GAM convergence issues.

Description

Sometimes the GAM estimation process fails to converge. There are two possible reasons for this (assuming that the obvious possibility of insufficient iterations being allowed by gam.control has been excluded). Firstly, a GAM is just a (penalized) GLM and the IRLS scheme used to estimate GLMs is not guaranteed to converge. Hence non convergence of a GAM may relate to a lack of stability in the basic IRLS scheme. Secondly, nesting GAM smoothing parameter estimation within the IRLS iteration loop (the mgcv default) can also make convergence more difficult to achieve. Of course, these two scenarios are not really completely distinct, but thinking about convergence in these terms does lead to a useful practical approach for achieving convergence.

Basically, the first step is to establish whether the IRLS iterations are capable of converging. To do this fit the problematic GAM with all smooth terms specified with fx=TRUE so that the smoothing parameters are all fixed at zero. If this `largest' model can converge then it is the smoothing parameter estimation that is causing the problem and convergence can usaully be achieved by setting the gam.control option spIterType="outer". This iterates the IRLS scheme to convergence for each trial set of smoothing parameters, and is usually much slower than the default spIterType="perf", but it avoids the smoothing parameter estimation de-stabilizing the IRLS iterations. Note that this approach does not in anyway change the model being fitted.

If, on the other hand, the model with all smoothing parameters set to zero does not converge, then the problem lies with the IRLS process itself. Having tried increasing maxit in gam.control, there are several other possibilities for stabilizing the iteration. All are based on the fact that too much model flexibility is almost always a major factor in non-convergence. It is possible to try (i) setting lower bounds on the smoothing parameters using the min.sp argument of gam: this may or may not change the model being fitted; (ii) reducing the flexibility of the model by reducing the basis dimensions k in the specification of s and te model terms: this obviously changes the model being fitted somewhat; (iii) introduce a small regularization term into the fitting via the irls.reg argument of gam.control: this option obviously changes the nature of the fit somewhat, since parameter estimates are pulled towards zero by doing this. Once the basic IRLS process has been stabilized, you can try calling gam using the stabilized model and its options. If the default fitting fails again, then retry using the spIterType="outer" option discussed previously.

In most cases model fitting can be made to converge in this way, and the stabilization often has little or no real affect on the estimates. However care and judgement are needed, since some of these techniques have the potential to have a large impact on estimates.

Often, of course, a major contributer to fitting difficulties is that the model is a very poor description of the data.

Author(s)

Simon N. Wood simon@stats.gla.ac.uk