Parametric Characterization of Multimodal Distributions with Non-gaussian Modes

Abstract

In statistics, mixture models are used to characterize datasets with multimodal distributions. A class of mixture models called Gaussian Mixture Models (GMMs) has gained immense popularity among practitioners because of its sound statistical foundation and an efficient learning algorithm, which scales very well with both the dimension and the size of a dataset. However, the underlying assumption, that every mixing component is normally distributed, can often be too rigid for several real life datasets. In this paper, we introduce a new class of parametric mixture models that are based on Copula functions. The goal is to relax the assumption about the normality of mixing components. We formulate a class of functions called Gaussian Mixture Copula functions for the characterization of multi-modal distributions. The parameters of the proposed Gaussian Mixture Copula Model (GMCM) can be obtained in a Maximum-Likelihood setting. For this purpose, an Expectation-Maximization (EM) and a Gradientbased optimization algorithm are proposed. Owing to the nonconvex log-likelihood function, only locally optimal solutions can be obtained. We also provide experimental evidence of the benefits of the GMCM over GMM using both synthetic and real-life datasets.

Robust Estimation of Value-at-Risk through Distribution-Free and Parametric Approaches Using the Joint Severity and Frequency Model: Applications in Financial, Actuarial, and Natural Calamities Domains

“…Copulas have been used extensively in various applications in QRM (Nelsen 2007;Tewari et al 2011;Sklar 1959;Kojadinovic and Yan 2010;Panchenko 2005;Kojadinovic and Yan 2011;Kole et al 2007). In this paper, we propose to use the copula model in a generalized QRM setting where the losses are broken down in the joint severity/frequency model.…”

Copulas in Machine Learning

“…For lack of space, additional works that, generally speaking, use copulas in a more plug-in manner, are not discussed. For the interested reader, these include copula-based independent component analysis [35], component analysis [27,2], mixture models (e.g., [14,51]), dependency seeking clustering [40]. Also of great interest but not presented here is the use of copulas as a particular instance within the cumulative distribution network model [17,45].…”

Due Time Driven Surgery Scheduling

“…Copulas provide a way to describe joint distributions by separating the estimation of the marginal distributions of the random variable from the dependencies between them. Unfortunately, copulas such as the Gaussian-or (Student) t-copula come with their own set of restrictions as they can perform less well on multimodal data [53].…”