gam formula using s and te terms.
Various smooth classes are available, for different modelling tasks, and users can add smooth classes
(see user.defined.smooth). What defines a smooth class is the basis used to represent
the smooth function and quadratic penalty (or multiple penalties) used to penalize
the basis coefficients in order to control the degree of smoothness. Smooth classes are
invoked directly by s terms, or as building blocks for tensor product smoothing
via te terms (only smooth classes with single penalties can be used in tensor products). The smooths
built into the mgcv package are all based one way or another on low rank versions of splines. For the full rank
versions see Wahba (1990).Note that smooths can be used rather flexibly in gam models. In particular the linear predictor of the GAM can
depend on (a discrete approximation to) any linear functional of a smooth term, using by variables and the
`summation convention' explained in linear.functional.terms.
The single penalty built in smooth classes are summarized as follows [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Broadly speaking the default penalized thin plate regression splines tend to give the best MSE performance, but they are a little slower to set up than the other bases. The knot based penalized cubic regression splines (with derivative based penalties) usually come next in MSE performance, with the P-splines doing just a little worse. However the P-splines are useful in non-standard situations.
All the preceding classes (and any user defined smooths with single penalties) may be used as marginal
bases for tensor product smooths specified via te terms, except for "re" terms. Tensor
product smooths are smooth functions
of several variables where the basis is built up from tensor products of bases for smooths of fewer (usually one)
variable(s) (marginal bases). The multiple penalties for these smooths are produced automatically from the
penalties of the marginal smooths. Wood (2006b) give the general recipe for this construction.
Tensor product smooths often perform better than isotropic smooths when the covariates of a smooth are not naturally
on the same scale, so that their relative scaling is arbitrary. For example, if smoothing with repect to time and
distance, an isotropic smoother will give very different results if the units are cm and minutes compared to if the units are
metres and seconds: a tensor product smooth will give the same
answer in both cases (see te for an example of this). Note that tensor product terms are knot based, and the
thin plate splines seem to offer no advantage over cubic or P-splines as marginal bases.
Some further specialist smoothers that are not suitable for use in tensor products are also available.
[object Object],Univariate and bivariate adaptive smooths are available (see adaptive.smooth).
These are appropriate when the degree of smoothing should itself vary with the covariates to be smoothed, and the
data contain sufficient information to be able to estimate the appropriate variation. Because this flexibility is
achieved by splitting the penalty into several `basis penalties' these terms are not suitable as components of tensor
product smooths, and are not supported by gamm.,[object Object],Smooth factor interactions are often produced using by variables (see gam.models), but a special smoother
class (see factor.smooth.interaction) is available for the case in which a smooth is required at each of a large number of factor levels (for example a smooth for each patient in a study), and each smooth should have the same smoothing parameter. The "fs" smoothers are set up to be efficient when used with gamm, and have penalties on each null sapce component (i.e. they are fully `random effects').
Wahba (1990) Spline Models of Observational Data. SIAM
Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95-114
Wood, S.N. (2006a) Generalized Additive Models: an introduction with R, CRC
Wood, S.N. (2006b) Low rank scale invariant tensor product smooths for generalized additive mixed models. Biometrics 62(4):1025-1036
s, te, tprs, cubic.regression.spline,
p.spline, adaptive.smooth, user.defined.smooth## see examples for gam and gammRun the code above in your browser using DataLab