Statistical Analysis of Boolean Models

 Assume that the pattern which is observed in the window $W \subset \RL^2$ is a sample from a Boolean model $\Xi$. The parameters of the Boolean model can be classified into two groups. The first group is formed by the individual (or microscopic) parameters. Typical examples are the expected area $\oA$ and the expected perimeter $\oU$ of the grain, and the intensity $\lambda$ of the Poisson point process of germs. The second group is formed by the aggregate (or macroscopic) parameters. Typical examples are the area fraction p, the capacity functional $T_{\Xi}(K)$ and the contact distribution function H_B^*(r). Since the individual parameters cannot be directly observed, they have to be estimated through the aggregate parameters. Hence we will focus on the estimation of the aggregate parameters first.

The simplest parameter is the area fraction p. An estimator is given by the proportion of the window W covered by $\Xi$, i.e., $\hat{p}_1 = \nu (\Xi \cap W)/ \nu (W)$.Another estimator, based on the fact that p is also the probability that an arbitrary point lies in $\Xi$, is given by $\hat{p}_2 = N( \Xi \cap W \cap \iz^2)/N(W \cap \iz^2)$,where $\iz^2$ is a lattice in the plane and N(A) counts the number of lattice points lying in the set A. The asymptotic properties of these estimators and further estimators for p can be found in Baddeley (1980), Mase (1982), Molchanov (1997), Stoyan et al. (1995), Weibel (1980).

The distribution of the Boolean model $\Xi$ is determined by its capacity functional $T_{\Xi}(K)$, the probability that $\Xi$ hits a fixed compact set K, defined in (4.11). It can be estimated by $\hat{T}_{\Xi}(K) = \nu \left( ( \Xi \oplus \check{K}) \cap (W \ominus K) \right)/ \nu (W \ominus K)$,for $K \in {\cal K}$,which was discussed in Ripley (1986).

The contact distribution function can now be estimated by  

$$\hat{H}^*_B(r) = 1 - \frac{1 - \hat{T}_{\Xi}(rB)}{1 - \hat{p}},$$ (24)

where $\hat{p}$ and $\hat{T}_{\Xi}(K)$ are estimators for the area fraction and the capacity functional respectively; see Heinrich (1993).

There are several approaches to estimate the individual parameters $\lambda$, $\oU$ and $\oA$.In Molchanov (1997) the minimum contrast method for contact distribution functions was applied to obtain estimators for these parameters, see also Diggle (1981): Let B be the unit disc and consider the spherical contact distribution function H^*(r) given in (4.12). Using (4.13) it can be shown that its logarithmic transform H^l(r) is given by $H^l(r) = \lambda \pi r^2 + \lambda r \oU$ for $r \geq 0$.The essence of the minimum contrast method lies in finding minimum contrast estimators $\hat{\lambda}$ and $\hat{\oU}$ which minimize the integral $\int_0^{r_0} \, \left(H^l(r) - \hat{H}^l(r)\right)^2 dr$for suitably chosen r₀, where $\hat{H}^l(r)$ is the empirical logarithmic spherical contact distribution function. It follows from (4.10) that $\oA$ can be estimated using $\hat{\lambda}$ and $\hat{p}$ as $\hat{\oA} = - \hat{\lambda}^{-1} \log ( 1 - \hat{p})$.Another method of estimating the parameters $\lambda$,$\oA$ and $\oU$ was proposed by Hall (1988) and is based on (4.15), (4.16) and (4.17). His idea is to construct a regular lattice inside the observation window W in which each edge has length l, each face is convex and has area a, and each face is bordered by m edges. If $\gamma_0$, $\gamma_1$ and $\gamma_2$ denote the proportionals of vertices, edges, and faces, respectively, that are not covered by the Boolean model $\Xi$, then estimators for $\lambda$, $\oA$ and $\oU$ can be calculated by solving the equations

Schmitt (1991) proposed a formula for estimating $\lambda$, assuming only sure boundedness of M₀. He suggested the estimator

$$\hat{\lambda} = \frac{1}{\varepsilon^2} \, \ln \, \frac{\Pro... ...G \cup K) = \emptyset) \Prob( \Xi \cap (G \cup L) = \emptyset)}$$

with suitable test sets G, K and L depending on the arbitrary, strictly positive number $\varepsilon$. Note that the probability $\Prob( \Xi \cap A = \emptyset)$ can be estimated by estimating $1 - (\mbox{area fraction of } \Xi \oplus \check{A})$.

Estimators for p, $\oU$ and $\oA$ based on the specific connectivity number (or specific Euler-Poincar'e characteristic), which can be viewed as the expected difference between the number of clumps and the number of holes per unit area, can be found e.g. in Molchanov (1997).

In order to test the hypothesis that an observed pattern is a realization of a Boolean model, we can use the polynomial expansion of contact distribution functions H_B^*(r). For a Boolean model it holds that the logarithm of 1 - H_B^*(r) is a polynomial of the form $\ln (1 - H_B^*(r)) = -a(B) r - b(B) r^2$ for $r \geq 0$,with $a(B),b(B) \geq 0$ for every convex set B; see e.g. Cressie (1991), Molchanov (1997), Stoyan et al. (1995). For various B, $\hat{H}_B(r)$ can be computed using (4.26). If $\ln (1 - \hat{H}_B^*(r))$is well approximated by quadratic polynomials, then the hypothesis cannot be rejected.

A different approach for testing the Boolean model assumption is based on the so-called Laslett's Theorem, which states that the tangent points after a simple transformation form a homogeneous Poisson process with the same intensity $\lambda$; see Cressie (1991), Molchanov (1997). Hence the problem of testing for a Boolean model reduces to testing for a homogeneous Poisson process which was discussed in Section 2.2.