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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).
In practice, smooth.construct is usually called via smooth.construct2 and the wrapper
function smoothCon, in order to handle by variables and
centering constraints (see the smoothCon documentation if
you need to handle these things directly, for a user defined smooth class).
smooth.construct(object,data,knots)
smooth.construct2(object,data,knots)is a smooth specification object, generated by an s or te term in a GAM
formula. Objects generated by s terms have class xx.smooth.spec where xx is given by the
bs argument of s (this convention allows the user to add their own smoothers).
If object is not class tensor.smooth.spec it will have the following elements:
The names of the covariates for this smooth, in an array.
Argument k of the s term generating the object. This is the dimension of the basis
used to represent the term (or, arguably, 1 greater than the basis dimension for cc terms).
bs.dim<0 indicates that the constructor should set this to the default value.
TRUE if the term is to be unpenalized, otherwise FALSE.
the number covariates of which this smooth is a function.
the order of the smoothness penalty or NA for autoselection of this. This is argument
m of the s term that generated object.
the name of any by variable to multiply this term as supplied as an argument to s.
"NA" if there is no such term.
A suitable label for use with this term.
An object containing information that may be needed for basis setup
(used, e.g. by "tp" smooths to pass optional information on big dataset
handling).
Any identity associated with this term --- used for linking bases
and smoothing parameters. NULL by default, indicating no linkage.
Smoothing parameters for the term. Any negative are estimated,
otherwise they are fixed at the supplied value. Unless NULL (default),
over-rides sp argument to gam.
If object is of class tensor.smooth.spec then it was generated by a te term in the GAM formula,
and specifies a smooth of several variables with a basis generated as a tensor product of lower dimensional bases.
In this case the object will be different and will have the following elements:
is a list of smooth specification objects of the type listed above, defining the bases which have their tensor product formed in order to construct this term.
is the array of names of the covariates that are arguments of the smooth.
is the name of any by variable, or "NA".
is an array, the elements of which indicate whether (TRUE) any of the margins in the
tensor product should be unpenalized.
A suitable label for use with this term.
is the number of covariates of which this smooth is a function.
TRUE if multiple penalties are to be used.
TRUE if 1-D marginal smooths are to be re-parameterized in terms of
function values.
Any identity associated with this term --- used for linking bases
and smoothing parameters. NULL by default, indicating no linkage.
Smoothing parameters for the term. Any negative are estimated,
otherwise they are fixed at the supplied value. Unless NULL (default),
over-rides sp argument to gam.
For smooth.construct a data frame or list containing the evaluation of the elements of object$term,
with names given by object$term. The last entry will be the by variable, if object$by
is not "NA". For smooth.construct2 data need only be an object within which object$term
can be evaluated, the variables can be in any order, and there can be irrelevant variables present as well.
an optional data frame or list containing the knots relating to object$term.
If it is NULL then the knot locations are generated automatically. The structure of knots should
be as for data, depending on whether smooth.construct or smooth.construct2 is used.
The input argument object, assigned a new class to indicate what type of smooth it is and with at least the
following items added:
The model matrix from this term. This may have an "offset"
attribute: a vector of length nrow(X) containing any contribution of
the smooth to the model offset term. by variables do not need to be
dealt with here, but if they are then an item by.done must be added to
the object.
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.
An array giving the ranks of the penalties.
The dimension of the penalty null space (before centering).
The following items may be added:
The matrix defining any identifiability constraints on the term, for use when fitting. If this is NULL then
smoothCon will add an identifiability constraint that each term should
sum to zero over the covariate values. Set to a zero row matrix if no
constraints are required. If a supplied C has an attribute "always.apply" then it is never ignored, even if any
by variables of a smooth imply that no constraint is actually needed. Code for creating C should check whether
the specification object already contains a zero row matrix, and leave this unchanged if it is (since this signifies
no constraint should be produced).
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.
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 gamm requires it).
Turning off rescaling is useful if the values of the smoothing parameters should be interpretable in a model,
for example because they are inverse variance components.
the degrees of freedom associated with this term (when
unpenalized and unconstrained). If this is null then smoothCon will set it to the basis
dimension. smoothCon will reduce this by the number of constraints.
0 if this term should not be used as a tensor product marginal, 1 if
it can be used and plotted, and 2 is it can be used but not plotted. Set to 1 if NULL.
Set to FALSE if this smooth should not be plotted by plot.gam. Set to TRUE if NULL.
Set to FALSE to ensure that the smooth is never subject to side constraints as a result of nesting.
smooths may depend on fewer `underlying' smoothing parameters than there are elements of
S. In this case L is the matrix mapping the vector of underlying log smoothing
parameters to the vector of logs of the smoothing parameters actually multiplying the S[[i]].
L=NULL signifies that there is one smoothing parameter per S[[i]].
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.
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.
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
# NOT RUN {
## 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)
# }
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