The Annals of Statistics

The topography of multivariate normal mixtures

Surajit Ray and Bruce G. Lindsay

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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.

Article information

Ann. Statist. Volume 33, Number 5 (2005), 2042-2065.

First available: 25 November 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E10: Characterization and structure theory 62H05: Characterization and structure theory
Secondary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]

Mixture modal cluster multivariate mode clustering dimension reduction topography manifold


Ray, Surajit; Lindsay, Bruce G. The topography of multivariate normal mixtures. The Annals of Statistics 33 (2005), no. 5, 2042--2065. doi:10.1214/009053605000000417.

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