Statistics of ambiguous rotations

TL;DR

This paper develops statistical methods for analyzing ambiguous rotations, which are rotations known only up to symmetry, with applications in biomechanics, crystallography, and seismology.

Contribution

It introduces new tests and models for ambiguous rotations, including uniformity tests, parametric models, and regression approaches.

Findings

A test for uniformity of ambiguous rotations is proposed.

Parametric models for ambiguous rotations are developed.

An illustrative example with diopside crystal orientations demonstrates the methods.

Abstract

The orientation of a rigid object can be described by a rotation that transforms it into a standard position. For a symmetrical object the rotation is known only up to multiplication by an element of the symmetry group. Such ambiguous rotations arise in biomechanics, crystallography and seismology. We develop methods for analyzing data of this form. A test of uniformity is given. Parametric models for ambiguous rotations are presented, tests of location are considered, and a regression model is proposed. A brief illustrative example involving orientations of diopside crystals is given.

Tables

Table 1. Table 1: Symmetry groups and frames.

Group	Name	Frame
$C_1$	trivial	$({𝐮}_1,{𝐮}_2,{𝐮}_3)$	${𝐮}_1,{𝐮}_2,{𝐮}_3$ orthonormal, ${𝐮}_3={𝐮}_1\times {𝐮}_2$
$C_2$	cyclic	$({𝐮}_0,\pm {𝐮}_1)$	${𝐮}_0,{𝐮}_1$ orthonormal
$C_r\left(r\ge 3\right)$	cyclic	$({𝐮}_1,\dots ,{𝐮}_r)$	${𝐮}_1,\dots ,{𝐮}_r$ coplanar,
			known up to cyclic order,
			${𝐮}_i^T{𝐮}_{i-1}=\cos \left(2\pi /r\right)$ for $i=2,\dots ,r$
$D_2$	dihedral	$(\pm {𝐮}_1,\pm {𝐮}_2)$	orthogonal axes
$D_r\left(r\ge 3\right)$	dihedral	$({𝐮}_1,\dots ,{𝐮}_r)$	${𝐮}_1,\dots ,{𝐮}_r$ coplanar,
			known up to cyclic order and reversal,
			${𝐮}_i^T{𝐮}_{i-1}=\cos \left(2\pi /r\right)$ for $i=2,\dots ,r$
$T=A_4$	tetrahedral	$\{{𝐮}_1,\dots ,{𝐮}_4\}$	${𝐮}_i^T{𝐮}_j=-1/3$ for $i\ne j$
$O={\Sigma }_4$	octahedral	$\{\pm {𝐮}_1,\pm {𝐮}_2,\pm {𝐮}_3\}$	orthogonal axes
	= cubic
$Y=A_5$	icosahedral	$\{\pm {𝐮}_1,\dots ,\pm {𝐮}_6\}$	$\left|{𝐮}_i^T{𝐮}_j\right|=5^{-1/2}$ for $i\ne j$
	= dodecahedral

Table 2. Table 2: Some embeddings 𝐭 K : S ​ O ​ ( 3 ) / K → E : subscript 𝐭 𝐾 → 𝑆 𝑂 3 𝐾 𝐸 {\mathbf{t}}_{K}:SO(3)/K\rightarrow E .

Group, $K$	${𝐭}_K$
$C_1$	${𝐭}_{C_1}({𝐮}_1,{𝐮}_2,{𝐮}_3)=({𝐮}_1,{𝐮}_2,{𝐮}_3)$
$C_2$	${𝐭}_{C_2}({𝐮}_0,\pm {𝐮}_1)=({𝐮}_0,{𝐮}_1{𝐮}_1^T-\left(1/3\right){𝐈}_3)$
$C_r\left(r\ge 3\right)$
$r$ odd	${𝐭}_{C_r}\left([{𝐮}_1,\dots ,{𝐮}_r]\right)=({𝐮}_0,{\sum }_{i=1}^r{\otimes }^r{𝐮}_i)$
$r$ even	${𝐭}_{C_r}\left([{𝐮}_1,\dots ,{𝐮}_r]\right)=({𝐮}_0,{\sum }_{i=1}^r{\otimes }^r{𝐮}_i-r/(r+1)\mathrm{symm}\left({\otimes }^{r/2}{𝐈}_3\right))$
$D_2$	${𝐭}_{D_2}(\pm {𝐮}_1,\pm {𝐮}_2)=({𝐮}_1{𝐮}_1^T-\left(1/3\right){𝐈}_3,{𝐮}_2{𝐮}_2^T-\left(1/3\right){𝐈}_3,{𝐮}_3{𝐮}_3^T-\left(1/3\right){𝐈}_3)$
$D_r\left(r\ge 3\right)$
$r$ odd	${𝐭}_{D_r}\left([{𝐮}_1,\dots ,{𝐮}_r]\right)={\sum }_{i=1}^r{\otimes }^r{𝐮}_i$
$r$ even	${𝐭}_{D_r}\left([{𝐮}_1,\dots ,{𝐮}_r]\right)={\sum }_{i=1}^r{\otimes }^r{𝐮}_i-r/(r+1)\mathrm{symm}\left({\otimes }^{r/2}{𝐈}_3\right)$
$T$	${𝐭}_T\left(\{{𝐮}_1,{𝐮}_2,{𝐮}_3,{𝐮}_4\}\right)={\otimes }^3{𝐮}_1+{\otimes }^3{𝐮}_2+{\otimes }^3{𝐮}_3+{\otimes }^3{𝐮}_4$
$O$	${𝐭}_O\left(\{\pm {𝐮}_1,\pm {𝐮}_2,\pm {𝐮}_3\}\right)={\otimes }^4{𝐮}_1+{\otimes }^4{𝐮}_2+{\otimes }^4{𝐮}_3-(3/5)\mathrm{symm}\left({\otimes }^2{𝐈}_3\right)$
$Y$	${𝐭}_Y\left(\{\pm {𝐮}_1,\dots ,\pm {𝐮}_6\}\right)={\sum }_{i=1}^6{\otimes }^{10}{𝐮}_i-(6/11)\mathrm{symm}\left({\otimes }^5{𝐈}_3\right)$

Table 3. Table 3: Inner products of transforms of symmetric frames.

Group, $K$	Inner product
$C_1$	$⟨{𝐭}_{C_1}({𝐮}_1,{𝐮}_2,{𝐮}_3),{𝐭}_{C_1}({𝐯}_1,{𝐯}_2,{𝐯}_3)⟩={𝐮}_1^T{𝐯}_1+{𝐮}_2^T{𝐯}_2+{𝐮}_3^T{𝐯}_3$
$C_2$	$⟨{𝐭}_{C_2}({𝐮}_0,\pm {𝐮}_1),{𝐭}_{C_2}({𝐯}_0,\pm {𝐯}_1)⟩={𝐮}_0^T{𝐯}_0+{\left({𝐮}_1^T{𝐯}_1\right)}^2-1/3$
$C_r\left(r\ge 3\right)$
$r$ odd	$⟨{𝐭}_{C_r}\left([{𝐮}_1,\dots ,{𝐮}_r]\right),{𝐭}_{C_r}\left([{𝐯}_1,\dots ,{𝐯}_r]\right)⟩={𝐮}_0^T{𝐯}_0+{\sum }_{i=1}^r{\sum }_{j=1}^r{\left({𝐮}_i^T{𝐯}_j\right)}^r$
$r$ even	$⟨{𝐭}_{C_r}\left([{𝐮}_1,\dots ,{𝐮}_r]\right),{𝐭}_{C_r}\left([{𝐯}_1,\dots ,{𝐯}_r]\right)⟩={𝐮}_0^T{𝐯}_0+{\sum }_{i=1}^r{\sum }_{j=1}^r{\left({𝐮}_i^T{𝐯}_j\right)}^r-r^2/\left(r+1\right)$
$D_2$	$⟨{𝐭}_{D_2}(\pm {𝐮}_1,\pm {𝐮}_2),{𝐭}_{D_2}\pm {𝐯}_1,\pm {𝐯}_2)⟩=\left({𝐮}_1^T{𝐯}_1\right){}^2+\left({𝐮}_2^T{𝐯}_2\right){}^2+\left({𝐮}_3^T{𝐯}_3\right){}^2-1$
$D_r\left(r\ge 3\right)$
$r$ odd	$⟨{𝐭}_{D_r}\left([{𝐮}_1,\dots ,{𝐮}_r]\right),{\underset{¯}{\text{t}}}_{D_r}\left([{𝐯}_1,\dots ,{𝐯}_r]\right)⟩={\sum }_{i=1}^r{\sum }_{j=1}^r{\left({𝐮}_i^T{𝐯}_j\right)}^r$
$r$ even	$⟨{𝐭}_{D_r}\left([{𝐮}_1,\dots ,{𝐮}_r]\right),{𝐭}_{D_r}\left([{𝐯}_1,\dots ,{𝐯}_r]\right)⟩={\sum }_{i=1}^r{\sum }_{j=1}^r{\left({𝐮}_i^T{𝐯}_j\right)}^r-r^2/\left(r+1\right)$
$T$	$⟨{𝐭}_T\left(\{{𝐮}_1,{𝐮}_2,{𝐮}_3,{𝐮}_4\}\right),{𝐭}_T\left(\{{𝐯}_1,{𝐯}_2,{𝐯}_3,{𝐯}_4\}\right)⟩={\sum }_{i=1}^4{\sum }_{j=1}^4{\left({𝐮}_i^T{𝐯}_j\right)}^3$
$O$	$⟨{𝐭}_O\left(\{\pm {𝐮}_1,\pm {𝐮}_2,\pm {𝐮}_3\}\right),{𝐭}_O\left(\{\pm {𝐯}_1,\pm {𝐯}_2,\pm {𝐯}_3\}\right)⟩={\sum }_{i=1}^3{\sum }_{j=1}^3{\left({𝐮}_i^T{𝐯}_j\right)}^4-9/5$
$Y$	$⟨{𝐭}_Y\left(\{\pm {𝐮}_1,\dots ,\pm {𝐮}_6\}\right),{𝐭}_Y\left(\{\pm {𝐯}_1,\dots ,\pm {𝐯}_6\}\right)⟩={\sum }_{i=1}^6{\sum }_{j=1}^6{\left({𝐮}_i^T{𝐯}_j\right)}^{10}-36/11$

Table 4. Table 4: Values of squared radius, ρ 2 superscript 𝜌 2 \rho^{2} , and dimension, ν 𝜈 \nu

Group	${\rho }^2$	$\nu$
$C_1$	3	9
$C_2$	5/3	8
$C_r\left(r\ge 3\right)$
$r$ odd	$1+2^{1-r}{r}^2$	$\left(r+2\right)\left(r+1\right)/2+3$
$r$ even	$1+r^2{2}^{1-r}\left\{1+2^{-1}\left(\frac{r}{r/2}\right)\right\}-r^2/\left(r+1\right)$	$\left(r+2\right)\left(r+1\right)/2+3$
$D_2$	$2$	10
$D_r\left(r\ge 3\right)$
$r$ odd	$2^{1-r}{r}^2$	$\left(r+2\right)\left(r+1\right)/2$
$r$ even	$r^2{2}^{1-r}\left\{1+2^{-1}\left(\frac{r}{r/2}\right)\right\}-r^2/\left(r+1\right)$	$\left(r+2\right)\left(r+1\right)/2$
$T$	32/9	10
$O$	6/5	9
$Y$	$18816/6875$	21

Table 5. Table 5: High-concentration asymptotic variance, 𝚺 𝚺 \Sigma , of κ 1 / 2 superscript 𝜅 1 2 \kappa^{1/2} .

Group	$𝚺$
$C_1$	$\left(1/2\right){𝐈}_3$
$C_2$	$\mathrm{diag}(1/2,1/4,1/6)$
$C_r\left(r\ge 3\right)$	$\mathrm{diag}[{\left(1+r{A}_r\right)}^{-1},{\left(1+r{A}_r\right)}^{-1},{\left\{2r{A}_r-r\left(r-1\right)A_{r-1}\right\}}^{-1}]$
$D_2$	$\left(1/4\right){𝐈}_3$
$D_r\left(r\ge 3\right)$	$\mathrm{diag}[{\left(r{A}_r\right)}^{-1},{\left(r{A}_r\right)}^{-1},{\left\{r{A}_r-r\left(r-1\right)A_{r-1}\right\}}^{-1}]$
$T$	$0.070{𝐈}_3$
$O$	$\left(1/8\right){𝐈}_3$
$Y$	$0.026{𝐈}_3$

Equations

$${\rho }^2=\Vert \mathbf{t}([\mathbf{U}]){\Vert }^2,$$

$$d={\rho }^2-\Vert \bar{\mathbf{t}}{\Vert }^2,$$

$$S=(\nu /{\rho }^2)n{\Vert \bar{\mathbf{t}}\Vert }^2=n\nu (1-d/{\rho }^2),$$

$$S=(1/3)S_R+(2/15)S_B,$$

$$({\nu }_C/{\rho }_C^2)S=(1/3)S_R+({\nu }_D/{\rho }_D^2)S_D,$$

$$T_G=-\sum _{i=1}^n\sum _{j=1}^n\sum _{\mathbf{R}\in K}{\left\{3-\mathrm{trace}\left({\mathbf{U}}_i^{\mathrm{T}}{\mathbf{U}}_j\mathbf{R}\right)\right\}}^{1/2},$$

$$f([\mathbf{U}];[\mathbf{M}],\kappa )=g\left(⟨\mathbf{t}([\mathbf{U}]),\mathbf{t}([\mathbf{M}])⟩;\kappa \right),$$

$$f([\mathbf{U}];[\mathbf{M}],\kappa )=c(\kappa {)}^{-1}\exp \{\kappa ⟨\mathbf{t}([\mathbf{U}]),\mathbf{t}([\mathbf{M}])⟩\}.$$

$$f([\mathbf{U}];[\mathbf{M}],\kappa )=c(\kappa {)}^{-1}{\left\{1+⟨\mathbf{t}([\mathbf{U}]),\mathbf{t}([\mathbf{M}])⟩\right\}}^{\kappa }.$$

$$f([\mathbf{U}];[\mathbf{M}],\kappa )=1+\kappa ⟨\mathbf{t}([\mathbf{U}]),\mathbf{t}([\mathbf{M}])⟩$$

$${\mathbf{A}}({\mathbf{v}})=\left(\begin{array}[]{ccc}o&-v_{3}&v_{2}\\ v_{3}&0&-v_{1}\\ -v_{2}&v_{1}&0\end{array}\right)$$

$$f([\mathbf{V}]\mid [\mathbf{U}];\mathbf{A},\kappa )=c(\kappa {)}^{-1}\exp \{\kappa ⟨{\mathbf{t}}_2([\mathbf{V}]),{\mathbf{t}}_1([\mathbf{A}\mathbf{U}])⟩\}.$$

$$\hat{\mathbf{A}}=\arg \underset{\mathbf{A}\in SO(3)}{\max }\sum _{i=1}^n⟨{\mathbf{t}}_2([{\mathbf{V}}_i]),{\mathbf{t}}_1([\mathbf{A}{\mathbf{U}}_i])⟩.$$

$$r=(n{\rho }_{12}{)}^{-1}\sum _{i=1}^n⟨{\mathbf{t}}_1([\hat{\mathbf{A}}{\mathbf{U}}_i]),{\mathbf{t}}_2([{\mathbf{V}}_i])⟩,$$

$$\sum _{i=1}^n\left\{{\rho }^2-⟨\mathbf{t}([{\mathbf{I}}_3]),\mathbf{t}([{\mathbf{V}}_i^{\mathrm{T}}\mathbf{A}{\mathbf{U}}_i])⟩\right\}$$

$$2\kappa \sum _{i=1}^n\left\{{\rho }^2-⟨\mathbf{t}([{\mathbf{I}}_3]),\mathbf{t}([{\mathbf{V}}_i^{\mathrm{T}}\mathbf{A}{\mathbf{U}}_i])⟩\right\}$$

$$2\kappa \sum _{i=1}^n\left\{{\rho }^2-⟨\mathbf{t}([{\mathbf{I}}_3]),\mathbf{t}([{\mathbf{V}}_i^{\mathrm{T}}\hat{\mathbf{A}}{\mathbf{U}}_i])⟩\right\}$$

$$2\kappa \sum _{i=1}^n\left\{⟨\mathbf{t}([{\mathbf{I}}_3]),\mathbf{t}([{\mathbf{V}}_i^{\mathrm{T}}\hat{\mathbf{A}}{\mathbf{U}}_i])⟩-⟨\mathbf{t}([{\mathbf{I}}_3]),\mathbf{t}([{\mathbf{V}}_i^{\mathrm{T}}\mathbf{A}{\mathbf{U}}_i])⟩\right\}$$

$$\left\{\mathbf{A}:2\hat{\kappa }\sum _{i=1}^n\left\{⟨\mathbf{t}([{\mathbf{I}}_3]),\mathbf{t}([{\mathbf{V}}_i^{\mathrm{T}}\hat{\mathbf{A}}{\mathbf{U}}_i])⟩-⟨\mathbf{t}([{\mathbf{I}}_3]),\mathbf{t}([{\mathbf{V}}_i^{\mathrm{T}}\mathbf{A}{\mathbf{U}}_i])⟩\right\}<{\chi }_{3;\alpha }^2\right\}.$$

Full text

Keywords: Frame, Orientation, Regression, Symmetry, Tensor, Test of uniformity.

1 Introduction

Data that are rotations of $\mathbb{R}^{3}$ occur in various areas of science, such as palaeo-magnetism (Pesonen et al., 2003; Villalaín et al., 2016; Koymans et al., 2016), plate tectonics and seismology (Stein & Wysession, 2003; Hardebeck, 2006; Arnold & Townend, 2007; Walsh et al., 2009; Khalil & McClay, 2016), biomechanics (Rivest, 2005; Lekadir et al., 2015; Spronck et al., 2016), crystallography (Hielscher et al., 2010; Griffiths et al., 2016) and texture analysis, i.e., analysis of orientations of crystalites (Kunze & Schaeben, 2004, 2005; Du et al., 2016). The sample space is the 3-dimensional rotation group, $SO(3)$, and methods for handling such data are now an established part of directional statistics; see §13.2 of Mardia & Jupp (2000). In some contexts the presence of symmetry means that the rotations are observed subject to ambiguity, so that it is not possible to distinguish a rotation ${\mathbf{X}}$ from ${\mathbf{X}}{\mathbf{R}}$ for any rotation ${\mathbf{R}}$ in some given subgroup $K$ of $SO(3)$. From the mathematical point of view, the sample space is the quotient $SO(3)/K$ of $SO(3)$ by $K$. Such spaces arise in many scientific contexts: the case in which $K$ is generated by the rotations through $180^{\circ}$ about the coordinate axes gives the orthogonal axial frames considered by Arnold & Jupp (2013), which can be used to describe aspects of earthquakes; many groups $K$ of low order occur as the symmetry groups of crystals; the icosahedral group is the symmetry group of some carborane molecules (Jemmis, 1982), of most closed-shell viruses (Harrison, 2013), of the natural quasicrystal, icosahedrite (Bindi et al., 2011), and of the blue phases of some liquid crystals (Seideman,1990, §6.1.2). The object of this paper is to give a unified account of some general tools for the analysis of data consisting of ambiguous rotations with a finite symmetry group.

2 Ambiguous rotations

2.1 Symmetry groups

The orientation of a rigid object in $\mathbb{R}^{3}$ can be described by a rotation that transforms it into some standard position. If the object is asymmetrical then this rotation is unique, so that the orientations of the object correspond to elements of the rotation group $SO(3)$. If the object is symmetrical then the set of rotations that have no visible effect on the object forms a subgroup $K$ of $SO(3)$. Then the orientations of the object correspond to elements of the homogeneous space $SO(3)/K$, i.e. the set of equivalence classes of elements of $SO(3)$ under the right action of $K$. We shall consider the cases in which $K$ is finite. In particular, the orientations of T-shaped, X-shaped and $+$-shaped objects in $\mathbb{R}^{3}$ are elements of $SO(3)/K$ with $K=C_{2},D_{2}$ and $D_{4}$, respectively. For ${\mathbf{U}}$ in $SO(3)$ we shall denote the equivalence class of ${\mathbf{U}}$ in $SO(3)/K$ by $[{\mathbf{U}}]$.

The finite subgroups of $SO(3)$ are known also as the point groups of the first kind. The classification result for these groups, given e.g. in Miller (1972), states that any such group is isomorphic to one of the following: the cyclic groups, $C_{r}$, for $r=1,2,\dots$, the dihedral groups, $D_{r}$, for $r=2,3,\dots$, the tetrahedral group, $T$, the octahedral group, $O$, and the icosahedral group, $Y$. These groups are listed in Table 1, together with the frames of vectors that will be used to represent elements of the sample spaces $SO(3)/K$. The group $C_{1}$ has one element, the identity, ${\mathbf{I}}_{3}$.

2.2 Frames and symmetric frames

For each point group $K$ of the first kind, every element of $SO(3)/K$ can be represented uniquely by a $K$-frame, i.e. an equivalence class of a frame, meaning a set of vectors or axes in $\mathbb{R}^{3}$. For $K=C_{r}$ with $r\geq 3$ or $K=D_{r}$ with $r\geq 3$, it is convenient to take the vectors of the frame to be unit normals to the sides of a regular $r$-gon; for $K=C_{2}$ we take a unit vector and an axis orthogonal to it; for $K=D_{2}$ we take a pair of orthogonal axes; for $K=T$, $O$ or $Y$, it is convenient to take the vectors to be unit normals to the sides of a regular tetrahedron, cube or dodecahedron, respectively. Permutation of the vectors of the frame by the action of $K$ leads to ambiguity. This ambiguity is removed by passing to the corresponding $K$-frame, i.e. the equivalence class of the frame under such permutations. The $K$-frames will be denoted by square brackets, e.g. for $K=C_{r}$, $[{\mathbf{u}}_{1},\dots,{\mathbf{u}}_{r}]$ denotes the $K$-frame arising from $({\mathbf{u}}_{1},\dots,{\mathbf{u}}_{r})$. By a symmetric frame, we shall mean a $K$-frame for some $K$. The frames that we consider are listed in Table 1, together with an indication of the ambiguities.

Special cases of Table 1 include the 7 crystal systems: triclinic, monoclinic, trigonal, tetragonal, orthorhombic, hexagonal and cubic with symmetry groups $C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$, $D_{2}$, $D_{6}$ and $O$, respectively.

3 Transforming symmetric frames to tensors

3.1 Embeddings of the sample spaces

In order to carry out statistics on $SO(3)/K$, we shall take the embedding approach used in, e.g.  §10.8 of Mardia & Jupp (2000). We shall embed $SO(3)/K$ in an inner-product space, $E$, on which $SO(3)$ acts. The embedding will be a well-defined equivariant one-to-one function ${\mathbf{t}}:SO(3)/K\rightarrow E$ such that ${\mathbf{t}}([{\mathbf{U}}])$ has expectation ${\mathbf{0}}$ if $[{\mathbf{U}}]$ is uniformly distributed on $SO(3)/K$. For $E=L^{2}(SO(3))$, the space of square-integrable functions on $SO(3)$, a very wide class of such embeddings can be obtained by averaging over $K$. Let ${\mathbf{t}}_{0}:SO(3)\to L^{2}(SO(3))$ be an embedding used in the Hilbert space approach to Sobolev tests of uniformity; see §§10.8, 13.2.2 of Mardia & Jupp (2000), Giné (1975) and §4 of Prentice (1978). Define ${\mathbf{t}}:SO(3)/K\to L^{2}(SO(3))$ by ${\mathbf{t}}([{\mathbf{U}}])=|K|^{-1}\sum_{{\mathbf{R}}\in K}{\mathbf{t}}_{0}({\mathbf{U}}{\mathbf{R}})$, where $|K|$ denotes the number of elements in $K$. If ${\mathbf{t}}$ is one-to-one then it is an embedding. In general, such ${\mathbf{t}}$ are quite complicated, so in this paper, for each $K$, we focus on a simple choice of embedding, ${\mathbf{t}}_{K}$, of $SO(3)/K$ into an appropriate space of symmetric tensors. These ${\mathbf{t}}_{K}$ are given in Table 2. Corresponding expressions for $\langle{\mathbf{t}}_{K}([{\mathbf{U}}]),{\mathbf{t}}_{K}([{\mathbf{W}}])\rangle$ with ${\mathbf{U}},{\mathbf{W}}\in SO(3)$ are given in Table 3. Here $\langle\cdot,\cdot\rangle$ is the standard inner product on the appropriate tensor product.

Define $\rho^{2}$ by

[TABLE]

which has the same value for all ${\mathbf{U}}$ in $SO(3)$. Then ${\mathbf{t}}$ embeds $SO(3)/K$ in the sphere of radius $\rho$ with centre the origin in the vector space $E$.

Each symmetric frame can be represented by an element ${\mathbf{U}}$ of $SO(3)$. In the triclinic case, where $K=C_{1}$, ${\mathbf{U}}$ is unique and ${\mathbf{t}}([{\mathbf{U}}])={\mathbf{U}}$. We have restricted our attention to point groups, $K$, of the first kind, i.e., excluding reflections. However, in situations where reflection symmetries are also present we can adopt a right-handed convention for all orientations, and then neglect reflections. For example, we can treat observations on $O(3)/\{{\mathbf{I}}_{3},-{\mathbf{I}}_{3}\}$ in the same way as those on $SO(3)=SO(3)/C_{1}$.

3.2 Sample mean

Observations $[{\mathbf{U}}_{1}],\dots,[{\mathbf{U}}_{n}]$ in $SO(3)/K$ can usefully be summarized by the sample mean $\bar{{\mathbf{t}}}$ of their images by ${\mathbf{t}}$, i.e., by $\bar{{\mathbf{t}}}=n^{-1}\sum_{i=1}^{n}{\mathbf{t}}([{\mathbf{U}}_{i}])$. The sample mean $[{\bar{\mathbf{U}}}]$ is defined as the $[{\mathbf{U}}]$ in $SO(3)/K$ that maximizes $\langle{\mathbf{t}}([{\bar{\mathbf{U}}}]),\bar{{\mathbf{t}}}\rangle$. Although $[{\bar{\mathbf{U}}}]$ is not necessarily unique, it follows from Theorem 3.2 of Bhattacharya & Patrangenaru (2003) that if $[{\mathbf{U}}_{1}],\dots,[{\mathbf{U}}_{n}]$ are generated by a continuous distribution then $[{\bar{\mathbf{U}}}]$ is unique with probability 1.

3.3 Sample dispersion

A sensible measure of dispersion is

[TABLE]

analogous to the quantity $1-\bar{R}^{2}$ used for spherical data; see p. 164 of Mardia & Jupp (2000). The dispersion satisfies the inequalities $0\leq d\leq\rho^{2}$, where $\rho^{2}$ is defined in (1). Since ${\mathbf{t}}$ is one-to-one, $d=0$ if and only if $[{\mathbf{U}}_{1}]=\dots=[{\mathbf{U}}_{n}]$. Transformation of $[{\mathbf{U}}_{1}],\dots,[{\mathbf{U}}_{n}]$ to $[{\mathbf{V}}{\mathbf{U}}_{1}],\dots,[{\mathbf{V}}{\mathbf{U}}_{n}]$ with ${\mathbf{V}}$ in $SO(3)$ leaves $d$ unchanged. If $K=C_{1}$ then $d=3-\mathrm{trace}\left({\bar{\mathbf{R}}}^{2}\right)$, where ${\bar{\mathbf{R}}}=\left({\bar{\mathbf{X}}}^{T}{\bar{\mathbf{X}}}\right)^{1/2}$ with ${\bar{\mathbf{X}}}={\bar{\mathbf{t}}}$, the sample mean of ${\mathbf{X}}=({\mathbf{u}}_{1},{\mathbf{u}}_{2},{\mathbf{u}}_{3})$, as in p. 290 of Mardia & Jupp (2000). If $K=D_{2}$ then $d=d_{1}$, where $d_{1}$ is one of the measures of dispersion defined in §2.3 of Arnold & Jupp (2013).

4 Tests of uniformity

4.1 A simple test

The uniform distribution on $SO(3)/K$ is the unique distribution that is invariant under the action of $SO(3)$ on $SO(3)/K$ in which ${\mathbf{V}}$ in $SO(3)$ maps $[{\mathbf{U}}]$ to $[{\mathbf{V}}{\mathbf{U}}]$. Since the embeddings ${\mathbf{t}}$ were chosen so that ${\rm E}\left\{{\mathbf{t}}([{\mathbf{U}}])\right\}={\mathbf{0}}$ for ${\mathbf{U}}$ uniformly distributed on $SO(3)/K$, it is intuitively reasonable to reject uniformity if ${\bar{\mathbf{t}}}$ is far from ${\mathbf{0}}$, i.e. if $n\left\|{\bar{\mathbf{t}}}\right\|^{2}$ is large. Significance can be assessed using simulation from the uniform distribution. For large samples, the following asymptotic result can be used.

Proposition 1

Given a random sample on $SO(3)/K$, define $S$ by*

[TABLE]

where $\rho^{2}$ and $d$ are given by (1) and (2), respectively, and $\nu$ is the dimension of $E$.

1. (i)

For $K=C_{1},D_{r}$ with $r\geq 2$, $T,O$ or $Y$, under uniformity, the asymptotic distribution of $S$ is $S\sim\chi_{\nu}^{2}$, as $n\rightarrow\infty$. 2. (ii)

For $K=C_{2}$,

[TABLE]

where $S_{R}=3n{\bar{R}}^{2}$ is the Rayleigh statistic for uniformity of ${\mathbf{u}}_{0}$ and $S_{B}=(15/2)n\left\{\mathrm{tr}({\bar{\mathbf{T}}}^{2})-(1/3)\right\}$ is the Bingham statistic for uniformity of $\pm{\mathbf{u}}_{1}$, ${\bar{R}}$ being the mean resultant length of ${\mathbf{u}}_{0}$ and ${\bar{\mathbf{T}}}$ being the sample scatter matrix of $\pm{\mathbf{u}}_{1}$. Under uniformity, $S_{R}$ and $S_{B}$ are asymptotically independent with asymptotic distributions $\chi_{3}^{2}$ and $\chi_{5}^{2}$, respectively. 3. (iii)

For $K=C_{r}$ with $r\geq 3$,

[TABLE]

where the subscripts $C$ and $D$ refer respectively to $C_{r}$-frames and the corresponding $D_{r}$-frames obtained by replacing the directed normal to the plane of a $C_{r}$-frame by the undirected normal, and $S_{R}=3n{\bar{R}}^{2}$ is the Rayleigh statistic for uniformity of ${\mathbf{u}}_{0}$. Under uniformity, $S_{R}$ and $S_{D}$ are asymptotically independent with asymptotic distributions $\chi_{3}^{2}$ and $\chi_{\nu_{D}}^{2}$, respectively.

Values of $\rho^{2}$ and $\nu$ are given in Table 4. In the case $K=C_{1}$, $S$ is the Rayleigh statistic (Mardia & Jupp, 2000, p. 287) for testing uniformity on $SO(3)$. In the case $K=D_{2}$, $S$ is the statistic given in §3 of Arnold & Jupp (2013) for testing uniformity on $O(3)/\mathbb{Z}_{2}^{3}$.

4.2 Some consistent tests

The test of uniformity based on $S$ is consistent only against alternatives for which ${\rm E}\{{\mathbf{t}}([{\mathbf{U}}])\}$ is non-zero. For example, in any equal mixture of two frame cardioid distributions with densities (8) having concentrations $\kappa$ and $-\kappa$, $E\{{\mathbf{t}}([{\mathbf{U}}])\}={\mathbf{0}}$, and so, in asymptotically large samples, $S$ cannot distinguish between such mixtures and the uniform distribution.

Tests of uniformity on $SO(3)/K$ that are consistent against all alternatives can be obtained as follows by averaging over $K$ Prentice’s generalization to $\mathbb{R}P^{3}$ of Giné’s $G_{n}$ test of uniformity; see §4 of Prentice (1978) and §13.2.2 of Mardia & Jupp (2000). Given $[{\mathbf{U}}_{1}],\dots,[{\mathbf{U}}_{n}]$ in $SO(3)/K$, with representatives ${\mathbf{U}}_{1},\dots,{\mathbf{U}}_{n}$ in $SO(3)$, put

[TABLE]

cf. the construction in §2 of Jupp & Spurr (1983). Uniformity is rejected if $T_{G}$ is large compared with the randomization distribution obtained by replacing ${\mathbf{U}}_{1},\dots,{\mathbf{U}}_{n}$ by ${\mathbf{R}}_{1}{\mathbf{U}}_{1},\dots,{\mathbf{R}}_{n}{\mathbf{U}}_{n}$, where ${\mathbf{R}}_{1},\dots,{\mathbf{R}}_{n}$ are independent random rotations obtained from the uniform distribution on $SO(3)$. The orthorhombic case, $K=D_{2}$, is considered in §3 of Arnold & Jupp (2013). It follows from Theorem 3.1 of Jupp & Spurr (1983) and the consistency of Giné’s test on $\mathbb{R}P^{3}$ (Mardia & Jupp, 2000, p. 289) that the test based on $T_{G}$ is consistent against all alternatives. More general Sobolev statistics on $SO(3)/K$ can be obtained from Sobolev statistics on $SO(3)$ by averaging over $K$, as in (4).

Permutational multi-sample tests, tests of symmetry, tests of independence, and goodness-of-fit tests for symmetric frames can be obtained by applying the machinery of Wellner (1979), Jupp & Spurr (1983), Jupp & Spurr (1985) and Jupp (2005), respectively, to the embedding ${\mathbf{t}}$. These tests of independence are considered in §7.

5 Distributions on $SO(3)/K$

5.1 A general class of distributions

An appealing class of distributions on $SO(3)/K$ consists of those with densities of the form

[TABLE]

where $g\left(\cdot;\kappa\right)$ is a suitable known function and $[{\mathbf{M}}]\in SO(3)/K$. The parameter $[{\mathbf{M}}]$ measures location and $\kappa$ measures concentration. If $g\left(\cdot;\kappa\right)$ is a strictly increasing function, as in (6) or León et al. (2006) with $\kappa>0$, then the mode is $[{\mathbf{M}}]$.

In the case $K=C_{1}$, the densities (5) depend on ${\mathbf{U}}$ only through $\mathrm{trace}({\mathbf{U}}{\mathbf{M}}^{\mathrm{\scriptscriptstyle T}})$ and the axes and the rotation angles of the random rotations are independent, with the axes being uniformly distributed. These distributions were introduced by Bingham et al. (2009) under the name of uniform axis-random spin distributions and by Hielscher et al. (2010) under the name of radially symmetric distributions. For $K\neq C_{1}$, elements of $SO(3)/K$ do not have well-defined axes and, in general, the distributions on $SO(3)$ with densities ${\tilde{f}}$ of the form ${\tilde{f}}({\mathbf{U}})=f([{\mathbf{U}}];[{\mathbf{M}}],\kappa)$ do not have uniformly distributed axes.

Taking $g(x;\kappa)$ proportional to $e^{\kappa x}$ in (5) gives the densities of the form

[TABLE]

For $\kappa>0$, the mode is $[{\mathbf{M}}]$ and the maximum likelihood estimate of $[{\mathbf{M}}]$ is the sample mean. The family (6) is a subfamily of the crystallographic exponential family introduced by Boogaart (2002, §3.2). For $K=C_{1}$, (6) is the density of the matrix Fisher distribution with parameter matrix $\kappa{\mathbf{M}}$ and $c(\kappa)={{}_{0}F_{1}}(3/2,(\kappa^{2}/4){\mathbf{I}}_{3})$ (Mardia & Jupp, 2000, §13.2.3). For $K=D_{2}$, (6) is the density of the equal concentration frame Watson distributions considered in Arnold & Jupp (2013, §6.1). Taking $g(x;\kappa)$ proportional to $\left(1+x\right)^{\kappa}$ in (5) gives the densities of the form

[TABLE]

For $K=C_{1}$, these densities are those of the de la Vallée Poussin distributions introduced by Schaeben (1997), and, under the name of Cayley distributions, by León et al. (2006).

Taking $g(x)=1+\kappa x$ with $0\leq\kappa\leq\rho^{-2}$ in (5) gives the densities

[TABLE]

of the frame cardioid distributions, which are analogous to the cardioid distributions on the circle (Mardia & Jupp, 2000, §3.5.5). Useful estimators of $[{\mathbf{M}}]$ and $\kappa$ in (8) are the moment estimators, $[{\hat{\mathbf{M}}}]$ and $\hat{\kappa}$, where $[{\hat{\mathbf{M}}}]$ is the sample mean defined in §3.2 and $\hat{\kappa}=(1-1/n)^{-1}s^{-2}\langle\bar{{\mathbf{t}}},{\mathbf{t}}([{\hat{\mathbf{M}}}])\rangle$ with $s^{2}$ the sample variance of $\langle{\mathbf{t}}([{\mathbf{U}}]),{\mathbf{t}}([{\hat{\mathbf{M}}}])\rangle$.

Distributions on $SO(3)/K$ can be identified with distributions on $SO(3)$ that are invariant under the action of $K$. One way of generating such distributions is to average a given distribution on $SO(3)$ over $K$. This averaging construction has been used by Walsh et al. (2009) in the orthorhombic case, Du et al. (2016) in the cubic case, and by Matthies (1982), Gorelova et al. (2014) and Niezgoda et al. (2016) in the general crystallographic case. Because the parameters of the distributions (5) are readily interpretable and the distributions (6), being exponential models, have pleasant inferential properties, we find these models more useful than many models obtained by averaging over $K$, especially as the latter can be quite demanding numerically.

5.2 Concentrated distributions

A standard coordinate system on $SO(3)$ is given by the inverse of a restriction of the exponential map ${\mathbf{S}}\mapsto\sum_{k=0}^{\infty}(k!)^{-1}{\mathbf{S}}^{k}$ from the space of skew-symmetric $3\times 3$ matrices to $SO(3)$. This can be modified to provide coordinate systems on $SO(3)/K$. Let $[{\mathbf{M}}]$ be an element of $SO(3)/K$. There are neighbourhoods $\mathcal{N}_{[{\mathbf{M}}]}$ of $[{\mathbf{M}}]$ in $SO(3)/K$ and $\mathcal{V}$ of ${\mathbf{0}}$ in $\mathbb{R}^{3}$ such that each $[{\mathbf{U}}]$ in $\mathcal{N}_{[{\mathbf{M}}]}$ can be written uniquely as $[{\mathbf{U}}]=[{\mathbf{M}}\exp\left\{{\mathbf{A}}({\mathbf{v}})\right\}]$, where

[TABLE]

with ${\mathbf{v}}=(v_{1},v_{2},v_{3})^{\mathrm{\scriptscriptstyle T}}$ in $\mathcal{V}$. Define $p_{[{\mathbf{M}}]}$ from $\mathcal{N}_{[{\mathbf{M}}]}$ to $\mathcal{V}$ by $p_{[{\mathbf{M}}]}([{\mathbf{U}}])={\mathbf{v}}$, where $[{\mathbf{U}}]=[{\mathbf{M}}\exp\left\{{\mathbf{A}}({\mathbf{v}})\right\}]$. Then $p_{[{\mathbf{M}}]}$ is a coordinate system on $\mathcal{N}_{[{\mathbf{M}}]}$. Second-order Taylor expansion about ${\mathbf{0}}$ of $[{\mathbf{U}}]$ as a function of ${\mathbf{v}}$, together with some computer algebra, gives the high-concentration asymptotic distribution of $[{\mathbf{U}}]$.

Proposition 2

For $[{\mathbf{U}}]$ near $[{\mathbf{M}}]$ in $SO(3)/K$ put $[{\mathbf{U}}]=[{\mathbf{M}}\exp\left\{{\mathbf{A}}({\mathbf{v}})\right\}]$ for ${\mathbf{v}}$ near ${\mathbf{0}}$ in $\mathbb{R}^{3}$. If $[{\mathbf{U}}]$ has density (6) with ${\mathbf{t}}={\mathbf{t}}_{K}$ as in Table 2 then the asymptotic distribution of $\kappa^{1/2}{\mathbf{v}}$ as $\kappa\rightarrow\infty$ is normal with mean ${\mathbf{0}}$ and variance $\Sigma$, where $\Sigma$ is given in Table 5. If $[{\mathbf{U}}]$ has density (7) with ${\mathbf{t}}={\mathbf{t}}_{K}$ then $(\kappa/2)^{1/2}{\mathbf{v}}$ has this asymptotic distribution. *

6 Tests of location

6.1 One-sample tests

Let $[{\mathbf{M}}]$ be an element of $SO(3)/K$ which is some measure of location of a distribution on $SO(3)/K$. There are various tests of the null hypothesis that $[{\mathbf{M}}]=[{\mathbf{M}}_{0}]$, where $[{\mathbf{M}}_{0}]$ is a given element of $SO(3)/K$. The case with $K=D_{2}$ was considered by Arnold & Jupp (2013, §8).

Permutation tests can be based on the following symmetries of $SO(3)/K$: For ${\mathbf{R}}$ in $K$, define $\rho_{[{\mathbf{M}}_{0}]}({\mathbf{R}})$ as the transformation that takes $[{\mathbf{U}}]$ to $\rho_{[{\mathbf{M}}_{0}]}({\mathbf{R}})[{\mathbf{U}}]=[{\mathbf{M}}_{0}{\mathbf{R}}{\mathbf{M}}_{0}^{\mathrm{\scriptscriptstyle T}}{\mathbf{U}}]$. Then $\rho_{[{\mathbf{M}}_{0}]}({\mathbf{R}})$ is well-defined and preserves $[{\mathbf{M}}_{0}]$.

For a sample summarized by the sample mean ${\bar{\mathbf{t}}}$ of ${\mathbf{t}}$, an appealing measure of the squared distance between the sample and $[{\mathbf{M}}_{0}]$ is $\|{\bar{\mathbf{t}}}-{\mathbf{t}}([{\mathbf{M}}_{0}])\|^{2}$. It is appropriate to reject the null hypothesis for large values of $\|{\bar{\mathbf{t}}}-{\mathbf{t}}([{\mathbf{M}}_{0}])\|^{2}$. If the distribution of $[{\mathbf{U}}]$ is symmetric under $\rho_{[{\mathbf{M}}_{0}]}$ then significance can be assessed by comparing the observed value of $\|{\bar{\mathbf{t}}}-{\mathbf{t}}([{\mathbf{M}}_{0}])\|^{2}$ with its randomization distribution, which can be obtained by replacing $[{\mathbf{U}}_{1}],\dots,[{\mathbf{U}}_{n}]$ by $\rho_{[{\mathbf{M}}_{0}]}({\mathbf{R}}_{1})[{\mathbf{U}}_{1}],\dots,\rho_{[{\mathbf{M}}_{0}]}({\mathbf{R}}_{n})[{\mathbf{U}}_{n}]$, where ${\mathbf{R}}_{1},\dots,{\mathbf{R}}_{n}$ are independent and distributed uniformly on $K$.

If $[{\mathbf{U}}_{1}],\dots,[{\mathbf{U}}_{n}]$ is a sample from a concentrated distribution with density (6) and mode $[{\mathbf{M}}]$ then it is sensible to test $H_{0}:[{\mathbf{M}}]=[{\mathbf{M}}_{0}]$ by applying Hotelling’s 1-sample $T^{2}$ test to $p_{[{\mathbf{M}}_{0}]}([{\mathbf{U}}_{1}]),\dots,p_{[{\mathbf{M}}_{0}]}([{\mathbf{U}}_{n}])$, where $p_{[{\mathbf{M}}_{0}]}$ is the projection onto the tangent space given in §5.2.

6.2 Two-sample tests

Suppose that two independent random samples $[{\mathbf{U}}_{1}],\dots,[{\mathbf{U}}_{n}]$ and $[{\mathbf{V}}_{1}],\dots,[{\mathbf{V}}_{m}]$ on $SO(3)/K$ are summarized by the sample means ${\bar{\mathbf{t}}}_{1}$ and ${\bar{\mathbf{t}}}_{2}$ of ${\mathbf{t}}([{\mathbf{U}}_{1}]),\dots,{\mathbf{t}}([{\mathbf{U}}_{n}])$ and ${\mathbf{t}}([{\mathbf{V}}_{1}]),\dots,{\mathbf{t}}([{\mathbf{V}}_{m}])$. Then the squared distance between the two samples can be measured by $\|{\bar{\mathbf{t}}}_{1}-{\bar{\mathbf{t}}}_{2}\|^{2}$. It is appropriate to reject the null hypothesis that the parent populations are the same if $\|{\bar{\mathbf{t}}}_{1}-{\bar{\mathbf{t}}}_{2}\|^{2}$ is large. Significance can be assessed by comparing the observed value of $\|{\bar{\mathbf{t}}}_{1}-{\bar{\mathbf{t}}}_{2}\|^{2}$ with its randomization distribution, obtained by sampling from the potential values corresponding to the partitions of the combined sample into samples of sizes $n$ and $m$.

Suppose that $[{\mathbf{U}}_{1}],\dots,[{\mathbf{U}}_{n}]$ and $[{\mathbf{V}}_{1}],\dots,[{\mathbf{V}}_{m}]$ are samples from concentrated distributions with density (6) on $SO(3)/K$. Let $[{\dot{\mathbf{M}}}]$ be the maximum likelihood estimate of the mode $[{\mathbf{M}}]$ under the null hypothesis that the parent populations are the same. Then the null hypothesis can be tested by applying Hotelling’s 2-sample $T^{2}$ test to $p_{[{\dot{\mathbf{M}}}]}([{\mathbf{U}}_{1}]),\dots,p_{[{\dot{\mathbf{M}}}]}([{\mathbf{U}}_{n}])$ and $p_{[{\dot{\mathbf{M}}}]}([{\mathbf{V}}_{1}]),\dots,p_{[{\dot{\mathbf{M}}}]}([{\mathbf{V}}_{m}])$, where $p_{[{\dot{\mathbf{M}}}]}$ is the projection onto the tangent space given in §5.2.

7 Independence, regression and misorientation

7.1 Independence

Let ${\mathbf{t}}_{j}:SO(3)/K_{j}\rightarrow E$, for $j=1,2$, be equivariant functions into some common inner-product space $E$ such that ${\mathbf{t}}_{j}([{\mathbf{U}}])$ has expectation ${\mathbf{0}}$ if $[{\mathbf{U}}]$ is uniformly distributed on $SO(3)/K_{j}$. Then association of random variables $[{\mathbf{U}}]$ on $SO(3)/K_{1}$ and $[{\mathbf{V}}]$ on $SO(3)/K_{2}$ can be measured in terms of association of ${\mathbf{t}}_{1}([{\mathbf{U}}])$ and ${\mathbf{t}}_{2}([{\mathbf{V}}])$.

The general approach of Jupp & Spurr (1985) leads to the following test of independence. Given pairs $([{\mathbf{U}}_{1}],[{\mathbf{V}}_{1}]),\dots,([{\mathbf{U}}_{n}],[{\mathbf{V}}_{n}])$ in $SO(3)/K_{1}\times SO(3)/K_{2}$, independence of ${\mathbf{U}}$ and ${\mathbf{V}}$ is rejected for large values of $\sum_{i=1}^{n}\sum_{j=1}^{n}\langle{\mathbf{t}}_{1}([{\mathbf{U}}_{i}]),{\mathbf{t}}_{1}([{\mathbf{U}}_{j}])\rangle\langle{\mathbf{t}}_{2}([{\mathbf{V}}_{i}]),{\mathbf{t}}_{2}([{\mathbf{V}}_{j}])\rangle$. The observed value of this statistic is compared with the randomization distribution given $[{\mathbf{U}}_{1}],\dots,[{\mathbf{U}}_{n}]$, $[{\mathbf{V}}_{1}],\dots,[{\mathbf{V}}_{n}]$. An alternative randomization test rejects independence for large values of the correlation coefficient $r$ defined in (11). For $K_{1}=K_{2}=C_{1}$ and ${\mathbf{t}}_{1}={\mathbf{t}}_{2}={\mathbf{t}}_{C_{1}}$ of Table 2, this is one of the tests considered by Rivest & Chang (2006).

7.2 Regression

A reasonable model for homoscedastic regression of $[{\mathbf{V}}]$ in $SO(3)/K_{2}$ on $[{\mathbf{U}}]$ in $SO(3)/K_{1}$ has regression function $[{\mathbf{U}}]\mapsto[{\mathbf{A}}{\mathbf{U}}]$ for some ${\mathbf{A}}$ in $SO(3)$ and error distribution that is a mild generalization of (6), so that the density of $[{\mathbf{V}}]$ given $[{\mathbf{U}}]$ is

[TABLE]

For $K_{1}=K_{2}=C_{1}$ and ${\mathbf{t}}_{1}={\mathbf{t}}_{2}={\mathbf{t}}_{C_{1}}$ of Table 2 , model (9) is a generalization of the spherical regression model of Chang (1986). It is the submodel ${\mathbf{A}}_{2}={\mathbf{I}}_{3}$ of the models with regression function ${\mathbf{U}}\mapsto{\mathbf{A}}{\mathbf{U}}{\mathbf{A}}_{2}$ that were introduced by Prentice (1989) and explored by Chang & Rivest (2001) and Rivest & Chang (2006). For $K_{1}\neq C_{1}$, it is not possible in general to extend the model (9) to have regression function of the form $[{\mathbf{U}}]\mapsto[{\mathbf{A}}{\mathbf{U}}{\mathbf{A}}_{2}$]. If $K_{1}\neq K_{2}$ then the ${\mathbf{t}}_{j}$ given in Table 2 are not suitable, since in many cases $\langle{\mathbf{t}}_{2}([{\mathbf{V}}]),{\mathbf{t}}_{1}([{\mathbf{A}}{\mathbf{U}}])\rangle=0$ for all values of $[{\mathbf{U}}]$, $[{\mathbf{V}}]$ and ${\mathbf{A}}$. Instead, it is sensible to take $E=L^{2}(SO(3))$ and the ${\mathbf{t}}_{j}$ obtained by the averaging construction described in the first paragraph of §3.1.

For $\kappa>0$, the maximum likelihood estimate of ${\mathbf{A}}$ is

[TABLE]

In general, $\hat{{\mathbf{A}}}$ is a well-defined element of $SO(3)$, rather than an element of some quotient.

Put $\rho_{12}=\max_{{\mathbf{U}}\in SO(3)}\langle{\mathbf{t}}_{1}([{\mathbf{U}}]),{\mathbf{t}}_{2}([{\mathbf{I}}_{3}])\rangle$. If $\rho_{12}>0$ then define

[TABLE]

where ${\hat{\mathbf{A}}}$ is given by (10). Then $-1\leq\,r\leq 1$ and $r$ can be regarded as a form of uncorrected sample correlation of $[{\mathbf{U}}]$ and $[{\mathbf{V}}]$. If $K_{1}=K_{2}=D_{2}$, $n=1$ and ${\mathbf{t}}_{1}={\mathbf{t}}_{2}={\mathbf{t}}$ is defined by ${\mathbf{t}}([{\mathbf{U}}])={\mathbf{U}}\,\mathrm{diag}(1,0,-1){\mathbf{U}}^{\mathrm{\scriptscriptstyle T}}$ then $r=\cos\omega$, where $\omega$ is the misorientation angle for $D_{2}$ introduced by Tape & Tape (2012). Application of Proposition 2 to the decomposition

[TABLE]

gives the following high-concentration asymptotic distributions.

Proposition 3

For $({\mathbf{U}}_{1},{\mathbf{V}}_{1}),\dots,({\mathbf{U}}_{n},{\mathbf{V}}_{n})$ from model (9),*

1. (i)

Asymptotically, for large $\kappa$,

[TABLE]

and the quantities in (12) and (13) are asymptotically independent. 2. (ii)

An approximate high-concentration $100(1-\alpha)\%$ confidence region for ${\mathbf{A}}$ is

[TABLE]

7.3 Misorientation

The relationship between ambiguous rotations $[{\mathbf{U}}]$ in $SO(3)/K_{1}$ and $[{\mathbf{V}}]$ in $SO(3)/K_{2}$ can be described by the misorientation, which is an element of the double coset space $K_{1}\mbox{\textbackslash}SO(3)/K_{2}$. Since $[{\mathbf{U}}]$ and $[{\mathbf{V}}]$ are images of ${\mathbf{U}}{\mathbf{R}}_{1}$ and ${\mathbf{V}}{\mathbf{R}}_{2}$ in $SO(3)$ for any ${\mathbf{R}}_{1}$ in $K_{1}$ and ${\mathbf{R}}_{2}$ in $K_{2}$, $({\mathbf{U}}{\mathbf{R}}_{1})^{T}{\mathbf{V}}{\mathbf{R}}_{2}$ determines a well-defined element of $K_{1}\mbox{\textbackslash}SO(3)/K_{2}$. See p. 274 of Morawiec (1997). In crystallography it is usual to identify $K_{1}\mbox{\textbackslash}SO(3)/K_{2}$ with an asymmetric domain, a neighbourhood of ${\mathbf{0}}$ in $\mathbb{R}^{3}$ that is in one-to-one correspondence, modulo null sets, with $K_{1}\mbox{\textbackslash}SO(3)/K_{2}$ under ${\mathbf{v}}\mapsto\exp\{{\mathbf{A}}({\mathbf{v}})\}$ followed by projection of $SO(3)$ to $K_{1}\mbox{\textbackslash}SO(3)/K_{2}$. Then the misorientation between $[{\mathbf{U}}]$ and $[{\mathbf{V}}]$ is taken as the element, ${\mathbf{P}}$, of the asymmetric domain that satisfies ${\mathbf{V}}={\mathbf{U}}{\mathbf{P}}$ and has smallest rotation angle among all such rotations in the domain. In the case in which the conditional distribution of $[{\mathbf{V}}]$ given $[{\mathbf{U}}]$ is uniform the distributions of the angle and axis of the misorientation are given in Morawiec (2004, Ch. 7). For general pairs $([{\mathbf{U}}_{1}],[{\mathbf{V}}_{1}]),\dots,([{\mathbf{U}}_{n}],[{\mathbf{V}}_{n}])$ in $SO(3)/K_{1}\times SO(3)/K_{2}$, we define the mean misorientation as the element $\hat{A}$ of $SO(3)$ defined in (10). An alternative definition of the mean misorientation is the element $(\hat{{\mathbf{A}}}_{1},[\hat{{\mathbf{A}}}_{2}])$ of $SO(3)\times(K\mbox{\textbackslash}SO(3))$ that maximizes $\sum_{i}\max_{{\mathbf{R}}_{i}\in K}\langle{\mathbf{t}}_{1}([{\mathbf{A}}_{1}{\mathbf{U}}_{i}]),{\mathbf{t}}_{2}([{\mathbf{V}}_{i}{\mathbf{R}}_{i}{\mathbf{A}}_{2}])\rangle$.

8 Example

To illustrate the estimators and tests introduced above, we consider some samples of orientations of diopside crystals. These crystals are monoclinic, so we can represent their orientations by $C_{2}$-frames.

The stereonet in Fig. 1 shows the $\ {\mathbf{u}}_{0}$ vectors and $\pm{\mathbf{u}}_{1}$ axes given by orientations of 100 diopside crystals. A randomization test of uniformity based on $S$ from (3) in §4.1 has $p-$value less than $0.001$, leading to decisive rejection of uniformity.

The stereonets in Fig. 2 show the $\ {\mathbf{u}}_{0}$ vectors and $\pm{\mathbf{u}}_{1}$ axes given by orientations of 34 crystals from one region of a specimen and 37 crystals from another region. The two-sample permutation test of §6.2 yields a $p$-value of $0.07$ for equality of the populations of the orientations in the two regions, so the hypothesis of equality is not rejected.

Acknowledgements

We thank David Mainprice for providing the diopside data.