Abstract
Euclidean Distance Matrix Analysis (EDMA) of form is a coordinate free approach to the analysis of form using landmark data. In this paper, the problem of estimation of mean form, variance-covariance matrix, and mean form difference under the Gaussian perturbation model is considered using EDMA. The suggested estimators are based on the method of moments. They are shown to be consistent, that is as the sample size increases these estimators approach the true parameters. They are also shown to be computationally very simple. A method to improve their efficiency is suggested. Estimation in the presence of missing data is studied. In addition, it is shown that the superimposition method of estimation leads to incorrect mean form and variance-covariance structure.
Similar content being viewed by others
References
Arnold, S. F., 1981, Theory of Linear Models and Multivariate Analysis: John Wiley and Sons, New York.
Bookstein, F., 1986, Size and Shape Spaces for Landmark Data in Two Dimensions: Stat. Sci., v. 1, p. 181–242.
Bookstein, P., and Sampson, P., 1990, Statistical Methods for the Geometric Components of Shape Change: Comm. Stat. Theory Methods, v. 19, p. 1939–1972.
Campbell, G., 1986, Comments on “Size and Shape Spaces for Landmark Data in Two Dimensions” by F. L. Bookstein: Stat. Sci., v. 1, p. 227–228.
Chung, K. L., 1974, A Course in Probability Theory: Academic Press, New York.
deGunst, M. C. M., 1987, On the Distribution of General Quadratic Functions in Normal Vectors: Statistica Neerlandica, p. 245–251.
Dik, J. J., and deGunst, M. C. M., 1985, The Distribution of General Quadratic Forms in Normal Variables: Statistica Neerlandica, p. 14–26.
Goodall, C., 1991, Procrustes Methods in the Statistics Analysis of Shape: J. Roy. Stat. Soc. Ser. B, v. 53, p. 285–339.
Goodall, C., and Bose, A., 1987, Models and Procrustes Methods for the Analysis of Shape Difference: Proc. 19th Symp. Interface Between Computer Science and Statistics, Philadelphia, PA, pp. 86–92.
Gower, J. C., 1966, Some Distance Properties of Latent Root and Vector Methods in Multivariate Analysis: Biometrika, v. 53, p. 315–328.
Gower, J., 1975, Generalized Procrustes Analysis: Psychometrika, v. 40, p. 33–50.
Johnson, N., and Kotz, S., 1970, Distributions in Statistical—Continuous Univariate Distributions 2: Houghton Mifflin Company, Boston.
Kiefer, J., and Wolfowitz, J., 1956, Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Nuisance Parameters: Ann. Math. Statist., v. 27, p. 887–906.
Langron, S. P., and Collins, A. J., 1985, Perturbation Theory for Generalized Procrustes Analysis: J. Roy. Statist. Soc. Ser. B, v. 47, p. 277–284.
Leakey, M. G., Leakey, R. E., Richtsmeier, J. T., Simons, E. L., and Walker, A. C., 1991, Similarities in Aegyptopithecus and Afropithecus Facial Morphology: Folia Primatol., v. 56, p. 65–85.
Lele, S., 1991a, Some Comments on Coordinate Free and Scale Invariant Method in Morphometrics: Am. J. Phys. Anthropol., v. 85, p. 407–418.
Lele, S., 1991b, Comments on Goodall's Paper: J. Roy. Stat. Soc. B, v. 53, p. 334.
Lele, S., 1992, A Quantitative Method for the Reconstruction of Missing Landmarks on Biological Objects: Unpublished manuscript.
Lele, S., and Richtsmeier, J. T., 1990, Statistical Models in Morphometrics: Are They Realistic: Syst. Zool., v. 39, n. 1, p. 60–69.
Lele, S., and Richtsmeier, J. T., 1991, Euclidean Distance Matrix Analysis: A Coordinate Free Approach for Comparing Biological Shapes: Am. J. Phys. Anthropol., v. 86, p. 415–427.
Lele, S., and Richtsmeier, J. T., 1992, On Comparing Biological Shapes: Detection of Influential Landmarks: Am. J. Phys. Anthropol., v. 87, p. 49–65.
Lindsay, B. G., 1983, The Geometry of Mixture Likelihoods, Part II: The Exponential Family: Ann. Stat., v. 11, p. 783–792.
Mardia, K. V., and Dryden, I., 1989, The Statistical Analysis of Shape Data: Biomtk., v. 76, p. 271–282.
Mardia, K. V., Kent, T., and Bibby, J. M., 1979, Multivariate Analysis: Academic Press, New York.
Neyman, J., and Scott, E. J., 1948, Consistent Estimates Based on Partially Consistent Observations: Econometrika v. 16, p. 1–32.
Rao, C. R., 1973, Linear Statistical Inference and Its Applications: John Wiley, New York.
Rohlf, J., and Bookstein, F., ed., 1990, Proceedings of the Michigan Morphometrics Workshop: Special Publication No. 2, Museum of Zoology, University of Michigan, Ann Arbor, Michigan.
Rohlf, F. J., and Slice, D., 1990, Extensions of the Procrustes Method for the Optimal Superimposition of Landmarks: Syst. Zool., v. 39, p. 40–59.
Roth, V. L., 1988, The Biological Basis of Homology,in C. J. Humphries (Ed.), Ontogeny and Systematics: Columbia University Press.
Serfling, R. J., 1980, Approximation Theorems of Mathematical Statistics: John Wiley and Sons, New York.
Van Valen, L. M., 1982, Homology and Causes: J. Morphol., v. 173, p. 305–312.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lele, S. Euclidean Distance Matrix Analysis (EDMA): Estimation of mean form and mean form difference. Math Geol 25, 573–602 (1993). https://doi.org/10.1007/BF00890247
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00890247