R: Prediction/Construction wrapper functions for GAM smooth...

smoothCon

R Documentation

Prediction/Construction wrapper functions for GAM smooth terms

Description

Wrapper functions for construction of and prediction from smooth
terms in a GAM. The purpose of the wrappers is to allow user-transparant
re-parameterization of smooth terms, in order to allow identifiability
constraints to be absorbed into the parameterization of each term, if required.
The routine also handles ‘by’ variables and construction of identifiability constraints automatically,
although this behaviour can be over-ridden.

is a smooth specification object or a smooth object.

data

A data frame, model frame or list containing the values of the
(named) covariates at which the smooth term is to be
evaluated. If it's a list then n must be supplied.

knots

An optional data frame supplying any knot locations to be
supplied for basis construction.

absorb.cons

Set to TRUE in order to have identifiability
constraints absorbed into the basis.

scale.penalty

should the penalty coefficient matrix be scaled to have
approximately the same ‘size’ as the inner product of the terms model matrix
with itself? This can improve the performance of gamm fitting.

n

number of values for each covariate, or if a covariate is a matrix,
the number of rows in that matrix: must be supplied explicitly if data is a list.

dataX

Sometimes the basis should be set up using data in data, but the model matrix
should be constructed with another set of data provided in dataX — n is assumed to
be the same for both. Facilitates smooth id's.

null.space.penalty

Should an extra penalty be added to the smooth which will penalize the
components of the smooth in the penalty null space: provides a way of penalizing terms out of the model altogether.

apply.by

set to FALSE to have basis setup exactly as in default case, but to return add an additional
matrix X0 to the return object, containing the model matrix without the by variable, if a by
variable is present. Useful for bam discrete method setup.

sparse.cons

If 0 then default sum to zero constraints are used. If -1 then sweep and
drop sum to zero constraints are used (default with bam). If 1 then one
coefficient is set to zero as constraint for sparse smooths. If 2 then sparse coefficient sum to zero
constraints are used for sparse smooths. None of these options has an effect if the smooth supplies its own
constraint.

diagonal.penalty

If TRUE then the smooth is reparameterized to turn the penalty into an identity matrix,
with the final diagonal elements zeroed (corresponding to the penalty nullspace). May result in a matrix diagRP in the returned object for use by PredictMat.

Details

These wrapper functions exist to allow smooths specified using
smooth.construct and Predict.matrix method
functions to be re-parameterized so that identifiability constraints are no
longer required in fitting. This is done in a user transparent
manner, but is typically of no importance in use of GAMs. The routine's
also handle by variables and will create default identifiability
constraints.

If a user defined smooth constructor handles by variables itself, then its
returned smooth object should contain an object by.done. If this does not exist
then smoothCon will use the default code. Similarly if a user defined Predict.matrix
method handles by variables internally then the returned matrix should have a
"by.done" attribute.

Default centering constraints, that terms should sum to zero over the covariates, are produced unless
the smooth constructor includes a matrix C of constraints. To have no constraints (in which case
you had better have a full rank penalty!) the matrix C should have no rows. There is an option to
use centering constraint that generate no, or limited infil, if the smoother has a sparse model matrix.

smoothCon returns a list of smooths because factor by variables result in multiple copies
of a smooth, each multiplied by the dummy variable associated with one factor level. smoothCon modifies
the smooth object labels in the presence of by variables, to ensure that they are unique, it also stores
the level of a by variable factor associated with a smooth, for later use by PredictMat.

The parameterization used by gam can be controlled via
gam.control.

Value

From smoothCon a list of smooth objects returned by the
appropriate smooth.construct method function. If constraints are
to be absorbed then the objects will have attributes "qrc" and
"nCons". "nCons" is the number of constraints. "qrc" is
usually the qr decomposition of the constraint matrix (returned by
qr), but if it is a single positive integer it is the index of the
coefficient to set to zero, and if it is a negative number then this indicates that
the parameters are to sum to zero.

For predictMat a matrix which will map the parameters associated with
the smooth to the vector of values of the smooth evaluated at the covariate
values given in object.

## example of using smoothCon and PredictMat to set up a basis
## to use for regression and make predictions using the result
library(MASS) ## load for mcycle data.
## set up a smoother...
sm <- smoothCon(s(times,k=10),data=mcycle,knots=NULL)[[1]]
## use it to fit a regression spline model...
beta <- coef(lm(mcycle$accel~sm$X-1))
with(mcycle,plot(times,accel)) ## plot data
times <- seq(0,60,length=200) ## creat prediction times
## Get matrix mapping beta to spline prediction at 'times'
Xp <- PredictMat(sm,data.frame(times=times))
lines(times,Xp%*%beta) ## add smooth to plot
## Same again but using a penalized regression spline of
## rank 30....
sm <- smoothCon(s(times,k=30),data=mcycle,knots=NULL)[[1]]
E <- t(mroot(sm$S[[1]])) ## square root penalty
X <- rbind(sm$X,0.1*E) ## augmented model matrix
y <- c(mcycle$accel,rep(0,nrow(E))) ## augmented data
beta <- coef(lm(y~X-1)) ## fit penalized regression spline
Xp <- PredictMat(sm,data.frame(times=times)) ## prediction matrix
with(mcycle,plot(times,accel)) ## plot data
lines(times,Xp%*%beta) ## overlay smooth