| smooth.terms {mgcv} | R Documentation |
Smooth terms are specified in a 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
bs="tp". These are low rank isotropic smoothers of any number of covariates. By isotropic is
meant that rotation of the covariate co-ordinate system will not change the result of smoothing. By low rank is meant
that they have far fewer coefficients than there are data to smooth. They are reduced rank versions of the thin plate splines and use the thin plate spline penalty. They are the default
smooth for s terms because there is a defined sense in which they are the optimal smoother of any given
basis dimension/rank (Wood, 2003). Thin plate regression splines do not have `knots'
(at least not in any conventional sense): a truncated eigen-decomposition is used to achieve the rank reduction. See tprs for further details.
bs="ts" is as "tp" but with a small ridge penalty added to the smoothing penalty, so that the whole term can be
shrunk to zero.
bs="cr".
These have a cubic spline basis defined by a modest sized
set of knots spread evenly through the
covariate values. They are penalized by the conventional intergrated square second derivative cubic spline penalty.
For details see cubic.regression.spline and e.g. Wood (2006a).
bs="cs" specifies a shrinkage version of "cr".
bs="cc" specifies a cyclic cubic regression splines. i.e. a penalized cubic regression splines whose ends match, up to second
derivative.
bs="ps".
These are P-splines as proposed by Eilers and Marx (1996). They combine a B-spline basis, with a discrete penalty
on the basis coefficients, and any sane combination of penalty and basis order is allowed. Although this penalty has no exact interpretation in terms of function shape, in the way that the derivative penalties do, P-splines perform almost as well as conventional splines in many standard applications, and can perform better in particular cases where it is advantageous to mix different orders of basis and penalty.
bs="cs" gives a cyclic version of a P-spline.
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. 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.
Finally univariate and bivariate adaptive (bs="ad") 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.
Simon Wood <simon.wood@r-project.org>
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
s, te, tprs, cubic.regression.spline,
p.spline, adaptive.smooth, user.defined.smooth
## see examples for gam and gamm