Simulation of Boolean Models

 We assume now that the grains are discs with bounded radii R_n. Let $r_0< \infty$ be such that $r_0 = \inf \{ r\, :\, \Prob(R_n \leq r)=1\}$.Then a Boolean model $\Xi$ in a bounded window $B \subset \RL^2$ can be simulated in the following way. First the Poisson process X of germs is simulated in the set $B \oplus b(o,r_0) = \{ x + y \, : \, x \in B, \, \vert y\vert \leq r_0\}$.Then, for each generated germ X_n, the random radius R_n is simulated according to its distribution. The sample of $\Xi$ in B is the union $\bigcup_n \, \left\{ X_n + M_n\right\} \cap B$,where M_n is the disc with radius R_n.

It is often necessary to simulate the typical clump of a Boolean model, e.g. for testing hypotheses where the exact distribution of the test statistic is unknown. But Monte Carlo simulation can be used in order to obtain an approximative empirical reference distribution; see Hermann et al. (1989), Hinde and Miles (1980), Møller (1994). In connection with this a simulation procedure is useful which has been developed in Møller (1994) and which is based on an algorithm due to Quine and Watson (1984) for radial simulation of a homogeneous Poisson process. Let $U_1,U_2,\ldots$ be independent random vectors uniformly distributed on the circle $\partial b(o,1)$ and let $V_1,V_2,\ldots$ be independent random variables uniformly distributed on the interval (0,1). Furthermore, assume that the sequences $\{U_n\}$ and $\{V_n\}$ are independent. For $\lambda \gt$ fixed, we put  

$$X_n = U_n \sqrt{\frac{Q_n}{\pi \lambda}}$$ (25)

where the Q_n are defined recursively by $Q_1 = - \ln V_1$ and $Q_{n+1} = Q_n - \ln V_{n+1}$.Then $X=\{X_1,X_2,\ldots\}$ is a homogeneous Poisson process with intensity $\lambda$ where the X_n are numbered in such a way that $0 < \vert X_1\vert < \vert X_2\vert < \ldots$with probability 1. In other words, this radial simulation algorithm is based on the following invariance property of Poisson processes: Let $\{T_1,T_2,\ldots\}$ be a homogeneous Poisson process with intensity $\lambda$ on the nonnegative halfline $\RL_+ = [0,\infty)$and let $Z_1,Z_2,\ldots$ be independent random variables uniformly distributed on the interval $[0,2\pi )$. Assume that the sequences $\{T_n\}$ and $\{Z_n\}$ are independent. Put $R_n = (T_n/ \pi)^{1/2}$.Then, the set of points $\{X_1,X_2,\ldots\}$ with polar coordinates X_n = (R_n,Z_n) forms a homogeneous Poisson process with intensity $\lambda$in the plane.

This radial simulation procedure of the homogeneous Poisson process X can be used for simulating the typical clump of the Boolean model. As we already mentioned, we can simulate the clump which contain the disc with center at the origin which is induced by the point process $X \cup \{o\}$, where a point at the origin has been added to X. As before, we assume that the radii of the discs are bounded by $r_0< \infty$. By $C_0(x_1,\ldots,x_n)\}$ we will denote the clump induced by the discs centered at $\{o,x_1,\ldots,x_n\}$and containing the disc with center at the origin. Let $x=\{x_1,x_2,\ldots\}$ be a realization of the radially generated Poisson process X and let $\{r_1,r_2,\ldots\}$be a realization of the sequence of radii $\{R_1,R_2,\ldots,\}$. Hall (1988) showed that the clump $C_0\{x_1,x_2,\ldots\}$which contains the disc centered at the origin is finite with probability 1 if $\lambda$ is smaller than a critical value $\lambda_c$;see Hall (1988) for estimated critical intensities for Boolean models generated by fixed-radius discs. Let l_n denote the distance of the farthest point of the clump $C_0\{x_1,\ldots,x_n\}$ to the origin and let $n_0 = \min \{ n \, : \, \vert x_{n+1}\vert \gt l_n + r_0\}$. Note that $C_0\{x_1,x_2,\ldots\} = C_0\{x_1,\ldots,x_n\}$for all $n \geq n_0$ where n₀ is finite with probability 1 if $\lambda < \lambda_c$. Thus, generate the realizations $x_1,\ldots,x_{n+1}$ of $X_1,\ldots,X_{n+1}$ until $n \geq n_0$.