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2021 Geodesic projection of the von Mises–Fisher distribution for projection pursuit of directional data
Sungkyu Jung
Author Affiliations +
Electron. J. Statist. 15(1): 984-1033 (2021). DOI: 10.1214/21-EJS1807

Abstract

We investigate geodesic projections of von Mises–Fisher (vMF) distributed directional data. The vMF distribution for random directions on the (p1)-dimensional unit hypersphere Sp1Rp plays the role of multivariate normal distribution in directional statistics. For one-dimensional circle S1, the vMF distribution is called von Mises (vM) distribution. Projections onto geodesics are one of main ingredients of modeling and exploring directional data. We show that the projection of vMF distributed random directions onto any geodesic is approximately vM-distributed, albeit not exactly the same. In particular, the distribution of the geodesic-projected score is an infinite scale mixture of vM distributions. Approximations by vM distributions are given along various asymptotic scenarios including large and small concentrations (κ,κ0), high-dimensions (p), and two important cases of double-asymptotics (p,κ, κpc or κpλ), to support our claim: geodesic projections of the vMF are approximately vM. As one of potential applications of the result, we contemplate a projection pursuit exploration of high-dimensional directional data. We show that in a high dimensional model almost all geodesic-projections of directional data are nearly vM, thus measures of non-vM-ness are a viable candidate for projection index.

Funding Statement

This work was supported by the National Research Foundation of Korea grant funded by the Korea government (MSIT) (No. 2019R1A2C2002256).

Citation

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Sungkyu Jung. "Geodesic projection of the von Mises–Fisher distribution for projection pursuit of directional data." Electron. J. Statist. 15 (1) 984 - 1033, 2021. https://doi.org/10.1214/21-EJS1807

Information

Received: 1 May 2020; Published: 2021
First available in Project Euclid: 16 March 2021

Digital Object Identifier: 10.1214/21-EJS1807

Subjects:
Primary: 62H11
Secondary: 62E20, 62R30

JOURNAL ARTICLE
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Vol.15 • No. 1 • 2021
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