Nonparametric Measure-Transportation-Based Methods for Directional Data

TL;DR

This paper proposes various nonparametric tools based on measure transportation for directional data and constructs a universally consistent test of uniformity and a class of fully distribution-free and universally consistent tests for directional MANOVA which, in simulations, outperform all their existing competitors.

Abstract

This paper proposes various nonparametric tools based on measure transportation for directional data. We use optimal transports to define new notions of distribution and quantile functions on the hypersphere, with meaningful quantile contours and regions and closed-form formulas under the classical assumption of rotational symmetry. The empirical versions of our distribution functions enjoy the expected Glivenko-Cantelli property of traditional distribution functions. They provide fully distribution-free concepts of ranks and signs and define data-driven systems of (curvilinear) parallels and (hyper)meridians. Based on this, we also construct a universally consistent test of uniformity and a class of fully distribution-free and universally consistent tests for directional MANOVA which, in simulations, outperform all their existing competitors. A real-data example involving the analysis of sunspots concludes the paper.

Figures (6)

• Figure 1: Left: the tangent-normal decomposition of ${\bf F}$ with respect to ${\bf F}({\pmb \theta}_{\rm M})$. Right: in the rotationally symmetric case, the optimal transport $\mathbf{F}$ reduces to an optimal univariate transport acting on the projection ${\bf z}^\top {\pmb \theta}_{\text{\rm M}}$ of $\mathbf{z}$ along the rotation axis ${\pmb \theta}_{\text{\rm M}}$.
• Figure 2: The grid used to define signs and ranks on ${\cal S}^{d-1}$ for $d=3$. The final grid is obtained as the product of a reference grid $\mathfrak{S}^{(n_S)}\coloneqq \{{\mathbf{s}_1,\ldots,\mathbf{s}_{n_S}}\}$ over ${\mathcal{S}}^{d-2}$ (here, $n_S$ equispaced points on the circle ${\mathcal{S}}^{1}$) and $n_R$ equispaced points on the unit interval.
• Figure 3: Upper panel: empirical quantile contours (probability contents 12.2%, 48.8%, and 70.7%, respectively) computed from $n=2001$ ($n_R = 40$, $n_S = 50$ and $n_0 =1$) points drawn from a von Mises-Fisher distribution (left), a mixture of two von Mises-Fisher distributions (middle) and a tangent von Mises-Fisher distribution (right). Lower panel: the corresponding empirical meridians; points with the same color have the same signs.
• Figure 4: Rejection rates (at $0.05$ nominal level) of the pseudo-vMF and rank-based tests with uniform, vMF-location, vMF-concentration, and vMF-location-concentration scores, for Cases (1) (top-left panel), (2) (top-right panel), (3) (bottom-left panel), and (4) (bottom-right panel).
• Figure 5: Sunspots data: plot of the sunspots of the 22nd solar cycle ($n_1 = 4551$ red points) and the 23rd ($n_2 = 5373$ green points) solar cycle.

Theorems & Definitions (18)

• definition 1
• definition 2
• proposition 1
• definition 3
• proposition 2
• definition 4
• proposition 3
• proposition 4
• proposition 5
• proposition 6