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(am="magic",gam="outer",outer="nlm",pearson=FALSE,family=NULL)"magic" if the Wood (2004) method
(magic) is to be used, or "mgcv" if the faster, but less"perf.magic" for the performance
iteration (see details) with magic as the basic estimation
eng"nlm" to use nlm with exact first derivatives
to optimize the smoothness selection criterion. "nlm.fd" to use
nlm witgam to check the supplied
method argument. In this circumstance the family argument is passed, to check
that it works with the specified method. Negative binomial families only work
wiThe 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.
Both disadvantages of performance iteration are surmountable by using what is
basically O'Sullivan's (1986) suggestion. Here the P-IRLS scheme is iterated
to convergence for a fixed set of smoothing parameters, with an appropriate
GCV/UBRE score evaluated at convergence. This score at convergence is
optimized in some way. This is termed "outer" optimization, since the
optimization is outer to the P-IRLS loop. Outer iteration is slower than
performance iteration.
The `appropriate GCV/UBRE' score in the previous paragraph can be defined in one of two ways either (i) the deviance, or (ii) the Pearson statistic can be used in place of the residual sum of squares in the GCV/UBRE score. (ii) makes the GCV/UBRE score correspond to the score for the working linear model at convergence of the P-IRLS, but in practice tends to result in oversmoothing, particularly with low n binomial data, or low mean counts. Hence the default is to use (i).
Several alternative optimisation methods can be used for outer
optimization. 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. Finally, a quasi-Newton
scheme with exact derivtives can be used instead, based on optim. In practice this usually
seems to be slower than the nlm method.
It is possible to iterate the performance iteration to convergence and then
improve the smoothing parameter estimates using outer iteration: only a few steps are
usually required in the outer iteration in this case, so it may be quite
efficient, but it is not recommended if the performance iteration itself is
non-convergent. When using `pure' outer iteration, a single step of the
performance iteration is in fact taken first, to obtain estimates of the scale
of the GCV/UBRE objective: starting values for the smoothing parameters are
obtained using initial.sp.
In summary: performance iteration is fast, but can fail to converge. Outer iteration is slower, but more reliable. At present only performance iteration is available for negative binomial families.
Wood, S.N. (2000) Modelling and Smoothing Parameter Estimation with Multiple Quadratic Penalties. J.R.Statist.Soc.B 62(2):413-428
Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95-114
Wood, S.N. (2004) Stable and efficient multiple smoothing parameter estimation for generalized additive models. J. Amer. Statist. Ass.
gam.control gam, gam.fit, glm.control