density.fd | R Documentation |
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
.
## S3 method for class 'fd' density(x, WfdParobj, conv=0.0001, iterlim=20, active=1:nbasis, dbglev=0, ...)
x |
a set observations, which may be one of two forms:
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 $W(t).$ |
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 |
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. |
... |
Other arguments to match the generic function 'density' |
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
.
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 |
intensity.fd
# 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 <- fd(matrix(0,11,1), basisobj) WfdParobj <- fdPar(Wfd0) # estimate density denslist <- density.fd(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))
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