A Mixture of Coalesced Generalized Hyperbolic Distributions

Tortora, Cristina; Franczak, Brian C.; Browne, Ryan P.; McNicholas, Paul D.

doi:10.1007/s00357-019-09319-3

Cited by 35 publications

(26 citation statements)

References 70 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, if the objective is dealing with outliers, it will be better to consider the PDQ approach with the multivariate contaminated normal distribution [25] and this will be a topic of future work. Other approaches for handling cluster concentration will also be considered (e.g., [9]) as will methods that accommodate asymmetric, or skewed, clusters (e.g., [18,19,21,22,32,34]). Let f (x i ; k , k ) be the generic symmetric unimodal multivariate density function of the random variable with parameter k and location parameter k then satisfies all the three properties and it is a dissimilarity measure for k = 1, … , K.…”

Section: Resultsmentioning

confidence: 99%

A Probabilistic Distance Clustering Algorithm Using Gaussian and Student-t Multivariate Density Distributions

2020

Self Cite

View full text Add to dashboard Cite

A new dissimilarity measure for cluster analysis is presented and used in the context of probabilistic distance (PD) clustering. The basic assumption of PD-clustering is that for each unit, the product between the probability of the unit belonging to a cluster and the distance between the unit and the cluster is constant. This constant is a measure of the classifiability of the point, and the sum of the constant over units is called joint distance function (JDF). The parameters that minimize the JDF maximize the classifiability of the units. The new dissimilarity measure is based on the use of symmetric density functions and allows the method to find clusters characterized by different variances and correlation among variables. The multivariate Gaussian and the multivariate Student-t distributions have been used, outperforming classical PD clustering, and its variation PD clustering adjusted for cluster size, on simulated and real datasets.

show abstract

Section: Resultsmentioning

confidence: 99%

A Probabilistic Distance Clustering Algorithm Using Gaussian and Student-t Multivariate Density Distributions

2020

Self Cite

View full text Add to dashboard Cite

show abstract

“…To overcome this issue, we could extend our MSCN distribution with the aim of introducing skewness; the resulting model could be used to define the components of a mixture. Examples of competing approaches in this directions are given in Franczak et al (2015) and Tortora et al (2018).…”

Section: Discussionmentioning

confidence: 99%

Multiple scaled contaminated normal distribution and its application in clustering

Punzo

Tortora

2019

Statistical Modelling

Self Cite

View full text Add to dashboard Cite

The multivariate contaminated normal (MCN) distribution represents a simple heavytailed generalization of the multivariate normal (MN) distribution to model elliptical contoured scatters in the presence of mild outliers, referred to as "bad" points. The MCN can also automatically detect bad points. The price of these advantages is two additional parameters, both with specific and useful interpretations: proportion of good observations and degree of contamination. However, points may be bad in some dimensions but good in others. The use of an overall proportion of good observations and of an overall degree of contamination is limiting. To overcome this limitation, we propose a multiple scaled contaminated normal (MSCN) distribution with a proportion of good observations and a degree of contamination for each dimension. Once the model is fitted, each observation has a posterior probability of being good with respect to each dimension. Thanks to this probability, we have a method for simultaneous directional robust estimation of the parameters of the MN distribution based on down-weighting and for the automatic directional detection of bad points by means of maximum a posteriori probabilities. The term "directional" is added to specify that the method works separately for each dimension. Mixtures of MSCN distributions are also proposed as an application of the proposed model for robust clustering. An extension of the EM algorithm is used for parameter estimation based on the maximum likelihood approach. Real and simulated data are used to show the usefulness of our mixture with respect to well-established mixtures of symmetric distributions with heavy tails.

show abstract

“…If the density is quasi-concave then it is unimodal. Tortora et al (2015) point out that the generalized hyperbolic distribution has a quasi-concave density and, if λ > (p + 1)/2, it has a log-concave density. Because the HTH distribution is obtained by integrating out a set of variables from the symmetric hyperbolic distribution, we can show that the HTH distribution is quasi-concave if the symmetric hyperbolic density is s-concave or log-concave -note that, in general, s ∈ R. Next, we will show that the symmetric generalized hyperbolic distribution is an s-concave density.…”

Section: S-concavity Of the Truncated Hyperbolic Distributionmentioning

confidence: 98%

“…Eberlein and Keller (1995) used the hyperbolic distribution to model returns of German equities. Tortora et al, 2015;Morris and McNicholas, 2016) applied the GHD to the context of model-based clustering.…”

Section: Generalized Hyperbolic Distributionmentioning

confidence: 99%

Hidden truncation hyperbolic distributions, finite mixtures thereof, and their application for clustering

Murray

Browne

McNicholas

2017

Journal of Multivariate Analysis

Self Cite

View full text Add to dashboard Cite

A hidden truncation hyperbolic (HTH) distribution is introduced and finite mixtures thereof are applied for clustering. A stochastic representation of the HTH distribution is given and a density is derived. A hierarchical representation is described, which aids in parameter estimation. Finite mixtures of HTH distributions are presented and their identifiability is proved. The convexity of the HTH distribution is discussed, which is important in clustering applications, and some theoretical results in this direction are presented. The relationship between the HTH distribution and other skewed distributions in the literature is discussed. Illustrations are providedboth of the HTH distribution and application of finite mixtures thereof for clustering.

show abstract

A Mixture of Coalesced Generalized Hyperbolic Distributions

Cited by 35 publications

References 70 publications

A Probabilistic Distance Clustering Algorithm Using Gaussian and Student-t Multivariate Density Distributions

A Probabilistic Distance Clustering Algorithm Using Gaussian and Student-t Multivariate Density Distributions

Multiple scaled contaminated normal distribution and its application in clustering

Hidden truncation hyperbolic distributions, finite mixtures thereof, and their application for clustering

Contact Info

Product

Resources

About