Abstract
We investigate geodesic projections of von Mises–Fisher (vMF) distributed directional data. The vMF distribution for random directions on the (p−1)-dimensional unit hypersphere Sp−1⊂Rp 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→∞,κ→0), high-dimensions (p→∞), and two important cases of double-asymptotics (p,κ→∞, κ∕p→c or κ∕p→λ), 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.
Highlights
The von Mises–Fisher distribution, sometimes referred to as the Fisher–von Mises–Langevin distribution, plays a central role in modeling and inference for the directional data (Mardia and Jupp, 2000; Ley and Verdebout, 2017).While the multivariate normal distribution is not defined on the sample space Sp−1 = {x ∈ Rp : x x = 1} of directions, the vMF distribution is the closest notion to the normal distribution
We show that PvM(p, κ, δ) is well approximated by vM distributions in various asymptotic scenarios, including large and small concentrations
To examine the potential of such an index, we show for a location mixture of vMF distributions the projection score is approximately vM-distributed in high dimensions
Summary
Von Mises–Langevin distribution, plays a central role in modeling and inference for the directional data (Mardia and Jupp, 2000; Ley and Verdebout, 2017). For the special asymptotic regime of p → λ ∈ (0, ∞), we provide a high dimensional approximation of the distribution by the projected normal distribution For directional data on Sp−1 and their projections onto a geodesic (intrinsically a circle S1), the vMF and vM distribution play the role of the normal distribution. We propose to use a measure of discrepancy from the vM distribution as a one-dimensional projection pursuit index for directional data. Special functions and their properties are listed in Appendix A, and most proofs are contained in Appendix B
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