Do species that inhabit a certain area form an ecosystem or are they a random collection? To answer this question, ecologists select appropriate numerical characteristics: average body mass, body length, and so on. Each species is described by a point in the corresponding d-dimensional space and all observed species form a set of points. We can then describe species diversity by computing a geometric measure of this set: the volume of the smallest bounding box or of the convex hull, the diameter of the smallest enclosing sphere, the mean pairwise distance between the points, and so on.

The computed value of the selected measure is then compared with the mean value of this measure over all possible random selections of points from the set of all species, taking into account the corresponding standard deviation. If at a given geographic location the value of the measure differs from the mean by more than two (or three) standard deviations, then we are reasonably confident that we have an ecosystem and not a random collection of species. The corresponding mean and standard deviation values are usually computed by Monte Carlo simulations.

The main limitation of this approach is that it takes too much computation time. The authors propose much more efficient algorithms for computing the mean and standard deviation. Even faster algorithms become possible if we take into account that for ecological purposes, it is sufficient to compute the mean and standard deviation with some accuracy.