2 Point Processes

2.5 Bivariate K-functions

In a bivariate stationary spatial point process, let λj:j=1,2 denote the intensity, or mean number of type j events per unit area.

Definition 2.10.

Bivariate K-functions: we define functions Ki⁢j⁢(s) such that

λj⁢Ki⁢j⁢(s) = expected number of further type j events within
distance s of an arbitrary type i event

There are (at least) two possible physical means by which bivariate point process data may arise:

  • •

    the two processes originate independently of one another

  • •

    locations of the two processes originate as part of a single process, and then are randomly marked (‘1’ or ‘2’)

Proposition 2.3.

Theoretical results relevant to interpretation of estimated K-functions:

  • •

    if type j events form a homogeneous Poisson process, then

    Kj⁢j⁢(s)=π⁢s2
  • •

    if type 1 and type 2 events form independent processes, then

    K12⁢(s)=π⁢s2
  • •

    if type 1 and type 2 events form a random labelling of a univariate process with K-function K⁢(s), then

    K11⁢(s)=K12⁢(s)=K22⁢(s)=K⁢(s)

From the above results, it can be seen that the generating hypotheses of independence and random labelling are equivalent if and only if both processes are homogeneous Poisson processes. In this case, K11⁢(s)=K12⁢(s)=K22⁢(s)=π⁢s2.

Example 2.2.

Displaced amacrine cells in the retina of a rabbit (Diggle, 1986), see Figure 2.4.

  • •

    type 1 events transmit information to the brain when a light goes on

  • •

    type 2 events transmit information to the brain when a light goes off

  • •

    interest is in discriminating between two developmental hypotheses:

    1. 1.

      on and off cells are initially generated in separate layers which later fuse to form the mature retina

    2. 2.

      on and off cells are initially undifferentiated in a single layer and acquire their distinct functionality at a later stage

Figure 2.4: Link, Caption: Locations of 294 displaced amacrine cells in the retina of a rabbit. Solid and open circles respectively identify ‘on’ and ‘off’ cells. While both processes appear to be regular in nature, it is difficult visually to understand whether there is a relationship between the two sets of points, and if so what the nature of that relationship is. Estimating and plotting the bivariate K-function is one way to gain some understanding.
Figure 2.5: Link, Caption: Second-order properties of the bivariate amacrine cells point pattern. Functions plotted are D^⁢(s)=K^⁢(s)-π⁢s2: the short dashes are the on cells only, the long dashes are all cells and the solid line is the bivariate case. For comparison, the parabola -π⁢s2 is also shown as a solid line.

From Figure 2.5, we conclude that the random labelling hypothesis does not seem tenable, but the independence hypothesis seems reasonable.