Generalized Additive Model fitting by `outer' iteration, requires extra derivatives of the variance and link functions to be added to family objects. The first 3 functions add what is needed. Model checking can be aided by adding quantile and random deviate generating functions to the family. The final two functions do this.
fix.family.link(fam)
fix.family.var(fam)
fix.family.ls(fam)
fix.family.qf(fam)
fix.family.rd(fam)A family.
A family object with extra component functions dvar,
d2var, d2link, d3link, d4link, ls, and possibly qf and rd,
depending on which functions are called. fix.family.var also adds a variable scale set to
negative to indicate that family has a free scale parameter.
Consider the first 3 function first.
Outer iteration GAM estimation requires derivatives of the GCV, UBRE/gAIC, GACV, REML or ML score, which are obtained by finding the derivatives of the model coefficients w.r.t. the log smoothing parameters, using the implicit function theorem. The expressions for the derivatives require the second and third derivatives of the link w.r.t. the mean (and the 4th derivatives if Fisher scoring is not used). Also required are the first and second derivatives of the variance function w.r.t. the mean (plus the third derivative if Fisher scoring is not used). Finally REML or ML estimation of smoothing parameters requires the log saturated likelihood and its first two derivatives w.r.t. the scale parameter. These functions add functions evaluating these quantities to a family.
If the family already has functions dvar, d2var, d3var, d2link,
d3link, d4link and for RE/ML ls, then these functions simply
return the family unmodified: this allows non-standard links
to be used with gam when using outer iteration (performance
iteration operates with unmodified families). Note that if you only need Fisher scoring then
d4link and d3var can be dummy, as they are ignored. Similalry ls is only needed for
RE/ML.
The dvar function is a function of a mean vector, mu, and returns
a vector of corresponding first derivatives of the family variance
function. The d2link function is also a function of a vector of mean
values, mu: it returns a vector of second derivatives of the link,
evaluated at mu. Higher derivatives are defined similarly.
If modifying your own family, note that you can often get away with supplying
only a dvar and d2var, function if your family only requires links that occur in
one of the standard families.
The second two functions are useful for investigating the distribution of residuals and are used by
qq.gam. If possible the functions add quantile (qf) or random deviate (rd) generating functions
to the family. If a family already has qf or rd
functions then it is left unmodified. qf functions are only available for some families, and for quasi families
neither type of function is available.