Statistical Analysis of Stationary Tessellations

 In Section 5.2 we saw that the fundamental parameters of a stationary planar tessellation $C=\{C_1,C_2,\ldots\}$ are $\lambda_0$, $\lambda_2$ and L_A. Assume that the tessellation is observed in a (bounded) window $B \subset \RL^2$. A natural unbiased estimator for $\lambda_0$ is  

$$\hat{\lambda}_0 = \frac{\mbox{number of nodes in }B}{\nu(B)}.$$ (6)

Furthermore, in the ordinary equilibrium state, i.e., when (5.2) holds, we can use (5.3) and (5.6) to obtain an unbiased estimator for $\lambda_2$: 

$$\hat{\lambda}_2^{(1)} = \frac{\hat{\lambda}_0}{2}.$$ (7)

However, in general, unbiased estimation of $\lambda_2$ is more difficult because of the presence of edge effects; see Brakke (1985), Stoyan et al. (1995). Note that from the observation of the tessellation in B, typically we do not know how many cell-centroids are in B. If the window B is convex and if the stationary tessellation C is isotropic, then an unbiased estimator for $\lambda_2$ is  

$$\hat{\lambda}_2^{(2)} = \frac{N(B) - 1 - \frac{1}{2} N_e(B)}{\nu(B)}$$ (8)

where N(B) is the number of cells intersecting B and N_e(B) is the number of edges intersecting the boundary $\partial B$ of B. A proof of the unbiasedness of $\hat{\lambda}_2^{(2)}$ can be found in Stoyan et al. (1995).

A third estimator for $\lambda_2$ was proposed by Saltykov (1974) in the case that B is a rectangle and the cells are so small that each intersecting cell has at most two intersections of edges with $\partial B$: 

$$\hat{\lambda}_2^{(3)} = \frac{z(B) + \frac{1}{2} w(B) + \frac{1}{4} u(B)}{ \nu(B)}$$ (9)

where z(B) is the number of cells lying completely in B, w(B) the number of cells that intersect $\partial B$ but do not cover the corners, and u(B) the number of cells that cover at least one of the corners of B.

Note that for each stationary tessellation in the ordinary equilibrium state, $\hat{\lambda}_2^{(1)} = \hat{\lambda}_2^{(2)} = \hat{\lambda}_2^{(3)}$.This was shown in Hahn (1995) where the estimation of variances has also been studied.

In order to estimate L_A, the expected total length of segments per unit area, one can proceed in the following way; see Stoyan et al. (1995). A natural unbiased estimator for L_A is $\hat{L}_A = \vert E(B)\vert/ \nu(B)$,where |E(B)| is the total length of all segments in B of the segment process E induced by the stationary tessellation C. Another possibility is to consider a test system $B_0 \subset B$consisting of $n \geq 1$ circles with given centers and fixed radius r. Then $\hat{L}_A^{(1)} = N_E(B_0)/(4nr)$is an unbiased estimator for L_A, where N_E(B₀) is the number of intersections of the line process E with the test system B₀. Further estimators for L_A and their estimation variances are considered in Bene$\check{\mbox{s}}$ et al. (1994).

If C is stationary and isotropic, then $\hat{L}_A^{(2)} = \pi N_E(B^{\prime})/(2 \nu_1(B^{\prime}))$is an unbiased estimator for L_A, where $B^{\prime}$ is a test system of segments in B of total length $\nu_1(B^{\prime})$.

Assume now that the parameters $\lambda_0$, $\lambda_2$ and L_A have been estimated and that each node of the observed tessellation is touched by not more than three polygons, i.e., we can assume that the underlying stationary tessellation is in the ordinary equilibrium state. Then we can study the question whether the observed realization is sampled from a Poisson-Voronoi tessellation, where the construction of a test statistic can be based on (5.4), i.e., on the fact that for a Poisson-Voronoi tessellation  

$$\lambda_0 = 2 \lambda_2 \, , \qquad L_A = 2 \sqrt{\lambda_2}.$$ (10)

Gradient estimators for characteristics (like area, number of edges, etc.) of the typical Voronoi cell generated by a stationary point process X in $\RL^2$ have been studied in Baccelli et al. (1995), Zuyev et al. (1997), where it is assumed that X is superimposed by an infinitesimally thin perturbation point process $X^{\prime}$ which is independent of X; see also Section 5.5.