Estimating Probability Density Using Smoothing Splines

Description

Estimate probability densities using smoothing spline ANOVA models. The symbolic model specification via formula follows the same rules as in lm, but with the response missing.

Usage

ssden(formula, type=NULL, data=list(), alpha=1.4, weights=NULL,
      subset, na.action=na.omit, id.basis=NULL, nbasis=NULL, seed=NULL,
      domain=as.list(NULL), quad=NULL, qdsz.depth=NULL, bias=NULL,
      prec=1e-7, maxiter=30, skip.iter=FALSE)

ssden1(formula, type=NULL, data=list(), alpha=1.4, weights=NULL,
       subset, na.action=na.omit, id.basis=NULL, nbasis=NULL, seed=NULL,
       domain=as.list(NULL), quad=NULL, prec=1e-7, maxiter=30)

Arguments

Details

The model specification via formula is for the log density. For example, ~x1*x2 prescribes a model of the form

log f(x1,x2) = g_{1}(x1) + g_{2}(x2) + g_{12}(x1,x2) + C

with the terms denoted by "x1", "x2", and "x1:x2"; the constant is determined by the fact that a density integrates to one.

The selective term elimination may characterize (conditional) independence structures between variables. For example, ~x1*x2+x1*x3 yields the conditional independence of x2 and x3 given x1.

Parallel to those in a ssanova object, the model terms are sums of unpenalized and penalized terms. Attached to every penalized term there is a smoothing parameter, and the model complexity is largely determined by the number of smoothing parameters.

The selection of smoothing parameters is through a cross-validation mechanism described in the references, with a parameter alpha; alpha=1 is "unbiased" for the minimization of Kullback-Leibler loss but may yield severe undersmoothing, whereas larger alpha yields smoother estimates.

A subset of the observations are selected as "knots." Unless specified via id.basis or nbasis, the number of "knots" q is determined by max(30,10n^{2/9}), which is appropriate for the default cubic splines for numerical vectors.

Value

ssden returns a list object of class "ssden". ssden1 returns a list object of class c("ssden1","ssden").

dssden and cdssden can be used to evaluate the estimated joint density and conditional density; pssden, qssden, cpssden, and cqssden can be used to evaluate (conditional) cdf and quantiles.

The method project.ssden can be used to calculate the Kullback-Leibler projection of "ssden" objects for model selection; project.ssden1 can be used to calculate the square error projection of "ssden1" objects.

Note

In ssden, default quadrature will be constructed for numerical vectors on a hyper cube, then outer product with factor levels will be taken if factors are involved. The sides of the hyper cube are specified by domain; for domain$x missing, the default is c(min(x),max(x))+c(-1,1)*(max(x)-mimn(x))*.05. In 1-D, the quadrature is the 200-point Gauss-Legendre formula returned from gauss.quad. In multi-D, delayed Smolyak cubatures from smolyak.quad are used on cubes with the marginals properly transformed; see Gu and Wang (2003) for the marginal transformations.

For reasonable execution time in higher dimensions, set skip.iter=TRUE in call to ssden.

ssden1 does not involve multi-D quadrature but does not perform as well as ssden. It can be used in very high dimensions where ssden is infeasible.

The results may vary from run to run. For consistency, specify id.basis or set seed.

Author(s)

Chong Gu, chong@stat.purdue.edu

References

Gu, C. (2002), Smoothing Spline ANOVA Models. New York: Springer-Verlag.

Gu, C. and Wang, J. (2003), Penalized likelihood density estimation: Direct cross-validation and scalable approximation. Statistica Sinica, 13, 811–826.

Examples

## 1-D estimate: Buffalo snowfall
data(buffalo)
buff.fit <- ssden(~buffalo,domain=data.frame(buffalo=c(0,150)))
plot(xx<-seq(0,150,len=101),dssden(buff.fit,xx),type="l")
plot(xx,pssden(buff.fit,xx),type="l")
plot(qq<-seq(0,1,len=51),qssden(buff.fit,qq),type="l")
## Clean up
## Not run: rm(buffalo,buff.fit,xx,qq)
dev.off()
## End(Not run)

## 2-D with triangular domain: AIDS incubation
data(aids)
## rectangular quadrature
quad.pt <- expand.grid(incu=((1:40)-.5)/40*100,infe=((1:40)-.5)/40*100)
quad.pt <- quad.pt[quad.pt$incu<=quad.pt$infe,]
quad.wt <- rep(1,nrow(quad.pt))
quad.wt[quad.pt$incu==quad.pt$infe] <- .5
quad.wt <- quad.wt/sum(quad.wt)*5e3
## additive model (pre-truncation independence)
aids.fit <- ssden(~incu+infe,data=aids,subset=age>=60,
                  domain=data.frame(incu=c(0,100),infe=c(0,100)),
                  quad=list(pt=quad.pt,wt=quad.wt))
## conditional (marginal) density of infe
jk <- cdssden(aids.fit,xx<-seq(0,100,len=51),data.frame(incu=50))
plot(xx,jk$pdf,type="l")
## conditional (marginal) quantiles of infe (TIME-CONSUMING)
## Not run: 
cqssden(aids.fit,c(.05,.25,.5,.75,.95),data.frame(incu=50))

## End(Not run)
## Clean up
## Not run: rm(aids,quad.pt,quad.wt,aids.fit,jk,xx)
dev.off()
## End(Not run)

## One factor plus one vector
data(gastric)
gastric$trt
fit <- ssden(~futime*trt,data=gastric)
## conditional density
cdssden(fit,c("1","2"),cond=data.frame(futime=150))
## conditional quantiles
cqssden(fit,c(.05,.25,.5,.75,.95),data.frame(trt="1"))
## Clean up
## Not run: rm(gastric,fit)

## Sampling bias
## (X,T) is truncated to T<X<1 for T~U(0,1), so X is length-biased
rbias <- function(n) {
  t <- runif(n)
  x <- rnorm(n,.5,.15)
  ok <- (x>t)&(x<1)
  while(m<-sum(!ok)) {
    t[!ok] <- runif(m)
    x[!ok] <- rnorm(m,.5,.15)
    ok <- (x>t)&(x<1)
  }
  cbind(x,t)
}
xt <- rbias(100)
x <- xt[,1]; t <- xt[,2]
## length-biased
bias1 <- list(t=1,wt=1,fun=function(t,x){x[,]})
fit1 <- ssden(~x,domain=list(x=c(0,1)),bias=bias1)
plot(xx<-seq(0,1,len=101),dssden(fit1,xx),type="l")
## truncated
bias2 <- list(t=t,wt=rep(1/100,100),fun=function(t,x){x[,]>t})
fit2 <- ssden(~x,domain=list(x=c(0,1)),bias=bias2)
plot(xx,dssden(fit2,xx),type="l")
## Clean up
## Not run: rm(rbias,xt,x,t,bias1,fit1,bias2,fit2)