Compute a Probability Density Function
density.fd
Language Reference for FDA Library

Compute a Probability Density Function

DESCRIPTION:

Like the regular S-PLUS function density, this function computes a probability density function for a sample of values of a random variable. However, in this case the density function is defined by a functional parameter object WfdParobj along with a normalizing constant C.

The density function p(x) has the form p(x) = C exp[W(x)] where function W(x) is defined by the functional data object WfdParobj.

USAGE:

density.fd(x, WfdParobj, conv=0.0001, iterlim=20,
           active=2:nbasis, dbglev=1)

REQUIRED ARGUMENTS:

x
a strictly increasing set variable values. These observations may be one of two forms:
  1. a vector of observatons x_i
  2. a two-column matrix, with the observations x_i in the first column, and frequencies f_i in the second.
The first option corresponds to all f_i = 1.
WfdParobj
a functional parameter object specifying the initial value, basis object, roughness penalty and smoothing parameter defining function W(t).

OPTIONAL ARGUMENTS:

conv
a positive constant defining the convergence criterion.
iterlim
the maximum number of iterations allowed.
active
a logical vector of length equal to the number of coefficients defining Wfdobj. If an entry is T, the corresponding coefficient is estimated, and if F, it is held at the value defining the argument Wfdobj. Normally the first coefficient is set to 0 and not estimated, since it is assumed that W(0) = 0.
dbglev
either 0, 1, or 2. This controls the amount information printed out on each iteration, with 0 implying no output, 1 intermediate output level, and 2 full output. If levels 1 and 2 are used, it is helpful to turn off the output buffering option in S-PLUS.

VALUE:

a named list of length 4 containing:
Wfdobj
a functional data object defining function W(x) that that optimizes the fit to the data of the monotone function that it defines.
C
the normalizing constant.
Flist
a named list containing three results for the final converged solution: (1) f: the optimal function value being minimized, (2) grad: the gradient vector at the optimal solution, and (3) norm: the norm of the gradient vector at the optimal solution.
iternum
the number of iterations.
iterhist
a iternum+1 by 5 matrix containing the iteration history.

DETAILS:

The goal of the function is provide a smooth density function estimate that approaches some target density by an amount that is controlled by the linear differential operator Lfdobj and the penalty parameter. For example, if the second derivative of W(t) is penalized heavily, this will force the function to approach a straight line, which in turn will force the density function itself to be nearly normal or Gaussian. Similarly, to each textbook density function there corresponds a W(t), and to each of these in turn their corresponds a linear differential operator that will, when apply to W(t), produce zero as a result.

To plot the density function or to evaluate it, evaluate Wfdobj, exponentiate the resulting vector, and then divide by the normalizing constant C.

SEE ALSO:

intensity.fd

EXAMPLES:

#  set up range for density
rangeval <- c(-3,3)
#  set up some standard normal data
x <- rnorm(50)
#  make sure values within the range
x[x < -3] <- -2.99
x[x >  3] <-  2.99
#  set up basis for W(x)
basisobj <- create.bspline.basis(rangeval, 11)
#  set up initial value for Wfdobj
Wfd0 <- create.fd(matrix(0,11,1), basisobj)
WfdParobj <- fdPar(Wfd0)
#  estimate density
denslist <- densityfd(x, WfdParobj)
#  plot density
xval <- seq(-3,3,.2)
wval <- eval.fd(xval, denslist$Wfdobj)
pval <- exp(wval)/denslist$C
plot(xval, pval, type="l", ylim=c(0,0.4))
points(x,rep(0,50))