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Title: Geodesic projection of directional distributions for projection pursuit Authors:  Sungkyu Jung - Seoul National University (Korea, South) [presenting]
Abstract: Geodesic projections of directional data are investigated, following either von Mises-Fisher (vMF) distribution or the angular Gaussian (AG) distribution. The vMF distribution for random directions on the $(p-1)$-dimensional unit hypersphere $\mathbb{S}^{p-1} \subset \mathbb{R}^{p}$ plays the role of multivariate normal distribution in directional statistics, and the rotationally symmetric AG distribution is very similar to the vMF. Projections onto geodesics are one of the 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, while the projection of any AG distribution onto subspheres, including geodesics, is AG-distributed. As one of the 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, and sometimes exactly AG, thus measures of non-vM-ness are a viable candidates for projection index.