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(gam="outer",outer="newton",gcv="deviance",
               family=NULL)

### Arguments

`gam` | Which method to use in the generalized case (i.e. all case other than gaussian with identity link). `"perf"` for the performance iteration (see details) with `magic` as the basic 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"` (default) for modified Newton method backed up by steepest descent, based on exact first and second derivatives. `"bfgs"` for a hybrid Newton- Quasi-Newton approach which can be faster for models with many smoothing parameters. `"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"` 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 (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 (see Wood, 2008).
`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.

### Details

The default methods used by `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.

### 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. (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

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

### See Also

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

[Package mgcv version 1.4-1.1 Index]