© Institute of Mathematical Statistics, 2005 THE TOPOGRAPHY OF MULTIVARIATE NORMAL MIXTURES 1

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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.

Subfields: Artificial Intelligence, Food Science, Statistics and Probability

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Citations in the corpus

Overcoming Catastrophic Forgetting by Incremental Moment Matching (2017, neutral): “According to Ray and Lindsay [17], all the modes of MoG with K clusters lie on Motivated by the above description, a mode-IMM approximate MoG with Laplacian approximation, in which the logarithm of the function is expressed by a Taylor expansion [9].”
A Review on Modal Clustering (2016, methodology): “It is worth to mention that similar ideas had been earlier studied by Ray & Lindsay (2005), who analysed the number and location of the modes of a mixture of normal densities by introducing the ridgeline manifold, having a strong potential impact on the clustering problem (Li et al., 2007).”
Asymptotic theory for density ridges (2014, background): “The concept of ridges in point clouds was introduced by Cheng, Hall and Hartigan (2004), Hall and Peng (2001), Hall, Qian and Titterington (1992), Wegman and Luo (2002). An algorithm for finding density ridges was given by Ozertem and Erdogmus (2011). Recently, Genovese et al. (2014) provided some f”
Maximum number of modes of Gaussian mixtures (2017, background): “Lindsay initiate a systematic study and ask interesting questions in [13].”
Finite mixture models and model-based clustering (2010, methodology): “In a k-component Gaussian mixture, finding these reduced modes is achieved by analyzing the values of the density lying on the so-called ridgeline surface [104] that are given for pair of components with distributions Np(μi,Σi) and Np(μj ,Σj), by x(α) = [ (1 − α)Σ i + αΣ −1 j ]−1 [ (1− α)Σ i μi + αΣ”
Statistical Methods for Distributional Analysis (2015, methodology): “08559) allows us to reproduce the kernel density estimation results in 26A more appropriate method could be to test the number of modes of the distribution, see Ray and Lindsay (2005) 27From f(x; Θ) = ∑K k=1 πk φ(x ;μk, σk), we have f(y; Θ) = ∑K k=1 πk Λ(y ;μk, σk) where x = log y.”