1. (Problem 9-Section 5.3.3)

Poisson processes can be defined in dimensions higher than 1. Such models are used in fields such as ecology when comparing observed locations of say, tree saplings, with what would result from a completely random process. In a homogeneous two-dimensional Poisson process, the distribution of points in a region is Poisson with rate proportional to its area, and the numbers of points in non-overlapping regions are independent of each other. Use the following code to plot a two-dimensional Poisson process of points having rate = 2 per unit area, in the square [0, 10] [0, 10].

lambda <- 2

regionlength <- 10

regionwidth <- 10

N <- rpois(1, lambda*regionlength*regionwidth)

U1 <- runif(N, min=0, max=regionwidth)

U2 <- runif(N, min=0, max=regionlength)

As usual, you can use the plot() function to display the spatial locations of the points.

plot(U1, U2)

Modify the code so that it is a Poisson process with rate = 4 per unit area, and plot the result, comparing the number of points with the = 2 case.

2. (Problem 10-Section 5.3.3)

Referring to the previous exercise, suppose tree saplings are observed in a 15 square meter region at a rate of 0.25 per square meter. Under the assumption that the sapling locations follow a homogeneous Poisson process, simulate a single realization and plot the resulting point pattern. From what you know about tree saplings, identify at least one problem with the Poisson process as a model for this phenomenon.