The topography of multivariate normal mixtures

@article{Ray2005TheTO,
  title={The topography of multivariate normal mixtures},
  author={Surajit Ray and Bruce G. Lindsay},
  journal={Annals of Statistics},
  year={2005},
  volume={33},
  pages={2042-2065}
}
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… Expand

Figures from this paper

Multivariate Logistic Mixtures
Logistic mixtures, unlike normal mixtures, have not been studied for their topography. In this paper we discuss analogs of some of the multivariate normal mixture results for the multivariateExpand
Modal Inference and Its Application to High-Dimensional Clustering
TLDR
This work constructs an extension of EM algorithm that can be used to find the modes of a mixture density, and turns kernel density estimation into clustering tool in which the data points become identified with each other by their association with a common mode of the density estimator. Expand
On the Number of Modes of Finite Mixtures of Elliptical Distributions
TLDR
The concept of the ridgeline from Ray and Lindsay is extended to finite mixtures of general elliptical densities with possibly distinct density generators in each component to obtain bounds for the number of modes of two-component mixture of t distributions in any dimension. Expand
On the upper bound of the number of modes of a multivariate normal mixture
TLDR
The main result of this article states that one can get as many as D+1 modes from just a two component normal mixture in D dimensions, which provides a clear guideline as to when one can use mixture analysis for clustering high dimensional data. Expand
Asymptotic theory for density ridges
The large sample theory of estimators for density modes is well understood. In this paper we consider density ridges, which are a higher-dimensional extension of modes. Modes correspond toExpand
Maximum number of modes of Gaussian mixtures
Gaussian mixture models are widely used in Statistics. A fundamental aspect of these distributions is the study of the local maxima of the density or modes. In particular, it is not known how manyExpand
On the choice of high-dimensional regression parameters in Gaussian random tomography
  • C. Rau
  • Mathematics
  • Results in Applied Mathematics
  • 2019
Abstract The stochastic Radon transform was introduced by Panaretos in order to estimate a biophysical particle, modelled as a three-dimensional probability density subject to random and unknownExpand
Mode Estimation for High Dimensional Discrete Tree Graphical Models
TLDR
Though the mode finding problem is generally intractable in high dimensions, this paper unveils that, if the distribution can be approximated well by a tree graphical model, mode characterization is significantly easier. Expand
Inference for multivariate normal mixtures
TLDR
It is shown that the maximum penalized likelihood estimator is strongly consistent when the number of components has a known upper bound. Expand
Dimensionality reduction for data of unknown cluster structure
TLDR
This work focuses on data coming from a mixture of Gaussian distributions and proposes a method that preserves the distinctness of the clustering structure, although this structure is assumed to be yet unknown. Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 23 REFERENCES
Some descriptive properties of normal mixtures
Abstract When two single normal densities are mixed together the shape of the resulting distribution will vary in the extent to which it indicates the underlying structure, depending on the values ofExpand
On the Number of Modes of a Gaussian Mixture
TLDR
It is demonstrated that the number of modes can exceed the numberof components when the components are allowed to have arbitrary and different covariance matrices. Expand
Robust mixture modelling using the t distribution
TLDR
The use of the ECM algorithm to fit this t mixture model is described and examples of its use are given in the context of clustering multivariate data in the presence of atypical observations in the form of background noise. Expand
On the Modes of a Mixture of Two Normal Distributions
This paper presents a procedure for answering the question of whether a mixture of two normal distributions, with five known parameters μ1, μ2, σ1, σ2, p, is unimodal or not. The approach of theExpand
Scale Space Methods in Computer Vision
TLDR
The key to solve the ambiguity is the investigation of both the scale space saddles and the iso-intensity manifolds (the extension of isophotes in scale space) through them. Expand
Morphology-driven simplification and multiresolution modeling of terrains
TLDR
An algorithm to compute the above decomposition and the critical net, and a TIN simplification algorithm that preserves them are described, which build a multiresolution terrain model, which provides a representation of critical features at any level of detail. Expand
The effect of unequal variance-covariance matrices on Fisher's linear discriminant function.
This paper undertakes an investigation of the effect of unequal variance-covariance matrices on Fisher's linear discriminant function when used for discrimination or risk estimation. The behavior ofExpand
ML Estimation of the MultivariatetDistribution and the EM Algorithm
Maximum likelihood estimation of the multivariatetdistribution, especially with unknown degrees of freedom, has been an interesting topic in the development of the EM algorithm. After a brief reviewExpand
Discrimination, Allocatory and Separatory, Linear Aspects
Publisher Summary This chapter discusses the twin goals of linear discrimination, that is, allocation and separation. It reviews linearity in the mutivariate normal case and discusses the extent toExpand
Finite Mixture Models
  • G. McLachlan, D. Peel
  • Computer Science, Mathematics
  • Wiley Series in Probability and Statistics
  • 2000
The important role of finite mixture models in the statistical analysis of data is underscored by the ever-increasing rate at which articles on mixture applications appear in the statistical and ge...
...
1
2
3
...