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Random walk as a null model for high-dimensional morphometrics of fossil series: geometrical considerations

Published online by Cambridge University Press:  08 April 2016

Fred L. Bookstein
Fred L. Bookstein*
Affiliation:
Faculty of Life Sciences, University of Vienna, A-1091 Vienna, Austria, and Department of Statistics, University of Washington, Seattle, Washington 98195, U.S.A. E-mail: [email protected]
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Abstract

Over the past quarter-century there has been considerable innovation in methods for assessing the tempo and mode of evolution in paleobiological data sets. The current literature of these methods centers on three competing hypotheses—stasis, random walk, and directional trend—corresponding to an increasing scaling of variance with time interval (unchanging, for stasis; linear, for random walk; quadratic, for trend). For applications to a single trait there are powerful methods for discriminating among these hypotheses; but for multivariate data sets, especially the very high-dimensional multivariate data arising in image-feature-based and morphometric studies, current statistical approaches appear to be of less help. This paper proves that in the limiting case of high-dimensional morphospaces, the principal component or principal coordinate ordination of every sufficiently lengthy isotropic random walk tends to the same geometrical shape, which is not that of an ellipsoid and for which the principal components or coordinates are not independent even though they are uncorrelated. Specifically, the “scatter” of PC1 against PC2 is just a parabolic curve. The quantitative characteristics of this specific shape are not described appropriately by the corresponding “covariance structure” or Gaussian model, and the discrepancy may be pertinent to much of the existing literature of methods for differentiating among those three models of evolutionary multivariate time series. From a close examination of this common geometry of the ideal random walk model as seen in its principal components, I suggest a test for stasis, along with a mixed model illustrated by a reanalysis of some data of Gunz et al., and a related test for directional trend. These comments are intended to apply to all high-dimensional morphospaces, not just those arising in geometric morphometrics. Applications of principal components in this context distort high-dimensional data in ways that have a tendency to mislead; but these distortions can be intercepted so that studies of tempo and mode can nevertheless proceed.

Type
Articles
Information
Paleobiology , Volume 39 , Issue 1 , Winter 2013 , pp. 52 - 74
Copyright
Copyright © The Paleontological Society 

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References

Literature Cited

Aguirre, W. E., and Bell, M. A. 2012. Twenty years of body shape evolution in a threespine stickleback population adapting to a lake environment. Biological Journal of the Linnean Society 105:817–831.CrossRefGoogle Scholar
Bell, M. A. 2001. Lateral plate evolution in the threespine stickleback: getting nowhere fast. Genetica 112–113:445–461.Google Scholar
Berg, H. C. 1993. Random walks in biology, 2nd ed. Princeton University Press, Princeton, N.J.Google Scholar
Bingham, N. H., and Doney, R. A. 1988. On higher-dimensional analogues of the Arc-sine Law. Journal of Applied Probability 25:120–131.CrossRefGoogle Scholar
Blackith, R. E., and Reyment, R. A. 1971. Multivariate morphometrics. Academic Press.Google Scholar
Bookstein, F. L. 1987. Random walk and the existence of evolutionary rates. Paleobiology 13:446–464.Google Scholar
Bookstein, F. L. 1988. Random walk and the biometrics of morphological characters. Evolutionary Biology 23:369–398.CrossRefGoogle Scholar
Codling, E. A., Plank, M. J., and Benhamou, S. 2008. Random walk models in biology. Journal of the Royal Society Interface 5:813–834.Google Scholar
Dean, D., Marcus, L., and Bookstein, F. L. 1996. Chi-square test of biological space curve affinities. Pp. 235–261inMarcus, L. F., Corti, M., Loy, A., Naylor, G. J. P., and Slice, D. E., eds. Advances in morphometrics. NATO ASI Series A, Life Sciences, Vol. 284. Plenum, New York.Google Scholar
Dryden, I. M., and Mardia, K. V. 1998. Statistical shape analysis. Wiley, Chichester, U.K.Google Scholar
Einstein, A. 1905. Über die von molekularischen Theorie der Wärme geforderte Bewegung von ruhenden Flüssigkeiten suspendierten Teilchen (Investigations on the theory of Brownian movement). Annalen der Physik 17:549–560. Edited R. Fürth, translated by A. D. Cowper. Dover Books, 1956.Google Scholar
Feller, W. 1968. An introduction to probability theory and its applications, Vol. 1, 3rd ed. Wiley, Chichester, U.K.Google Scholar
Felsenstein, J. 1988. Phylogenies and quantitative characters. Annual Review of Ecology and Systematics 19:445–471.Google Scholar
Felsenstein, J. 2004. Inferring phylogenies. Sinauer, Sunderland, Mass.Google Scholar
Fisk, P. R. 1970. A note on a characterization of the multivariate normal distribution. Annals of Mathematical Statistics 41:486–494.Google Scholar
Gingerich, P. D. 1993. Quantification and comparison of evolutionary rates. American Journal of Science 293A:453–478.Google Scholar
Gould, S. J. 1989. Wonderful life: the Burgess Shale and the nature of history. W. W. Norton, New York.Google Scholar
Gould, S. J., and Eldredge, N. 1977. Punctuated equilibria: the tempo and mode of evolution reconsidered. Paleobiology 3:115–151.CrossRefGoogle Scholar
Gower, J. C. 1966. Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53:325–338.Google Scholar
Gunz, P., Mitteroecker, P., Stadlmayr, A., Weber, G. W., Seidler, H., and Bookstein, F. L. 2009. The origins of modern human diversity: multi-wave and multi-pathway migration during Late Pleistocene. Proceedings of the National Academy of Sciences USA 106:6094–6098.CrossRefGoogle Scholar
Hannisdal, B. 2006. Phenotypic evolution in the fossil record: numerical experiments. Journal of Geology 114:133–153.Google Scholar
Hannisdal, B. 2007. Inferring phenotypic evolution in the fossil record by Bayesian inversion. Paleobiology 33:98–115.CrossRefGoogle Scholar
Hunt, G. 2006. Fitting and comparing models of phyletic evolution: random walks and beyond. Paleobiology 32:578–601.Google Scholar
Hunt, G. 2007. The relative importance of directional change, random walks, and stasis in the evolution of fossil lineages. Proceedings of the National Academy of Sciences USA 104:18404–18408.Google Scholar
Kahneman, D. 2011. Thinking, fast and slow. Farrar Straus Giroux, New York.Google Scholar
Kendall, D. G. 1984. Shape manifolds, Procrustean metrics and complex projective spaces. Bulletin of the London Mathematical Society 16:81–121.Google Scholar
Kimura, M. 1983. The neutral theory of molecular evolution. Cambridge University Press, Cambridge.Google Scholar
Lande, R. 1979. Quantitative genetic analysis of multivariate evolution, applied to brain:body size allometry. Evolution 33:402–426.Google ScholarPubMed
Lynch, M., and Hill, W. G. 1986. Phenotypic evolution by neutral mutation. Evolution 40:915–935.Google Scholar
Mardia, K. V., Bookstein, F. L., Kent, J. T., and Meyer, C. R. 2006. Intrinsic random fields and image deformations. Journal of Mathematical Imaging and Vision 26:59–71.Google Scholar
Mitteroecker, P., and Bookstein, F. L. 2009. The ontogenetic trajectory of the phenotypic covariance matrix, with examples from craniofacial shape in rats and humans. Evolution 63:727–737.Google Scholar
Nei, M. 2007. The new mutation theory of phenotypic evolution. Proceedings of the National Academy of Sciences USA 104:12235–12242.Google Scholar
Pearson, K., and Lee, A. 1903. On the laws of inheritance in man. I. Inheritance of physical characters. Biometrika 2:357–462.CrossRefGoogle Scholar
Perrin, J. 1909. Mouvement Brownien et réalité moléculaire. Annales de Chimie et de Physique XVIII:5–113.Google Scholar
Perrin, J. 1923. Atoms, 2nd ed. Translated by D. L. Hamnick. Constable, London.Google Scholar
Podani, J., and Miklós, I. 2002. Resemblance coefficients and the horseshoe effect in principal coordinates analysis. Ecology 83:3331–3343.CrossRefGoogle Scholar
Polly, P. D. 2008. Developmental dynamics and G-matrices: can morphometric spaces be used to model phenotypic evolution? Evolutionary Biology 35:83–96.Google Scholar
Raup, D. M. 1977. Stochastic models in evolutionary palaeobiology. Pp. 59–78inHallam, A., ed. Patterns of evolution as illustrated by the fossil record. Elsevier, Amsterdam.Google Scholar
Reyment, R. A. 1991. Multidimensional palaeobiology. Pergamon, Oxford.Google Scholar
Roopnarine, P. D. 2003. Analysis of rates of morphologic evolution. Annual Review of Ecology, Evolution, and Systematics 34:605–632.Google Scholar
Rudnick, J., and Gaspari, G. 2004. Elements of the random walk: an introduction for advanced students and researchers. Cambridge University Press, Cambridge.Google Scholar
Sheets, H. D., and Mitchell, C. E. 2001. Why the null matters: statistical tests, random walks, and evolution. Genetica 112–113:105–125.Google Scholar
Torgerson, W. 1958. Theory and methods of scaling. Wiley, Chichester, U.K.Google Scholar
Wartenberg, D., Ferson, S., and Rohlf, F. J. 1987. Putting things in order: a critique of detrended correspondence analysis. American Naturalist 129:434–448.Google Scholar
Wood, A. R., Zelditch, M. L., Rountrey, A. N., Eiting, T. P., Sheets, H. D., and Gingerich, P. D. 2007. Multivariate stasis in the dental morphology of the Paleocene–Eocene condylarth Ectocion. Paleobiology 33:248–260.Google Scholar