inferential statistics of ebsd data from individual crystalline...

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Inferential statistics of EBSD data from individualcrystalline grains

Florian Bachmann1, Ralf Hielscher2, Peter E. Jupp3,

Wolfgang Pantleon4, Helmut Schaeben1, Elias Wegert5

1Geoscience Mathematics and Informatics,

TU Bergakademie Freiberg, Germany

2Applied Functional Analysis

TU Chemnitz, Germany

3School of Mathematics and Statistics

University of St. Andrews, Fife, Scotland

4Center for Fundamental Research

Metal Structures in Four Dimensions

Risø National Laboratory for Sustainable Energy,

Materials Research Division

Technical University of Denmark, Roskilde, Denmark

5Applied Analysis, TU Bergakademie Freiberg, Germany

(Received 0 XXXXXXX 0000; accepted 0 XXXXXXX 0000)

Abstract

Highly concentrated distributed crystallographic orientation measurements with re-

spect to individual crystalline grains are analysed here by means of ordinary statistics

neglecting their spatial reference. Since crystallographic orientations are modeled as

PREPRINT: Journal of Applied Crystallography A Journal of the International Union of Crystallography

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left cosets of a given subgroup of SO(3), non-spatial statistical analysis adapts ideas

borrowed from the Bingham quaternion distribution on S3. Special emphasis is put on

the mathematical definition and the numerical determination of a “mean orientation”

characterizing the crystallographic grain as well as on distinguishing between the case

of symmetry with respect to the mean orientation and systematic deviations like the

prolate or oblate case. Applications to simulated as well as to experimental data are

presented.

Keywords: Texture analysis, orientation distribution, individual orientation mea-

surements, EBSD data, crystallographic orientation distribution, crystallographic ori-

entation distribution per grain, highly concentrated Bingham quaternion distribution,

crystallographic symmetry, software toolbox MTEX for texture analysis

1. Introduction and motivation

The subject of this communication is the non–spatial statistical analysis of highly con-

centrated crystallographic orientations measured with respect to individual crystallo-

graphic grains. Crystallographic orientations are explicitly considered as left cosets of

crystallographic symmetry groups, i.e., classes of crystallographically symmetrically

equivalent rotations. Its objective is twofold.

Primarily, it is an attempt to clarify a lasting confusion concerning the “mean

orientation” characterizing a grain, i.e., its mathematical definition and its numerical

determination, cf. (Humbert et al., 1996; Morawiec, 1998; Barton and Dawson 2001a;

2001b; Glez and Driver, 2001; Humphreys et al., 2001; Pantleon, 2005; Cho et al.,

2005; He et al., 2008; Krog-Pedersen et al., 2009; Pantleon et al., 2008).

The second objective is to verify these theoretical findings with fabricated and ex-

perimental data using the versatile features of the free and open source Matlab R©

software toolbox MTEX (Hielscher, 2007; Hielscher and Schaeben, 2008; Schaeben et

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al., 2007) providing a unifying approach to texture analysis with individual (“EBSD”)

or integral (“pole figure”) orientation measurements. For the time being, the functions

implemented in MTEX require the assumption that the rotation or orientation mea-

surements are independent even if they are spatially indexed and likely to be spatially

dependent. Whenever this artificial assumption is essential, it is made explicit. Readers

are referred to (Boogaart, 1999; 2002) for fundamental spatial statistics of orientations,

and to forthcoming publications reporting the additional implementation of functions

in MTEX to apply spatial statistics numerically.

The next section provides preliminaries concerning quaternions, notions of descrip-

tive spherical statistics, and the relationship of individual orientation measurements

and left cosets of some rotational subgroups induced by crystallographic symmetry.

The third section applies the Bingham quaternion distribution to highly concentrated

individual orientation measurements as sampled e.g., by electron back scatter diffrac-

tion (EBSD). In particular, it is shown that quaternion multiplication of the measure-

ments by a given reference orientation acts analogously to centering the measurements

with respect to this reference orientation. Therefore, replacing the initial orientation

measurements by their corresponding disorientations, i.e., orientation differences with

respect to the reference orientation, is obsolete. The orientation tensor derived from

the initial measurements contains all ingredients necessary for the characterization

of details of the highly concentrated distribution of orientations. The final section

completes this communication with practical examples of simulated and experimental

EBSD data sets.


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2. Preliminaries

2.1. Quaternions

The group SO(3) may be considered in terms of H, the skew field of real quaternions.

In particular, a unit quaternion q ∈ S3 ⊂ H determines a rotation. For unit quaternions

the inverse equals the conjugate, q−1 = q∗. The unit quaternion q and its negative

−q describe the same rotation. Thus, it would make sense to identify the pair (q,−q)

with a unique rotation. Working with pairs may be quite cumbersome; for general

statistical purposes the proper entity to work with is the (4× 4) symmetric matrix

Q =

q2

1 q1q2 q1q3 q1q4

q1q2 q22 q2q3 q2q4

q1q3 q2q3 q23 q3q4

q1q4 q2q4 q3q4 q24

.

Thinking of q = (q1, q2, q3, q4)T as elements of S3 ⊂ R4 we could write Q = qqT.

The product of two quaternions is defined as

pq =Sc(p)Sc(q)−Vec(p) ·Vec(q)

+ Sc(p)Vec(q) + Sc(q)Vec(p) + Vec(p)×Vec(q),(1)

where Sc(p) and Vec(p) denote the scalar and vector part of the quaternion p, respec-

tively. The quaternion product can be rewritten as

pq = Lpq =

p0 −p1 −p2 −p3

p1 p0 −p3 p2

p2 p3 p0 −p1

p3 −p2 p1 p0

q , (2)

and analogously

qp = Rpq =

p0 −p1 −p2 −p3

p1 p0 p3 −p2

p2 −p3 p0 p1

p3 p2 −p1 p0

q , (3)

where the matrices Lp and Rp, respectively, have the properties

Lp∗ = LTp , Rp∗ = RT

p ,

Lp∗Lp = Lp∗p = L1 = E ,

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i.e., they are orthonormal (cf. Gurlebeck et al, 2006, p. 33).

The scalar product of two quaternions is defined as

p · q = Sc(pq∗) = Sc(p)Sc(q) + Vec(p) ·Vec(q) = cos ∠(p, q). (4)

The scalar product of the unit quaternions p and q agrees with the inner (scalar)

product of p and q when recognized as elements of S3 ⊂ R4, i.e., the scalar product

provides the canonical measure for the distance of unit quaternions (cf. Gurlebeck et

al., 2006, p. 21).

2.2. Descriptive spherical statistics

Even though spherical statistics is not the topic here, the entities considered are

borrowed from spherical statistics (Watson, 1983, Mardia and Jupp, 2000). The fol-

lowing issues have been clarified in the context of texture analysis by Morawiec (1998).

In spherical statistics a distinction is made between vectors q and pairs (q,−q) of

antipodal vectors, i.e., axes.

Vectorial data

The key summary statistic of vectorial data is the normalized sample mean vector

r =1

‖∑n

`=1 q`‖

n∑`=1

q` ,

where r provides a measure of location and ‖∑q`‖ provides a measure of dispersion

(concentration). The normalized mean vector solves the following minimization prob-

lem. Given vectorial data q1, . . . , qn ∈ S3 ⊂ R4 find r ∈ S3 minimizing the “mean


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angular” deviation. More precisely, with θ` = ∠(q`, x) the entity to be minimized is

FR(x; q1, . . . , qn) =1n

n∑`=1

(1− cos θ`) = 1− 1n

n∑`=1

cos θ`

= 1− 1n

n∑`=1

x · q` = 1− x ·( 1n

n∑`=1

q`

)= 1− x · q .

Thus, the solution x0 = r is provided by the normalized mean vector

r =q

‖q‖,

and

FR(r; q1, . . . , qn) = 1− ‖q‖

is a measure for the spherical dispersion of the data. The normalized mean vector is

also the solution of the problem of minimizing the sum of squared Euclidean distances

as

1n

n∑`=1

‖q` − x‖2 =1n

n∑`=1

(q` − x) · (q` − x) =1n

n∑`=1

(2− 2 cos θ`)

= 2FR(x; q1, . . . , qn),

cf. (Humbert et al., 1996). However, the normalized mean vector does not provide

statistical insight into a sample of quaternions used to parametrize rotations (Watson,

1983; Mardia and Jupp, 2000, §13.2.1).

Axial data

The key summary statistic of axial data is the orientation tensor

T =1n

n∑`=1

q`qT` .

Its spectral decomposition into the set of eigenvectors a1, . . . , a4 and the set of cor-

responding eigenvalues λ1, . . . , λ4 provides a measure of location and a corresponding

measure of dispersion, respectively. Since the orientation tensor T and the tensor of


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inertia I are related by

I = E − T,

where E denotes the unit matrix, the eigenvectors of T provide the principal axes of

inertia and the eigenvalues of T provide the principal moments of inertia. From the

definition of T as a quadratic form, T is positive definite, implying that all eigenvalues

are real and non–negative; moreover, they add to 1.

The spectral decomposition solves the following minimization problem. Given axial

data q1, . . . , qn ∈ S3 find a ∈ S3 minimizing the mean squared orthogonal distance.

With θ` = ∠(q`, x) as before, the entity to be minimized is

FT (x; q1, . . . , qn) =1n

n∑`=1

sin2 θ` =1n

n∑`=1

(1− cos2 θ`

)= 1− 1

n

∑cos2 θ`

= 1− 1n

n∑`=1

(xTq`

)2= 1− xT

( 1n

n∑`=1

q`qT`

)x

= 1− xTTx . (5)

Obviously, the solution x0 = a1 is provided by the eigenvector a1 of T corresponding

to the largest eigenvalue denoted λ1. Then 1−λ1 is a measure of the spherical disper-

sion of the axial data with respect to the quaternion ±a1. The eigenvalues are related

to the shape parameters of the Bingham (quaternion) distribution (Bingham, 1964;

1974; Kunze and Schaeben, 2004; 2005) by a system of algebraic equations involving

partial derivatives of the hypergeometric function 1F1 of a (4× 4) matrix argument.

In general, a single eigenvector and its eigenvalue are insufficient to provide a reason-

able characterization of the data. Therefore, often the ratios of the eigenvalues are

analyzed and interpreted.


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Normalized mean vector vs. first principal axis of inertia

It is emphasized that the mean vector for axial data is always identically zero. De-

spite this mathematical fact, it may sometimes be tempting to treat axes like vectors.

If axes are treated as vectors by some “trick”, e.g., moving all data to the “upper hemi-

sphere”, the statistical inference will generally be misleading. A uniform distribution

on the upper hemisphere will appear as preferred distribution in terms of vectors, but

as uniform distribution in terms of axes.

In the special case of a highly preferred unimodal distribution of axes, and after addi-

tional provision as sketched above, the normalized mean vector may be a sufficiently

good approximation to the eigenvector corresponding to the largest eigenvector. This

approximation may be considered sufficiently good as long as cos is considered as a

sufficiently good approximation to cos2, i.e., for (very) small angles.

In the case of highly unimodal preferred orientation, the largest eigenvalue λ1 is

much larger than the other eigenvalues and provides a first measure of the dispersion

just by itself. The other eigenvalues and their ratios provide insight into the shape of

the distribution, e.g., the extent to which it is symmetrical or not.

This special case seems to apply to individual orientation measurements within a crys-

talline grain, where the EBSD measurements should differ only very little. However,

careful attention should be paid to the way of defining individual grains, i.e., to the

way of defining grain boundaries. A common practice in defining the boundaries of

grains is based on a measurement-by-measurement comparison of orientations, i.e.,

two measurement q0 = q(x) and q1 = q(x + δx) located next to one another (sepa-

rated by a lag of size δx) are compared. If the orientation difference is smaller than a

user–defined threshold δq, the two positions x, x+ δx are assigned to the same grain.

Then q1 = q(x+ δx) and q2 = q(x+ 2δx) are compared. If the orientation difference is


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larger than the threshold, a grain boundary is constructed, e.g., (Lloyd et al., 1997),

(Heilbronner, 2000).

Determined in this way the size and the shape of a crystallographic grain depend

rather on the threshold δq defined by the user than by the amount of available sampling

controlled by the lag size δx. Thus, a grain is referred to as a definition– rather

than a sampling–limited spatial object (cf. Bonham–Carter, 1996, p. 29–30). Then

the orientation gradient along the line x, x + mδx, m ∈ N, may be large enough to

result occasionally in surprisingly large dispersions which may be prohibitive of any

reasonable interpretation in terms of “grain” properties or prevent a simple comparison

of “grains” defined in this way.

2.3. Considering crystallographic symmetry

In case of crystallographic symmetry GLaue ∩ S3 = GLaue ⊂ S3 to be considered

in terms of the restriction GLaue of the Laue group GLaue to rotations, an additional

difficulty has to be mastered, as crystal orientations that are equivalent under crystal

symmetries must be considered to determine a mean orientation. In fact, the initial

orientation measurements are of the form q`gm` , q` ∈ S3, gm` ∈ GLaue, i.e., they are

some elements of the corresponding left cosets q`GLaue.

If qi and qj are very close, qig1 and qjg2, g1, g2 ∈ GLaue, g1 6= g2, may appear to

be mathematically different though they are physically close. The practical problem

with the computation of the orientation tensor T and its spectral decomposition,

respectively, is to transform the initial elements of the sample such that they are not

only physically but also mathematically highly concentrated.

Thus, explicitly considering crystallographic symmetry, the entity to be minimized

(cf. Eq. (5)) actually is

FT (x; q1, . . . , qn, GLaue) = 1− xT( 1n

n∑`=1

(q`GLaue)(q`GLaue)T)x


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given crystallographic orientations q1, . . . , qn, and the rotational symmetry group

GLaue, i.e.,

argminx∈S3,gm`∈GLaue

{1− xT

( 1n

n∑`=1

(q`gm`)(q`gm`)T)x}

(6)

with gm` ∈ GLaue,m` = 1, . . . , s, ` = 1, . . . , n, where s denotes the cardinality of

GLaue.

While the map S3 → S3/GLaue is continuous, its inverse is not. Thus, assigning to

each measured orientation q`gm` the element of q`GLaue with the smallest angle of

rotation does not lead to a solution of Eq. (6).

Therefore, we apply the following heuristics to resolve problem Eq. (6). To this end,

an initial reference orientation is required. Given the data set q1, . . . , qn, we may choose

q1 as reference orientation. For every ` = 1, . . . , n, determine argmaxg∈GLaue

q`g · q1 =

g(0)0,` ∈ GLaue and replace q` by q(0)

` = q`g(0)0,` . Then we evaluate FT (x; q(0)

1 , . . . , q(0)n ) for

determining the largest eigenvalue λ(0)1 and its corresponding eigenvector a(0)

1 .

Then we choose a(0)1 as the reference orientation, determine for every ` = 1, . . . , n,

argmaxg∈GLaue

q(0)` g · a(0)

1 = g(1)0,` ∈ GLaue, and replace q

(0)` by q

(1)` = q

(0)` g

(1)0,` . Then

we evaluate FT (x; q(1)1 , . . . , q

(1)n ) for determining the largest eigenvalue λ

(1)1 and its

corresponding eigenvector a(1)1 .

Now let i = 1, . . . , I, enumerate the successive iterations. For every ` = 1, . . . , n, we

replace q(i−1)` by q(i)

` . Next we evaluate FT (x; q(i)1 , . . . , q

(i)n ), and determine the largest

eigenvalue λ(i)1 and its corresponding eigenvector a

(i)1 , which will be the reference

orientation in the next step of this iterative approximation. We continue until λ(i)1

does not increase any longer and its corresponding eigenvector a(i)1 does not change

any more.

This iterative procedure to determine the proper eigenvector corresponding to the

largest eigenvalue is similar to the one suggested in (Pantleon, 2005; Pantleon et al.,

2008; He et al., 2008; Krog-Pedersen et al. 2009) to be applied to the normalizedIUCr macros version 2.0β5: 2001/06/20

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mean vector. It converges only if the measurements are sufficiently well clustered,

i.e., if their dispersion is sufficiently small, and if it is appropriately initialized. The

modal orientation of the kernel-estimated orientation density function may be more

appropriate than an arbitrary measured orientation.

3. Characterizing sets of highly preferred orientation

Neglecting this problem of how to define grains and their boundaries, respectively, the

mixture and confusion about the normalized mean vector and the spectral decompo-

sition can be avoided if it is assumed that

• the measurements q1, . . . , qn are highly preferred, i.e., the dispersion with respect

to an individual grain is small, and

• a mean orientation a1 has been assigned to the grain by means of the eigenvector

a1 corresponding to the largest eigenvector λ1 according to Eq. (5), which has

been interpreted itself as a measure of the extent of preferred orientation, i.e.,

(λ1, a1) are no longer the statistics of primary interest.

Then the focus is on (λ`, a`), ` = 2, 3, 4, and their ratios, respectively. Therefore the

spectral decomposition of the symmetric positive definite matrix T is considered in

detail and related to the spectral decomposition suggested earlier (Pantleon, 2005;

Pantleon et al., 2008; He et al., 2008; Krog-Pedersen et al. 2009).

If T is a (4× 4) positive definite symmetric matrix, then the decomposition

T = V ΛV T

exists where

Λ =

λ1 0 0 00 λ2 0 00 0 λ3 00 0 0 λ4


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is the (4×4) diagonal matrix of eigenvalues λ1, . . . , λ4, and where V = (a1, a2, a3, a4) is

the (4× 4) matrix comprising the corresponding column eigenvectors a1, . . . , a4 ∈ R4.

Equivalently,

V TTV = Λ .

Here, the matrix T is given in terms of a set of unit vectors q1, . . . , qn ∈ S3 ⊂ R4 as

T =1n

n∑`=1

q`qT` =

1nQnQ

Tn ,

where Qn is the (4 × n) matrix composed of column vectors q1, . . . , qn. Hence, T is

symmetric and positive definite.

Let p∗ denote the conjugate of an arbitrary quaternion p. Considering the set

q1p∗, . . . , qnp

∗ ∈ S3, i.e., the set of disorientations with respect to p, and the cor-

responding

T ∗ =1n

n∑`=1

(q`p∗)(q`p∗)T =1n

n∑`=1

(Rp∗q`)(Rp∗q`)T = Rp∗ T RTp∗ , (7)

the decomposition is

V TT ∗V = (Rp∗V )T (Rp∗ T RTp∗) (Rp∗V ) = V TTV = Λ (8)

with V = RTp∗V composed of column eigenvectors a` = a`p

∗, ` = 1, . . . , 4, with respect

to eigenvalues λ`, ` = 1, . . . , 4. The analogue is true for the multiplication with p∗ from

the left.

In particular, if p = a1, then a1 = (1, 0, 0, 0)T, i.e., multiplication with a∗1 from the

right acts like centering the measurements at the identity rotation (1, 0, 0, 0)T. In this

case V is of the form

V =

1 0 0 00 w11 w21 w31

0 w12 w22 w32

0 w13 w23 w33

=(

1 0T

0 W

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with the (3× 3) matrix W composed of the (3× 1) column vectors

w` = a`+1a∗1|R3 , ` = 1, 2, 3, (9)

where a`+1a∗1 = a`+1 are restricted to R3 by omitting the zero scalar part of the

quaternions a`+1, ` = 1, 2, 3. Then

V T =(

1 0T

0 WT

).

Accordingly, T ∗ may be rewritten explicitly as

T ∗ =

t∗11 t∗12 t∗13 t∗14

t∗21 t∗22 t∗23 t∗24

t∗31 t∗32 t∗33 t∗34

t∗41 t∗42 t∗43 t∗44

=

t∗11 t∗12 t∗13 t∗14

t∗21

t∗31 T ∗

t∗41

with

T ∗ =

t∗22 t∗23 t∗24

t∗32 t∗33 t∗34

t∗42 t∗43 t∗44

.

The (3× 3) lower right submatrix T ∗ of T ∗ corresponds to the matrix Q (Eq. (13)

of Pantleon, 2005).

Eventually

V TT ∗V =t∗11

∑t∗1,j+1 w1,j

∑t∗1,j+1 w2,j

∑t∗1,j+1 w3,j∑

t∗i+1,1 w1,i∑t∗i+1,2 w2,i WTT ∗W∑t∗i+1,3 w3,i

= Λ

or, equivalently

WTT ∗W = Λ =

λ2 0 00 λ3 00 0 λ4

.

Thus, if a`, ` = 1, . . . , 4, are the eigenvectors of T with respect to eigenvalues λ1, . . . , λ4,

then w` = a`+1a∗1|R3 , ` = 1, 2, 3, Eq. (9), are the eigenvectors of the (3× 3) lower right

submatrix T ∗ of T ∗ with respect to eigenvalues λ2, λ3, λ4.


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4. Inferential statistics

4.1. Inferential statistics with respect to the Bingham quaternion distribution

Neglecting the spatial dependence of EBSD data from individual crystalline grains,

the Bingham quaternion distribution

f(±q;A) =(

1F1(12, 2, A)

)−1exp(qTAq)

with the hypergeometric function 1F1(12 , 2, ◦) of matrix argument seems appropriate

(Bingham, 1964; 1974; Schaeben, 1993; Kunze and Schaeben, 2004; 2005). It is em-

phasized that the densities f(±q;A) and f(±q;A + tI), with t ∈ R and the (4 × 4)

identity matrix I, define the same distribution. Furthermore, these densities form a

transformation model under the group O(4). For U ∈ O(4)

f(±Uq;UAUT) = f(±q;A)

implying that

1F1(12, 2, UAUT) = 1F1(

12, 2, A).

In particular, if

A = UTKU

with U orthonormal and K = diag(κ1, . . . , κ4), then

1F1(12, 2, A) = 1F1(

12, 2,K)

and, therefore, the estimates of the parameters of the Bingham quaternion distribution

based on the sample q1, . . . , qn are given by

U = V T, (10)

∂ log 1F1(12 , 2,K)

∂κi

∣∣∣K=K

= λi, i = 1, . . . , 4. (11)

It should be noted that Eq. (11) determines κi, i = 1, . . . , 4, only up to an additive con-

stant, because κi and κi+t, i = 1, . . . , 4, result in the same λi, i = 1, . . . , 4. Uniqueness

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Provided that κ1 > κ2, κ3, κ4 calculation shows that

qTAq = qTUTKUq = xTKx = κ1 −12

4∑j=2

x2j

σ2j

,

where x = (x1, x2, x3, x4)T = Uq and σ2j = (2[κ1 − κj ])−1 for j = 2, 3, 4. Thus if q has

the Bingham distribution with parameter matrix A and κ1 − κj → ∞ for j = 2, 3, 4,

then ±x tends to ±(1, 0, 0, 0)T and the distribution of (x2/σ2, x3/σ3, x4/σ4)T tends

to that of three independent standard normal random variables. Then for j = 2, 3, 4,

λj ' E[x2j ] ' σ2

j , and so

κj − κ1 ' −(2λj)−1. (12)

Since we expect that κ1−κj and κ1− κj , j = 1, . . . , 4, are very large, the interesting

statistical issue is to test rotational symmetry, i.e., to test the null hypothesis of

spherical symmetry that κ2 = κ3 = κ4. If this hypothesis can be rejected we would

be interested in distinguishing the “prolate” case κ2 > κ3 = κ4 and the “oblate” case

κ2 = κ3 > κ4, respectively.

Assuming rotational symmetry, i.e., degeneracy of the Bingham to the Watson dis-

tribution, the test statistic

TBinghams = n

4∑`=2

(κ` − κ)(λ` − λ) ∼ χ25, n→∞, (13)

where

κ =13

4∑`=2

κ`, λ =13

4∑`=2

λ`,

(Mardia and Jupp, 2000, p. 234). For a given sample of individual orientation mea-

surements q1, . . . , qn we compute the value ts and the corresponding

p = Prob(Ts > ts),

and conclude that the null hypothesis of rotational symmetry may be rejected for any

significance level α > p.


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If the hypothesis of spherical symmetry is rejected, we test for prolateness using

TBinghamp =

n

2(κ3 − κ4)(λ3 − λ4) ∼ χ2

2, n→∞, (14)

or, analogously, for oblateness using

TBinghamo =

n

2(κ2 − κ3)(λ2 − λ3) ∼ χ2

2, n→∞, (15)

respectively.

According to Eq. (12), for 1 ≈ λ1 >> λ2 ≥ λ3 ≥ λ4 we may apply the asymptotics

κj ' κj = −(2λj)−1 (j = 2, 3, 4), (16)

assuming κ1 ' κ1 = 0. Then the test statistics TBingham simplify to

TBinghams ' T asymp Bingham

s =n

6

∑i=2,3,4

∑j=2,3,4

(λiλj− 1)

(17)

TBinghamp ' T asymp Bingham

p =n

4

(λ3

λ4+λ4

λ3− 2)

(18)

TBinghamo ' T asymp Bingham

o =n

4

(λ2

λ3+λ3

λ2− 2). (19)

Note that three summands in Eq. (17) vanish and that the λj are arranged in de-

creasing order. These formulas agree well with the common sense interpretation of a

spherical, prolate and oblate shape, respectively.

4.2. Inferential statistics without parametric assumptions

Dropping the assumption that ±q has a Bingham distribution and applying large

sample approximation the tedious numerics with respect to 1F1 can be avoided; only

cij =1n

n∑`=1

(qT` ai)2(qT` aj)

2

is required, where ai, i = 1, . . . , 4, are the eigenvectors of T corresponding to the

eigenvalues λ1 ≥ λ2 ≥ λ3 ≥ λ4. Analogously to results by (Prentice, 1984, §6; 1986,


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§5; Mardia and Jupp, 2000, §10.7.2) the test statistics are

for the spherical case

Ts = 15nλ2

2 + λ23 + λ2

4 − (1− λ1)2)/32(1− 2λ1 + c11)

∼ χ25, n→∞, (20)

for the prolate case

Tp = 8nλ2

3 + λ24 + (1− λ1 − λ2)2/2

2(1− 2(λ1 + λ2) + c11 + 2c12 + c22)∼ χ2

2, n→∞, (21)

and for the oblate case

To = 8nλ2

2 + λ23 + (1− λ1 − λ4)2/2

2(1− 2(λ1 + λ4) + c11 + 2c14 + c44)∼ χ2

2, n→∞. (22)

For details concerning inferential statistics the reader is referred to the Appendix

A. Some details of the computation and asymptotics are given in Appendix B.

5. Practical applications

Several samples of individual orientation measurements are analyzed. The first sample

referred to as simIOM is a set of simulated measurements, the following three samples

are actual measurements on oxygen-free high-conductivity copper, deformed to 25%

in tension (Krog-Pedersen et al., 2009).

The data are displayed as

• an orientation map applying the colour coding given be the inverse pole figure

colour bar as shown in Figure 1,

• and applying RGB colour coding of Euler angles (ϕ1, φ, ϕ2) according to Bunge’s

zxz convention, where the Euler angle ϕ1 is associated with red, φ with green,

and ϕ2 with blue,

• a 3d scatter plot with respect to axis–angle parametrisation,


18

• pole points q` GLaue h G∗Laue q

∗` , ` = 1, . . . , n, for some crystallographic forms

h ∈ S2, i.e., crystallographic directions and their symmetrically equivalents

GLaue h G∗Laue, and

• plots of the orientation density and

• pole density function.

Fig. 1. Inverse pole figure colour bar

Here, orientation maps are used for a rough indication of spatial dependence only.

The orientation density function is computed by non–parametric kernel density esti-

mation with the de la Vallee Poussin kernel, and corresponding pole density functions

are determined as superpositions of the totally geodesic Radon transform of the de la

Vallee Poussin kernel.

In all cases the orientation tensor T and its eigenvalues and eigenvectors are com-

puted, and the shape parameters are derived. Then the spherical, oblate, and prolate

case, respectively, are tested, if reasonable.

All computations have been done with our free and open Matlab R© toolbox MTEX

3.0, which can be downloaded from http://code.google.com/p/mtex/.

5.1. Simulated EBSD data

The first data set was fabricated by simulation of 900 (pseudo-)independent spa-

tially indexed individual orientation measurements from a given Bingham quaternion

distribution with modal orientation qmodal = (0.78124, 0.26042, 0.15035, 0.54703)


19

((65, 35, 5) in Euler angles according to Bunge’s zxz convention), and dispersion pa-

rameters K = diag(340, 0, 0, 0) in a 30× 30 grid. The simulated data are displayed as

orientation maps and as a 3d scatter plot with respect to axis–angle parametrisation

in Figure 2, as pole point plots of crystallographic forms GLaueh (100), (110), (111),

and (113) in Figure 3, and as pole and orientation densities in Figure 4

Fig. 2. 900 simulated spatially indexed individual orientations from theBingham quaternion distribution with modal orientation qmodal =(0.78124, 0.26042, 0.15035, 0.54703) and dispersion parameters K =diag(340, 0, 0, 0) in a 30 × 30 grid as colour coded orientation map accordingto inverse pole figure colour bar (left) and RGB colours (center), respectively, andas 3d scatter plot (right).

Fig. 3. Pole point plots of 900 simulated spatially indexed individual orientations fromthe Bingham quaternion distribution for crystallographic forms {100}, {110}, {111},and {113}, augmented with the mean orientation.


20

Fig. 4. Orientation density function (left) and pole density functions (right) of 900 sim-ulated spatially indexed individual orientations according to the Bingham quater-nion distribution for crystallographic forms {100}, {110}, {111}, and {113}, andorientation density function (center), augmented with the mean orientation.

For these simulated data the estimated texture index I = 61.2971, and entropy

E = −3.6491. The eigenvalues are for i = 1, . . . , 4, λi = 0.9954, 0.0016, 0.0015, 0.0014,

respectively, and their ratios are λ2/λ3 = 1.1043, λ3/λ4 = 1.0517.

5.2. Experimental EBSD data

Experimental orientation data are collected on a deformed oxygen-free high conduc-

tivity (OFHC) copper with a purity of 99.99 %. The starting material is recrystallized

to a grain size (mean linear intercept length excluding twin boundaries obtained by

EBSD) of about 30 µm. A cylindrical dog-bone ASTM standard tensile specimen with

a gauge area of ø4 mm×20 mm is prepared by spark-cutting and loaded in tension on

an INSTRON testing machine with a cross-head speed of 0.5 mm min−1 till failure.

The ultimate tensile strength is reached at 25 % strain. Part of the gauge section of

the deformed specimen has been cut out and half the sample is ground away so that

the longitudinal midsection is exposed. After further polishing, the sample is finally

electro-polished in preparation for the EBSD investigation. Using a scanning electron

microscope Zeiss SUPRA 35 equipped with a field emission gun and a Nordlys2 de-

tector, lattice orientations on a central section of the specimen (comprising the tensile


21

direction) are recorded with Channel5 software from HKL technology. The investi-

gated regions are well outside the neck and only the homogeneously deformed part is

analyzed.

Orientations are determined on the longitudinal midsection in a square grid of

1100 × 800 points with a distance of 1 µm between measuring points. The disorien-

tations between all neighbouring points are calculated taking into account the cubic

symmetry of the face-centred cubic crystalline lattice of copper. Grains are identified

as contiguous areas fully surrounded by boundaries with disorientation angles larger

than 7◦ separating them from neighbouring grains. Three grains are selected to repre-

sent different shapes of orientation distributions; their orientation maps are presented

in Figure 5, Figure 8, and Figure 11 displaying the tensile axis vertically. Due to the

medium staking-fault energy of copper, many twins are present which can be recog-

nized in the orientation maps as twin lamellae (e.g., the horizontal white cut in the

grain shown in Figure 8).

Next we shall analyse the distribution of crystallographic orientations of three in-

dividual grains, labeled grain 40, grain 147, and grain 109.

Data of grain 40 are depicted as colour coded orientation maps and as 3d scatter

plot in Figure 5, as pole point plots in Figure 6, and pole density and orientation

densities in Figure 7.

Fig. 5. Grain 40 with 3068 spatially indexed individual orientations as colour codedorientation map according to inverse pole figure colour bar (left) and RGB colours(center), respectively, and as 3d scatter plot (right).


22

Fig. 6. Pole point plots of grain 40 for crystallographic forms {100}, {110}, {111}, and{113}, augmented with the mean orientation.

Fig. 7. Orientation density function (left) and pole density functions (right) ofspatially indexed individual orientations of grain 40 for crystallographic forms{100}, {110}, {111}, and {113}, augmented with the mean orientation.

In the same way, data of grain 147 are depicted in Figure 8, Figure 9, and Figure 10.



23

Fig. 9. Pole point plots of grain 147 for crystallographic forms {100}, {110}, {111},and {113}, augmented with the mean orientation.


Figure 11, Figure 12, and Figure 13 display the data of grain 109.



24

Fig. 12. Pole point plots of grain 109 for crystallographic forms {100}, {110}, {111},and {113}, augmented with the mean orientation.


The analyses of the orientation data, in particular the spectral analyses of the

orientation matrices T , are summarized numerically in the following tables.


25

Eigenvalues λi of T , corresponding shape parameters κi of Bingham distribution

simIOM grain 40 grain 147 grain 109

sample size 900 3068 4324 2253

texture index 61.2971 337.7395 308.9108 178.4238entropy -3.6491 -5.3425 -5.1136 -4.7782

λ1 0.9954e+000 0.9965e+000 0.9983e+000 0.9956e+000λ2 0.1645e–002 0.2928e–002 0.8713e–003 0.2166e–002λ3 0.1489e–002 0.3000e–003 0.4989e–003 0.1906e–002λ4 0.1416e–002 0.2645e–003 0.3513e–003 0.3120e–003λ2/λ3 1.1043 9.7599 1.7465 1.1364λ3/λ4 1.0517 1.1345 1.4199 6.1100

visual inspection “spherical” “prolate” “spherical” “oblate”

computation involving 1F1, Eq. (11)

κ1 3.5344e+002 1.8908e+003 1.4234e+003 1.6029e+003κ2 4.9037e+001 1.7195e+003 8.4916e+002 1.3716e+003κ3 1.7354e+001 2.2410e+002 4.2083e+002 1.3401e+003κ4 0.0 0.0 0.0 0.0κ2 − κ3 31.6829 1.4954e+003 4.2832e+002 31.4763κ3 − κ4 17.3542 2.2410e+002 4.2083e+002 1.3401e+003

computation applying large concentration approximation, Eq. (16)

κ1 3.5293e+002 1.8903e+003 1.4229e+003 1.6024e+003κ2 4.9037e+001 1.7196e+003 8.4916e+002 1.3716e+003κ3 1.7354e+001 2.2410e+002 4.2083e+002 1.3401e+003κ4 0.0 0.0 0.0 0.0κ2 − κ3 31.6830 1.4956e+003 4.2832e+002 31.4763κ3 − κ4 17.3542 2.2410e+002 4.2083e+002 1.3401e+003

Table 1. Summary of the spectral analysis of the orientation tensor T for grain 40, grain

147, and grain 109, respectively, comprising texture index, entropy, largest eigenvalue λ1

corresponding to principal axes a1 of smallest inertia, remaining eigenvalues λi and their

ratios, estimated shape parameters κi of the Bingham quaternion distribution, involving

either Eq. (11), or large concentration approximation, Eq. (16).


26

Statistical tests

simIOM grain 40 grain 147 grain 109

sample size 900 3068 4324 2253

Bingham statistics involving 1F1, Eqs. (13), (14), (15)

pspherical 0.3895 0 0 0spherical “no rejection” reject reject reject

poblate − 0 0 0.0099oblate − reject reject reject for α > p

pprolate − 0.0022 0 0prolate − reject for α > p reject reject

Bingham statistics involving large concentration approximation,Eqs. (17), (18), (19)




Large sample approximation statistics without parametric assumptions,Eqs. (20), (21), (22)




Table 2.

Summary of the spectral analysis of the orientation tensor T for grain 40, grain 147, and

grain 109, respectively, comprising tests for the spherical, oblate, and prolate Bingham case

applying shape parameters estimated with either Eq. (11), or large concentration

approximation, Eqs. (17), (19), (18), respectively, and large-sample approximation tests for

spherical, oblate and prolate symmetry, respectively, without parametric assumptions,

Eqs. (20), (22), (21);

a “–” indicates that the test does not apply, “no rejection” indicates that rejection at any

reasonable level of significance is not possible.


27

Fig. 14. Scatter plots of orientations in axis–angle system centered with respect toeigenvector corresponding to largest eigenvalue; grain 40 – prolate (left), grain 147– spherical (center), grain 109 – oblate (right).

The orientation maps of grain 40 (Figure 5) exhibit only gentle colour variations,

corresponding to an unimodal orientation density function (Figure 7). Its pole point

plots (Figure 6) reveal best that the distribution is not spherically symmetric with

respect to its mean. Only the pole point mode in the centre of the (110)–pole point

plot is almost spherical indicating that this is the direction of the axis of rotation or

bending, respectively. This observation agrees with the {111} < 110 > glide system

of copper.

The orientation maps of grain 147 (Figure 8) are almost uniformly colour coded,

all plots seem to indicate spherical symmetry with respect to the unique mode of the

orientation density function.

The orientation maps of grain 109 (Figure 11) indicate that it comprises two parts

with distinct orientations, clearly visible as bimodality in the pole point, pole and

orientation density plots (Figure 13). The centered scatter plots (Figure 14) agree

roughly with the analyses of the ratios of eigenvalues and the statistcially tests.

6. Conclusions

Conclusions are restricted to methodological aspects only.

The proper statistic of individual orientation measurements is the orientation ten-

sor T , if independence is assumed. For sufficiently concentrated distributions the first


28

principal axis of inertia (corresponding to the largest eigenvalue of T , i.e., thesmall-

est principal moment of inertia) is fairly well approximated by the normalized mean

quaternion (and its negative), if special provisions are taken, or by the modal quater-

nion (and its negative) of some kernel-estimated orientation density. However, neither

the normalized mean quaternion nor the modal quaternion leads to inferential statis-

tics.

The ratios of the three remaining small eigenvalues may indicate the geometrical

shape of the sample. Special cases are oblateness, prolateness or a spherical shape.

They may infer texture generating processes, e.g., different deformation regimes.

Oblate, prolate, and spherical shape of the distribution can hardly be distinguished

by means of pole point, pole or orientation density plots. Three dimensional scatter

plots may give a first hint, but are usually distorted and therefore hard to read.

The ratios of the three small eigenvalues of the orientation tensor T may be in-

terpreted as describing a tendency towards one of the three symmetric cases, if we

exclude some special cases by visual inspection of all available plots. From descriptive

statistics one proceeds to inferential statistics either applying parametric assumptions

with reference to the Bingham quaternion distribution on S3 or using large sample

approximations avoiding any parametric assumptions. Then significance tests prove

the absence of a symmetry, if they lead to the rejection of the null hypothesis with

respect to a given significance.

Comparing the (x, y)–plots of the fabricated and the experimental data suggests

that one should proceed to methods of spatial statistics as introduced by Boogaart

and Schaeben (2002a; 2002b), e.g., to capture the spatially induced correlation.

All computations are done with our free and open Matlab R© toolbox MTEX 3.0.


29

7. Acknowledgments

Special thanks are due to Raymond Kan, Joseph L. Rotman School of Management,

University of Toronto, Toronto, Ontario, Canada, for contributing a Matlab R© script

file to evaluate 1F1(12 , 2,K), Plamen Koev, Department of Mathematics, San Jose

State University, San Jose, CA, USA, for introducing us to Raymond Kan, and point-

ing out the particularly simple form of 1F1(12 , 2,K) in terms of top–order zonal poly-

nomials.

One of the authors (WP) gratefully acknowledges the Danish National Research

Foundation for supporting the Center for Fundamental Research: Metal Structures in

Four Dimensions, within which part of this work was performed.

Appendix A.Tests of rotational symmetry of axial distributions

A.1. A useful proposition

Proposition If X = (X1, . . . , Xp)T is a random vector with

X ∼ N(

0, αIp − β11T)

then

α−1p∑i=1

(Xi − X

)2 ∼ χp−12.

Proof

First note that

var(X − X1

)=

(Ip − p−111T

)(αIp − β11T

)(Ip − p−111T

)= α

(Ip − p−111T

).


30

Take V independent of X with V ∼ N(0, α/p). Then

X − X1 + V 1 ∼ N (0, αIp) ,

and so

α−1p∑i=1

(Xi − X + V

)2 = α−1p∑i=1

(Xi − X

)2 +p

αpV 2 ∼ χp2,

from which the result follows by Cochran’s theorem.

A.2. Testing O(p− 2)-symmetry of Bingham distributions

Consider the Bingham distributions on RP p−1 with densities

f(±x;A) = 1F1(1/2, p/2, A)−1 exp{xTAx},

where x = (x1, . . . , xp)T is a random unit vector and A is a symmetric p × p matrix.

We shall use the spectral decomposition

A =p∑i=1

κiµiµTi

of A, where µ1, . . . , µp are orthonormal vectors and κ1 ≥ · · · ≥ κp. We are interested

in the hypothesis

Hprolate : κ3 = · · · = κp

of O(p− 2)-symmetry of the distribution about the plane spanned by µ1 and µ2, and

in the hypothesis

Hoblate : κ2 = · · · = κp−1

of O(p− 2)-symmetry of the distribution about the plane spanned by µ1 and µp.

Observations ±x1, . . . ,±xn on RP p−1 can be summarised by the scatter matrix T

about the origin, defined [see (9.2.10) of Mardia & Jupp (2000)] by

T =1n

n∑i=1

xixTi .


31

Let t1, . . . , tp (with t1 ≥ · · · ≥ tp) denote the eigenvalues of T . From (10.3.37) of

Mardia & Jupp (2000), the maximum likelihood estimates κ1, . . . , κp of κ1, . . . , κp

under the full Bingham model are given by

∂ log 1F1(1/2, p/2,K)∂κi

∣∣∣∣K=K

= ti i = 1, . . . , p,

where K = diag(κ1, . . . , κp). [Recall that κ1, . . . , κp can be determined only

up to addition of a constant, since, for any real constant c, the matrices A

and A + cIp give the same distribution.] The maximum likelihood estimates

κ1, κ2, κ, . . . , κ of κ1, . . . , κp under Hprolate are given (up to addition of a constant)

by

∂ log 1F1(1/2, p/2,K)∂κi

∣∣∣∣K=K

= ti i = 1, 2 (A.1)

∂ log 1F1(1/2, p/2,K)∂κ3

∣∣∣∣K=K

= t, (A.2)

where K = diag(κ1, κ2, κ, . . . , κ) and

t =1

p− 2

p∑i=3

ti.

Similarly, the maximum likelihood estimates κ1, κ, . . . , κ, κp of κ1, . . . , κp

under Hoblate are given (up to addition of a constant) by

∂ log 1F1(1/2, p/2,K)∂κi

∣∣∣∣K=K

= ti i = 1, p (A.3)

∂ log 1F1(1/2, p/2,K)∂κ2

∣∣∣∣K=K

= t, (A.4)

where K = diag(κ1, κ, . . . , κ, κp) and

t =1

p− 2

p−1∑i=2

ti.

It follows from standard results on regular exponential models that the likelihood

ratio statistics wprolate and woblate of Hprolate and Hprolate satisfy the large-sample


32

approximations

wprolate ' wprolate = n

p∑i=3

(κi − κ)(ti − t

)(A.5)

woblate ' woblate = n

p−1∑i=2

(κi − κ)(ti − t

). (A.6)

In the case p = 4, (A.5) and (A.6) can be rewritten as

wprolate ' wprolate =n

2(κ3 − κ4) (t3 − t4) (A.7)

woblate ' woblate =n

2(κ2 − κ3) (t2 − t3) . (A.8)

The parameter matrix A of a Bingham distribution in Hprolate is specified by the

pair of orthogonal p-vectors κ1µ1 and κ2µ2, and so the dimension of Hprolate is p +

(p − 1). Similarly, the parameter matrix A of a Bingham distribution in Hoblate is

specified by the pair of orthogonal p-vectors κ1µ1 and κpµp, and so the dimension of

Hoblate is p+(p−1). The parameter space of the Bingham distributions has dimension

p(p+ 1)/2− 1. Thus the codimension of both Hprolate and Hoblate is

[p(p+ 1)/2− 1]− [p+ (p− 1)] =p(p− 3)

2.

It follows from standard results on likelihood ratio statistics that the (large-sample)

asymptotic distributions of wprolate and woblate (and wprolate and woblate) under the

null hypotheses Hprolate and Hoblate are

wprolate.∼. χ2

p(p−3)/2

woblate.∼. χ2

p(p−3)/2

(provided that κ1 > κ2 > κ3 in the prolate case and κ1 > κ2 and κp−1 > κp in the

oblate case).

A.3. A large-sample test of SO(p− r)-symmetry

In this section we drop the assumption that ±x has a Bingham

distribution. We are interested in the hypothesis of O(p − r)-symmetry of theIUCr macros version 2.0β5: 2001/06/20

33

distribution about the r-dimensional subspace spanned by r orthonormal vectors

µ1, . . . , µr. Without loss of generality, we can take the unit eigenvectors of E[xxT] to

be µ1, . . . , µp with µi = (0, . . . , , 0︸︷︷︸i−1

, 1, 0, . . . , 0) for i = 1, . . . , p.

Put

cij = E[x2ix

2j ] 1 ≤ i, j ≤ p.

Under O(p− r)-symmetry, the matrices (E[xixj ]) and (cij) have the form

(E[xixj ]) =(

diag(λ1, . . . , λr) 00 λIp−r

)and

(cij) =

c11 . . . c1r c1,r+11T

c21 . . . c2r c2,r+11T

.... . . ,

. . ....

cr1 . . . crr cr,r+11T

c1,r+11 . . . cr,r+11 (cr+1,r+1 − cr+1,r+2) Ip−r + cr+1,r+211T

,

where 1 = (1, . . . , 1)T.

From the definition of Sp−1,p∑i=1

xi2 = 1. (A.9)

Taking the expectation of (A.9) givesr∑i=1

λi + (p− r)λ = 1. (A.10)

Multiplying (A.9) by xj2and taking the expectation givesr∑i=1

cji + (p− r)cj,r+1 = λj j = 1, . . . , r. (A.11)

Similarly, multiplying (A.9) by xr+12and taking the expectation gives

r∑j=1

cr+1,j + cr+1,r+1 + (p− r − 1)cr+1,r+2 = λ. (A.12)

Further,

{(cos θ)xr + sin θ)xr+1}4

= (cos θ)4x4r + 4(cos θ)3(sin θ)x3

rxr+1 + 6(cos θ)2(sin θ)2x2rx

2r+1

+4(cos θ)(sin θ)3xrx3r+1 + +(sin θ)4x4

r+1,


34

and so

crr = (cos θ)4crr + 0 + 6(cos θ)2(sin θ)2cr,r+1 + 0 + (sin θ)4crr

= (cos2 θ + sin2 θ)2crr + 2 cos2 θ sin2 θ(3cr,r+1 − crr), ∀θ,

leading to

crr = 3cr,r+1. (A.13)

Substituting (A.10), (A.11) and (A.13) into (A.12) yields

r∑j=1

1p− r

(λj −

r∑i=1

cij

)+ (p− r + 2)cr,r+1 =

1−∑r

i=1 λip− r

,

and so

cr,r+1 =1− 2

∑ri=1 λi +

∑ri=1

∑rj=1 cij

(p− r)(p− r + 2). (A.14)

Simple calculations show that (tr+1,r+1, . . . , tpp; tr+1,r+2, . . . , tp−1,p)T has variance

matrix((cr+1,r+1 − cr+1,r+2) Ip−r +

(cr+1,r+2 − λ2

)11T 0

0 cr+1,r+2I(p−r)(p−r−1)/2

).

It follows from (A.13) that this is equal to

cr+1,r+2

(2Ip−r +

(1− λ2

cr+1,r+2

)11T 0

0 I(p−r)(p−r−1)/2

). (A.15)

Partition T = (tij) (according to the subspaces spanned by µ1, . . . , µr and

µr+1, . . . , µp, respectively) as

T =(T11 T12

T21 T22

),

where

T11 =

t11 . . . t1r...

. . ....

tr1 . . . trr

and

T22 =

tr+1,r+1 . . . tr+1,p...

. . ....

tp,r+1 . . . tp,p

.


35

Since the unit eigenvectors of T are O(n−1/2) close to µ1, . . . , µp, we have∥∥∥∥∥T22 −tr(T22

)p− r

Ip−r

∥∥∥∥∥2

=p∑

i=r+1

(ti − t·1)2 + 2∑

r+1≤i<j≤pt2ij +O(n−1/2),

where t1, . . . , tp denote the eigenvalues of T and t·1 = tr(T22

)/(p− r). It follows from

(A.15) and the Proposition at the beginning of this Appendix that

12cr+1,r+2

∥∥∥∥∥T22 −tr(T22

)p− r

Ip−r

∥∥∥∥∥2

.∼. χ2(p−r−1)(p−r+2)/2 n→∞.

From (A.14), a suitable estimate of cr+1,r+2 is

cr+1,r+2 =1− 2

∑ri=1 λi +

∑ri=1

∑rj=1 cij

(p− r)(p− r + 2), (A.16)

where

λi = ti i = 1, . . . , r

cij =1n

n∑k=1

(xTk ti

)2 (xTk tj

)2i, j = 1, . . . , r,

with t1, . . . , tp being the unit eigenvectors of T .

The case r = 0

This gives Bingham’s (1974) test of uniformity. (See p. 232 of Mardia & Jupp, 2000.)

Uniformity is rejected for large values of

p(p+ 2)2

n∥∥T − p−1Ip

∥∥2 =p(p+ 2)

2n

(p∑i=1

t2i −1p

),

where t1, . . . , tp denote the eigenvalues of T . Under uniformity,

p(p+ 2)2

n

(p∑i=1

t2i −1p

).∼. χ2

(p+2)(p−1)/2 n→∞.

The case r = 1

This gives Prentice’s (1984) test of symmetry about an axis. (See p. 235 of Mardia

& Jupp, 2000.) Symmetry about the major axis µ1 is rejected for large values of

12c23

∥∥∥∥∥T22 −tr(T22

)p− 1

Ip−1

∥∥∥∥∥2

=n(p2 − 1)

{∑pi=2 t

2i − (1− t1)2/(p− 1)

}2(1− 2t1 + c11)

,


36

where t1, . . . , tp (with t1 ≥ · · · ≥ tp) denote the eigenvalues of T corresponding to

eigenvectors t1, . . . , tp and

c11 =1n

n∑i=1

(xTi t1

)4.

Under symmetry,

n(p2 − 1){∑p

i=2 t2i − (1− t1)2/(p− 1)

}2(1− 2t1 + c11)

.∼. χ2(p+1)(p−2)/2 n→∞.

Similarly, symmetry about the minor axis µp is rejected for large values of

12c23

∥∥∥∥∥T22 −tr(T22

)p− 1

Ip−1

∥∥∥∥∥2

=n(p2 − 1)

{∑p−1i=1 t

2i − (1− tp)2/(p− 1)

}2(1− 2tp + cpp)

,

where

cpp =1n

n∑i=1

(xTi tp

)4.

Under symmetry,

n(p2 − 1){∑p−1

i=1 t2i − (1− tp)2/(p− 1)

}2(1− 2tp + cpp)

.∼. χ2(p+1)(p−2)/2 n→∞.

The case r = 2

(a) Prolate case

O(p− 2)-symmetry of the distribution about the plane spanned by µ1 and µ2 is

rejected for large values of

12c34

∥∥∥∥∥T22 −tr(T22

)p− 2

Ip−2

∥∥∥∥∥2

=np(p− 2)

{∑pi=3 t

2i − (1− t1 − t2)2/(p− 2)

}2(1− 2[t1 + t2] + c11 + 2c12 + c22)

,

where t1, . . . , tp (with t1 ≥ · · · ≥ tp) denote the eigenvalues of T corresponding

to eigenvectors t1, . . . , tp and

cij =1n

n∑k=1

(xTk ti

)2 (xTk tj

)2.


37

Under symmetry,

np(p− 2){∑p

i=3 t2i − (1− t1 − t2)2/(p− 2)

}2(1− 2[t1 + t2] + c11 + 2c12 + c22)

.∼. χ2p(p−3)/2 n→∞

and this statistic is asymptotically equal to wprolate of (A.5).

(b) Oblate case

Similarly, O(p− 2)-symmetry of the distribution about the plane spanned by µ1

and µp is rejected for large values of

12c23

∥∥∥∥∥T22 −tr(T22

)p− 2

Ip−2

∥∥∥∥∥2

=np(p− 2)

{∑p−1i=2 t

2i − (1− t1 − tp)2/(p− 2)

}2(1− 2[t1 + tp] + c11 + 2c1p + cpp)

.

Under symmetry,

np(p− 2){∑p−1

i=2 t2i − (1− t1 − tp)2/(p− 2)

}2(1− 2[t1 + tp] + c11 + 2c1p + cpp)

.∼. χ2p(p−3)/2 n→∞

and this statistic is asymptotically equal to woblate of (A.6).

Appendix B.Computations and Asymptotics

B.1. Numerical computations

In order to solve the nonlinear equation Eq. (11) we used a modified Newton method.

For brevity we write this equation as f(κ) = λ, with κ = (κ1, κ2, κ3, κ4) and λ =

(λ1, λ2, λ3, λ4).

Recall that a solution exists only if 0 < λj < 1 (j = 1, 2, 3, 4) and∑4

1 λj = 1,

and then κ1, κ2, κ3, κ4 are determined up to an additive constant. This implies thatIUCr macros version 2.0β5: 2001/06/20

38

the Jacobian Df of f is not invertible, and thus we modified Newton’s method by

using the pseudo–inverse of Df . The initial values for Newton’s iteration were chosen

according to Eq. (12), i.e.,

κ1 = 0, κj = −(2λj)−1 (j = 2, 3, 4). (B.1)

During the computations the approximate solutions were normalized such that∑41 κj = 0.

The Jacobian Df involves not only the hypergeometric function 1F1(12 , 2,K) but

also its first and second derivatives. When we performed the experiments were not

aware of the Kume–Wood formula (Kume and Wood, 2007), which expresses the

derivatives of 1F1(12 , 2,K) by the values of a higher-dimensional hypergeometric func-

tion, and thus we approximated the derivatives numerically.

The range of λ we were mainly interested in is

1− 3ε ≤ λ1 < 1, 0 < λ2, λ3, λ4 ≤ ε (B.2)

with small ε. In this range the function f is rather flat. For example, at λ = (1 −

3ε, ε, ε, ε) we observed the asymptotic behaviour

Df ≈ 12ε2

3 −1 −1 −1−1 1 0 0−1 0 1 0−1 0 0 −1

as ε→ +0. In practice, computations had to be performed in the range of κj between

−1000 and 1000. Since it was impossible to meet the required accuracy with existing

Matlab R© code (we are grateful to Raymond Kan) we transfered parts of this code into

Mathematica R© and used high precision arithmetic.

Note that, in the present case of highly concentrated distributions, it may be advan-

tageous to evaluate 1F1(12 , 2,K) using the asymptotic formula, Eq. (9) of (Bingham et

al., 1992), for the normalising constant of concentrated Bingham distributions, whichIUCr macros version 2.0β5: 2001/06/20

39

gives for p = 1, 2, . . .

1F1(12, 2, A) = eκ1π−1/2|T0|−1/2

2F0(12,12, T−1

0 )

= eκ1π−1/2|T0|−1/2

{p−1∑k=0

[(1/2)k]2C(k)(T−10 )

k!+O(τ−p)

}

as τ → ∞, where κ1, . . . , κ4 are the eigenvalues (assumed to be distinct) of A, T0 =

diag(κ1 − κ2, κ1 − κ3, κ1 − κ4), C(k) is the top-order zonal polynomial corresponding

to the single-element partition of k, and τ = min{κ1 − κ2, κ1 − κ3, κ1 − κ4}. Here it

is assumed that κ1, κ2, κ3, κ4 are distinct, an extension to the general case is given in

Appendix 2 of (Bingham et al., 1992).

Taking κ1 = 0 (which we may do without loss of generality) and p = 2 gives

1F1(1/2; 2;A) = π−1/2 (−κ2κ3κ4)−1/2

1− 12

4∑j=2

1κj

+O(τ−2)

. (B.3)

Then∂1F1(1/2; 2;A)

∂κj= − 1

2κj+

12κj2

+O(τ−3) j = 2, 3, 4. (B.4)

Combining Eq. (B.4) with Eq. (11) shows that the maximum likelihood estimates

κ2, κ3 and κ4 of κ2, κ3 and κ4 satisfy

λj = − 12κj

(1− 1

κj

)+O(τ−3) j = 2, 3, 4. (B.5)

Hence

κj

(1− 1

κj

)−1

= − 12λj

+O(τ−1) j = 2, 3, 4,

and so

κj + 1 = − 12λj

+O(τ−1) j = 2, 3, 4. (B.6)

Define κ2, κ3 and κ4 by

κj = − 12λj− 1 j = 2, 3, 4. (B.7)


40

Then Eq. (B.6) shows that

κj = κj +O(τ−1) j = 2, 3, 4. (B.8)

From∑4

j=1 λj = 1 and Eq. (B.5) we have

λ1 = 1 +4∑j=2

12κj

(1− 1

κj

)+O(τ−3),

so that1

2λ1=

12− 1

4

4∑j=2

1κj

+O(τ−2). (B.9)

Setting

κ1 = 0, κj = −(2λj)−1 − (2λ1)−1 (j = 2, 3, 4), (B.10)

and combining Eq. (B.9) with Eq. (B.6) gives

κj = − 12λj− 1

2λ1= κj +

12

+O(τ−1) j = 2, 3, 4. (B.11)

Let κ2, κ3 and κ4 be the estimators (motivated by the arguments leading to Eq. (12)

and given in Eq. (B.1) defined by

κj = − 12λj

j = 2, 3, 4. (B.12)

Combining Eq. (B.8), Eq. (B.11), Eq. (B.12) and Eq. (B.7) gives

κj = κj +O(τ−1)

κj = κj +12

+O(τ−1)

κj = κj + 1 +O(τ−1).

Thus, for concentrated distributions, κj will tend to be closer to κj than κj is, which

in turn will tend to be closer to κj than κj is.


41

B.2. Asymptotic formulas

In order to check the quality of the approximation Eq. (16) we made 1000 tests

with randomly chosen values of λ in the range given by Eq. (B.2), computed the

corresponding approximate values κ by Eq. (16) and checked the corresponding values

λ = f(κ) against λ. Since we are only interested in λj and κj for j = 2, 3, 4, we measure

the discrepancy in terms of the absolute and relative errors

∆λ = max |λj − λj |, δλ =max |λj − λj |

max |λj |.

For ε = 0.003 (which was satisfied for the data in Table 1) we got the upper bounds

∆λ < 0.000041, δλ < 0.015, and for ε = 0.01 we still had ∆λ < 0.00058, δλ < 0.058.

Moreover, we confirmed empirically that the solution to Eq. (11) is even better

approximated by Eq. (B.10). instead of Eq. (B.1). Here 1000 random tests with ε =

0.003 gave ∆λ < 1.5 · 10−7, δλ < 5.0 · 10−5, for ε = 0.01 we got ∆λ < 6.1 · 10−6, δλ <

6.1 · 10−4, and even for ε = 0.1 the estimated errors were ∆λ < 0.0076, δλ < 0.092.

Since the test statistics Eq. (13), Eq. (14), Eq. (15) involve only differences of

κ2, κ3, κ4 and λ2, λ3, λ4, which are not influenced by the additional term (2λ1)−1 in

Eq. (B.10), we get the same approximations Eq. (17), Eq. (18), Eq. (19) also for the

modified values of κj . Thus treating our experimental data with the simple approx-

imate formulas Eq. (B.1) amounts to a smaller effect than perturbing the measured

values of λ by 0.005%.

References

[1] Barton, N.R., Dawson, P.R., 2001a, On the spatial arrangement of lattice orientations inhot-rolled multiphase titanium: Modelling Simul. Mater. Sci. Eng. 9, 433-463

[2] Barton, N.R., Dawson, P.R., 2001b, A methodology for determining average lattice ori-entation and its application: Metallurgical and Materials Transactions 32, 1967–1975

[3] Bingham, C., 1964, Distributions on the sphere and on the projective plane: PhD Thesis,Yale University, 80 p

[4] Bingham, C., 1974, An antipodally symmetric distribution on the sphere: Ann. Statist.2, 1201-1225.


42

[5] Bingham, C., Chang, T., Richards, D., 1992, Approximating the matrix Fisher and Bing-ham distribution: Applications to spherical regression and Procrustes analysis: J. Mult.Anal. 41, 314–337

[6] Bonham–Carter, G.F., 1996, Geographic Information Systems for Geoscientists: Perga-mon

[7] Boogaart, K.G.v.d., 1999, Spatial statistics for individual orientation measurements, in:Szpunar, J.A., (Ed.), Proceedings of the Twelfth International Conference on Textures ofMaterials (ICOTOM-12), August 9-13,1999, Montreal, Quebec, Canada, NRC ResearchPress, Ottawa, Ontario, Canada, Vol. 1, 29–33

[8] Boogaart, K.G. v.d., 2002 Statistics for individual crystallographic orientation measure-ments: PhD Thesis, Technische Universitat Freiberg, Germany, Shaker-Verlag, Aachen

[9] Boogaart, K.G.v.d., Schaeben, H., 2002a, Kriging of regionalized directions, axes, andorientations: I: Directions and axes: Math. Geol. 34, 479-503

[10] Boogaart, K.G.v.d., Schaeben, H., 2002b, Kriging of regionalized directions, axes, andorientations: II: Orientations: Math. Geol. 34, 671-677

[11] Cho, J.-H., Rollett, A.D., Oh, K.H., 2005, Determination of a Mean Orientation in Elec-tron Backscatter Diffraction Measurements: Metallurgical and Materials Transactions A36A, 3427–3438

[12] Glez, J.C., Driver, J., 2001, Orientation distribution analysis in deformed grains: J. Appl.Cryst. 34, 280–288

[13] Gurlebeck, K., Habetha, K., Sproßig, W., 2006, Funktionentheorie in der Ebene und imRaum: Birkhauser

[14] He, W., Ma, W., Pantleon, W., 2008, Microstructure of individual grains in cold-rolledaluminium from orientation inhomogeneities resolved by electron backscattering diffrac-tion: Materials Science and Engineering A 494, 21-27

[15] Heilbronner, R., 2000, Automatic grain boundary detection and grain size analysis usingpolarization micrographs or orientation images: Journal of Structural Geology 22, 969–981

[16] Hielscher, R., 2007, The Radon Transform on the Rotation Group – Inversion and Ap-plication to Texture Analysis: Ph.D. thesis, TU Bergakademie Freiberg, Germany

[17] Hielscher, R., Schaeben, H., 2008, Novel Pole Figure Inversion Method: Specification ofthe MTEX Algorithm: J.Appl.Cryst. 41 1024–1037

[18] Hillier, G., Kan, R., Wang, X., 2009, Computationally Efficient Recursions for Top-OrderInvariant Polynomials with Applications: Econometric Theory 25, 211–242

[19] Humbert, M., Grey, N., Muller, J., Esling, C., 1996, Determination of a mean orientationfrom a cloud of orientations: J.Appl.Cryst 29, 662-666

[20] Humphreys, F.J., Bate, P.S., Hurley, P.J., 2001, Orientation averaging of electronbackscattered diffraction data: Journal of Microscopy 201, 50–58

[21] Koev, P., Edelman, A., 2006, The efficient evaluation of the hypergeometric function ofa matrix argument: Mathematics of Computation 75, 833–846

[22] Krog-Pedersen, S., Bowen, J.R., Pantleon, W., 2009, Quantitative characterization of theorientation spread within individual grains in copper after tensile deformation: Interna-tional Journal of Materials Research 2009(3), 433–438


43

[23] Kume, A., Wood, A.T.A., 2007, On the derivatives of the normalising constant of theBingham distribution: Stat. Probab. Lett. 77, 832–837

[24] Kunze, K., Schaeben, H., 2004, The Bingham distribution of rotations and its sphericalRadon transform in texture analysis: Math. Geol. 36, 917–943

[25] Kunze, K., Schaeben, H., 2005, Ideal Patterns of preferred crystallographic orientationand their representation by the von Mises Fisher matrix or Bingham quaternion distri-bution: in van Houtte, P. and Kestens, L., (eds.), Proceedings ICOTOM14, Leuven, Jul11 15, 2005, Materials Science Forum 495 497, 295–300

[26] Lloyd, G.E., Farmer, A.B., Mainprice, D., 1997, Misorientation analysis and the formationand orientation of subgrain and grain boundaries: Tectonophysics 279, 55–78

[27] Mardia, K.V., Jupp, P.E., 2000, Directional Statistics: J.Wiley & Sons

[28] Morawiec, A., 1998, A note on mean orientation: J.Appl.Cryst. 31, 818–819

[29] Pantleon, W., 2005, Retrieving orientation correlations in deformation structures fromorientation maps: Materials Science and Technology 21, 1392–1396

[30] Pantleon, W., He, W., Johansson, T.P., Gundlach, C., 2008, Orientation inhomogeneitieswithin individual grains in cold-rolled aluminium resolved by electron backscatter diffrac-tion: Materials Science and Engineering A 483484, 668-671

[31] Prentice, M.J., 1984, A distribution-free method of interval estimation for unsigned di-rectional data: Biometrika 71, 147–154

[32] Prentice, M.J., 1986, Orientation statistics without parametric assumptions: J.R. Stat.Soc. Ser. B 48, 214-222

[33] Schaeben, H., Hielscher, R., Fundenberger, J.-J., Potts, D., Prestin, J., 2007, Orientationdensity function-controlled pole probability density function measurements: automatedadaptive control of texture goniometers: J. Appl. Cryst. 40, 570-579

[34] Watson, G.S., 1983, Statistics on Spheres: J.Wiley & Sons


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