smooth.terms: Smooth terms in GAM

Description

Smooth terms are specified in a gam formula using s, te, ti and t2 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, ti or t2 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],[object Object]

Broadly speaking the default penalized thin plate regression splines tend to give the best MSE performance, but they are 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, ti or t2 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) and Wood, Scheipl and Faraway (2012), give the general recipe for these constructions. The te construction results in fewer, more interpretable, penalties, while the t2 construction is more natural if you are interested in functional ANOVA decompositions. t2 works with the gamm4 package.

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 te 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],[object Object]

Arguments

References

Eilers, P.H.C. and B.D. Marx (1996) Flexible Smoothing with B-splines and Penalties. Statistical Science, 11(2):89-121

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

Wood S.N., F. Scheipl and J.J. Faraway (2012) Straightforward intermediate rank tensor product smoothing in mixed models. Statistical Computing.

Examples

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Description

Arguments

References

See Also

Examples