Point Process Methods
5.3 Complete spatial randomness .1 Introduction.1Introduction
Figure 5.5. Completely random point patterns: 10 simulated realisations of the Poisson point process with intensity 50 in the unit square.
A point process which deserves to be called ‘completely random’ is illustrated in Figure 5.5.
Each panel is a realisation of the homogeneous Poisson point process, also called complete spatial randomness (CSR). The process is characterised by two key properties:
homogeneity: the points have no preference for any spatial location;
independence: information about the outcome in one region of space has no influence on the out-come in other regions.
CSR is important in many ways. It is a realistic model of some physical phenomena, such as radioactivity, rare events, and extreme events. It serves as a benchmark or standard reference model of completely random patterns, against which other patterns can be compared. In many statistical tests, CSR serves as the null hypothesis. Many other models are built starting from CSR, and many mathematical concepts are defined relative to CSR.
Unlike the binomial point process illustrated in Figure 5.4 which has a fixed total number of points, the completely random process shown in Figure 5.5 has a random number of points. This is emphasised by Figure 5.6 which shows realisations of CSR with the same average number of points as the patterns in Figure 5.4. Note the variation in the number of points in each panel of Figure 5.6.
Statisticians call this model the ‘Poisson process’ because the number of points falling in any region follows a Poisson distribution (as explained below). This can be confusing for some readers.
Why should we assume that the number of points should follow a particular distribution? Actually we do not assume it. It turns out that the two properties of homogeneity and independence, described above, effectively imply that the distribution must be Poisson. We begin by explaining how this comes about.
5.3.2 Derivation from basic principles
Homogeneity (in this sense) means that the expected number of points falling in a region B should be proportional to its area |B| on average, that is,
En(X∩ B) =λ|B| (5.3)
Figure 5.6. Completely random point patterns: 10 simulated realisations of the Poisson point process with intensity 8 in the unit square. Note different numbers of points in each panel.
whereλ is constant. In effectλis the average number of random points per unit area, and is known as the intensity of the point process.
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Figure 5.7.Concept of independence between outcomes in disjoint regions (light grey shading).
The concept of spatial independence is sketched in Figure 5.7. The three panels show three different possible realisations (outcomes) of the random point process. In each panel, the same two regions of space — say, A and B — are highlighted in light grey, and the numbers of random points falling in A and in B are displayed. A consequence of the independence assumption is that these counts n(X ∩A) and n(X∩B) must be independent random variables. The value of n(X∩A) carries no information affecting the probabilities of different possible values of n(X ∩ B).
The binomial point process described in Section 5.2.4 does not have the property of spatial independence. Since there are known to be exactly n points altogether, the information that there are (say) 5 points in a region A implies that there are exactly n − 5 points in the complement of A, violating independence.
The independence assumption applies to any choice of disjoint regions A and B, and to any number of such regions. In particular, independence implies that quadrat counts are independent, for any size of quadrat. The left panel of Figure 5.8 shows a division of the rectangle into a 5 × 5 array of quadrats. The independence assumption implies that the numbers of points falling in each of the 25 quadrats must be independent random variables. The right panel of Figure 5.8 shows a finer division, and this too must have the same independence property.
Taking finer and finer divisions of space into squares, the same independence property must continue to hold in each case. When the squares are extremely small, most of the squares will not
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Figure 5.8. The assumption of independence implies that quadrat counts are independent, for any size of quadrat.
contain a random point, and a few squares will contain exactly one random point. Some squares could conceivably contain more than one point, so we need to impose a third assumption:
orderliness: there is negligible probability2that a region contains more than one point, when the region is small.
This implies that no two points can coincide, and also excludes fractal-like behaviour.
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Figure 5.9. Poisson limit. The number of points falling in a given region (indicated by the curved contour) is the result of a large number of trials (indicated by numerals), each having a small probability of success. If the trials are independent then the total number of points has a Poisson distribution.
Figure 5.9 sketches a region of space A which has been subdivided into tiny squares. The number n(X ∩ A) of random points falling in A is equal to the sum of the numbers falling in the squares inside A. Assuming, as above, that the outcomes in different squares are independent, and that there is negligible chance that some squares have more than one point, n(X ∩ A) is the number
2To be precise, this probability, divided by the area of the region, must go to zero as the area goes to zero.
of successes in a large number of independent trials, each trial having a small probability of success.
By a famous theorem in probability theory, this means that n(X∩A) has a Poisson distribution [383].
The Poisson probability distribution is the classical law of the frequency of rare events. It is often used to model accidents, cases of rare diseases, and other rare events. Under the Poisson distribution, the probability of obtaining exactly k rare events is
P{N = k} = e−µµk
k! (5.4)
for any k = 0,1,2,.... As usual in statistics, Greek letters represent parameters: here the Greek letterµ (‘mu’, cognate to ‘m’) is the mean of the Poisson distribution. For example, Figure 5.10 shows bar charts of the probabilities of the Poisson distribution with meansµ= 0.6 andµ= 1.7, respectively.
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Count Probability 0.10.30.5
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Count Probability 0.10.30.5
Figure 5.10. Probabilities for the Poisson distributions with meanµ = 0.6 (Left) and µ= 1.7 (Right).
Since the expected number of points falling in the test region A is En(X∩A) =λ|A|, we have just found that the random number of points n(X ∩ A) has a Poisson distribution with meanµ=λ|A|.
This finding is truly remarkable. Starting with two general assumptions (homogeneity and in-dependence) we have come to a very specific conclusion: the number of random points falling in a test region follows a Poisson distribution. This may help to explain the importance of the Poisson distribution in point processes.
In summary, the homogeneous Poisson point process (or ‘complete spatial randomness’, CSR) with intensityλ> 0 is a locally finite point process with the properties of
homogeneity: the number n(X ∩ B) of random points falling in a test region B has mean value En(X∩ B) =λ|B|;
independence: for test regions B1, B2, . . . , Bmwhich do not overlap, the counts n(X∩B1), . . . , n(X∩
Bm) are independent random variables;
Poisson distribution: the number n(X∩B) of random points falling in a test region B has a Poisson distribution (5.4).
See Section 9.2.1 for more detail.
5.3.3 Useful properties of CSR
CSR has several important and useful properties which we mention briefly here.
For a test region B, suppose we are given the information that n(X ∩ B) = n, that is, exactly n points of the Poisson process fell in B. The conditional property of CSR is that these n points
exactly 8 points is statistically equivalent to one of the panels in Figure 5.4.
Thinningmeans deleting some of the points from a point pattern. Under ‘completely random thinning’ each point of the point pattern is randomly deleted or retained, with probability p of retention, independently of the fate of other points. The thinning property of CSR is that if we start with a homogeneous Poisson process with intensityλ, and apply completely random thinning with retention probability p, the points retained after thinning constitute a homogeneous Poisson process with intensity pλ. See Figure 5.11.
Figure 5.11. Thinning property of Poisson process. Left: Poisson process X. Middle: points of X are randomly deleted or retained.Right: points that are retained constitute a Poisson process.
Superimposingtwo point processes X and Y means that we combine the points from both pro-cesses into a new point process Z = X ∪ Y. The superposition property of the Poisson process is that if X and Y are homogeneous Poisson processes with intensitiesλX andλY, and if they are independent of each other, then their superposition Z is also a homogeneous Poisson process, with intensityλX=λX+λY. See Figure 5.12.
Figure 5.12. Superposition property of Poisson process. When two independent Poisson point processes X and Y (Left and Middle) are superimposed, the result Z is also a Poisson process (Right).
5.3.4 Simulation of CSR
It is easy to simulate the Poisson process directly, using the properties above. Given a region B where the realisation is to be generated, and an intensity valueλ, we first determine the total number of points by generating a random number N according to a Poisson distribution (5.4) with mean µ=λ|B|. These N points are then placed independently in B with a uniform distribution. In spatstat, use the command rpoispp. Each panel of Figure 5.5 was generated by the command
> plot(rpoispp(50))
To develop some intuition about completely random patterns, it is useful to repeat the command plot(rpoispp(100)) several times (use the up-arrow key or Ctrl-P to recall the previous com-mand line) so that you see several replicates of the Poisson process. Alternatively
> plot(rpoispp(100, nsim=9))
would produce a 3 ×3 array of different realisations of the same constant intensity Poisson process.
In particular you will notice that the points in a homogeneous Poisson process are not ‘uniformly spread’: there are empty gaps and clusters of points. A typical realisation of the process does not show a uniform spread of points: after all, the points are independent of one another. A typical realisation will include some large gaps and some clusters of points.
There is even a small chance of having no random points in a region B: the probability that a Poisson variable with meanµ=λ|B| will yield the value 0 is e−λ |B|according to (5.4).
The command rpoispp has arguments lambda (the intensity) and win (the window in which to simulate). The default window is the unit square. Figure 5.13 shows a realisation of CSR inside the letter ‘R’, generated by the command
> plot(rpoispp(100, win=letterR))
Figure 5.13.Homogeneous Poisson process realisation inside the letter R.
To simulate a Poisson process conditionally on a fixed number of points, use the command runifpoint, which generates a realisation of the binomial point process.