Complete spatial randomness (CSR)
CSR is a hypothesis to verify before proceeding with further analysis on a point pattern (PP). It describes a point process in which point events occur within a given study area in a completely random fashion. It is synonymous with a homogeneous spatial Poisson process.
CSR is seen as the “intermediate” situation between regular spatial structures and clustered spatial structures. It is therefore necessary to:
- quantify the CSR situation
- assess the "deviations" from this situation in one direction or the other
- if the "deviation" is large, the hypothesis is rejected
Types of tests:
- Tests based on quadrat counts
- Tests based on distance
- graphical tests
- Monte Carlo tests
Test based on quadrat counts
We identify \(A_1, ..., A_m\) by performing a tessellation of \(W\) through a collection of square cells of area a
\(N(A_i)\): random number of events in \(A_i\). CSR implies that \(N(A_i) \sim Pois(\lambda a)\).
The test compares the observed number \(n_i\) of events in \(A_i\) with the expected (constant) number under \(H_0\)
The test statistic contains the dispersion index, i.e. the ratio between the variance and the mean:
\[ X^2 = \sum_{i=1}^m \frac{(n_i- \overline n)^2}{\overline n} = (m-1) \space \space I \] Asymptotically, if \(\overline n\) is not too small, it can be approximated by:
\[X^2|H_0 \sim \chi^2_{m-1}\]
Where \(n_i\) is the number of events in the i-th cell, \(\overline n = \frac{\sum n_i}{m}\), and \(I\) = dispersion index, that is:
\[I= \frac{S^2}{\overline n}\]
- If \(I=1\) then the mean equals the sample variance, and therefore there is evidence of a Poisson distribution (CSR)
- If \(I \gg 1\) then we are in a situation of over-dispersion and events tend to attract each other, arranging themselves in clusters
- If \(I \ll 1\) then we are in a situation of under-dispersion and events tend to repel each other, arranging themselves in a regular pattern (this situation is difficult to diagnose)
Tests based on distances
Comparison between the sample (empirical) distribution function of some distance related to the PP points and the one expected under CSR (i.e. under \(H_0\)).
Types of distance between points of the point pattern
- Inter event distances
- Nearest neighbour distance
- Point to Nearest Event distance
Nearest neighbour distance
Distribution function of R under \(H_0\) (CSR hypothesis):
\[ G(r)= 1-\exp(- \lambda \pi r^2) \]
Empirical distribution function of R evaluated at r:
\[ \hat G_0(r) = \sum_{i=1}^n I(r_i \le r)/n \]
where \(r_i\): distance between event \(s_i\) and its nearest event in \(W\)
Graphical test
In the graphical test, a comparison is made between \(G(t)\) and \(\hat G_0(r)\)
Graphical test with Monte Carlo envelopes
We perform \(B\gg0\) replications of \(\hat G_0(r)\) and check whether (and for which values) the empirical curve falls outside the envelope.
The procedure is not a significance test in the usual sense, but it is very useful in the case of complex processes for which the distribution function may not be known analytically but one is able to simulate trajectories.
An empirical curve that falls outside the envelope under \(H_0\) for large stretches indicates that the observed PP provides little support for the CSR hypothesis.
Advantages
- flexibility in the choice of the test statistic
- valid test for samples of any size (numerical approximation tied to the number K of iterations performed)
Disadvantages
- low power: the test has a generic alternative hypothesis (for example, possible alternatives to CSR include both inhomogeneous PPs as well as regular or clustered ones)
- arbitrariness in the choice of K