Modeling with a large class of unimodal multivariate distributions

Abstract

In this paper we introduce a new class of multivariate unimodal distributions, motivated by Khintchine's representation. We start by proposing a univariate model, whose support covers all the unimodal distributions on the real line. The proposed class of unimodal distributions can be naturally extended to higher dimensions, by using the multivariate Gaussian copula. Under both univariate and multivariate settings, we provide MCMC algorithms to perform inference about the model parameters and predictive densities. The methodology is illustrated with univariate and bivariate examples, and with variables taken from a real data-set. example, [29].The traditional approach is to model the data via a mixture model, see [31] for finite mixture distributions, and, more recently, [13]. The idea, which carries through to infinite mixture models, is that each component in the mixture is represented by a parametric density function, typically from the same family, which is usually the normal distribution. This assumption, often made for convenience, supposes each cluster or group can be adequately modeled via a normal distribution. For if a group requires two such normals, then it is deemed there are in fact two groups. Yet two normals could be needed even if there is one group and it happens to be skewed, for example.On the other hand, if we start by thinking about what a cluster could be represented by in terms of probability density functions, then a unimodal density is the most obvious choice. Quite simply, with lack of further knowledge, i.e. only observing the data, a bimodal density would indicate two clusters. This was the motivation behind [25], which relies heavily on the unimodal density models being defined on the real line, for which there are no obvious extensions to the multivariate setting.There are representations of unimodal density functions on the real line (e.g. [18] and [10]) and it is not difficult to model such a density adequately. See, for example,[3], [23] and [25]. Clearly, the aim is to provide large classes of unimodal densities and hence infinite dimensional models are often used.However, while there is an abundance of literature on unimodal density function modeling on the real line, there is a noticeable lack of work in two and higher dimensions. The reasons are quite straightforward; first there is a lack of a representation for unimodal distributions outside of the real line and, second, the current approaches to modeling unimodal densities on the real line do not naturally extend to higher dimensions.The aim in this paper is to introduce and demonstrate a class of unimodal densities for a multivariate setting. While marginal density functions will be modeled using the [18] representation on the real line, the dependence structure will be developed using a Gaussian copula model. To elaborate, using an alternative form of representation of

Biclustering via Semiparametric Bayesian Inference

“…The third choice consists of noise modeled as a unimodal distribution based on the Rayleigh distribution. More precisely, by employing the unimodal distributions representation derived by Khintchine (1938), and following the work of Paez and Walker (2018), we find that any unimodal density f Y (y) can be expressed as…”

A New Robust Multivariate Mode Estimator for Eye-tracking Calibration

“…Most common techniques include: i) the Silverman bandwidth test (Silverman 1981), which uses a normal kernel density estimate with increasing bandwidth; ii) the excess mass test proposed in (Müller and Sawitzki 1991); and iii) the Hartigan's Dip Test (Hartigan and Hartigan 1985), which has been widely adopted due to its low computational complexity, its high statistical power, and the absence of tuning parameter. The extension of these tests to multivariate distributions is not straightforward, and several definitions of unimodality have been suggested for the multidimensional setting (Paez and Walker 2018;Kouvaras and Kokolakis 2007). The Hartigan's SPAN and RUNT statistic (Hartigan and Mohanty 1992), and the MAP test (Rozál and Hartigan 1994) are often cited as multivariate alternatives, but these types of procedures are usually far more complex than univariate ones, both conceptually and computationally, relying for instance on the construction of several spamming trees (Siffer et al 2018).…”

Selective Clustering Annotated using Modes of Projections

“…To cluster the data, a parametric model, such as the multivariate normal [Fraley & Raftery [2002]], uniform normal [Banfield & Raftery [1993]], multivariate t [Peel & McLachlan [2000]], or a skew variant of one of these distributions [Lee & McLachlan [2013]], is typically assumed to describe the component distributions F m . Non-parametric approaches are also possible: Li et al [2007] suppose the mixture is made up of non-parametric components and estimates them with Gaussian kernels; Rodríguez & Walker [2014] and Paez & Walker [2017] take a Bayesian approach to estimating mixture models based on unimodal distributions; Kosmidis & Karlis [2016] take a copula-based approach that influenced the model formulation used in this work. The EM algorithm of Dempster et al [1977] (or one of its extensions) is then used to find local maximizers of the likelihood for a fixed number of clusters g. The number of components g * can be determined in terms of the model by picking the g * that optimizes some criterion (such as BIC) over a user-specified range of g, and subsequently refined by merging clusters with, for example, the method of Baudry et al [2010] or Tantrum et al [2003].…”

“…SCAMP takes a perspective similar to Paez & Walker [2017]: clusters are defined in terms of unimodality. In many scientific contexts, unimodality of observations along a measurement coordinate of the data matrix X reflects physical homogeneity of interest.…”