2 Point Processes 2.2 The Poisson Process 2.4 The

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2.3 Point Process Intensities

How should we define point process analogues of the mean and covariance structure for real-valued spatial processes?

Definition 2.5.

Let $\delta x$ denote a small region containing the point $x$ . The first-order intensity function of a spatial point process is

\lambda(x)=\lim_{|\delta x|\rightarrow 0}\left\{\frac{\mathbb{E}[N(\delta x)]}% {|\delta x|}\right\}

Definition 2.6.

Let $\delta x$ and $\delta y$ denote small regions containing the points $x$ and $y$ respectively. The second-order intensity function of a spatial point process is

\lambda_{2}(x,y)=\lim_{\stackrel{{\scriptstyle\scriptstyle|\delta x|% \rightarrow 0}}{{\scriptstyle|\delta y|\rightarrow 0}}}\left\{\frac{\mathbb{E}% [N(\delta x)N(\delta y)]}{|\delta x||\delta y|}\right\}

Definition 2.7.

The covariance density of a spatial point process is:

\gamma(x,y)=\lambda_{2}(x,y)-\lambda(x)\lambda(y).

Proposition 2.2.

For a stationary, isotropic spatial point process, we have:

(i): $\lambda(x)\equiv\lambda=\mathbb{E}[N(A)]/|A|$ , which is constant, for all $A$ .
(ii): $\lambda_{2}(x,y)\equiv\lambda_{2}(\|x-y\|)\equiv\lambda_{2}(u)$ i.e., the second-order intensity function depends only on distance
(iii): $\gamma(x,y)\equiv\gamma(u)=\lambda_{2}(u)-\lambda^{2}$ .

Physical interpretation:

•

$\lambda=$ expected number of events per unit area.
•

$\lambda_{2}(u)=??$