Some recent ‘refinements’ of the Unitary Association Method: a short discussion

Abstract

In a recent Lethaia paper,Monnet et al. (2011) elaborated an excel-lentanalysis ofthe Devonianammonoid biochronology fromMor-occo. In their paper, they proposed a series of refinementsintended to ameliorate the performances of the Unitary Associa-tion Method (UAM) and of the UAgraph program. Their mainproposals and criticisms are welcome, but need to be evaluated.UAM is a graph theoretical model designed for the constructionof concurrent range zones using a fully deterministic approach.The basic idea is to construct a discrete sequence of coexistenceintervals of species. Each interval, corresponding to one UA, is ofminimal duration while consisting of a maximal set of intersectingranges. Each UA is characterized by a set of species allowing itsidentification in the stratigraphical sections. The chronological sig-nificance of the units generated by the UA software (UAgraph byHammer Guex Savary, http://folk.uio.no/ohammer/uagraph)depends on two important parameters: (1) Their lateral traceability(what we call ‘reproducibility’); and (2) The superpositional con-trol between two adjacent units. The major difficulties related tothe construction of such units are related to the conflicting inter-taxa relationships (=cycles in the biostratigraphical graph G*) andthe inter-maximal cliques contradictions (strongly connected com-ponents in the maximal clique graph Gk). Such contradictions aregenerally eliminated by adding virtual coexistences (=edges) in theG* graph. The goal of the method is not to reduce the number ofvirtual coexistences at all cost, but to produce a robust solutionthat reflects the quality of the data, i.e. a solution optimizing thereproducibility andinter-unitssuperpositional control. UAs shouldbe considered as intervals of uncertainty on which we have estab-lished an order relation: the UA sequences represent units whichare ordered as older, sub-contemporaneous, or younger. The orderof the event within a single particular unit is not known (details inGuex & Davaud 1984; Guex 1991; see also Mailliot et al. 2006 foran application) because the virtual coexistences between some taxarepresent intertaxa exclusions, which cannot be solved by means ofstatistical techniques; they can only be solved by the constructionof composite stratigraphicalsections, whenever possible. For exam-ple: if you tell a neutral observer that three different sectionscontain, respectively, an ammonite, trilobite and nummulite inter-calated between Precambrian and Quaternary sediments, theobserver will be unable to tell you that the true order is trilobite->ammonite-> nummulite, because there is no superpositional con-trol between the three taxa. The observer can only enumerate thenine possible sequences, which is an useless exercise. The presenceof any one of the three taxa just indicates that the sedimentis post-Precambrian and pre-Quaternary.Monnet et al. (2011) were first to suggest the use of the statisti-cal technique known as ‘bootstrapping’ to evaluate the robustnessofthe UAgraphsolutions by meansof ‘confidence intervals’.‘Boot-strapping’ is a well-known technique of re-sampling using itera-tions resulting from the application of a Monte-Carlo method.Monnet et al. propose to use that technique, but they do not sayhow the re-sampling should be done: for instance, should onebootstrap the occurrencesof the taxa,the samples andthe sections?The identity and ordering of the UAs will of course partially breakdown in the bootstrap replicates – how should the ‘confidenceintervals’then be reported? Inourview, the use of such a techniquewould add nothing to current results of the UAs: the robustness ofthe solutions, as mentioned before, depends strictly on the repro-ducibility and inter-units superpositional control. They are theonly criteria allowing a serious evaluation of the internal coherenceof the UAresults.Concerning the strongly connected components of the Gkgraph, Monnet et al. (p. 15) write : ‘UA-graph currently solvesthese contradictions by using the ‘weakest link’ rule (i.e. the cliquesuperposition supported by the fewest inter-taxon relationships isdestroyed; see Guex 1991, p. 82). Given the uncertainties related tothis type of resolution, the unique result produced by the UAM islikely to be partly wrong. Figure 9 reports an imaginary example(Fig. 9A) containing cycles between its maximal cliques, where theautomatic resolution by the software UA-graph yields a result(Fig. 9B), which is clearly not the most parsimonious comparedwith what can be found empirically (Fig. 9C). This example clearlyillustrates and demonstrates that the ‘weakest link’ rule is not ade-quate in such cases.’ Figure 9A of Monnet et al. (2011) is redrawn