Description Usage Arguments Details Value See Also Examples
The intensity $mu$ of a series of event times that obey a
homogeneous Poisson process is the mean number of events per unit time.
When this event rate varies over time, the process is said to be
nonhomogeneous, and $mu(t)$, and is estimated by this function
intensity.fd
.
1 2 | intensity.fd(x, WfdParobj, conv=0.0001, iterlim=20,
dbglev=1, returnMatrix=FALSE)
|
x |
a vector containing a strictly increasing series of event times. These event times assume that the the events begin to be observed at time 0, and therefore are times since the beginning of observation. |
WfdParobj |
a functional parameter object estimating the log-intensity function
$W(t) = log[mu(t)]$ .
Because the intensity function $mu(t)$ is necessarily positive,
it is represented by |
conv |
a convergence criterion, required because the estimation process is iterative. |
iterlim |
maximum number of iterations that are allowed. |
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, turn off the output buffering option. |
returnMatrix |
logical: If TRUE, a two-dimensional is returned using a special class from the Matrix package. |
The intensity function $I(t)$ is almost the same thing as a
probability density function $p(t)$ estimated by function
densify.fd
. The only difference is the absence of
the normalizing constant $C$ that a density function requires
in order to have a unit integral.
The goal of the function is provide a smooth intensity function
estimate that approaches some target intensity by an amount that is
controlled by the linear differential operator Lfdobj
and
the penalty parameter in argument WfdPar
.
For example, if the first derivative of
$W(t)$ is penalized heavily, this will force the function to
approach a constant, which in turn will force the estimated Poisson
process itself to be nearly homogeneous.
To plot the intensity function or to evaluate it,
evaluate Wfdobj
, exponentiate the resulting vector.
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. |
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 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 | # Generate 101 Poisson-distributed event times with
# intensity or rate two events per unit time
N <- 101
mu <- 2
# generate 101 uniform deviates
uvec <- runif(rep(0,N))
# convert to 101 exponential waiting times
wvec <- -log(1-uvec)/mu
# accumulate to get event times
tvec <- cumsum(wvec)
tmax <- max(tvec)
# set up an order 4 B-spline basis over [0,tmax] with
# 21 equally spaced knots
tbasis <- create.bspline.basis(c(0,tmax), 23)
# set up a functional parameter object for W(t),
# the log intensity function. The first derivative
# is penalized in order to smooth toward a constant
lambda <- 10
Wfd0 <- fd(matrix(0,23,1),tbasis)
WfdParobj <- fdPar(Wfd0, 1, lambda)
# estimate the intensity function
Wfdobj <- intensity.fd(tvec, WfdParobj)$Wfdobj
# get intensity function values at 0 and event times
events <- c(0,tvec)
intenvec <- exp(eval.fd(events,Wfdobj))
# plot intensity function
plot(events, intenvec, type="b")
lines(c(0,tmax),c(mu,mu),lty=4)
|
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