2 Point Processes 2.3 Point Process Intensities 2.5 Bivariate

K

2.4 The $K$ -Function

To get a more easily interpretable quantity than $\lambda_{2}(u)$ , proceed as follows:

Definition 2.8.

The reduced second moment function of a stationary, isotropic spatial point process is

K(s)=2\pi\lambda^{-2}\int^{s}_{0}\lambda_{2}(r)r\mathrm{d}r.

Theorem 2.1.

For a stationary, isotropic, orderly process,

K(s)=\lambda^{-1}\mathbb{E}[\text{number of further events within distance $s$% of an arbitrary event}]

•

gives a tangible interpretation of $K(s)$ ,
•

suggests a method of estimating $K(s)$ from data,
•
hints at why an estimate of $K(s)$ would be a useful descriptor of an observed spatial pattern:
- –
  
  for clustered patterns, each event is likely to be surrounded by further members of the same cluster and, for small values of $s$ , $K(s)$ will be relatively large.
- –
  
  conversely, if events are regularly spaced, each one is likely to be surrounded by empty space and, for small values of $s$ , $K(s)$ will be relatively small.

A benchmark to determine what we mean by relatively large or small is provided by the following theorem:

Theorem 2.2.

For a homogeneous, planar Poisson process,

K(s)=\pi s^{2}

Proof will be given in the lecture.
$\Box$

Definition 2.9.

A random thinning, $P^{\prime}$ , of a point process $P$ , is a point process whose events are a subset of the events of $P$ generated by retaining or deleting the events of $P$ in a series of mutually independent Bernoulli trials.

We can now establish the following result:

Theorem 2.3.

$K(s)$ is invariant to random thinning.

Proof will be given in the lecture.
$\Box$

Conclusion: the interpretation of an estimated $K$ -function is robust to incomplete ascertainment of cases, provided the incompleteness is spatially neutral.

2.4.1 Estimating the $K$ -function

Confronted with a point pattern, we need to estimate $\lambda(s)$ and $K(s)$ in order to examine the first-order and second-order properties of the process that may have generated it. The data are of the form $x_{i}\in A:i=1,\dots,n$ , for some planar region $A$ .

Estimation of $\lambda$

Here we assume the process is homogeneous, so $\lambda(s)=\lambda$ for all $s$ . Because $\lambda$ is the expected number of events per unit area, we define the following simple estimator:

\hat{\lambda}=\frac{n}{|A|}

Estimation of $K(s)$

Similarly, because

\lambda K(s)=\mathbb{E}[\text{number of further events within distance $s$ of % an arbitrary event}],

we can construct an estimator of $K(s)$ as follows.

1.

Define $E(s)=\lambda K(s)$ and let $d_{ij}$ be the distance between the events $x_{i}$ and $x_{j}$ . Define

$\tilde{E}(s)=\frac{1}{n}\sum^{n}_{i=1}\sum_{j\neq i}I(d_{ij}\leq s),$ (2.1)

where $I(\cdot)$ denotes the indicator function.
2.

The estimator $\tilde{E}(s)$ is negatively biased because we do not observe events outside $A$ , so the observed counts from events $x_{i}$ close to the boundary of $A$ will be artificially low.
3.

Introduce weights, $w_{ij}$ = reciprocal of proportion of circumference of circle, centre $x_{i}$ and radius $d_{ij}$ , which is contained in $A$ .

Unnumbered Figure: Link
4.

An edge-corrected estimator for $E(s)$ is

$\hat{E}(s)=\frac{1}{n}\sum^{n}_{i=1}\sum_{j\neq i}w_{ij}I(d_{ij}\leq s).$
5.

Since $K(s)=E(s)/\lambda$ , define

$\displaystyle\hat{K}(s)$ $\displaystyle=$ $\displaystyle\hat{E}(s)/\hat{\lambda}$ (2.2)

$\displaystyle=$ $\displaystyle\frac{|A|}{n^{2}}\sum_{i=1}^{n}\sum_{j\neq i}w_{ij}I(d_{ij}\leq s)$ (2.3)

Explicit formulae for the $w_{ij}$ can be computed if $A$ is a rectangle or a circle. An algorithm for an arbitrary polygon $A$ is used in Rowlingson and Diggle (1993).

Notes:

1.

Typically, ${\rm Var}\{\hat{K}(s)\}$ tends to increase with $s$ .
2.

As the sampling variance of $\hat{K}(s)$ increases with $s$ , estimates for large $s$ tend to be unreliable. For data on a unit square, it is advisable to estimate only for $s\leq 0.25$ .
3.

The sampling distribution of $\hat{K}(s)$ is largely intractable. See Diggle (2002) for discussion of ways of dealing with this.
4.

There is some technical advantage in using $n(n-1)$ rather than $n^{2}$ as the divisor in the expression (3) for $\hat{K}(s)$ , and it is this version which is implemented in the spatstat software.
5.

To test the hypothesis of complete spatial randomness, $\hat{K}(s)$ can be compared to the value expected under this assumption, $\pi s^{2}$ (see figure below).

2.4.2 Examples of Estimated $K$ -functions

Unnumbered Figure: Link

Caption:Examples of Point processes. Top left: a homogeneous Poisson process - here the points appear randomly within the observation window. Top right: a clustered point process - here the points have a tendency to cluster together. Bottom: a regular Point process - here there is inhibition between the points i.e. no pair of points is close together.

Figure 2.3: First Link, Second Link, Caption: Left hand side: example

K

-functions from Poisson, clustered and regular point processes. The solid line is the

K

function of a simulated Poisson process, the dotted line is the

K

-function of a cluster process and the dashed line is the

K

-function of a regular process. Right hand side: it can sometimes be useful to plot

K(S)-\pi s^{2}

instead of just

K(s)

. Since

K(s)=\pi s^{2}

for a Poisson process, we’d expect

K(S)-\pi s^{2}\approx 0

for Poisson processes,

K(S)-\pi s^{2}\gtrapprox 0

for clustered processes and

K(S)-\pi s^{2}\lessapprox 0

for regular (inhibitory) processes. Of course, in reality, the situation may be more complex, for instance, the process may have similar properties to a Poisson Process when

s

is small, but may behave more like a cluster process for higher values of

s

	$\displaystyle\hat{K}(s)$	$\displaystyle=$	$\displaystyle\hat{E}(s)/\hat{\lambda}$		(2.2)
		$\displaystyle=$	$\displaystyle\frac{\|A\|}{n^{2}}\sum_{i=1}^{n}\sum_{j\neq i}w_{ij}I(d_{ij}\leq s)$		(2.3)

2.4 The K-Function