On the Spherical Dirichlet Distribution: Corrections and Results

TL;DR

This paper introduces a new Spherical-Dirichlet Distribution tailored for data on the positive hypersphere, correcting previous errors, deriving properties, and demonstrating its application in data mining and gene expression analysis.

Contribution

It presents the first detailed development and application of the Spherical-Dirichlet Distribution, including properties, estimators, and real-world data fitting.

Findings

Distribution effectively models data on the positive hypersphere.

Estimators like MLE and method of moments are derived.

Applications show good fit to simulated and real data.

Abstract

This note corrects a technical error in Guardiola (2020, Journal of Statistical Distributions and Applications), presents updated derivations, and offers an extended discussion of the properties of the spherical Dirichlet distribution. Today, data mining and gene expressions are at the forefront of modern data analysis. Here we introduce a novel probability distribution that is applicable in these fields. This paper develops the proposed Spherical-Dirichlet Distribution designed to fit vectors located at the positive orthant of the hypersphere, as it is often the case for data in these fields, avoiding unnecessary probability mass. Basic properties of the proposed distribution, including normalizing constants and moments are developed. Relationships with other distributions are also explored. Estimators based on classical inferential statistics, such as method of moments and maximum likelihood estimators are obtained. Two applications are developed: the first one uses simulated data, and the second uses a real text mining example. Both examples are fitted using the proposed Spherical-Dirichlet Distribution and their results are discussed.