Point Process

Definition

Notation

  • $\mathcal{S}$ : a metric space with metric $d$

  • $X$ : point process on $\mathcal{S}$

  • $x$ : realization of a point process $X$

  • $x$ is said to be locally finite if $n(x_B)<\infty$ whenever $B\subset\mathcal{S}$ is bounded. $n(x_B)$ is the number of points in $x_B=x\cap B$

Definition 2 (Point Process). A point process $X$ defined on $\mathcal{S}$ is a measurable mapping defined on some probability space $(\Omega,\mathcal{F},P)$ and taking values in \((\mathbb{N}_{lf},\mathcal{N}_{lf})\)

  • $\mathbb N_{lf}=\lbrace x\subset\mathcal{S}:n(x_B)<\infty,\;\;\forall\;\mathrm{bounded}\;\; B\subset\mathcal{S}\rbrace $

  • \(\mathcal{N}_{lf}\) is a \(\sigma\)-algebra on \(\mathbb N_{lf}\) such that

\[\mathcal{N}_{lf}=\sigma(\lbrace x\in\Bbb N_{lf}:n(x_B)=m)\rbrace :B\in\mathcal{B}_0,m\in\Bbb{N}_0)\]

where $\mathcal{B}_0$ is the class of bounded Borel sets of Borel $\sigma$-algebra on $\mathcal{S}$

The distribution $P_X$ of point process $X$ is given by

\[P_X(F) = P(\lbrace \omega\in\Omega:X(\omega)\in F\rbrace )\quad\forall F\in\mathcal{N}_{lf}\]

Remarks 1.

  • $X$ is measurable is equivalent to : count function $N(B):=n(x_B)$ is a random variable for any $B\in\mathcal{B}_0$

  • The distribution of a point process $X$ is determined by the finite dimensional distributions of its count function.

  • \(\mathcal{N}_{lf}=\sigma(\mathcal{N}^\circ_{lf})\) where \(\mathcal{N}^\circ_{lf} = \lbrace \lbrace x\in\mathcal{N}_{lf}:n(x_B)=0\rbrace :B\in\mathcal{B}_0\rbrace\) is the class of void events

  • $v(B):=P(N(B)=0)$ for $B\in\mathcal{B}_0$ is a void probability

  • If $\mathcal{S}$ is polish space, $\mathcal{N}_{lf}$ is separable

Definition 3 (Marked Point Process). Marked point process $X$ with points in $T$ and mark space $\mathcal{M}\subset\mathbb R^p$ (or marked point pattern) means a point process with extra information attached to each point.

$X=\lbrace (\xi,m_\xi):\xi\in Y\rbrace $, where $Y$ is a point process on $T$ and $m_\xi\in \mathcal{M}$ is called a mark.

Poisson Point process

Binomial point process

Definition 4 (Binomial point process). Binomial point process $X\sim binomial(B,n,f)$ for a density function $f$ on $B\subset\mathcal{S}$ is defined when it consists of $n$ iid points with density $f$ on $B$.

Poisson point process

  • Poisson point process serves as a model for CSR

  • Also it serves as a reference point process for advanced models

  • Poisson point process $X$ on $\mathcal{S}\subset\mathbb R^d$ is specified by locally integrable intensity function $\rho:\mathcal{S}\to[0,\infty)$

Definition 5 (Poisson point process). Poisson point process $X\sim\textrm{poisson}(\mathcal{S},\rho)$ is defined if

  1. *$\forall B\subset\mathcal{S}$ with $\mu(B)=\int_B\rho(\xi)d\xi<\infty$

    \(N(B)\sim\textrm{Poisson}(\mu(B))\)*

  2. $\forall n\in\mathbb N$ and $\forall B\subset\mathcal{S}$ with $0<\mu(B)<\infty$, conditional on $N(B)=n$, $x_B\sim binomial(B,n,f)$ with $f(\xi)=\rho(\xi)/\mu(B)$

$\mu(\cdot)$ above is called an intensity measure.

Remarks 2.

  • $\Bbb{E}N(B) = \mu(B)$ : intensity measure determines the expected number of points

  • When $\rho(\xi)\equiv\rho$, the process is called homogeneous. Otherwhise, the poisson point process is inhomogeneous.

  • When $\rho(\xi)\equiv 1$, it is called as standard Poisson point process.

  • If the distribution of process is invariant under translation, the process is called as stationary

    \[X+s=\lbrace \xi+s:\xi\in X\rbrace \equiv X\quad\forall s\in\mathbb R^d\]

  • If the distribution of process is invariant under rotations about the origin in $\mathbb R^d$ then the process is called as isotropic.

    \[\mathcal{O}X=\lbrace \mathcal{O}\xi:\xi\in X\rbrace \equiv X\quad\forall\mathcal{O}\text{ is rotation }\]

Properties of Poisson point process

Proposition 1.

  1. $X\sim poisson(\mathcal{S},\rho)$ if and only if $\forall B\subset\mathcal{S}$ with $\mu(B)=\int_B\rho(\xi)d\xi<\infty$ and $\forall F\in\mathcal{N}_{lf}$

    \[P(x_B\in F) = \sum_{n=0}^\infty\frac{\exp(-\mu(B))}{n!}\int_B\cdots\int_B\mathbf{1}[\lbrace \xi_1,\ldots,\xi_n\rbrace \in F]\times\prod_{i=1}^n\rho(\xi_i)d\xi_1\cdots d\xi_n\]

  2. If $X\sim poisson(\mathcal{S},\rho)$, then for functions $h:\Bbb N_{lf}\to[0,\infty)$ and $B\subset \mathcal{S}$ with $\mu(B)<\infty$,

    \[\Bbb Eh(x_B) = \sum_{n=0}^\infty\frac{\exp(-\mu(B))}{n!}\int_B\cdots\int_Bh(\lbrace \xi_1,\ldots,\xi_n\rbrace )\times\prod_{i=1}^n\rho(\xi_i)d\xi_1\cdots d\xi_n\]

Theorem 1 (Existence). $X\sim poisson(\mathcal{S},\rho)$ exists and is uniquely determined by its void probability \(v(B)=\exp(-\mu(B))\) for bounded $B\subset \mathcal{S}$.

Proposition 2 (Independent scattering). If $X$ is a Poisson process on $\mathcal{S}$, then $x_{B_1},\cdots$ are independent for disjoint sets $B_1,B_2,\cdots\subset \mathcal{S}$

Proposition 3 (Generating functional). If $X\sim poisson(\mathcal{S},\rho)$,

\[G_X(u)=\Bbb E\bigg[\prod_{\xi\in X}u(\xi)\bigg] = \exp\bigg(\int_\mathcal{S}(1-u(\xi))\rho(\xi)d\xi\bigg)\]

for functions $u:\mathcal{S}\to[0,1]$

Proposition 4 (Construction of stationary Poisson point process). Let $s_1,u_1,s_2,u_2,\ldots$ be mutually independent, where each $u_i$ is uniformly distributed on $\lbrace u\in\Bbb R^d:\Vert u\Vert = 1\rbrace $ and $s_i\sim Exp(\rho w_d)$ with mean $1/(\rho w_d)$ for $\rho>0$. $w_d=\pi^{d/2}/\Gamma(1+d/2)$ is the volume of the $d$-dimensional unit ball. Let $R_0=0$ and $R_i^d=R^d_{i-1}+s_i, i=1,2,\cdots$. Then,

\(X=\lbrace R_1u_1,R_2u_2,\cdots\rbrace \sim poisson(\mathbb R^d,\rho)\)*\

Theorem 2 (Slivnyak-Mecke). If $X\sim poisson(\mathcal{S},\rho)$, for function $h:\mathcal{S}\times\Bbb N_{lf}\to[0,\infty)$,

\[\Bbb E\bigg[\sum_{\xi\in X}h(\xi,X\backslash\xi)\bigg] = \int_\mathcal{S}\Bbb E[h(\xi,X)]\rho(\xi)d\xi\]

Theorem 3 (Extended Slivnyak-Mecke’s Theorem). If $X\sim poisson(\mathcal{S},\rho)$, for any $n\in\Bbb N$ and any function $h:\mathcal{S}^n\times\Bbb N_{lf}\to[0,\infty)$,

\[\Bbb E\bigg[\sum_{\xi_1,\cdots,\xi_n\in X}^{\neq} h(\xi_1,\cdots,\xi_n,X\backslash\lbrace \xi_1,\cdots,\xi_n\rbrace )\bigg] = \int_\mathcal{S} \Bbb E[h(\xi_1,\cdots,\xi_n,X]\prod_{i=1}^n\rho(\xi_i)d\xi_1,\cdots d\xi_n\]

where $\neq$ means the $n$ points $\xi_1,\cdots,\xi_n$ are pairwise distinct.

Two basic operations for point processes

  • Superposition : A disjoint union $\cup_{i=1}^\infty X_i$ of point processes

    Proposition 5. If $X_i\sim poisson(\mathcal{S},\rho_i), i=1,2,\cdots$ are mutually independent and $\rho=\sum_i\rho_i$ is locally integrable, then with probability one, $X=\sum_{i=1}^\infty X_i$ is a disjoint union and $X\sim poisson(\mathcal{S},\rho)$

  • Thinning

    • $X_{thin}$, independent thinning of $X$ with retention probability $p(\xi)$, is obtained by including $\xi\in X$ in $X_{thin}$ with probability $p(\xi)$, where points are included/excluded independently each other.

    • $X_{thin}=\lbrace \xi\in X:R(\xi)\leq p(\xi)\rbrace $ where $R(\xi)\sim uniform(0,1), \xi\in\mathcal{S}$ are mutually independent and independent of $X$.

    Theorem 4. Suppose $X_i\sim poisson(\mathcal{S},rho_i)$. Then, $X_{thin}$ with retention probability $p(\xi)$, and $X\backslash X_{thin}$ are independent Poisson point process with intensity functions $\rho_{thin}(\xi)=p(\xi)\rho(\xi)$ and $(\rho-\rho_{thin})(\xi)$ respectively.

  • Inhomogeneous Poisson point process from homogeneous Poisson point process by thinning

    Corollary 1. Suppose that $X\sim poisson(\Bbb R^d,\rho)$ with $|\rho(\xi)|<c$ for some $0<c<\infty$. Then, $X$ is distributed as an independent thinning of $poisson(\Bbb R^d,c)$ with retention probability $p(\xi)=\rho(\xi)/c$

Simulation of Poisson point process

Homogeneous case

Simulation of $X\sim poisson(\Bbb R^d,\rho)$ within bounded $B$

  1. if $B=b(0,r)$ :

    Use proposition 4 : i.e. generate \(s_1,\ldots,s_m\sim Exp(\rho w_d)\) and \(u_1,\ldots,u_m\sim uniform(\lbrace u:\Vert u\Vert = 1\rbrace )\), where $m$ is given by $R_{m-1}\leq r\leq R_m$. Then, return

    \[x_B = \lbrace R_1u_1,\cdots,R_{m-1}u_{m-1}\rbrace\]

  2. if $B=[0,a_1]\times\cdots\times[0,a_d]$ :

    generate $N(B)=Poisson(\rho a_1\cdots a_d)$, and generate $N(B)$ points uniformly in $B$.

  3. if $B$ is none of 1 and 2 :

    Simulate $X$ on a ball or box $B_0$ containing $B$ and disregard the points falling outside of the ball or box

Inhomogeneous case

When $X$ is inhomogeneous with $\rho(\xi),\xi\in B$ and $\rho(\xi)\leq \rho_0$ for a constant $\rho_0>0$, by Corollary 1,

  1. Generate a homogeneous Poisson point process $Y$ on $B$ with intensity function $\rho_0$.

  2. Obtain $X_B$ as an independent thinning of $Y_B$ with retention probability $p(\xi)=\rho(\xi)/\rho_0,\xi\in B$

Density of Poisson point process

  • Suppose that $X_1, X_2$ are two point processes on $\mathcal{S}$ and $X_1$ is absolutely continuous w.r.t. $X_2$. i.e.

    \[P(X_2\in F)=0\Rightarrow P(X_1\in F)=0\quad \forall F\in\mathcal{N}_{lf}\]

  • By Radon-Nikodym theorem, there exists a function $f:\Bbb N_{lf}\to[0,\infty]$ such tat $\forall F\in\mathcal{N}_{lf}$

    \[P(X_1\in F) = \Bbb E[I(X_2\in F)f(X_2)]\]

  • We call $f$ a density of $X_1$ w.r.t. $X_2$.

Proposition 6.

  1. For any numbers $\rho_1,\rho_2>0$, $poisson(\Bbb R^d,\rho_1)$ is absolutely continuous w.r.t. $poisson(\Bbb R^d,\rho_2)$ iff $\rho_1=\rho_2$.

  2. *For $i=1,2,$, suppose that $\rho_i:\mathcal{S}\to[0,\infty)$ such that $\mu_i(\mathcal{S})=\int_\mathcal{S}\rho_i(\xi)d\xi$ is finite and $\rho_2(\xi)>0$ whenever $\rho_1(\xi)>0$. Then $poisson(\mathcal{S},\rho_1)$ is absolutely continuous w.r.t. $poisson(\mathcal{S},\rho_2)$ with density

    \[f(x)=\exp(\mu_2(\mathcal{S})-\mu_1(\mathcal{S}))\prod_{\xi\in x}\frac{\rho_1(\xi)}{\rho_2(\xi)}\]

    for finite point configurations $x\subset \mathcal{S}$.*

From 2 at Proposition, we can let $\rho_2\equiv 1$ and suppose $\mathcal{S}$ is bounded. Then, such Poisson point process, say, $X_1$ is always absolutely continuous w.r.t. $poisson(\mathcal{S},1)$ and

\[f(x)=\exp(|\mathcal{S}|-\mu_1(\mathcal{S}))\prod_{\xi\in X}\rho_1(\xi)\]

Marked Poisson Point process

Definition 6. $X=\lbrace (\xi,m_\xi):\xi\in Y\rbrace $, $(\xi,m_\xi)\in\mathcal{T\times M}$ is a marked Poisson point process if $Y\sim poisson(\mathcal{T},\phi)$ where $\phi$ is locally integrable intensity function, and the *marks $\lbrace m_\xi,\xi\in Y\rbrace $ are mutually independent condional on $Y$.*

  • If the marks are identically distributed with a common distribution $Q$, then $Q$ is called the mark distribution.

  • If $\mathcal{M}=\lbrace 1,\cdots,K\rbrace $, it is called multitype Poisson point process.

Proposition 7. Let $X$ be a marked Poisson point process with $\mathcal{M}\subset \Bbb R^p$, where conditional on $Y$, each mark $m_\xi$ has a discrete or continuous density $p_\xi$, which doesn’t depend on $Y\backslash\xi$. Let $\rho(\xi,m)=\phi(\xi)p_\xi(m)$. Then,

  1. $X\sim poisson(\mathcal{T\times M},\rho)$

  2. If $\kappa(m)=\int_\mathcal{T}\rho(\xi,m)d\xi$ is locally integrable, then

\[\lbrace m_\xi:\xi\in Y\rbrace \sim poisson(\mathcal{M},\kappa)\]

Multivariate Poisson process and random labeling

  • Multitype point process $X$ with $\mathcal{M}=\lbrace 1,\ldots,K\rbrace $ is equivalent to multivariate point process $(X_1,\ldots,X_K)$

  • The following two statements are equivalent.

    1. $P(m_\xi=i\Vert Y=y)=p_\xi(i)$ depends only on $\xi$ for realizations $y$ of $Y$ and $\xi\in y$

    2. $(X_1,\cdots,X_K)$ is a multivariate Poisson point process with independent components $X_i\sim poisson(\mathcal{T},\rho_i)$, where \(\rho_i(\xi)=\phi(\xi)p_\xi(i),\quad i=1,\cdots,K\)

  • Random labeling means that conditional on $Y$, the marks $m_{\xi}$ are mutually independent and the distribution of $m_\xi$ does not depend on $Y$.

References

  • 서울대학교 공간자료분석 강의노트
  • Handbook of Spatial Statistics

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