The topography of multivariate normal mixtures. (English) Zbl 1086.62066

Summary: Multivariate normal mixtures provide a flexible method of fitting high-dimensional data. It is shown that their topography, in the sense of their key features as a density, can be analyzed rigorously in lower dimensions by use of a ridgeline manifold that contains all critical points, as well as the ridges of the density. A plot of the elevations on the ridgeline shows the key features of the mixed density. In addition, by use of the ridgeline, we uncover a function that determines the number of modes of the mixed density when there are two components being mixed. A followup analysis then gives a curvature function that can be used to prove a set of modality theorems.


62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62A01 Foundations and philosophical topics in statistics
62E10 Characterization and structure theory of statistical distributions
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[1] Behboodian, J. (1970). On the modes of a mixture of two normal distributions. Technometrics 12 131–139. · Zbl 0195.20304
[2] Bryan, J. G. (1951). The generalized discriminant function: Mathematical foundations and computational routine. Harvard Educational Review 21 90–95.
[3] Carreira-Perpiñán, M. Á. and Williams, C. K. I. (2003). On the number of modes of a Gaussian mixture. Scale-Space Methods in Computer Vision. Lecture Notes in Comput. Sci. 2695 625–640. Springer, New York. · Zbl 1067.68724
[4] Danovaro, E., De Floriani, L., Magillo, P., Mesmoudi, M. M. and Puppo, E. (2003). Morphology-driven simplification and multiresolution modeling of terrains. In Proc. Eleventh ACM International Symposium on Advances in Geographic Information Systems 63–70. ACM Press, New York.
[5] de Helguero, F. (1904). Sui massimi delle curve dimorfiche. Biometrika 3 84–98.
[6] Eisenberger, I. (1964). Genesis of bimodal distributions. Technometrics 6 357–363.
[7] Fisher, R. A. (1936). The use of multiple measurements in taxonomic problems. Annals of Eugenics 7 179–188.
[8] Geisser, S. (1977). Discrimination, allocatory and separatory, linear aspects. In Classification and Clustering (J. Van Ryzin, ed.) 301–330. Academic Press, New York.
[9] Gilbert, E. S. (1969). The effect of unequal variance–covariance matrices on Fisher’s linear discriminant function. Biometrics 25 505–515.
[10] Kakiuchi, I. (1981). Unimodality conditions of the distribution of a mixture of two distributions. Math. Sem. Notes Kobe Univ. 9 315–325. · Zbl 0485.62014
[11] Kemperman, J. H. B. (1991). Mixtures with a limited number of modal intervals. Ann. Statist. 19 2120–2144. JSTOR: · Zbl 0756.62008
[12] Lindsay, B. G. (1983). The geometry of mixture likelihoods. II. The exponential family. Ann. Statist. 11 783–792. JSTOR: · Zbl 0534.62002
[13] Liu, C. (1997). ML estimation of the multivariate \(t\) distribution and the EM algorithm. J. Multivariate Anal. 63 296–312. · Zbl 0884.62059
[14] McLachlan, G. and Peel, D. (2000). Finite Mixture Models . Wiley, New York. · Zbl 0963.62061
[15] Milnor, J. (1963). Morse Theory . Princeton Univ. Press, Princeton, NJ. · Zbl 0108.10401
[16] Morse, M. and Cairns, S. (1969). Critical Point Theory in Global Analysis and Differential Topology . Academic Press, New York. · Zbl 0177.52102
[17] Olsen, O. (2003). The scale structure of the gradient magnitude. Technical report, IT Univ. Copenhagen. Available at www.itu.dk/pub/Reports/ITU-TR-2003-29.pdf.
[18] Peel, D. and McLachlan, G. J. (2000). Robust mixture modelling using the \(t\) distribution. Statist. Comput. 10 339–348.
[19] Rao, C. R. (1948). The utilization of multiple measurements in problems of biological classification (with discussion). J. Roy. Statist. Soc. Ser. B 10 159–203. · Zbl 0034.07902
[20] Robertson, C. A. and Fryer, J. G. (1969). Some descriptive properties of normal mixtures. Skand. Aktuarietidskr. 1969 137–146. · Zbl 0205.46603
[21] Thomson, A. and Maciver, D. R. (1905). The Ancient Races of the Thebaid . Oxford Univ. Press.
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