Analysis of clade shape using queueing theory and the fast Fourier transform

Abstract

Using queueing theory and the fast Fourier transform, we develop a new approach to analyzing the diversity paths of fossil taxa. The methods employed focus on calculating the probabilities, for each taxonomic group, of arriving at particular values of the diversity at particular times, assuming that diversity paths are random walks. These probabilities, which are represented in probability matrices, allow us to establish a number of null hypotheses that can be used to test for randomness in the diversity paths of the taxonomic groups themselves. As the model employed here is conceptually similar to that employed by Gilinsky and Bambach (1986) in bootstrapping, one consequence of the approach is, in effect, an analytic solution of the bootstrap. A comparison of bootstrapping and analytic results in 15 taxa of marine invertebrates gives nearly identical results. The analytic approach is more general, however, and allows for the calculation of probability matrices under other model assumptions not treated here. We have also applied the probability matrices to a series of tests for nonrandomness as a whole, in contrast to the more usual method of testing particular clade statistics. The familial diversity paths of all 36 extinct orders of marine invertebrates with ten or more families were analyzed. In no case was the null hypothesis of randomness rejected at the 0.05 level. Given the model, and assuming that extinct taxa are representative of all taxa, the familial diversity paths of marine orders appear to behave as random walks.