ssden: Estimating Probability Density Using Smoothing Splines

In gss: General Smoothing Splines

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

formula	Symbolic description of the model to be fit.
type	List specifying the type of spline for each variable. See mkterm for details.
data	Optional data frame containing the variables in the model.
alpha	Parameter defining cross-validation score for smoothing parameter selection.
weights	Optional vector of bin-counts for histogram data.
subset	Optional vector specifying a subset of observations to be used in the fitting process.
na.action	Function which indicates what should happen when the data contain NAs.
id.basis	Index of observations to be used as "knots."
nbasis	Number of "knots" to be used. Ignored when id.basis is specified.
seed	Seed to be used for the random generation of "knots." Ignored when id.basis is specified.
domain	Data frame specifying marginal support of density.
quad	Quadrature for calculating integral. Mandatory if variables other than factors or numerical vectors are involved.
qdsz.depth	Depth to be used in smolyak.quad for the generation of quadrature.
bias	Input for sampling bias.
prec	Precision requirement for internal iterations.
maxiter	Maximum number of iterations allowed for internal iterations.
skip.iter	Flag indicating whether to use initial values of theta and skip theta iteration. See ssanova for notes on skipping theta iteration.

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.

If you get an error message from ssden stating "Newton iteration diverges", try to use a larger qdsz.depth which will execute slower, or switch to ssden1. The default values of qdsz.depth for dimensions 4, 5, 6+ are 12, 11, 10.

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, [e-mail]

References

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

Gu, C., Jeon, Y., and Lin, Y. (2013), Nonparametric density estimation in high dimensions. Statistica Sinica, 23, 1131–1153.

Gu, C. (2013), Smoothing Spline ANOVA Models (2nd Ed). New York: Springer-Verlag.

Gu, C. (2014), Smoothing Spline ANOVA Models: R Package gss. Journal of Statistical Software, 58(5), 1-25. URL http://www.jstatsoft.org/v58/i05/.

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=as.factor("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)

gss documentation built on Aug. 16, 2023, 9:07 a.m.