×

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.

MSC:

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
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.