## 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
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### References:

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