Curved factor analysis with the Ellipsoid-Gaussian distribution

TL;DR

This paper introduces the Ellipsoid-Gaussian distribution, a flexible and relatively simple model for capturing nonlinear, curved dependencies in multivariate data, using a Bayesian approach for inference.

Contribution

The paper proposes a new Ellipsoid-Gaussian distribution model that captures nonlinear relationships with lower complexity, and develops a Bayesian sampling method for inference.

Findings

Successfully models curved relationships in multivariate data

Demonstrates effectiveness on simulated and real datasets

Provides an R package for practical implementation

Abstract

There is a need for new models for characterizing dependence in multivariate data. The multivariate Gaussian distribution is routinely used, but cannot characterize nonlinear relationships in the data. Most non-linear extensions tend to be highly complex; for example, involving estimation of a non-linear regression model in latent variables. In this article, we propose a relatively simple class of Ellipsoid-Gaussian multivariate distributions, which are derived by using a Gaussian linear factor model involving latent variables having a von Mises-Fisher distribution on a unit hyper-sphere. We show that the Ellipsoid-Gaussian distribution can flexibly model curved relationships among variables with lower-dimensional structures. Taking a Bayesian approach, we propose a hybrid of gradient-based geodesic Monte Carlo and adaptive Metropolis for posterior sampling. We derive basic properties and illustrate the utility of the Ellipsoid-Gaussian distribution on a variety of simulated and real data applications. An accompanying R package is also available.