_{k=1}{n} w[k](y[k] - f(x[k])^2 + \lambda J(f)

   is minimized, where J(f) := \int f''(t)^2 dt is the quadratic roughness
   functional use. Since J(f) is quadratic one can find a matrix \Sigma such
   that J(g) = c^T{\Sigma}c where c is the vector of spline coefficients.
   With J(f) defined as above the elements of \Sigma becomes

     \Sigma_{ij} = \int \beta_i''(t)\beta_j''(t) dt

   where \beta(t) is the vector of B-spline base functions. Finally, writing
   the matrix W as W := diag(\sqrt{w}) one can write L(f) as

     L(f) = (y - f)^T W^2 (y - f) + \lambda c^T{\Sigma}c

   which is the form used in help(smooth.spline). So back to my question,
   using smooth.spline(), how can I get 

     1) the value of L(f(x)) for a my fitted (x,y) data,
     2) the value of roughness penalty c^T{\Sigma}c,
     3) the value of (y - f)^T W^2 (y - f)?

   Does smooth.spline() return any of these directly or indirectly?

   Thanks

   Henrik Bengtsson

   Dept. of Mathematical Statistics @ Centre for Mathematical Sciences 
   Lund Institute of Technology/Lund University, Sweden (+2h UTC)
   Office: P316, +46 46 222 9611 (phone), +46 46 222 4623 (fax)
   h b @ m a t h s . l t h . s e
   http://www.maths.lth.se/bioinformatics/


B-splines are NOT natural splines, technically.

You could use ssanova(y~x) in the gss package to calculate natural
cubic spline, but you have to use one of the automatic smoothing
parameter selection methods implemented.

The penalty (with the multiple lambda) is given in
summary(ssanova(y~x))$penalty, the residual sum of squares is in $rss.

Chong Gu
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