mgcv which allows
selection of the numerical method used to optimize the smoothing parameter
estimation criterion for a gam. It is used to set argument method of gam.gam.method(gam="outer",outer="newton",gcv="deviance",
family=NULL)"perf" for the performance
iteration (see details) with magic as the basic estimation
engine. <"newton" (default) for modified Newton method backed up by steepest
descent, based on exact first and second derivatives. "bfgs" for a hybri"deviance" or "GACV",
specifying the flavour of GCV to use with outer iteration. "deviance"
simply replaces the residual sum of squares term in a GCV score with the
deviance, following Hastie and Tibshirani (1gam to check the supplied
method argument. In this circumstance the family argument is passed, to check
that it works with the specified method.gam are based on Newton type optimization of
GCV/UBRE/AIC scores with respect to smoothing parameters, as described in Wood (2004)
and Wood (2008) for the additive and generalized additive model cases respectively.
The smoothing criteria (GCV etc) are evaluated for the fitted model itself (rather than some working
approximate model). Since this involves optimizing the criteria `outside' the PIRLS or LS method used
for fitting, it is referred to as `outer' iteration. In the generalized case several alternative optimisation methods can be used for outer optimization. Usually the fastest and most reliable approach is to use a modified Newton optimizer with exact first and second derivatives, and this is the default. However if there are large numbers of smoothing parameters then it can be faster to use a hybrid Newton-BFGS quasi-Newton approach, which avoids the expense of frequent full second derivative evaluation.
nlm can be used with finite differenced first derivatives. This is not ideal theoretically, since
it is possible for the finite difference estimates of derivatives to be very
badly in error on rare
occasions when the P-IRLS convergence tolerance is close to being matched
exactly, so that two components of a finite differenced derivative require
different numbers of iterations of P-IRLS in their evaluation. An alternative
is provided in which nlm uses numerically exact first derivatives, this
is faster and less problematic than the other scheme. A further alternative is to use a quasi-Newton
scheme with exact derivtives, based on optim. In practice this usually
seems to be slower than the nlm method.
The alternative approach of `performance oriented iteration' was suggested by Gu (and is rather similar to the PQL method in generalized linear mixed modelling). At each step of the P-IRLS (penalized iteratively reweighted least squares) iteration, by which a gam is fitted, the smoothing parameters are estimated by GCV or UBRE applied to the working penalized linear modelling problem. In most cases, this process converges and gives smoothness estimates that perform well (see e.g. Wood 2004).
The performance iteration has two disadvantages. (i) in the presence of co-linearity or concurvity (a frequent problem when spatial smoothers are included in a model with other covariates) then the process can fail to converge. Suppose we start with some coefficient and smoothing parameter estimates, implying a working penalized linear model: the optimal smoothing parameters and coefficients for this working model may in turn imply a working model for which the original estimates are better than the most recent estimates. This sort of effect can prevent convergence.
Secondly it is often possible to find a set of smoothing parameters that result in a lower GCV or UBRE score, for the final working model, than the final score that results from the performance iterations. This is because the performance iteration is only approximately optimizing this score (since optimization is only performed on the working model). The disadvantage here is not that the model with lower score would perform better (it usually doesn't), but rather that it makes model comparison on the basis of GCV/UBRE score rather difficult.
In summary: performance iteration can fail to converge. It may occasionally be faster than outer iteration, but since the Wood (2008) method `outer' iteration is faster in most cases.
Wood, S.N. (2004) Stable and efficient multiple smoothing parameter estimation for generalized additive models. J. Amer. Statist. Ass.
Wood, S.N. (2008) Fast stable direct fitting and smoothness selection for generalized additive models. J.R.Statist.Soc.B 70(3):495-518
gam.control gam, gam.fit, glm.control