gam.method {mgcv}R Documentation

Setting GAM fitting method

Description

This is a function of package 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.

Usage

gam.method(am="magic",gam="outer",outer="newton",gcv="deviance",
           family=NULL)

Arguments

am Which method to use for a pure additive model (i.e. identity link, gaussian errors). Either "magic" if the Wood (2004) method (magic) is to be used, or "mgcv" if the faster, but less stable and general Wood (2000) method (mgcv) should be used.
gam Which method to use in the generalized case (i.e. all case other than gaussian with identity link). "perf.magic" for the performance iteration (see details) with magic as the basic estimation engine. "perf.mgcv" for performance iteration with mgcv as the underlying estimation engine. "perf.outer" for magic based performance iteration followed by outer iteration (see details). "outer" for pure outer iteration.
outer The optimization approach to use to optimize log smoothing parameters by outer iteration."newton" for modified Newton method backed up by steepest descent, based on exact first and second derivatives. "nlm" to use nlm with exact first derivatives to optimize the smoothness selection criterion. "nlm.fd" to use nlm with finite differenced first derivatives (slower and less reliable). "optim" to use the "L-BFGS-B" quasi-Newton method option of routine optim, with exact first derivatives.
gcv One of "deviance", "GACV" or "pearson", 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 (1990, section 6.9). "GACV" (only available with outer method "newton") uses a varient of Gu and Xiang's (2001) generalized approximate cross validation, modified to deal with arbitrary link-error combinations. "pearson" uses a suggestion from the literature of replacing the RSS in a GCV score with the Pearson statistic: this is a very poor choise (see Wood, 2006, section 4.5.4). This latter option is not available with outer methods "newton" and "nlm".
family The routine is called by gam 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 with performance iteration, and the method is reset to this if necessary.

Details

The performance 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. It is usually very fast, since smoothing parameters are estimated alongside other model coefficients in a single P-IRLS process.

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.

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.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. 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. An 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.

In summary: performance iteration is fast, but can fail to converge. Outer iteration is a little slower, but more reliable. At present only performance iteration is available for negative binomial families.

Author(s)

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

References

Gu and Wahba (1991) Minimizing GCV/GML scores with multiple smoothing parameters via the Newton method. SIAM J. Sci. Statist. Comput. 12:383-398

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.

http://www.maths.bath.ac.uk/~sw283/

See Also

gam.control gam, gam.fit, glm.control


[Package mgcv version 1.3-23 Index]