\HeaderA{gam.fit2}{P-IRLS GAM estimation with GCV \& UBRE derivative calculation}{gam.fit2}
\keyword{models}{gam.fit2}
\keyword{smooth}{gam.fit2}
\keyword{regression}{gam.fit2}
\begin{Description}\relax
Estimation of GAM smoothing parameters is most stable if
optimization of the UBRE or GCV score is outer to the penalized iteratively
re-weighted least squares scheme used to estimate the model given smoothing 
parameters.

This routine estimates a GAM given log smoothing paramaters, and evaluates the
first derivative of the GCV and UBRE scores of the model with respect to the
log smoothing parameters. Calculation of exact derivatives is generally faster
than approximating them by finite differencing, as well as generally improving
the reliability of GCV/UBRE score minimization.

Not normally called directly, but rather a service routine for \code{\LinkA{gam}{gam}}.
\end{Description}
\begin{Usage}
\begin{verbatim}
gam.fit2(x, y, sp, S=list(),rS=list(),off, H=NULL, 
         weights = rep(1, nobs), start = NULL, etastart = NULL, 
         mustart = NULL, offset = rep(0, nobs), family = gaussian(), 
         control = gam.control(), intercept = TRUE,deriv=TRUE,
         gamma=1,scale=1,pearson=FALSE)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] The model matrix for the GAM.
\item[\code{y}] The response variable.
\item[\code{sp}] The log smoothing parameters.
\item[\code{S}] A list of penalty matrices. Typically penalty matrices contain only a
smallish square sub-matrix which is non-zero: this is what is actually
stored. \code{off[i]} indicates which parameter is the first one penalized
by \code{S[[i]]}.
\item[\code{rS}] List of square roots of penalty matrices, each having as few columns
as possible, but as many rows as there are parameters.
\item[\code{off}] \code{off[i]} indicates which parameter \code{S[[i]][1,1]} relates to.
\item[\code{H}] The fixed penalty matrix for the model.
\item[\code{weights}] prior weights for fitting.
\item[\code{start}] optional starting parameter guesses.
\item[\code{etastart}] optional starting values for the linear predictor.
\item[\code{mustart}] optional starting values for the mean.
\item[\code{offset}] the model offset
\item[\code{family}] the family - actually this routine would never be called with \code{gaussian()}
\item[\code{control}] control list as returned from \code{\LinkA{glm.control}{glm.control}}
\item[\code{intercept}] does the model have and intercept, \code{TRUE} or
\code{FALSE}
\item[\code{deriv}] Should derivatives of the GCV and UBRE scores be calculated?
\code{TRUE} or \code{FALSE}
\item[\code{gamma}] The weight given to each degree of freedom in the GCV and UBRE
scores can be varied (usually increased) using this parameter.
\item[\code{scale}] The scale parameter - needed for the UBRE score.
\item[\code{pearson}] The GCV/UBRE score can be based either on the Pearson statistic
or the deviance. The latter is generally to be preferred, as it is less prone
to severe undersmoothing.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
This routine is basically \code{\LinkA{glm.fit}{glm.fit}} with some
modifications to allow (i) for quadratic penalties on the log likelihood;
(ii) derivatives of the model coefficients with respect to
log smoothing parameters to be obtained (by updating alongside the P-IRLS
iteration) and (iii) derivatives of the GAM GCV and UBRE scores to be
evaluated at convergence. 

In addition the routine applies step halving to any step that increases the
penalized deviance substantially.

The most costly parts of the calculation are performed by calls to compiled C
code (which in turn calls LAPACK routines) in place of the compiled code that
would usually perform least squares estimation on the working model in the
IRLS iteration. 

Estimation of smoothing parameters by optimizing GCV scores obtained at
convergence of the P-IRLS iteration was proposed by O'Sullivan et al. (1986),
and is here termed `outer' iteration.
\end{Details}
\begin{Author}\relax
Simon N. Wood \email{simon.wood@r-project.org}

The routine has been modified from \code{glm.fit} in R 2.0.1, written
by the R core (see \code{\LinkA{glm.fit}{glm.fit}} for further credits).
\end{Author}
\begin{References}\relax
O 'Sullivan, Yandall \& Raynor (1986) Automatic smoothing of regression
functions in generalized linear models. J. Amer. Statist. Assoc. 81:96-103.

\url{http://www.stats.gla.ac.uk/~simon/}
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{gam.fit}{gam.fit}},  \code{\LinkA{gam}{gam}}, \code{\LinkA{mgcv}{mgcv}}, \code{\LinkA{magic}{magic}}
\end{SeeAlso}

