require(ggplot2)
## Loading required package: ggplot2
The homogeneous Poisson process is simulated on \([0, 1]\) by simulating a Poisson distributed number of events and by simulating the event times from the uniform distribution on \([0, 1]\).
t <- 1
lambda <- 100
N <- rpois(1, lambda * t)
events <- sort(runif(N))
qplot(events, seq(1, N)) +
geom_abline(slope = lambda)
From the figure above we see that the cumulative number of events (the counting process) follows closely the straight line with slope \(\lambda\). The compensated counting process (in this example it means subtracting the expected number of events) \[X_t = N_t - \lambda t\] can also be plotted. First we plot it at the event times only.
Xt <- seq(1, N) - lambda * events
qplot(events, Xt)
We can get a more accurate visualization of the sample path of the compensated process by using
a little ggplot2 magic. Specifically using the geom segment for showing the piecewise negative linear
drift of the process in between jumps. It requires the computation of the left limits \(X_{t-}\) of \(X_t\).
Xtminus <- seq(1, N-1) - lambda * events[-1]
qplot(x = events[-N],
y = Xt[-N],
xend = events[-1],
yend = Xtminus,
geom = "segment") +
geom_point()
