Estimation of GAM smoothing parameters is most stable if optimization of the UBRE/AIC, GCV, GACV, REML or ML score is outer to the penalized iteratively re-weighted least squares scheme used to estimate the model given smoothing parameters.

This routine estimates a GAM (any quadratically penalized GLM) given log smoothing paramaters, and evaluates derivatives of the smoothness selection scores of the model with respect to the log smoothing parameters. Calculation of exact derivatives is generally faster than approximating them by finite differencing, as well as generally improving the reliability of GCV/UBRE/AIC/REML score minimization.

The approach is to run the P-IRLS to convergence, and only then to iterate for first and second derivatives.

Not normally called directly, but rather service routines for `gam`

.

```
gam.fit3(x, y, sp, Eb ,UrS=list(),
weights = rep(1, nobs), start = NULL, etastart = NULL,
mustart = NULL, offset = rep(0, nobs), U1 = diag(ncol(x)),
Mp = -1, family = gaussian(), control = gam.control(),
intercept = TRUE,deriv=2,gamma=1,scale=1,
printWarn=TRUE,scoreType="REML",null.coef=rep(0,ncol(x)),
pearson.extra=0,dev.extra=0,n.true=-1,Sl=NULL,...)
```

x

The model matrix for the GAM (or any penalized GLM).

y

The response variable.

sp

The log smoothing parameters.

Eb

A balanced version of the total penalty matrix: usd for numerical rank determination.

UrS

List of square root penalties premultiplied by transpose of orthogonal basis for the total penalty.

weights

prior weights for fitting.

start

optional starting parameter guesses.

etastart

optional starting values for the linear predictor.

mustart

optional starting values for the mean.

offset

the model offset

U1

An orthogonal basis for the range space of the penalty --- required for ML smoothness estimation only.

Mp

The dimension of the total penalty null space --- required for ML smoothness estimation only.

family

the family - actually this routine would never be called with `gaussian()`

control

control list as returned from `glm.control`

intercept

does the model have and intercept, `TRUE`

or
`FALSE`

deriv

Should derivatives of the GCV and UBRE/AIC scores be calculated? 0, 1 or 2, indicating the maximum order of differentiation to apply.

gamma

The weight given to each degree of freedom in the GCV and UBRE scores can be varied (usually increased) using this parameter.

scale

The scale parameter - needed for the UBRE/AIC score.

printWarn

Set to `FALSE`

to suppress some warnings. Useful in
order to ensure that some warnings are only printed if they apply to the final
fitted model, rather than an intermediate used in optimization.

scoreType

specifies smoothing parameter selection criterion to use.

null.coef

coefficients for a model which gives some sort of upper bound on deviance. This allows immediate divergence problems to be controlled.

pearson.extra

Extra component to add to numerator of pearson statistic in P-REML/P-ML smoothness selection criteria.

dev.extra

Extra component to add to deviance for REML/ML type smoothness selection criteria.

n.true

Number of data to assume in smoothness selection criteria. <=0 indicates that it should be the
number of rows of `X`

.

Sl

A smooth list suitable for passing to gam.fit5.

...

Other arguments: ignored.

This routine is basically `glm.fit`

with some
modifications to allow (i) for quadratic penalties on the log likelihood;
(ii) derivatives of the model coefficients with respect to
log smoothing parameters to be obtained by use of the implicit function theorem and
(iii) derivatives of the GAM GCV, UBRE/AIC, REML or ML scores to be
evaluated at convergence.

In addition the routines apply step halving to any step that increases the penalized deviance substantially.

The most costly parts of the calculations are performed by calls to compiled C code (which in turn calls LAPACK routines) in place of the compiled code that would usually perform least squares estimation on the working model in the IRLS iteration.

Estimation of smoothing parameters by optimizing GCV scores obtained at convergence of the P-IRLS iteration was proposed by O'Sullivan et al. (1986), and is here termed `outer' iteration.

Note that use of non-standard families with this routine requires modification
of the families as described in `fix.family.link`

.

Wood, S.N. (2011) Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. Journal of the Royal Statistical Society (B) 73(1):3-36

O 'Sullivan, Yandall & Raynor (1986) Automatic smoothing of regression functions in generalized linear models. J. Amer. Statist. Assoc. 81:96-103.