On the Number of Modes of Finite Mixtures of Elliptical Distributions

Abstract

We extend the concept of the ridgeline from Ray and Lindsay (2005) to finite mixtures of general elliptical densities with possibly distinct density generators in each component. This can be used to obtain bounds for the number of modes of two-component mixtures of t distributions in any dimension. In case of proportional dispersion matrices, these have at most three modes, while for equal degrees of freedom and equal dispersion matrices, the number of modes is at most two. We also give numerical illustrations and indicate applications to clustering and hypothesis testing.

Maximum number of modes of Gaussian mixtures

“…We observe that the construction strategies used in our lower bound can be extended to elliptical distributions, not only Gaussians, and this could be pursued further. For example, an extension of Ray and Lindsay's concept of ridgeline for mixtures of elliptical distributions is done in [1], including a study of modes for mixtures of two t-distributions.…”

“…In Figure 4 we give an example of a Gaussian mixture that has this number of modes. The configuration relies on the deformation of 3 lines arranged in an equilateral triangle, and taking means as the middle points on the 3 sides, with all weights 1 3 . Apart from the modes coming from the means, the other 3 modes lie near the corresponding triangle vertices.…”

“…The d-dimensional Gaussian distribution N (µ, Σ) can be defined by its probability density function (1) φ…”

“…A natural conjecture could be that m(d, k) = k for all d, k, that is, a mixture with k Gaussian components can have at most k modes. 1 However, this fails already when d = k = 2, since a mixture of two bivariate Gaussians can have three distinct modes (and actually, m(2, 2) = 3).…”