gam.fit3                package:mgcv                R Documentation

P-IRLS GAM estimation with GCV & UBRE/AIC derivative calculation

Description:

     Estimation of GAM smoothing parameters is most stable if
     optimization of the UBRE/AIC 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 (any quadratically penalized GLM)
     given log  smoothing paramaters, and evaluates derivatives of the
     GCV and UBRE/AIC 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/AIC
     score minimization.

     The approach is to run the P-IRLS to convergence, and only then to
     iterate for first and second derivatives. 

     Not normally called directly, but rather service routines for
     'gam'.

Usage:

     gam.fit3(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=2,use.svd=TRUE,
              gamma=1,scale=1,printWarn=TRUE,...) 

Arguments:

       x: The model matrix for the GAM (or any penalized GLM).

       y: The response variable.

      sp: The log smoothing parameters.

       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. 'off[i]' indicates which
          parameter is the first one penalized by 'S[[i]]'.

      rS: List of square roots of penalty matrices, each having as few
          columns as possible, but as many rows as there are
          parameters.

     off: 'off[i]' indicates which parameter 'S[[i]][1,1]' relates to.

       H: The fixed penalty matrix for the model.

 weights: prior weights for fitting.

   start: optional starting parameter guesses.

etastart: optional starting values for the linear predictor.

 mustart: optional starting values for the mean.

  offset: the model offset

  family: the family - actually this routine would never be called with
          'gaussian()'

 control: control list as returned from 'glm.control'

intercept: does the model have and intercept, 'TRUE' or 'FALSE'

   deriv: Should derivatives of the GCV and UBRE/AIC scores be
          calculated? 0, 1 or 2, indicating the maximum order of
          differentiation to apply.

 use.svd: Should the algorithm use SVD ('TRUE') or the cheaper QR
          ('FALSE') as the second matrix decomposition of the final
          derivative/coefficient evaluation method? 

   gamma: The weight given to each degree of freedom in the GCV and
          UBRE scores can be varied (usually increased) using this
          parameter.

   scale: The scale parameter - needed for the UBRE/AIC score.

printWarn: Set to 'FALSE' to suppress some warnings. Useful in order to
          ensure that some warnings are only printed if they apply to
          the final fitted model, rather than an intermediate used in
          optimization.

     ...: Other arguments: ignored.

Details:

     This routine is basically '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 an extra iteration  and 
     (iii) derivatives of the GAM GCV and UBRE/AIC scores to be
     evaluated at convergence. 

     In addition the routines apply step halving to any step that
     increases the penalized deviance substantially.

     The most costly parts of the calculations 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. 

     Note that use of non-standard families with this routine requires
     modification of the families as described in 'fix.family.link'.

Author(s):

     Simon N. Wood simon.wood@r-project.org

     The routine has been modified from 'glm.fit' in R 2.0.1, written
     by the R core (see 'glm.fit' for further credits).

References:

     O 'Sullivan, Yandall & Raynor (1986) Automatic smoothing of
     regression functions in generalized linear models. J. Amer.
     Statist. Assoc. 81:96-103.

     Wood, S.N. (2008) Fast stable direct fitting and smoothness
     selection for generalized additive models. J.R.Statist. Soc. B
     70(3):495-518

     <URL: http://www.maths.bath.ac.uk/~sw283/>

See Also:

     'gam.fit',  'gam', 'mgcv', 'magic'

