Intensity Function for Point Process
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.
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 |
# 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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