Constructor functions for smooth terms in a GAM
Smooth terms in a GAM formula are turned into smooth specification objects of
class xx.smooth.spec during processing of the formula. Each of these objects is
converted to a smooth object using an appropriate smooth.construct function. New smooth classes
can be added by writing a new smooth.construct method function and a corresponding
Predict.matrix method function (see example code below).
smooth.construct(object,data,knots) smooth.construct2(object,data,knots)
object |
is a smooth specification object, generated by an
If
|
data |
For |
knots |
an optional data frame or list containing the knots relating to |
There are built in methods for objects with the following classes:
tp.smooth.spec (thin plate regression splines: see tprs);
ts.smooth.spec (thin plate regression splines with shrinkage-to-zero);
cr.smooth.spec (cubic regression splines: see cubic.regression.spline;
cs.smooth.spec (cubic regression splines with shrinkage-to-zero);
cc.smooth.spec (cyclic cubic regression splines);
ps.smooth.spec (Eilers and Marx (1986) style P-splines: see p.spline);
cp.smooth.spec (cyclic P-splines);
ad.smooth.spec (adaptive smooths of 1 or 2 variables: see adaptive.smooth);
re.smooth.spec (simple random effect terms);
mrf.smooth.spec (Markov random field smoothers for smoothing over discrete districts);
tensor.smooth.spec (tensor product smooths).
There is an implicit assumption that the basis only depends on the knots and/or the set of unique covariate combinations; i.e. that the basis is the same whether generated from the full set of covariates, or just the unique combinations of covariates.
Plotting of smooths is handled by plot methods for smooth objects. A default mgcv.smooth method
is used if there is no more specific method available. Plot methods can be added for specific smooth classes, see
source code for mgcv:::plot.sos.smooth, mgcv:::plot.random.effect, mgcv:::plot.mgcv.smooth
for example code.
The input argument object, assigned a new class to indicate what type of smooth it is and with at least the
following items added:
X |
The model matrix from this term. This may have an |
S |
A list of positive semi-definite penalty matrices that apply to this term. The list will be empty if the term is to be left un-penalized. |
rank |
An array giving the ranks of the penalties. |
null.space.dim |
The dimension of the penalty null space (before centering). |
The following items may be added:
C |
The matrix defining any identifiability constraints on the term, for use when fitting. If this is |
Cp |
An optional matrix supplying alternative identifiability constraints for use when predicting. By default the fitting constrants are used. This option is useful when some sort of simple sparse constraint is required for fitting, but the usual sum-to-zero constraint is required for prediction so that, e.g. the CIs for model components are as narrow as possible. |
no.rescale |
if this is non-NULL then the penalty coefficient matrix of the smooth will not be
rescaled for enhaced numerical stability (rescaling is the default, because |
df |
the degrees of freedom associated with this term (when
unpenalized and unconstrained). If this is null then |
te.ok |
|
plot.me |
Set to |
side.constrain |
Set to |
L |
smooths may depend on fewer ‘underlying’ smoothing parameters than there are elements of
|
Usually the returned object will also include extra information required to define the basis, and used by
Predict.matrix methods to make predictions using the basis. See
the Details section for links to the information included for the built in smooth classes.
tensor.smooth returned objects will additionally have each element of
the margin list updated in the same way. tensor.smooths also
have a list, XP, containing re-parameterization matrices for any 1-D marginal terms
re-parameterized in terms of function values. This list will have NULL
entries for marginal smooths that are not re-parameterized, and is only long
enough to reach the last re-parameterized marginal in the list.
User defined smooth objects should avoid having attributes names
"qrc" or "nCons" as these are used internally to provide
constraint free parameterizations.
Simon N. Wood simon.wood@r-project.org
Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95-114
Wood, S.N. (2006) Low rank scale invariant tensor product smooths for generalized additive mixed models. Biometrics 62(4):1025-1036
The code given in the example is based on the smooths advocated in:
Ruppert, D., M.P. Wand and R.J. Carroll (2003) Semiparametric Regression. Cambridge University Press.
However if you want p-splines, rather than splines with derivative based penalties, then the built in "ps" class is probably a marginally better bet. It's based on
Eilers, P.H.C. and B.D. Marx (1996) Flexible Smoothing with B-splines and Penalties. Statistical Science, 11(2):89-121
## Adding a penalized truncated power basis class and methods
## as favoured by Ruppert, Wand and Carroll (2003)
## Semiparametric regression CUP. (No advantage to actually
## using this, since mgcv can happily handle non-identity
## penalties.)
smooth.construct.tr.smooth.spec<-function(object,data,knots) {
## a truncated power spline constructor method function
## object$p.order = null space dimension
m <- object$p.order[1]
if (is.na(m)) m <- 2 ## default
if (m<1) stop("silly m supplied")
if (object$bs.dim<0) object$bs.dim <- 10 ## default
nk<-object$bs.dim-m-1 ## number of knots
if (nk<=0) stop("k too small for m")
x <- data[[object$term]] ## the data
x.shift <- mean(x) # shift used to enhance stability
k <- knots[[object$term]] ## will be NULL if none supplied
if (is.null(k)) # space knots through data
{ n<-length(x)
k<-quantile(x[2:(n-1)],seq(0,1,length=nk+2))[2:(nk+1)]
}
if (length(k)!=nk) # right number of knots?
stop(paste("there should be ",nk," supplied knots"))
x <- x - x.shift # basis stabilizing shift
k <- k - x.shift # knots treated the same!
X<-matrix(0,length(x),object$bs.dim)
for (i in 1:(m+1)) X[,i] <- x^(i-1)
for (i in 1:nk) X[,i+m+1]<-(x-k[i])^m*as.numeric(x>k[i])
object$X<-X # the finished model matrix
if (!object$fixed) # create the penalty matrix
{ object$S[[1]]<-diag(c(rep(0,m+1),rep(1,nk)))
}
object$rank<-nk # penalty rank
object$null.space.dim <- m+1 # dim. of unpenalized space
## store "tr" specific stuff ...
object$knots<-k;object$m<-m;object$x.shift <- x.shift
object$df<-ncol(object$X) # maximum DoF (if unconstrained)
class(object)<-"tr.smooth" # Give object a class
object
}
Predict.matrix.tr.smooth<-function(object,data) {
## prediction method function for the `tr' smooth class
x <- data[[object$term]]
x <- x - object$x.shift # stabilizing shift
m <- object$m; # spline order (3=cubic)
k<-object$knots # knot locations
nk<-length(k) # number of knots
X<-matrix(0,length(x),object$bs.dim)
for (i in 1:(m+1)) X[,i] <- x^(i-1)
for (i in 1:nk) X[,i+m+1] <- (x-k[i])^m*as.numeric(x>k[i])
X # return the prediction matrix
}
# an example, using the new class....
require(mgcv)
set.seed(100)
dat <- gamSim(1,n=400,scale=2)
b<-gam(y~s(x0,bs="tr",m=2)+s(x1,bs="ps",m=c(1,3))+
s(x2,bs="tr",m=3)+s(x3,bs="tr",m=2),data=dat)
plot(b,pages=1)
b<-gamm(y~s(x0,bs="tr",m=2)+s(x1,bs="ps",m=c(1,3))+
s(x2,bs="tr",m=3)+s(x3,bs="tr",m=2),data=dat)
plot(b$gam,pages=1)
# another example using tensor products of the new class
dat <- gamSim(2,n=400,scale=.1)$data
b <- gam(y~te(x,z,bs=c("tr","tr"),m=c(2,2)),data=dat)
vis.gam(b)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.