Spatial cluster point processes related to Poisson–Voronoi tessellations

Abstract

We discuss how to construct models for cluster point processes within ‘territories’ modelled by $d$-dimensional Voronoi cells whose nuclei are generated by a latent Poisson process (where the planar case $d=2$ is of our primary interest). Conditional on the territories/cells, the clusters are independent Poisson processes whose points may be aggregated around or away from the nuclei and along or away from the boundaries of the cells. Observing the superposition of clusters within a bounded region, we discuss how to account for edge effects. Bayesian inference for a particular flexible model is discussed in connection to a botanical example.

Appendices

Appendices

1.1 Appendix 1: Proof of Theorem 1

Let the situation be as in Sect. 2. By the Slivnyak-Mecke Theorem (Møller and Waagepetersen (2004) and the references therein), $M$ is equal to

$$\begin{aligned} \kappa \int _{W_{\rm ext}^c} {\rm P}(C(y;{\mathbf{Y}}\cup \{y\})\cap W\not =\emptyset )\,{\rm d}y \end{aligned}$$

which by stationarity of ${\mathbf{Y}}$ can be rewritten as

$$\begin{aligned} \kappa \int _{W_{\rm ext}^c} {\rm P}(C(o;{\mathbf{Y}}\cup \{o\})\cap W_{-y}\not =\emptyset )\,{\rm d}y \end{aligned}$$

(12)

where $W_{\rm ext}^c={\mathbb{R}}^{d}\setminus W_{\rm ext}$ is the complement of $W_{\rm ext}$ and $W_{-y}$ is $W$ translated by $-y$. Note that $C(o;{\mathbf{Y}} \cup \{o\})$ is the so-called typical Poisson–Voronoi cell (see e.g. (Møller (1989))). Denoting $T$ the distance from $o$ to the furthest vertex of $C(o;{\mathbf{Y}} \cup \{o\})$ and $d(o,W_{-y})$ the distance from $o$ to $W_{-y}$ (which is well-defined if e.g. $W_{-y}$ is compact),

$$\begin{aligned} \kappa \int _{W_{\rm ext}^c} {\rm P}(T > d(o,W_{-y}))\,{\rm d}y \end{aligned}$$

(13)

is an upper bound for (12) and hence also for $p$.

In order to bound (13), we start by deriving a lower bound on the cumulative distribution function (cdf) of $T$. Denote $\sigma _d=2\pi ^{d/2}/\Gamma (d/2)$ the surface area of the unit ball in ${\mathbb{R}}^{d}$, and $F_{d}$ the cdf of the Gamma-distribution with shape parameter $d$ and scale parameter 1.

Lemma 1

If $W$ is compact, then

$$\begin{aligned} {\rm P}(T>t)\le c_d\left[ 1-F_{d}(\kappa \omega _dt^{d})\right] ,\quad t\ge 0. \end{aligned}$$

(14)

Proof of Lemma 1: We shall ignore nullsets. With probability one, for all pairwise distinct points $y_1,\dots ,y_d\in {\mathbf{Y}}$, the $d$-dimensional closed ball $B(o,y_1,\dots ,y_d)$ containing $o,y_1,\dots ,y_d$ in its boundary is well-defined. Denote $R(o,y_1,\dots ,y_d)$ the radius of $B(o,y_1,\dots ,y_d)$. Then ${\rm P}(T>t)$ is at most

$$\begin{aligned} \frac{1}{d!} {\rm E}\sum ^{\not =}_{y_1,\dots ,y_d\in {\mathbf{Y}}} 1[&\,B(o,y_1,\dots ,y_d)\cap {\mathbf{Y}}\setminus \{y_1,\ldots ,y_d\}=\emptyset ,\\&\,R(o,y_1,\dots ,y_d)>t] \end{aligned}$$

where $\not =$ over the summation sign means that $y_1,\dots ,y_d$ are pairwise distinct, and noting that the sum is almost surely $d!$ times the number of vertices in $C(o;{\mathbf{Y}}\cup \{o\})$ with distance at least $t$ to $o$. Therefore, by repeated use of the Slivnyak-Mecke theorem, ${\rm P}(T>t)$ is at most

$$\begin{aligned} \frac{\kappa ^{d}}{d!} \int \cdots \int {\rm P}(&\,B(o,y_1,\dots ,y_d)\cap {\mathbf{Y}}=\emptyset ,\\&\,R(o,y_1,\dots ,y_d)>t)\,{\rm d}y_1\cdots \,{\rm d}y_d \end{aligned}$$

and hence, since ${\mathbf{Y}}$ is a stationary Poisson process and $B(o,y_1,\dots ,y_d)$ has volume $\omega _dR(o,y_1,\dots ,y_d)^{d}$, ${\rm P}(T>t)$ is at most

$$\begin{aligned}\frac{\kappa ^{d}}{d!} \int \cdots \int 1[R(o,y_1,\dots ,y_d)>t], \exp \left( -\kappa \omega _d R(o,y_1,\dots ,y_d)^{d}\right) \,{\rm d}y_1\cdots \,{\rm d}y_d =\frac{\kappa ^{d}}{d!}\frac{1}{|A|} \int \int \cdots \int 1[y_0\in A,\,R(y_0,y_1,\dots ,y_d)>t] \exp \left( -\kappa \omega _d R(y_0,y_1,\dots ,y_d)^{d}\right) {\rm d}y_0{\rm d}y_1\cdots \,{\rm d}y_d \end{aligned}$$

where $A\subset {\mathbb{R}}^{d}$ is an arbitrary Borel with volume $0<|A|<\infty$, and where $R=R(y_0,y_1,\dots ,y_d)$ is the radius of the $d$-dimensional closed ball $B(y_0,y_1,\dots ,y_d)$ containing $y_0,y_1,\dots ,y_d$ in its boundary (which is well-defined for Lebesgue almost all $(y_0,y_1,\dots ,y_d)\in {\mathbb{R}}^{d(d+1)}$). Denote $z=z(y_0,y_1,\dots ,y_d)$ the centre of $B(y_0,y_1,\dots ,y_d)$, $u_i=u_i(y_0,y_1,\dots ,y_d)$ the unit vector such that $y_i=z+Ru_i$ ($i=0,1,\ldots ,d$), $\Delta (u_0,u_1,\ldots ,u_d)$ the volume of the convex hull of $u_0,u_1,\ldots ,u_d$, and $\nu$ surface measure on the unit sphere in ${\mathbb{R}}^{d}$. Then, by Blasche-Petkantschin’s formula (e.g. Miles (1971)), ${\rm P}(T>t)$ is at most

$$\begin{aligned}&\frac{\kappa ^{d}}{|A|} \int \int \int \int \cdots \int 1[z+Ru_0\in A,\,R>t]R^{d^2-1}\\&\exp \left( -\kappa \omega _dR^{d}\right) \Delta (u_0,u_1,\ldots ,u_d)\\&{\rm d}z\,{\rm d}R\,\nu ({\rm d}u_0)\, \nu ({\rm d}u_1)\cdots \,\nu ({\rm d}u_d) \end{aligned}$$

which reduces to

$$\begin{aligned}&\kappa ^{d}\int _t^\infty R^{d^2-1}\exp \left( -\kappa \omega _dR^{d}\right) \,{\rm d}R\\&\int \int \cdots \int \Delta (u_0,u_1,\ldots ,u_d)\,\nu ({\rm d}u_0) \,\nu ({\rm d}u_1)\cdots \,\nu ({\rm d}u_d). \end{aligned}$$

Thereby, since

$$\begin{aligned}\int _t^\infty R^{d^2-1}\exp \left( -\kappa \omega _dR^{d}\right) \,{\rm d}R =\, \frac{(d-1)!}{d(\kappa \omega _d)^{d}} \left[ 1-F(_{d}(\kappa \omega _dt^{d}))\right] \end{aligned}$$

and

$$\begin{aligned} \int \int \cdots \int \Delta (u_0,u_1,\ldots ,u_d)\,\nu ({\rm d}u_0) \,\nu ({\rm d}u_1)\cdots \,\nu ({\rm d}u_d) =\,\frac{2^{d+1}\pi ^{(d^2+d-1)/2}\Gamma ((d^2+1)/2)}{d!\Gamma (d^2/2) \Gamma ((d+1)/2)^{d}} \end{aligned}$$

(see Theorem 2 in Miles (1971)), we obtain (14) after a straightforward calculation.

Proof of Theorem 1: It suffices to consider the case where $W=b(z,r_1)$. Then $p$ is at most

$$\begin{aligned} \kappa \int _{\Vert z-y\Vert \ge r_2}{\rm P} (T>d(o,b(z-y,r_1)))\,{\rm d}y \le \kappa c_d \int _{\Vert y\Vert > r_2}\int _{\kappa \omega _d(\Vert y\Vert -r_1)^{d}}^\infty f_{d}(t)\,{\rm d}t\,{\rm d}y \end{aligned}$$

where the inequality follows from Lemma 1. Hence, using Fubini’s theorem, a shift for $y$ to hyperspherical coordinates in ${\mathbb{R}}^{d}$, and the fact that $\omega _d=\sigma _d/d$, we easily deduce the result.

1.2 Appendix 2: Moment results

Since ${\mathbf{X}}$ is a Cox process driven by (2), moment results for ${\mathbf{X}}$ are inherited from the distribution of the primary point process ${\mathbf{Y}}$. In particular, ${\mathbf{X}}$ has intensity $\rho ={\rm E}\varLambda (o)$ and pair correlation function $g(x)={\rm E}\left[ \varLambda (o)\varLambda (x)\right] /\rho ^2$, $x\in {\mathbb{R}}^{d}$ (provided these expectations exist), see e.g. Møller and Waagepetersen (2004). This appendix discusses the expressions of $\rho$ and $g$.

Recall the notion of the typical Voronoi cell: Let $\Pi$ denote the space of compact convex polytopes $C\subset {\mathbb{R}}^{d}$ with $|C|>0$ and $o\in {\rm {int}}C$ (we equip $\Pi$ with the usual $\sigma$-algebra for closed subsets of ${\mathbb{R}}^{d}$ restricted to $\Pi$, i.e. the $\sigma$-algebra generated by the sets $\{C\in \Pi :C\cap K=\emptyset \}$ for all compact $K\subset {\mathbb{R}}^{d}$).

The typical Voronoi cell is a random variable ${\mathcal{C}}$ with state space $\Pi$ and distribution

$$\begin{aligned} P({\mathcal{C}}\in F)={\rm E}\sum _i1[y_i\in B,\,C_i-y_i\in F]/(\kappa |B|) \end{aligned}$$

(15)

where $B\subset {\mathbb{R}}^{d}$ is an arbitrary (Borel) set with $0<|B|<\infty$ (by stationarity of ${\mathbf{Y}}$, the right hand side in (15) does not depend on the choice of $B$). Intuitively, ${\mathcal{C}}$ is a randomly chosen cell viewed from its nucleus; formally, (15) is the Palm distribution of a Voronoi cell. It follows by standard measure theoretical considerations that

$$\begin{aligned} {\rm E}\sum _if(y_i,C_i-y_i)=\kappa {\rm E}\int f(y,{\mathcal{C}})\,{\rm d}y \end{aligned}$$

(16)

for any nonnegative (measurable) function $f$, and letting $\mathcal A=|{\mathcal{C}}|$, then ${\rm E}\mathcal A=1/\kappa$. See e.g. (Møller (1989)).

Since ${\mathbf{Y}}$ is a stationary Poisson process, by the Slivnyak-Mecke formula (see e.g. Møller and Waagepetersen (2004)),

$$\begin{aligned} {\rm E}\sum _if(y_i,{\mathbf{Y}})=\kappa {\rm E}\int f(y,{\mathbf{Y}}\cup \{y\})\,{\rm d}y \end{aligned}$$

(17)

for any nonnegative (and measurable) function $f$. By (16)-(17) and stationarity of ${\mathbf{Y}}$, we can then take

$$\begin{aligned} {\mathcal{C}}=C(o;{\mathbf{Y}}\cup \{o\}). \end{aligned}$$

(18)

Proposition 1

If ${\rm E} k(\mathcal A)<\infty$, then

$$\begin{aligned} \rho =\alpha +\beta \kappa {\rm E} k(\mathcal A) \end{aligned}$$

(19)

is finite.

Proof

By (2),

$$\begin{aligned} {\rm E} \varLambda _{\rm ext}(o)= \alpha +\beta {\rm E}\sum _i1[-y_i\in C_i-y_i]k(A_i)h(-y_i,C_i-y_i). \end{aligned}$$

Combining this with (16) and the facts that $\rho ={\rm E}\varLambda _{\rm ext}(o)$ and $|C_i|=|C_i-y_i|$, we obtain that $\rho$ is equal to

$$\begin{aligned}&=\,\alpha + \beta {\rm E}\sum _i1[-y_i\in C_i-y_i]k(|C_i-y_i|)h(-y_i,C_i-y_i)\\&=\,\alpha + \beta \kappa {\rm E}\int 1[-y\in {\mathcal{C}}]k(\mathcal A) h(-y_i,{\mathcal{C}}) \,{\rm d}y\\&=\,\alpha +\beta \kappa {\rm E} k(\mathcal A) \end{aligned}$$

whereby the assertion follows.

The pair correlation function $g$ is more complicated to evaluate. For example, let $k$ be the identity function. Then by similar arguments as in the proof of Proposition 1 and by using (17) and (18), we obtain

$$\begin{aligned}{\rm E}\left[ \varLambda (o)\varLambda (x)\right] =\,\alpha ^2+2\alpha \rho +\beta ^2 \kappa {\rm E}\int 1[\{y,x+y\}\subset {\mathcal{C}}] |{\mathcal{C}}|^2h(y,{\mathcal{C}})\,h(x+y,{\mathcal{C}})\,{\rm d}y + (\beta \kappa )^2{\rm E}\int \int \,1[o\in C(y_1;{\mathbf{Y}} \cup \{y_1,y_2\}),\,x\in C(y_2;{\mathbf{Y}}\cup \{y_1,y_2\})]\,|C(y_1;{\mathbf{Y}}\cup \{y_1,y_2\})| |C(y_2;{\mathbf{Y}}\cup \{y_1,y_2\})| \,h(-y_1;C(y_1;{\mathbf{Y}}\cup \{y_1,y_2\})-y_1)\,h(x-y_2;C(y_2;{\mathbf{Y}}\cup \{y_1,y_2\})-y_2) \,{\rm d}y_1\,{\rm d}y_2. \end{aligned}$$

Here the first mean value corresponds to the case where two secondary points with locations $o$ and $x$ belong to the same cell, and the second mean value corresponds to the case where they belong to two different cells. We are not aware of any analytic methods for evaluating these mean values, even if $h(\cdot ;C)$ is uniform on $C$, in which case

$$\begin{aligned}{\rm E}\left[ \varLambda (o)\varLambda (x)\right]=\alpha ^2+2\alpha \rho +\beta ^2\kappa \int P(\{y,x+y\}\subset C(o;\Phi \cup \{o\})) \,{\rm d}y +\,(\beta \kappa )^2\int \int P(o\in C(y_1;{\mathbf{Y}}\cup \{y_1,y_2\}),\,x\in C(y_2;{\mathbf{Y}}\cup \{y_1,y_2\})) \,{\rm d}y_1\,{\rm d}y_2=\alpha ^2+2\alpha \rho +\beta ^2\kappa \int \exp \left( -\kappa |b(o,\max \{\Vert y\Vert ,\Vert x+y\Vert \})|\right) {\rm d}y+ (\beta \kappa )^2\int \int 1[\Vert y_1\Vert \le \Vert y_2\Vert ,\,\Vert y_2-x\Vert \le \Vert y_1-x\Vert ]\, \exp \left( -\kappa |b(o,\Vert y_1\Vert )\cup b(x,\Vert y_2-x\Vert )|\right) \,{\rm d}y_1\,{\rm d}y_2 \end{aligned}$$

where the integrals may be evaluated by numerical methods.