Spatial quantiles on the hypersphere

Abstract

We propose a concept of quantiles for probability measures on the unit hypersphere ${\mathcal{S}}^{\mathit{d}-1}$ of ${\mathbb{R}}^{\mathit{d}}$. The innermost quantile is the Fréchet median, that is, the ${\mathit{L}}_1$-analog of the Fréchet mean. The proposed quantiles ${\mathit{\mu }}_{\mathit{\alpha },\mathit{u}}^{\mathit{m}}$ are directional in nature: they are indexed by a scalar order $\mathit{\alpha }\in [0,1]$ and a unit vector u in the tangent space ${\mathit{T}}_{\mathit{m}}{\mathcal{S}}^{\mathit{d}-1}$ to ${\mathcal{S}}^{\mathit{d}-1}$ at m. To ensure computability in any dimension d, our quantiles are essentially obtained by considering the Euclidean (Chaudhuri (J. Amer. Statist. Assoc. 91 (1996) 862–872)) spatial quantiles in a suitable stereographic projection of ${\mathcal{S}}^{\mathit{d}-1}$ onto ${\mathit{T}}_{\mathit{m}}{\mathcal{S}}^{\mathit{d}-1}$. Despite this link with Euclidean spatial quantiles, studying the proposed spherical quantiles requires understanding the nature of the (Chaudhuri (1996)) quantiles in a version of the projective space where all points at infinity are identified. We thoroughly investigate the structural properties of our quantiles and we further study the asymptotic behavior of their sample versions, which requires controlling the impact of estimating m. Our spherical quantile concept also allows for companion concepts of ranks and depth on the hypersphere. We illustrate the relevance of our construction by considering two inferential applications, related to supervised classification and to testing for rotational symmetry.