# intensity.fd

##### 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`

.

- Keywords
- smooth

##### Usage

```
intensity.fd(x, WfdParobj, conv=0.0001, iterlim=20,
dbglev=1, returnMatrix=FALSE)
```

##### Arguments

- 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

`mu(x) = exp[W(x)]`

.- 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.

##### Details

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.

##### Value

a named list of length 4 containing:

a functional data object defining function $W(x)$ that that optimizes the fit to the data of the monotone function that it defines.

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.

the number of iterations.

a `iternum+1`

by 5 matrix containing the iteration
history.

##### See Also

##### Examples

```
# NOT RUN {
# 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)
# }
```

*Documentation reproduced from package fda, version 2.4.7, License: GPL (>= 2)*