QUANTILES AND DEPTH FOR DIRECTIONAL DATA FROM ELLIPTICALLY SYMMETRIC DISTRIBUTIONS A P REPRINT

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QUANTILES AND DEPTH FOR DIRECTIONAL DATA FROM

ELLIPTICALLY SYMMETRIC DISTRIBUTIONS

A PREPRINT

Konstantin Hauch

Department of Mathematics,

Technische Universität Kaiserslautern,

Kaiserslautern, Germany

hauch@mathematik.uni-kl.de

Claudia Redenbach

Department of Mathematics,

Technische Universität Kaiserslautern,

Kaiserslautern, Germany

redenbach@mathematik.uni-kl.de

October 13, 2022

ABSTRACT

We present canonical quantiles and depths for directional data following a distribution which is el-

liptically symmetric about a direction µon the sphere Sd−1. Our approach extends the concept of

Ley et al. [1], which provides valuable geometric properties of the depth contours (such as convex-

ity and rotational equivariance) and a Bahadur-type representation of the quantiles. Their concept

is canonical for rotationally symmetric depth contours. However, it also produces rotationally sym-

metric depth contours when the underlying distribution is not rotationally symmetric. We solve this

lack of ﬂexibility for distributions with elliptical depth contours. The basic idea is to deform the

elliptic contours by a diffeomorphic mapping to rotationally symmetric contours, thus reverting to

the canonical case in Ley et al. [1]. A Monte Carlo simulation study conﬁrms our results. We use

our method to evaluate the ellipticity of depth contours and for trimming of directional data. The

analysis of ﬁbre directions in ﬁbre-reinforced concrete underlines the practical relevance.

Keywords directional statistics ·contour ·differential geometry ·angular Mahalanobis depth ·trimming

1 Introduction

The classes of rotationally symmetric distributions and elliptically symmetric distributions in Rdhave been well inves-

tigated by Kelker [2], Cambanis et al. [3] and Fang et al. [4]. A random vector V∈Rdhas a rotationally symmetric

distribution if VD

=OV for all O∈SO(d)where D

=refers to equality in distribution. Furthermore, every random vec-

tor V∈Rdfollowing a rotationally symmetric distribution can be represented as VD

=RU , where U∼Unif(Sd−1)

is independent of the real-valued random variable R∼FR.Ugives the direction of Vand Ris the length of V. Rota-

tionally symmetric distributions are often regarded as the most natural non-uniform distributions in Rd. For instance,

the charge distribution of an electric ﬁeld is rotationally symmetric around its source. However, not all phenomena

observed in practice can be represented by symmetric models.

Elliptically symmetric distributions extend the class of rotationally symmetric distributions. The distribution of a

random vector Wis elliptically symmetric if and only if WD

=RΣ1/2Uwith U∼Unif (Sd−1), real-valued R∼FR

independent of U, and Σ∈Rd×da symmetric, positive deﬁnite matrix. A random vector Wwith an elliptically

symmetric distribution can be transformed into a random vector V=RU with a rotationally symmetric distribution

via

Σ−1/2WD

=RΣ−1/2Σ1/2U=RU =V. (1.1)

These concepts of symmetry transfer to the unit sphere Sd−1, i.e., the case of directional data. Distributions on Sd−1

which are rotationally symmetric about a direction µ∈ Sd−1are also often regarded as the natural non-uniform

arXiv:2210.06098v1 [math.ST] 12 Oct 2022

Quantiles and depth for directional data from elliptically symmetric distributions A PREPRINT

distributions on Sd−1[5]. In most cases, rotationally symmetric distributions have tractable normalising constants.

Note that the density of a rotationally symmetric distribution is proportional to a function f(xTµ). Thus, a projection

onto µenables a one-dimensional analysis of the distribution, for example, its concentration around µ. The class of

distributions with rotational symmetry about µ∈ Sd−1is denoted by Rµ.

In practice, symmetric models are often too restrictive. For instance, Leong and Carlile [6] illustrated that rotational

symmetry about a direction is a too strong assumption for a directional data set from neurosciences. Kent [7] has ﬁtted

his elliptical model to a data set of 34 measurements of the directions of magnetisation for samples from the Great

Whin Sill (Northumberland, England). As in Rd, distributions that are elliptically symmetric about a direction µon

Sd−1are an extension of the rotationally symmetric distributions. The contours are more ﬂexible to form elliptical

shapes. Due to the curved shape of the sphere, the transition from distributions which are rotationally symmetric about

µ∈ Sd−1to distributions which are elliptically symmetric about µis not obvious. A remedy to this problem is to

linearise Sd−1at some base point µby considering the tangent space TµSd−1at µ. By using the theory for Rd, a

transformation between the two distributions can then be deﬁned in the tangent space.

Ley et al. [1] introduced a concept of quantiles and depth for directional data. They showed that their quantiles are

asymptotically normal and established a Bahadur-type representation [8] for directional data X∼FX∈ Rµ. A

Monte Carlo simulation study corroborated their theoretical results. Statistical tools, like directional DD- and QQ-

plots and a quantile-based goodness-of-ﬁt test, were deﬁned and illustrated on a data set of cosmic rays. Their results

are canonical for rotationally symmetric distributions. But their concept suffers from the disadvantage of producing

rotationally symmetric depth contours, even if the underlying distribution has elliptical contours [9].

In this paper, we present a procedure solving the latter issue if the underlying distribution has elliptical contours.

The paper is organised as follows. In Section 2, we ﬁrst introduce basics about the distributions under consideration,

extend the Mahalanobis transformation to Sd−1, and summarise the ﬁndings of Ley at al. [1]. Section 3contains

our main contribution. The idea is to map the unit vectors into the tangent space TµSd−1where µis the median

direction of the observed sample. The mapped vectors are elliptically symmetric around the origin in TµSd−1. The

multivariate Mahalanobis transformation [10,11] is then applied in TµSd−1to obtain a rotationally symmetric sample

in TµSd−1. Mapping it back to Sd−1, we obtain a sample of unit vectors which are rotationally symmetric about

µ. Thus, we can exploit the results from [1]. All transformations are diffeomorphic such that we can trace back the

results to the original unit vectors. Section 4afﬁrms our ﬁndings by a Monte Carlo study. Furthermore, we apply our

approach to a real-world data set from [12]: Directions of short steel ﬁbres crossing a crack in ultra-high performance

ﬁbre-reinforced concrete (UHPFRC).

2 Basics

2.1 Rotational and elliptical symmetry about a direction on Sd−1

Deﬁnition 2.1 (Rotational symmetry about a direction).Let X∈ Sd−1be a random vector and µ∈ Sd−1. The

distribution of Xis rotationally symmetric about µon Sd−1if and only if XD

=OX for every O∈SO(d)fulﬁlling

Oµ =µ.

Let Rµbe the class of distributions which are rotationally symmetric about µ∈ Sd−1. Projecting Xonto a vector

space orthogonal to µyields rotationally symmetric contours. Distributions FX∈ Rµare characterised by densities

of the form

fµ(x) = cdf(xTµ), x ∈ Sd−1,(2.1)

where f: [−1,1] →R≥0is absolutely continuous and cda normalising constant [1]. The distribution of XTµis

absolutely continuous w.r.t the Lebesgue measure on [−1,1] [13]. The density of XTµreads

f(t) = ωd−1cd(1 −t2)d−3

2f(t),(2.2)

where ωd−1is the surface area of Sd−2[5]. A widely known distribution in Rµis the von Mises-Fisher distribution

where f(t) = exp(κt).

Deﬁnition 2.2 (von Mises-Fisher distribution Md(µ, κ)[13]).The probability density function of the von Mises–Fisher

distribution is given by

fvMF,µ,κ(x) = cdexp (κxTµ),(2.3)

where κ≥0is a concentration parameter, µ∈ Sd−1the mean direction, and cdthe normalising constant.

Quantiles and depth for directional data from elliptically symmetric distributions A PREPRINT

The concentration around µincreases with κ. The von Mises-Fisher distribution is unimodal for κ > 0. For κ= 0,

we get the uniform distribution on the sphere.

A generalisation of the von Mises-Fisher distribution is the Fisher-Bingham distribution [13], where a general quadratic

equation is added in the exponent of the density in (2.1). An example is the Kent distribution [7].

Deﬁnition 2.3 (Kent distribution K(µ, κ, A)).The probability density function of the Kent distribution is given by

fK,µ,A(x) = cdexp (κxTµ+xTAx),(2.4)

where κ≥0is a concetration parameter, µ∈ Sd−1the mean direction, A∈Sym(d)with Aµ = 0da shape

parameter, and cdthe normalising constant. The concentration around µincreases with κ, while A∈Sym(d)

controls the shape of the density contours.

For large κ, the Kent distribution has a mode at µand density contours which are elliptical [14, p.177].

Let Fµbe the class of distributions on Sd−1with a bounded density that admit a unique modal direction µ. We further

assume that µcoincides with the Fisher spherical median [15], that is

µ= arg min

γ∈Sd−1E(arccos(XTγ)).(2.5)

For i.i.d. random vectors X, X1, . . . , Xn∈ Sd−1with X∼F∈ Fµ, we estimate µby the root-nconsistent empirical

Fisher spherical median [15]

ˆµ= arg min

γ∈S2

i=1

arccos(XT

iγ).(2.6)

Note that the deﬁnition of the class Rµdoes not include that µis the unique modal direction. Here, we restrict attention

to distributions in Rµ∩ Fµfrom now on. E.g., Md(µ, κ)∈ Rµ∩ Fµfor κ > 0, and K(µ, κ, A)∈ Fµfor κ > 0

under suitable conditions on A∈Sym(d)given in the next section.

2.2 Differential geometry

Differential geometry examines smooth manifolds using differential and integral calculus as well as linear and multi-

linear algebra. It originates in studying spherical geometries related to astronomy and the geodesy of the earth. For an

introduction to differential geometry, see e.g. [16].

We saw in (1.1) that a linear transformation Σtransforms a random vector with a rotationally symmetric distribution

into a random vector with an elliptically symmetric distribution in Rd. We want to proceed analogously for distribu-

tions on the sphere. However, in general, the linear transformation Σdoes not necessarily map Sd−1onto itself. A

remedy is provided by linearising the sphere Sd−1at a base point µ∈ Sd−1.

The tangent space TµSd−1to Sd−1at base point µ∈ Sd−1is the collection of all tangent vectors to Sd−1at µ. It is

a local Euclidean vector space with local origin in µ. Given µ∈ Sd−1and a tangent vector v∈TµSd−1, there is a

unique geodesic from µ∈ Sd−1to some x∈ Sd−1given as a mapping

cµ,v : [0,1] → Sd−1,(2.7)

starting at cµ,v(0) = µwith initial velocity ˙cµ,v(0) = vand ending in cµ,v (1) = x[17].

In the following, we deﬁne mappings between the tangent space and the sphere.

Deﬁnition 2.4 (Riemannian exponential map).The Riemannian exponential map

Expµ:TµSd−1→ Sd−1(2.8)

maps a vector v∈TµSd−1to Sd−1along the geodesic cµ,v such that x=Expµ(v) = cµ,v (1).

The exponential map is locally diffeomorphic onto V(µ) = Sd−1\ {−µ}, where −µis called cut point and the set

{−µ}is called cut locus. Within V(µ)the exponential map Expµhas an inverse, the Riemannian logarithmic map.

Deﬁnition 2.5 (Riemannian logarithmic map).The Riemannian logarithmic map

Logµ:V(µ)→TµSd−1(2.9)

maps a vector x∈ Sd−1into TµSd−1with Expµ(Logµ(x)) = x.

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QUANTILESANDDEPTHFORDIRECTIONALDATAFROMELLIPTICALLYSYMMETRICDISTRIBUTIONSAPREPRINTKonstantinHauchDepartmentofMathematics,TechnischeUniversitätKaiserslautern,Kaiserslautern,Germanyhauch@mathematik.uni-kl.deClaudiaRedenbachDepartmentofMathematics,TechnischeUniversitätKaiserslautern,Kaiserslautern,Germ...

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