mgcv (version 1.1-8)

smooth.construct: Constructor functions for smooth terms in a GAM

Description

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).

Usage

smooth.construct(object,data,knots)

Arguments

object

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

term

is the array of names of the covariates that are arguments of the smooth.

by

is the name of any by variable, or "NA".

fx

is an array, the elements of which indicate whether (TRUE) any of the margins in the tensor product should be unpenalized.

full.call

The full version of the s term, with all defaults expanded explicitly.

label

A suitable label for use with this term.

dim

is the number of covariates of which this smooth is a function.

null.space.dim

The dimension of the null space of the wiggliness penalty.

Value

• 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.
• C
  The matrix defining any constraints on the term - usually a one row matrix giving the column sums of the model matrix, which defines the constraint that each term should sum to zero over the covariate values.
• 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.
• df
  the degrees of freedom associated with this term (at least when unpenalized).
• 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 the infomation 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.

item

• data
• knots

code

tprs.smooth

Details

The returned objects for the built in smooth classes have the following extra elements. cr.smooth objects (generated using bs="cr") have an additional array xp giving the knot locations used to generate the basis.

cyclic.smooth objects (generated using bs="cc") have an array xp of knot locations and a matrix BD used to define the basis (BD transforms function values at the knots to second derivatives at the knots).

tprs.smooth objects require several items to be stored in order to define the basis. These are:

• shift

{A record of the shift applied to each covariate in order to center it around zero and avoid any co-linearity problems that might otehrwise occur in the penalty null space basis of the term. } Xu{A matrix of the unique covariate combinations for this smooth (the basis is constructed by first stripping out duplicate locations).} UZ{The matrix mapping the t.p.r.s. parameters back to the parameters of a full thin plate spline.} null.space.dimension{The dimension of the space of functions that have zero wiggliness according to the wiggliness penalty for this term.}

References

Wood, S.N. (2000) Modelling and Smoothing Parameter Estimation with Multiple Quadratic Penalties. J.R.Statist.Soc.B 62(2):413-428

Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95-114

Wood, S.N. (in press) Stable and efficient multiple smoothing parameter estimation for generalized additive models. J. Amer. Statist. Ass.

The p-spline code given in the example is based on:

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

http://www.stats.gla.ac.uk/~simon/

See Also

get.var, gamm, gam, Predict.matrix

Examples

# adding "p-spline" classes and methods

smooth.construct.ps.smooth.spec<-function(object,data,knots)
# a p-spline constructor method function
{ require(splines)
  if (length(object$p.order)==1) m<-rep(object$p.order,2) 
  else m<-object$p.order  # m[1] - basis order, m[2] - penalty order
  nk<-object$bs.dim-m[1]  # number of interior knots
  if (nk<=0) stop("basis dimension too small for b-spline order")
  x <- get.var(object$term,data)  # find the data
  xl<-min(x);xu<-max(x);xr<-xu-xl # data limits and range
  xl<-xl-xr*0.001;xu<-xu+xr*0.001;dx<-(xu-xl)/(nk-1) 
  if (!is.null(knots)) k <- get.var(object$term,knots) 
  else k<-NULL
  if (is.null(k)) 
  k<-seq(min(x)-dx*(m[1]+1),max(x)+dx*(m[1]+1),length=nk+2*m[1]+2)   
  if (length(k)!=nk+2*m[1]+2) 
  stop(paste("there should be ",nk+2*m[1]+2,"supplied knots"))
  object$X<-spline.des(k,x,m[1]+2,x*0)$design # get model matrix
  if (!object$fixed)       
  { S<-diag(object$bs.dim);if (m[2]) for (i in 1:m[2]) S<-diff(S)
    object$S<-list(t(S)%*%S)  # get penalty
    object$S[[1]] <- (object$S[[1]]+t(object$S[[1]]))/2 # exact symmetry
  }
  object$rank<-object$bs.dim-m[2]  # penalty rank 
  object$null.space.dim <- m[2]  # dimension of unpenalized space  
  object$knots<-k;object$m<-m      # store p-spline specific info.
  object$C<-matrix(colSums(object$X),1,object$bs.dim) #constraint
  object$df<-ncol(object$X)-1      # maximum DoF
  if (object$by!="NA")  # deal with "by" variable 
  { by <- get.var(object$by,data) # find by variable  
    if (is.null(by)) stop("Can't find by variable")
    object$X<-by*object$X # form diag(by)\%*\%X
  }
  class(object)<-"pspline.smooth"  # Give object a class
  object
}

Predict.matrix.pspline.smooth<-function(object,data)
# prediction method function for the p.spline smooth class
{ require(splines)
  x <- get.var(object$term,data)
  spline.des(object$knots,x,object$m[1]+2,x*0)$design
}

# an example, using the new class....
set.seed(0);n<-400;
x0 <- runif(n, 0, 1);x1 <- runif(n, 0, 1)
x2 <- runif(n, 0, 1);x3 <- runif(n, 0, 1)
f <- 2 * sin(pi * x0)
f <- f + exp(2 * x1) - 3.75887
f <- f+0.2*x2^11*(10*(1-x2))^6+10*(10*x2)^3*(1-x2)^10-1.396
e <- rnorm(n)*2
y <- f + e
b<-gam(y~s(x0,bs="ps",m=2)+s(x1,bs="ps",m=c(1,3))+
         s(x2,bs="ps",m=2)+s(x3,bs="ps",m=2))
plot(b,pages=1)
# another example using tensor products of the new class

test1<-function(x,z,sx=0.3,sz=0.4)
{ (pi**sx*sz)*(1.2*exp(-(x-0.2)^2/sx^2-(z-0.3)^2/sz^2)+
  0.8*exp(-(x-0.7)^2/sx^2-(z-0.8)^2/sz^2))
}
n<-400
x<-runif(n);z<-runif(n);
f <- test1(x,z)
y <- f + rnorm(n)*0.1
b <- gam(y~te(x,z,bs=c("ps","ps"),m=c(2,2)))
vis.gam(b)