Self-Weighted Optimization: Tree Searches and Character State Reconstructions Under Implied Transformation Costs
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Cladistics 13, 225–245 (1997) WWW http://www.apnet.com Self-Weighted Optimization: Tree Searches and Character State Reconstructions under Implied Transformation Costs Pablo A. Goloboff Consejo Nacional de Investigaciones Científicas y Técnicas, Instituto “Miguel Lillo,” Miguel Lillo 205, 4000 S.M. de Tucumán, Argentina Accepted 17 June 1997 A method to assess the cost of character state transfor- known as “optimization” (Farris, 1970). During opti- mations based on their congruence is proposed. mization, minimizing some character state Measuring the distortion of different transformations transformations may be preferable to minimizing oth- with a convex increasing function of the number of trans- ers; the relative costs of different character state formations, and choosing those reconstructions which transformations determine the character state recon- minimize the distortion for all transformations, may pro- structions, the tree costs, and which trees are chosen. vide a better optimality criterion than the linear functions implemented in currently used methods for The optimization methods for additive (Farris, 1970) optimization. If trees are optimized using such a meas- and non-additive (Fitch, 1971) characters are the two ure, transformation costs are dynamically determined most commonly used; in non-additive characters all during reconstructions; this leads to selecting trees changes are considered equally informative, but in implying that the possible state transformations are as additive characters (lineal or branched), the impor- reliable as possible. The present method is not iterative tance of transformations between different pairs of (thus avoiding the concern of different final results for states is different. Usually, there is no empirical basis different starting points), and it has an explicit optimal- for more complex models of transformation, such as ity criterion. It has a high computational cost; algorithms those allowed by the “generalized parsimony” to lessen the computations required for optimizations approach of Sankoff and Rousseau (1975), imple- and searches are described. © 1997 The Willi Hennig Society mented in PAUP (Swofford, 1993) and SPA (Goloboff, 1996a). According to some authors (e.g. Wheeler, 1992), this is the case even for DNA data (where it is possible to establish equivalencies between states of INTRODUCTION different characters, allowing in principle extrapola- tion from one character or data set to another). For morphology (where establishing such equivalencies In cladistics, the fit of trees to data is measured as a between character states is impossible), this is much function of the number of independent originations of more obviously so. Therefore, the consequence of character states required—found with a process using “generalized parsimony” is often that the results 0748-3007/97/030225+21/$25.00/0/cl970043 Copyright © 1997 by The Willi Hennig Society All rights of reproduction in any form reserved 225 226 Goloboff are determined more by the assumptions than by the relationships are even roughly approximate, homolo- evidence itself (Carpenter, 1994), and cladists tend to gizing a wing absence at the expense of considering avoid that approach. some wing presences as non-homologous is clearly Two main kinds of arguments have been suggested counterindicated. to decide which transformations it is preferable to min- As already pointed out, those two possible argu- imize. One argument proposes to use the direct ments to determine costs are complementary, and observation of the morphological relationships using one does not preclude using the other. Lip- between the states themselves; transformations scomb’s (1992) considering as alternatives her own between more similar states should be less costly than homology analysis and congruence-based methods transformations between radically different states. such as TSA would suggest the opposite, but the two This often allows defining relative degrees of apparent approaches focus on different, logically independent homology, and considering a character as additive components of the transformation costs. In principle, (Lipscomb, 1992). By its very nature, this approach is evaluating costs of transformations according to con- difficult to formalize, and it is not always applicable. gruence could be more easily formalized than Nonetheless, it seems obvious that, whenever the mor- evaluating them according to degrees of similarity. A phology clearly indicates degrees of homology, proper analytical method should accomplish this auto- nothing can be gained by discarding that information, matically, once the data set (including prior costs, or and Lipscomb’s approach should be followed. additivities, for each character) is given. However, The other possible argument for assigning costs to none of the methods proposed to date has been gener- transformations is their congruence: transformations ally accepted. The aim of this paper is to propose a way which are more incongruent with a tree are implied to to evaluate the relative informativeness of different be less reliable. This is additional information that can transformations dynamically, during the optimization be used, not instead of, but rather combined with, the process (and therefore during tree searches). Unlike information provided by the degrees of similarity other methods proposed to evaluate informativeness, among the states. Furthermore, the relative congru- the present method is not iterative, and therefore does ence of a transformation in the two possible directions not depend on initial hypotheses of implied costs. The (i.e. 0→1 or 1→0) may give information of asymmetries method is an extension of Goloboff’s (1993a) method in costs. Observation of morphology alone cannot pro- for comparing trees under the weights they imply vide that information, unless coupled with specific (implemented in the Ms-Dos program Pee-Wee; assumptions (such as “complex organs are less likely to Goloboff, 1993b), but applied to character state trans- evolve twice”). Phylogenetic conclusions, however, formations. I will first present a description of the often imply without ambiguity that groups character- method and its basic rationale, followed by a discus- ized by a transformation in one direction are more sion of specific fitting functions and algorithms for reliably defined than groups characterized by the optimization and tree searches. Finally, I will briefly transformation in the opposite direction. A clear exam- discuss the basic properties and implications of the ple is the presence of thoracic wings, defining the method, as compared to other methods proposed to pterygote insects (and occurring in no other group), as evaluate the relative informativeness of character state opposed to wing loss, observed in numerous unrelated transformations. insect groups, characterized by independent wing losses. That a given arthropod has thoracic wings is enough to make us certain that it belongs to the Ptery- SELF-WEIGHTED OPTIMIZATION gota, but the lack of wings tells us very little of its possible affinities with other arthropods—we know of secondarily apterous roaches, mantids, grasshoppers, The algorithms developed by Sankoff and collabora- flies, bugs, beetles, etc. An alternative formulation is tors (Sankoff and Rousseau, 1975; Sankoff and that sharing some of the states seems more likely due Cedergreen, 1983) allow finding most parsimonious to homology (i.e. traceable to a common node) than is reconstructions for any tree, under fixed transforma- sharing other states; if current hypotheses of insect tion costs. However, if one is interested in assessing Copyright © 1997 by The Willi Hennig Society All rights of reproduction in any form reserved Self-Weighted Optimization 227 costs based on congruence, the costs should be measured with concave functions (see Farris, 1969; tree-dependent, because different trees have different Goloboff, 1993a). The maximum possible value of that implications in terms of costs. On first thought, it function will vary for different trees, and those trees seems that one cannot choose among phylogenetic which allow attribution of higher reliability to the char- hypotheses unless the costs have been previously acter(s) are to be preferred. The most important aspect defined, and the costs cannot be chosen in the absence of cladistics is considering the evidence as the only fac- of a hypothesis. However, every topology evaluated tor determining the choice of a tree; as noted by during a search should be evaluated according to its Goloboff (1993a), choosing those trees which imply own implications of reliability. But the costs, even for a that evidence is most reliable is in direct agreement particular tree, are not determined automatically: with that basic tenet. many different reconstructions (parsimonious or not) Optimizations are usually described in terms of min- are usually possible, and each implies a certain number imizations. Maximizing a concave function such as 1/ → of steps, and hence a cost, for the different transforma- (sij+1) (where sij=number of i j transformations; note tions. Many of the possible reconstructions, however, that sij>=0) produces the same results as minimizing can be rejected on the grounds that they are not opti- its complement: mal under the weights they themselves imply. Those reconstructions, that is, postulate additional transfor- 1 sij dij = 1 – -------------- = -------------- . (1) mations for the types of change which (according to