Inferential statistics of electron backscatter diffraction data from within individual crystalline grains

Transcription

1 Journal of Applied Crystallography ISSN Editor: Anke R. Kaysser-Pyzalla Inferential statistics of electron backscatter diffraction data from within individual crystalline grains Florian Bachmann, Ralf Hielscher, Peter E. Jupp, Wolfgang Pantleon, Helmut Schaeben and Elias Wegert J. Appl. Cryst. (2). 43, Copyright c International Union of Crystallography Author(s) of this paper may load this reprint on their own web site or institutional repository provided that this cover page is retained. Republication of this article or its storage in electronic databases other than as specified above is not permitted without prior permission in writing from the IUCr. For further information see Many research topics in condensed matter research, materials science and the life sciences make use of crystallographic methods to study crystalline and non-crystalline matter with neutrons, X-rays and electrons. Articles published in the Journal of Applied Crystallography focus on these methods and their use in identifying structural and diffusioncontrolled phase transformations, structure property relationships, structural changes of defects, interfaces and surfaces, etc. Developments of instrumentation and crystallographic apparatus, theory and interpretation, numerical analysis and other related subjects are also covered. The journal is the primary place where crystallographic computer program information is published. Crystallography Journals Online is available from journals.iucr.org J. Appl. Cryst. (2). 43, Florian Bachmann et al. Inferential statistics of EBSD data

2 Journal of Applied Crystallography ISSN Inferential statistics of electron backscatter diffraction data from within individual crystalline grains Received 4 April 2 Accepted 29 July 2 Florian Bachmann, a Ralf Hielscher, b Peter E. Jupp, c Wolfgang Pantleon, d Helmut Schaeben a * and Elias Wegert e a Geoscience Mathematics and Informatics, TU Bergakademie Freiberg, Germany, b Applied Functional Analysis, TU Chemnitz, Germany, c School of Mathematics and Statistics, University of St Andrews, Fife, Scotland, d Center for Fundamental Research, Metal Structures in Four Dimensions, Risø National Laboratory for Sustainable Energy, Materials Research Division, Technical University of Denmark, Roskilde, Denmark, and e Applied Analysis, TU Bergakademie Freiberg, Germany. Correspondence schaeben@geo.tu-freiberg.de # 2 International Union of Crystallography Printed in Singapore all rights reserved Highly concentrated distributed crystallographic orientation measurements within individual crystalline grains are analysed by means of ordinary statistics neglecting their spatial reference. Since crystallographic orientations are modelled as left cosets of a given subgroup of SO(3), the non-spatial statistical analysis adapts ideas borrowed from the Bingham quaternion distribution on S 3. 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 several types of symmetry of the orientation distribution with respect to the mean orientation, like spherical, prolate or oblate symmetry. Applications to simulated as well as to experimental data are presented. All computations have been done with the free and open-source texture toolbox MTEX.. Introduction and motivation The subject of this communication is the non-spatial statistical analysis of highly concentrated crystallographic orientations measured within individual crystallographic grains. Crystallographic orientations are explicitly considered as left cosets of crystallographic symmetry groups, i.e. classes of crystallographically symmetrically equivalent rotations. Our objective is twofold. Primarily, this work is an attempt to clarify a lasting confusion concerning the mean orientation characterizing a grain, i.e. its mathematical definition and its numerical determination (Humbert et al., 996; Morawiec, 998; Barton & Dawson, 2a,b; Glez & Driver, 2; Humphreys et al., 2; Pantleon, 25; Cho et al., 25; He et al., 28; Krog-Pedersen et al., 29; Pantleon et al., 28), and then to proceed to its meaning for inferential statistics. The second objective is to verify these theoretical findings with fabricated and experimental data from within individual grains. Our approach here is diametrically different to the more commonly used approach in electron backscatter diffraction (EBSD) to the misorientation within individual grains, which seeks to explain the orientation gradients in terms of dislocation slip systems using the Nye tensor (El- Dasher et al., 23; Field et al., 25; Nye, 953; Pantleon, 28; Wheeler et al., 29; Wilkinson et al., 26). For all computations we use the versatile features of the free and open-source Matlab (MathWorks, Natick, MA, USA) software toolbox MTEX (Hielscher, 27; Hielscher & Schaeben, 28; Schaeben et al., 27), 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 measurements 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 van den Boogaart (999, 22) 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 descriptive spherical statistics, and the relationship of individual orientation measurements and left cosets of some rotational subgroups induced by crystallographic symmetry, i.e. classes of rotations equivalent by crystallographic symmetry, which cannot be distinguished physically but should be distinguished mathematically. The third section applies the Bingham quaternion distribution to highly concentrated individual orientation measurements as sampled, e.g., by EBSD. In particular, it is shown that quaternion multiplication of the measurements by a given reference orientation acts analogously to centring the measurements with respect to this reference orientation. Therefore, replacing the initial orientation measurements by 338 doi:.7/s x J. Appl. Cryst. (2). 43,

3 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. 2. Preliminaries 2.. Quaternions The group SO(3) may be considered in terms of H, the skew field of real quaternions. In particular, a unit quaternion q 2 S 3 H determines a rotation. For unit quaternions the inverse equals the conjugate, q ¼ q. The unit quaternion q and its negative q describe the same rotation. Thus, it would make sense to identify the pair ðq; qþ with a unique rotation. Working with pairs may be quite cumbersome; for general statistical purposes the proper entity to work with is the ð4 4Þ symmetric matrix q 2 q q 2 q q 3 q q 4 q Q ¼ q 2 q 2 2 q 2 q 3 q 2 q 4 B q q 3 q 2 q 3 q 2 3 q 3 q 4 A : ðþ q q 4 q 2 q 4 q 3 q 4 q 2 4 Thinking of q ¼ðq ; q 2 ; q 3 ; q 4 Þ T as elements of S 3 R 4 we could write Q ¼ qq T. The product of two quaternions is defined as pq ¼ ScðpÞScðqÞ VecðpÞVecðqÞ þ ScðpÞVecðqÞþScðqÞVecðpÞþVecðpÞVecðqÞ; where ScðpÞ and VecðpÞ denote the scalar and vector part of the quaternion p, respectively. The quaternion product can be rewritten as pq ¼ L p q ¼ p p p 2 p 3 p p p 3 p 2 p 2 p 3 p p p 3 p 2 p p C A q; and analogously p p p 2 p 3 p qp ¼ R p q ¼ p p 3 p 2 B p 2 p 3 p p A q; p 3 p 2 p p where the matrices L p and R p, respectively, have the properties ð2þ ð3þ ð4þ L p ¼ L T p; R p ¼ R T p; L p L p ¼ L p p ¼ L ¼ E; R p R p ¼ R pp ¼ R ¼ E; ð5þ i.e. they are orthonormal (cf. Gürlebeck et al., 26, p. 33). The scalar product of two quaternions is defined as p q ¼ Scðpq Þ¼ScðpÞScðqÞþVecðpÞVecðqÞ ¼cos ffðp; qþ: ð6þ 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 S 3 R 4, i.e. the scalar product provides the canonical measure for the distance of unit quaternions (cf. Gürlebeck et al., 26, p. 2) Descriptive spherical statistics Even though spherical statistics is not the topic here, the entities considered are borrowed from spherical statistics (Watson, 983; Mardia & Jupp, 2). The following issues have been clarified in the context of texture analysis by Morawiec (998). In spherical statistics a distinction is made between vectors q and pairs ðq; qþ of antipodal vectors, i.e. axes Vectorial data. The key summary statistic of vectorial data is the normalized sample mean vector r ¼ k P n q ¼ q k ; ð7þ ¼ where r provides a measure of location and k P q k provides a measure of dispersion (concentration). The normalized mean vector solves the following minimization problem. Given vectorial data q ;...; q n 2 S 3 R 4 find r 2 S 3 minimizing the mean angular deviation. More precisely, with ¼ffðq ; xþ the entity to be minimized is F R ðx; q ;...; q n Þ¼ ð cos n Þ¼ n ¼ ¼ n ¼ xq ¼ x n ¼ ¼ cos q! ¼ xq: ð8þ Thus, the solution x ¼ r is provided by the normalized mean vector and r ¼ q kqk ; F R ðr; q ;...; q n Þ¼ kqk ð9þ ðþ 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 kq n xk 2 ¼ ðq n xþðq xþ ¼ ð2 2 cos n Þ ¼ ¼ ¼ ¼ 2F R ðx; q ;...; q n Þ ðþ (Humbert et al., 996). However, the normalized mean vector does not provide statistical insight into a sample of quaternions used to parametrize rotations (Watson, 983; Mardia & Jupp, 2, x3.2.) Axial data. The key summary statistic of axial data is the orientation tensor T ¼ n ¼ q q T : ð2þ J. Appl. Cryst. (2). 43, Florian Bachmann et al. Inferential statistics of EBSD data 339

4 Its spectral decomposition into the set of eigenvectors a ;...; a 4 and the set of corresponding eigenvalues ;...; 4 provides a measure of location and a corresponding measure of dispersion, respectively. Since the orientation tensor T and the tensor of inertia I are related by I ¼ E T; ð3þ 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 sum to. The spectral decomposition solves the following minimization problem. Given axial data q ;...; q n 2 S 3 find a 2 S 3 minimizing the mean squared orthogonal distance. With ¼ffðq ; xþ as before, the entity to be minimized is F T ðx; q ;...; q n Þ¼ n ¼ sin 2 ¼ n ¼ cos 2 ¼ X cos 2 n ¼ ðx T q n Þ 2 ¼! ¼ x T q n q T x ¼ ¼ x T Tx: ð4þ Obviously, the solution x ¼ a is provided by the eigenvector a of T corresponding to the largest eigenvalue denoted. Then is a measure of the spherical dispersion of the axial data with respect to the quaternion a. The eigenvalues are related to the shape parameters of the Bingham (quaternion) distribution (Bingham, 964, 974; Kunze & Schaeben, 24, 25) by a system of algebraic equations involving partial derivatives of the hypergeometric function F of a ð4 4Þ matrix argument. In general, a single eigenvector and its eigenvalue are insufficient to provide a reasonable characterization of the data. Therefore, often the ratios of the eigenvalues are analysed and interpreted Normalized mean vector versus first principal axis of inertia. It is emphasized that the mean vector for axial data is always identically zero. Despite 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 hemisphere, 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 additional 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 cos 2, i.e. for (very) small angles. In the case of highly unimodal preferred orientation, the largest eigenvalue 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 crystalline 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 measurements q ¼ qðxþ and q ¼ qðx þ xþ located next to one another (separated by a grid step of size x) are compared. If the orientation difference is smaller than a userdefined threshold!, the two positions x; x þ x are assigned to the same grain. Then q ¼ qðx þ xþ and q 2 ¼ qðx þ 2xÞ are compared. If the orientation difference is larger than the threshold, a grain boundary is constructed (e.g. Lloyd et al., 997; Heilbronner, 2). Determined in this way the size and the shape of a crystallographic grain depend rather on the threshold! defined by the user than on the amount of available sampling controlled by the grid step size x. Thus, a grain is referred to as a definition rather than a sampling-limited spatial object (cf. Bonham-Carter, 996, p. 29 3). Then the orientation gradient along the line x; x þ m x, m 2 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 Considering crystallographic symmetry In the case of crystallographic symmetry G Laue \ S 3 ¼ ~G Laue S 3 to be considered in terms of the restriction ~G Laue of the Laue group G Laue 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 g m, q 2 S 3, g m 2 ~G Laue, i.e. they are some elements of the corresponding left cosets q ~G Laue. If q i and q j are very close, q i g and q j g 2, g ; g 2 2 ~G Laue, g 6¼ g 2, 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. equation (4)] actually is " # F T ðx; q ;...; q n ; ~G Laue Þ¼ x T ðq n ~G Laue Þðq ~G Laue Þ T x ¼ ð5þ 34 Florian Bachmann et al. Inferential statistics of EBSD data J. Appl. Cryst. (2). 43,

5 given crystallographic orientations q ;...; q n, and the rotational symmetry group ~G Laue, i.e. ( " # ) argmin x2s 3 ;g m 2 ~G Laue x T ðq n g m Þðq g m Þ T x ð6þ with g m 2 ~G Laue ; m ¼ ;...; s; ¼ ;...; n, where s denotes the cardinality of ~G Laue. While the map S 3! S 3 = ~G Laue is continuous, its inverse is not. Thus, assigning to each measured orientation q g m the element of q ~G Laue with the smallest angle of rotation does not lead to a solution of equation (6). Therefore, we apply the following heuristics to resolve the problem of equation (6). To this end, an initial reference orientation is required. Given the data set q ;...; q n,wemay choose q as reference orientation. For every ¼ ;...; n, determine argmax g2 ~G Laue q g q ¼ g ðþ ; 2 ~G Laue and replace q by q ðþ ¼ q g ðþ ;. Then we evaluate F T½x; q ðþ ;...; q ðþ n Š for determining the largest eigenvalue ðþ and its corresponding eigenvector a ðþ. Then we choose a ðþ as the reference orientation, determine for every ¼ ;...; n, argmax g2 ~G Laue q ðþ g aðþ ¼ gðþ ; 2 ~G Laue, and replace q ðþ by q ðþ ¼ q ðþ gðþ ;. Then we evaluate F T ½x; q ðþ ;...; qðþ n Š for determining the largest eigenvalue ðþ and its corresponding eigenvector a ðþ. Now let i ¼ ;...; I enumerate the successive iterations. For every ¼ ;...; n, we replace q ði Þ by q ðiþ. Next we evaluate F T ½x; q ðiþ ;...; q ðiþ n Š, and determine the largest eigenvalue ðiþ and its corresponding eigenvector a ðiþ, which will be the reference orientation in the next step of this iterative approximation. We continue until ðiþ does not increase any longer and its corresponding eigenvector a ðiþ does not change any more. This iterative procedure to determine the proper eigenvector corresponding to the largest eigenvalue is similar to the one suggested by Pantleon (25), Pantleon et al. (28), He et al. (28) and Krog-Pedersen et al. (29) to be applied to the normalized 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, i.e. the orientation for which the kernelestimated orientation density function is maximum, may be a more appropriate initial reference orientation than an arbitrary measured orientation. 3. Characterizing sets of highly preferred orientation Neglecting this problem of how to define grains and their boundaries, the confusion about the normalized mean vector and the spectral decomposition can be avoided if it is assumed that (a) the measurements q ;...; q n are highly preferred, i.e. the dispersion within an individual grain is small, and (b) a mean orientation a has been assigned to the grain by means of the eigenvector a corresponding to the largest eigenvector according to equation (4), which has been interpreted itself as a measure of the extent of preferred orientation, i.e. ð ; a Þ are no longer the statistics of primary interest. ¼ Then the focus is on ð ; a Þ; ¼ 2; 3; 4, and their ratios. Therefore the spectral decomposition of the symmetric positive definite matrix T is considered in detail and related to the spectral decomposition suggested earlier (Pantleon, 25; Pantleon et al., 28; He et al., 28; Krog-Pedersen et al., 29). If T is a ð4 4Þ positive definite symmetric matrix, then the decomposition T ¼ VV T ð7þ exists where ¼ C A ð8þ is the ð4 4Þ diagonal matrix of eigenvalues ;...; 4, and where V ¼ða ; a 2 ; a 3 ; a 4 Þ is the ð4 4Þ matrix comprising the corresponding column eigenvectors a ;...; a 4 2 R 4. Equivalently, V T TV ¼ : ð9þ Here, the matrix T is given in terms of a set of unit vectors q ;...; q n 2 S 3 R 4 as T ¼ n ¼ q q T ¼ n Q nq T n; ð2þ where Q n is the ð4 nþ matrix composed of column vectors q ;...; q n. Hence, T is symmetric and positive definite. Let p denote the conjugate of an arbitrary quaternion p. Considering the set q p ;...; q n p 2 S 3, i.e. the set of disorientations with respect to p, and the corresponding T ¼ n ¼ ðq p Þðq p Þ T ¼ n the decomposition is ¼ ðr p q ÞðR p q Þ T ¼ R p TR T p ; ev T T ev ¼ðR p VÞ T ðr p TR T p ÞðR p VÞ ¼VT TV ¼ ð2þ ð22þ with ev ¼ R T pv composed of column eigenvectors ea ¼ a p ; ¼ ;...; 4, with respect to eigenvalues ; ¼ ;...; 4. The analogue is true for the multiplication with p from the left. In particular, if p ¼ a, then ea ¼ð; ; ; Þ T, i.e. multiplication with a from the right acts like centring the measurements at the identity rotation ð; ; ; Þ T. In this case ev is of the form ew ev ¼ ew 2 ew 3 B ew 2 ew 22 ew 32 A ¼ T ð23þ ew ew 3 ew 23 ew 33 with the ð3 3Þ matrix ew composed of the ð3 Þ column vectors ew ¼ a þ a j R 3; ¼ ; 2; 3; ð24þ J. Appl. Cryst. (2). 43, Florian Bachmann et al. Inferential statistics of EBSD data 34

6 where a þ a ¼ ea þ are restricted to R 3 by omitting the zero scalar part of the quaternions ea þ ; ¼ ; 2; 3. Then ev T ¼ T ew T : ð25þ Accordingly, T may be rewritten explicitly as t t2 t3 t4 t t2 t3 t4 T t2 ¼ t22 t23 t24 B t3 t32 t33 t34 A ¼ t B t t4 t42 t43 t44 3 et C A with t 4 ð26þ t et 22 t23 t24 t32 t33 t34 A: ð27þ t42 t43 t44 The ð3 3Þ lower-right submatrix et of T corresponds to the matrix Q [equation (3) of Pantleon, 25]. Eventually ev T T ev ¼ t P t iþ; ew ;i B P iþ;2 ew 2;i P t iþ;3 ew 3;i P P P t ;jþ ew ;j t ;jþ ew 2;j t ;jþ ew 3;j ew T et C ew A ¼ ð28þ or, equivalently, ew T et 2 ew ¼ e 3 A: ð29þ 4 Thus, if a ; ¼ ;...; 4, are the eigenvectors of T with respect to eigenvalues ;...; 4, then ew ¼ a þ a j R 3; ¼ ; 2; 3, equation (24), are the eigenvectors of the ð3 3Þ lower-right submatrix et of T with respect to eigenvalues 2 ; 3 ; Inferential statistics 4.. Inferential statistics with respect to the Bingham quaternion distribution Neglecting the spatial dependence of EBSD data from individual crystalline grains, the Bingham quaternion distribution expðq fðq; AÞ ¼ F ð=2; 2; AÞ T AqÞ ð3þ with a random q 2 S 3, with a symmetric ð4 4Þ matrix A, and with the hypergeometric function F ð=2; 2; Þ of matrix argument seems appropriate (Bingham, 964, 974; Schaeben, 993; Kunze & Schaeben, 24, 25). It is emphasized that the densities f ðq; AÞ and f ðq; A þ tiþ, witht 2 R and the ð4 4Þ identity matrix I, define the same distribution. Furthermore, these densities form a transformation model under the group Oð4Þ. ForU 2 Oð4Þ f ðuq; UAU T Þ¼fðq; AÞ ð3þ implying that F =2; 2; UAU T ¼ F ð=2; 2; AÞ: ð32þ In particular, if A ¼ U T KU ð33þ with U orthonormal and K ¼ diagð ;...; 4 Þ, then F ð=2; 2; AÞ ¼ F ð=2; 2; KÞ ð34þ and, therefore, the estimates of the parameters of the Bingham quaternion distribution based on the sample q ;...; q n are given by bu ¼ V T ; log F ð=2; 2; KÞ ¼ i; i ¼ ;...; 4: ð36þ i ^K It should be noted that equation (36) determines ^ i, i ¼ ;...; 4, only up to an additive constant, because ^ i and ^ i þ t, i ¼ ;...; 4, result in the same i, i ¼ ;...; 4. Uniqueness could be conventionally imposed by setting ^ 4 ¼. Provided that > 2 ; 3 ; 4 calculation shows that q T Aq ¼ q T U T KUq ¼ x T Kx ¼ 2 X 4 x 2 j 2 j¼2 j ; ð37þ where x ¼ðx ; x 2 ; x 3 ; x 4 Þ T ¼ Uq and 2 j ¼½2ð j ÞŠ for j ¼ 2; 3; 4. Thus if q has the Bingham distribution with parameter matrix A and j!for j ¼ 2; 3; 4, then x tends to ð; ; ; Þ T and the distribution of ðx 2 = 2 ; x 3 = 3 ; x 4 = 4 Þ T tends to that of three independent standard normal random variables. Then for j ¼ 2; 3; 4, j Eðx 2 j Þ 2 j, and so j ð2 j Þ : ð38þ Since we expect that j and ^ ^ j ; j ¼ 2;...; 4, are very large, the interesting statistical issue is to test rotational symmetry, i.e. to test the null hypothesis of spherical symmetry that 2 ¼ 3 ¼ 4. If this hypothesis can be rejected we would be interested in distinguishing the prolate case, 2 > 3 ¼ 4, and the oblate case, 2 ¼ 3 > 4. Assuming rotational symmetry, i.e. degeneracy of the Bingham to the Watson distribution, the test statistic where T Bingham s ¼ n P4 ðb Þð Þ 2 5; n!; ð39þ ¼2 ¼ 3 X 4 ¼2 ^ ; ¼ 3 X 4 ¼2 ð4þ (Mardia & Jupp, 2, p. 234). For a given sample of individual orientation measurements q ;...; q n we compute the value t s and the corresponding p ¼ Prob ðt s > t s Þ; ð4þ 342 Florian Bachmann et al. Inferential statistics of EBSD data J. Appl. Cryst. (2). 43,

7 and conclude that the null hypothesis of rotational symmetry may be rejected for any significance level >p. If the hypothesis of spherical symmetry is rejected, we test for prolateness using is required, where a i ; i ¼ ;...; 4, are the eigenvectors of T corresponding to the eigenvalues Analogously to results (Prentice, 984, x6, 986, x5; Mardia & Jupp, 2, x.7.2) the test statistics are, for the spherical case, T Bingham p ¼ n 2 ð ^ 3 ^ 4 Þð 3 4 Þ 2 2; n!; ð42þ or, analogously, for oblateness using T Bingham o ¼ n 2 ð ^ 2 ^ 3 Þð 2 3 Þ 2 2; n!: ð43þ According to equation (38), for we may apply the asymptotics j e j ¼ ð2 j Þ ðj ¼ 2; 3; 4Þ; ð44þ assuming e ¼. Then the test statistics T Bingham simplify to T Bingham s T Bingham p T Bingham o asymp Bingham Ts ¼ n 6 X X i¼2;3;4 j¼2;3;4 i j ; ð45þ asymp Bingham Tp ¼ n 3 þ 4 2 ; ð46þ asymp Bingham To ¼ n 2 þ 3 2 : ð47þ Note that three summands in equation (45) vanish and that the j are arranged in decreasing order. These formulae agree well with the common-sense interpretation of a spherical, prolate and oblate shape, respectively 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 F can be avoided; only c ij ¼ n ¼ ðq T a i Þ 2 ðq T a j Þ 2 ð48þ Figure Inverse pole figure colour bar. Figure 2 9 simulated spatially indexed individual orientations from the Bingham quaternion distribution with modal orientation q modal ¼ (.7824,.2642,.535,.5473) and dispersion parameters K ¼ diag ð34; ; ; Þ in a 3 3 grid as a colour-coded orientation map according to the ðþ inverse pole figure colour bar (a) and RGB colours (b) and as a three-dimensional axis angle scatter plot centred at the mean orientation (c). J. Appl. Cryst. (2). 43, Florian Bachmann et al. Inferential statistics of EBSD data 343

8 T s ¼ 5n 2 2 þ 2 3 þ 2 4 ð Þ 2 =3 2 2ð 2 þ c Þ 5; n!; ð49þ for the prolate case, 2 3 þ 2 4 ð T p ¼ 8n 2 Þ 2 =2 2½ 2ð þ 2 Þþc þ 2c 2 þ c 22 Š 2 2; n!; and for the oblate case, ð5þ (b) applying RGB colour coding of Euler angles ð ;; 2 Þ according to Bunge s zxz convention, where the Euler angle is associated with red, with green and 2 with blue. For the simulated EBSD data, the units along both axes of the orientation map are just pixels; for the experimental data, the unit is mm for both axes of the orientation map. Thus each pixel of the map corresponds to one experimental orientation. The labels along the axes refer to the origin of the coordinate 2 2 þ 2 3 ð T o ¼ 8n 4 Þ 2 =2 2½ 2ð þ 4 Þþc þ 2c 4 þ c 44 Š 2 2; n!: ð5þ For details concerning inferential statistics the reader is referred to Appendix A. Some details of the computation and asymptotics are given in Appendix B. 5. Practical applications Several samples of individual orientation measurements are analysed. 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., 29). The data are displayed as follows: (a) Orientation maps applying the colour coding given by the inverse pole figure colour bar as shown in Fig. to the specimen x direction, and Figure 3 Pole point plots for crystallographic forms fg; fg; fg and f3g of 9 simulated spatially indexed individual orientations from the Bingham quaternion distribution, augmented with the mean orientation marked by a red dot emphasizing a small dispersion. Figure 4 -Sections of orientation density function (a) and pole density functions for crystallographic forms fg; fg; fg and f3g (b) of 9 simulated spatially indexed individual orientations according to the Bingham quaternion distribution, augmented with the mean orientation marked by a red dot. 344 Florian Bachmann et al. Inferential statistics of EBSD data J. Appl. Cryst. (2). 43,

9 system attached to the specimen comprising not just the analysed grain. (c) A three-dimensional scatter plot with respect to axis angle parametrization, i.e.! n ¼! ðn x ; n y ; n z Þ T with! 2 ½; 8Š and n 2 S 2 are displayed in a Cartesian ðx; y; zþcoordinate system. No provision whatsoever is made for the spherical metric of the axis angle parametrization. (d) Pole points q ~G Laue h ~G Laue q ; ¼ ;...; n, for some crystallographic forms h 2 S 2, i.e. crystallographic directions and their symmetrical equivalents ~G Laue h ~G Laue, and (e) plots of the orientation density and ( f) pole density function. Here, orientation maps are used for a rough indication of spatial dependence only. The orientation density function is computed by nonparametric kernel density estimation with the de la Vallée Poussin kernel, and corresponding pole density functions are determined as superpositions of the totally geodesic Radon transform of the de la Vallée Poussin kernel. In all cases the orientation tensor T and its eigenvalues and eigenvectors are computed, and the shape parameters are derived such that the smallest estimated shape parameter eventually equals whatever formula was used for its computation. Since the Bingham test statistics involve differences of estimated shape parameters they are independent of the way the shape parameters are determined as the differently estimated shape parameters differ by additive terms only. Then the spherical, oblate and prolate cases are tested, if reasonable. All computations have been done with our free and opensource Matlab toolbox MTEX 3., which can be downloaded from Simulated EBSD data The first data set was fabricated by simulation of 9 (pseudo-)independent spatially indexed individual orientation measurements from a given Bingham quaternion distribution with modal orientation q modal ¼ (.7824,.2642,.535,.5473) [(65,35,5 ) in Euler angles according to Bunge s zxz convention] and dispersion parameters K ¼ diagð34; ; ; Þ in a 3 3 grid. Assuming cubic crystal symmetry the 9 simulated orientation data are displayed as orientation maps of 3 3 pixels and as a three-dimensional scatter plot with respect to axis angle parametrization in Fig. 2, as pole point plots of crystallographic forms ~G Laue h (), (), () and (3) in Fig. 3, and as pole and orientation densities in Fig. 4. Figure 5 Grain 4 with 368 spatially indexed individual orientations as a colourcoded orientation map according to the ðþ inverse pole figure colour bar (a) and RGB colours (b) and as a three-dimensional axis angle scatter plot centred at the mean orientation (c). Figure 6 Pole point plots of grain 4 for crystallographic forms fg; fg; fg and f3g, augmented with the mean orientation marked by a red dot emphasizing a small dispersion. J. Appl. Cryst. (2). 43, Florian Bachmann et al. Inferential statistics of EBSD data 345

10 For these simulated data the estimated texture index I ¼ 6:297 and entropy E ¼ 3:649. The eigenvalues are for i ¼ ;...; 4; i ¼ :9954; :6; :5; :4, respectively, and their ratios are 2 = 3 ¼ :43; 3 = 4 ¼ : Experimental EBSD data Experimental orientation data were collected on a deformed oxygen-free high-conductivity copper with a purity of 99.99%. The starting material was recrystallized to a grain size (mean linear intercept length excluding twin boundaries obtained by EBSD) of about 3 mm. A cylindrical dog-bone ASTM standard tensile specimen with a gauge area of ø4 2 mm was prepared by spark-cutting and loaded in tension on an INSTRON testing machine with a cross-head speed of.5 mm min till failure. The ultimate tensile strength was reached at 25% strain. Part of the gauge section of the deformed specimen was cut out and half the sample was ground away so that the longitudinal midsection was exposed. After further polishing, the sample was 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 detector, lattice orientations on a central section of the specimen (comprising the tensile direction) were recorded with the Channel5 software from HKL Technology (Hobro, Denmark). The investigated regions are well outside the neck and only the homogeneously deformed part is analysed (Krog-Pedersen et al., 29). Orientations were determined on the longitudinal midsection in a square grid of 8 points with a distance of mm between measuring points. The disorientations between Figure 7 -Sections of orientation density function (a) and pole density functions for crystallographic forms fg; fg; fg and f3g (b) of spatially indexed individual orientations of grain 4, augmented with the mean orientation marked by a red dot. Figure 8 Grain 47 with 4324 spatially indexed individual orientations as a colourcoded orientation map according to the ðþ inverse pole figure colour bar (a) and RGB colours (b) and as a three-dimensional axis angle scatter plot centred at the mean orientation (c). 346 Florian Bachmann et al. Inferential statistics of EBSD data J. Appl. Cryst. (2). 43,

11 all neighbouring points were 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 large grains were selected to represent different shapes of orientation distributions; their orientation maps are presented in Figs. 5, 8 and, with the tensile axis displayed vertically. Owing to the medium stacking-fault energy of copper, many twins are present which can be recognized in the orientation maps as twin lamellae (e.g. the horizontal white cut in the grain shown in Fig. 8). Next we shall analyse the distribution of crystallographic orientations of three individual grains, labelled grain 4, grain 47 and grain 9. The data of grain 4 are depicted as colour-coded orientation maps and as a three-dimensional axis angle scatter plot centred at the identity in Fig. 5, as pole point plots in Fig. 6, and as pole density and orientation densities in Fig. 7. In the same way, the data of grain 47 are depicted in Figs. 8, 9 and. Figs., 2 and 3 display the data of grain 9. The analyses of the orientation data, in particular the spectral analyses of the orientation matrices T, are summarized numerically in Tables and 2. The orientation maps of grain 4 (Fig. 5) exhibit only gentle colour variations, corresponding to a unimodal orientation density function (Fig. 7). Its pole point plots (Fig. 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 ðþ pole point plot is almost spherical, indicating that this is the direction of the axis of internal rotation or bending within the grain, which is confirmed by the vector components of the eigenvector Vecða 2 Þ¼ð:499455; :545993; :8363Þ T corresponding to the second largest eigenvalue. The orientation maps of grain 47 (Fig. 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 9 (Fig. ) indicate that it comprises two parts with distinct orientations, clearly visible as Figure 9 Pole point plots of grain 47 for crystallographic forms fg; fg; fg and f3g, augmented with the mean orientation marked by a red dot emphasizing a small dispersion. Figure -Sections of orientation density function (a) and pole density functions for crystallographic forms fg; fg; fg and f3g (b) of spatially indexed individual orientations of grain 47, augmented with the mean orientation marked by a red dot. J. Appl. Cryst. (2). 43, Florian Bachmann et al. Inferential statistics of EBSD data 347

12 bimodality in the pole point, pole and orientation density plots (Fig. 3). The centred scatter plots agree roughly with the analyses of the ratios of eigenvalues and the statistical tests. These differences in the symmetry of the orientation density function originate from the peculiarities of plastic deformation. Crystal plasticity is achieved by glide on crystallographic glide systems: fg hi glide systems in the case of copper. Depending on the number of activated glide systems, intragrain disorientations develop with different degrees of freedom for the main rotation axis. For instance, a prolate distribution with a single dominating rotation axis indicates a single degree of freedom in selecting slip systems, whereas spherical symmetric distributions require three degrees of freedom in selecting slip systems (Krog-Pedersen et al., 29). 6. Conclusions Conclusions are restricted to rather methodological aspects. The proper statistic of individual orientation measurements is the orientation tensor T, if independence is assumed. For Table Summary of the spectral analysis of the orientation tensor T for grain 4, grain 47 and grain 9. Data comprise texture index, entropy, largest eigenvalue corresponding to principal axes a of smallest inertia, remaining eigenvalues i and their ratios, and estimated shape parameters i of the Bingham quaternion distribution, involving either equation (36) or the large-concentration approximation, equation (44). simiom Grain 4 Grain 47 Grain 9 Sample size Texture index Entropy = = Visual inspection Spherical Prolate Spherical Oblate Computation involving F, equation (36) Computation applying large-concentration approximation, equation (44) Figure Grain 9 with 2253 spatially indexed individual orientations as a colourcoded orientation map according to the ðþ inverse pole figure colour bar (a) and RGB colours (b) and as a three-dimensional axis angle scatter plot centred at the mean orientation (c). Figure 2 Pole point plots of grain 9 for crystallographic forms fg; fg; fg and f3g, augmented with the mean orientation marked by a red dot emphasizing a small dispersion. 348 Florian Bachmann et al. Inferential statistics of EBSD data J. Appl. Cryst. (2). 43,

13 Table 2 Summary of the spectral analysis of the orientation tensor T for grain 4, grain 47 and grain 9. Data comprise tests for the spherical, oblate and prolate Bingham case applying shape parameters estimated with either equation (36) or the largeconcentration approximation, equations (45), (47), (46), respectively, and large-sample approximation tests for spherical, oblate and prolate symmetry, respectively, without parametric assumptions, equations (49), (5), (5). a indicates that the test does not apply, no rejection indicates that rejection at any reasonable level of significance is not possible. simiom Grain 4 Grain 47 Grain 9 Sample size Bingham quaternion distribution on S 3 or using large-sample approximations avoiding any parametric assumptions. Then significance tests prove the absence of symmetry if they lead to the rejection of the null hypothesis with respect to a given significance. Comparing the ðx; yþ plots of the fabricated and the experimental data suggests that one should proceed to methods of spatial statistics as introduced by van den Boogaart & Schaeben (22a,b), e.g. to capture the spatially induced correlation. Bingham statistics involving F, equations (39), (42), (43) p spherical.3895 Spherical No rejection Reject Reject Reject p oblate.99 Oblate Reject Reject Reject for >p p prolate.22 Prolate Reject for >p Reject Reject Bingham statistics involving large-concentration approximation, equations (45), (46), (47) p spherical.3895 Spherical No rejection Reject Reject Reject p oblate. Oblate Reject Reject Reject for >p p prolate.22 Prolate Reject for >p Reject Reject Large-sample approximation statistics without parametric assumptions, equations (49), (5), (5) p spherical.43 Spherical No rejection Reject Reject Reject p oblate.43 Oblate Reject Reject Reject for >p p prolate.48 Prolate Reject for >p Reject Reject sufficiently concentrated distributions the first principal axis of inertia (corresponding to the largest eigenvalue of T, i.e. the smallest 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 quaternion (and its negative) of some kernel-estimated orientation density. However, neither the normalized mean quaternion nor the modal quaternion leads to inferential statistics. The ratios of the three remaining small eigenvalues may indicate the geometric shape of the sample of individual orientations. Special cases are oblateness, prolateness or a spherical shape. They may reveal details of the texturegenerating processes, as different deformation regimes or differently activated sets of slip systems. While a prolate shape of the distribution may be identified by means of pole point, pole or orientation density plots, oblate and spherical shapes can hardly be distinguished. 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 interpreted 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 Figure 3 -Sections of orientation density function (a) and pole density functions for crystallographic forms fg; fg; fg and f3g (b) of spatially indexed individual orientations of grain 9, augmented with the mean orientation marked by a red dot. J. Appl. Cryst. (2). 43, Florian Bachmann et al. Inferential statistics of EBSD data 349

14 APPENDIX A Tests of rotational symmetry of axial distributions A. A useful proposition Proposition. IfX ¼ðX ;...; X p Þ T is a random vector with then Proof. First note that X Nð;I p T Þ Pp ðx i XÞ 2 2 p : i¼ varðx XÞ ¼ðI p p T ÞðI p T ÞðI p p T Þ ¼ ði p p T Þ: Take V independent of X with V Nð;=pÞ. Then and so Xp i¼ X X þ V Nð;I p Þ; ðx i X þ VÞ 2 Xp ¼ i¼ ðx i XÞ 2 þ p pv2 2 p; from which the result follows by Cochran s theorem. A2. Testing O(p 2) symmetry of Bingham distributions ð52þ ð53þ ð54þ ð55þ ð56þ Consider the Bingham distributions on RP p with densities f ðx; AÞ ¼ F ð=2; p=2; AÞ expðx T AxÞ; ð57þ where x ¼ðx ;...; x p Þ T is a random unit vector and A is a symmetric p p matrix. We shall use the spectral decomposition A ¼ Pp i¼ i i T i ð58þ of A, where ;...; p are orthonormal vectors and p. We are interested in the hypothesis H prolate : 3 ¼¼ p ð59þ of Oðp 2Þ symmetry of the distribution about the plane spanned by and 2, and in the hypothesis H oblate : 2 ¼¼ p ð6þ of Oðp 2Þ symmetry of the distribution about the plane spanned by and p. Observations x ;...; x n on RP p can be summarized by the scatter matrix T about the origin, defined (see x9.2. of Mardia & Jupp, 2) by T ¼ n i¼ x i x T i : ð6þ Let t ;...; t p (with t t p ) denote the eigenvalues of T. From x.3.37 of Mardia & Jupp (2), the maximum likelihood estimates ^ ;...; ^ p of ;...; p under the full Bingham model are given log F ð=2; p=2; i ¼ t i ; i ¼ ;...; p; ð62þ K¼ ^K where ^K ¼ diagð ^ ;...; ^ p Þ. (Recall that ^ ;...; ^ p can be determined only up to addition of a constant, since, for any real constant c, the matrices A and A þ ci p give the same distribution.) The maximum likelihood estimates ~ ; ~ 2 ; ~;...; ~ of ;...; p under H prolate are given (up to addition of a constant) log F ð=2; p=2; ¼ t i ; i ¼ ; 2; ð63þ i K¼ log F ð=2 semip=2; ¼ ~t; ð64þ 3 K¼ ~K where ~K ¼ diagð ~ ; ~ 2 ; ~;...; ~Þ and ~t ¼ X p t p 2 i : ð65þ i¼3 Similarly, the maximum likelihood estimates ; ;...; ; p of ;...; p under H oblate are given (up to addition of a constant) log F ð=2; p=2; KÞ K¼ ¼ t i; i ¼ ; p; ð66þ log F ð=2; p=2; KÞ K¼ ¼ t; ð67þ 2 where K ¼ diagð ; ;...; ; p Þ and t ¼ X p t p 2 i : ð68þ i¼2 It follows from standard results on regular exponential models that the likelihood ratio statistics w prolate and w oblate of H prolate and H oblate satisfy the large-sample approximations w prolate ~w prolate ¼ n Pp ð ^ i ~Þðt i ~tþ; i¼3 w oblate w oblate ¼ n Pp ð ^ i Þðt i tþ: i¼2 ð69þ ð7þ In the case p ¼ 4, equations (69) and (7) can be rewritten as w prolate ~w prolate ¼ n 2 ð ^ 3 ^ 4 Þðt 3 t 4 Þ; w oblate w oblate ¼ n 2 ð ^ 2 ^ 3 Þðt 2 t 3 Þ: ð7þ ð72þ The parameter matrix A of a Bingham distribution in H prolate is specified by the pair of orthogonal p vectors and 2 2, 35 Florian Bachmann et al. Inferential statistics of EBSD data J. Appl. Cryst. (2). 43,

15 and so the dimension of H prolate is p þðp Þ. Similarly, the parameter matrix A of a Bingham distribution in H oblate is specified by the pair of orthogonal p vectors and p p, and so the dimension of H oblate is p þðp Þ. The parameter space of the Bingham distributions has dimension pðp þ Þ=2. Thus the codimension of both H prolate and H oblate is pðp 3Þ ½ pðp þ Þ=2 Š ½pþðp ÞŠ ¼ : ð73þ 2 It follows from standard results on likelihood ratio statistics that the (large-sample) asymptotic distributions of ~w prolate and w oblate (and w prolate and w oblate ) under the null hypotheses H prolate and H oblate are ~w prolate : 2 pðp 3Þ=2; ð74þ w oblate : 2 pðp 3Þ=2; ð75þ where : means is asymptotically distributed as (provided that > 2 > 3 in the prolate case and > 2 and p > p in the oblate case). A3. 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 Oðp rþ symmetry of the distribution about the r-dimensional subspace spanned by r orthonormal vectors ;...; r. Without loss of generality, we can take the unit eigenvectors of Eðxx T Þ to be ;...; p with i ¼ð; fflfflfflffl{zfflfflfflffl}...; ; ; ;...; Þ for i ¼ ;...; p. Put i c ij ¼ Eðx 2 i x 2 j Þ; i; j p: ð76þ Under Oðp rþ symmetry, the matrices ½Eðx i x j ÞŠ and ðc ij Þ have the form ½Eðx i x j ÞŠ ¼ diagð ;...; r Þ ð77þ I p r and ðc ij Þ¼ c... c r c ;rþ T c 2... c 2r c 2;rþ T c r... c rr c r;rþ T c ;rþ... c r;rþ ðc rþ;rþ c rþ;rþ2 ÞI p r þ c rþ;rþ2 T C A ð78þ P r i¼ i þðp rþ ¼ : ð8þ Multiplying equation (79) by x 2 j and taking the expectation gives P r i¼ c ji þðp rþc j;rþ ¼ j ; j ¼ ;...; r: ð8þ Similarly, multiplying equation (79) by x 2 rþ and taking the expectation gives P r j¼ c rþ;j þ c rþ;rþ þðp r Þc rþ;rþ2 ¼ : Further, ðcos Þxr þðsin Þx rþ 4 ¼ðcos Þ 4 x 4 r þ 4ðcos Þ 3 ðsin Þx 3 rx rþ and so þ 6ðcos Þ 2 ðsin Þ 2 x 2 rx 2 rþ þ 4ðcos Þðsin Þ 3 x r x 3 rþ þðsin Þ 4 x 4 rþ; c rr ¼ðcos Þ 4 c rr þ þ 6ðcos Þ 2 ðsin Þ 2 c r;rþ þ þðsin Þ 4 c rr ð82þ ð83þ ¼ðcos 2 þ sin 2 Þ 2 c rr þ 2 cos 2 sin 2 ð3c r;rþ c rr Þ; 8; ð84þ leading to c rr ¼ 3c r;rþ : ð85þ Substituting equations (8), (8) and (85) into (82) yields! X r p r j Xr c ij þðp rþ2þc r;rþ ¼ P r i¼ i ; p r j¼ i¼ ð86þ and so c r;rþ ¼ 2 P r i¼ i þ P r P r i¼ j¼ c ij : ð87þ ðp rþðp r þ 2Þ Simple calculations show that ðt rþ;rþ ;...; t pp ; t rþ;rþ2 ;..., t p ;p Þ T has variance matrix " # ðc rþ;rþ c rþ;rþ2 ÞI p r þðc rþ;rþ2 2 Þ T : c rþ;rþ2 I ðp rþðp r Þ=2 ð88þ It follows from equation (85) that this is equal to 2I c p r þð 2 =c rþ;rþ2 Þ T rþ;rþ2 : ð89þ I ðp rþðp r Þ=2 where ¼ð;...; Þ T. From the definition of S p, P p i¼ x 2 i ¼ : Taking the expectation of equation (79) gives ð79þ Partition T ¼ðt ij Þ (according to the subspaces spanned by ;...; r and rþ ;...; p, respectively) as T ¼ T T 2 ; ð9þ T 2 T 22 where J. Appl. Cryst. (2). 43, Florian Bachmann et al. Inferential statistics of EBSD data 35

16 t... t r B T ¼..... A ð9þ t r... t rr and t rþ;rþ... t rþ;p B T 22 C A: ð92þ t p;rþ... t p;p Since the unit eigenvectors of T are Oðn =2 Þ close to ;...; p, we have T 22 trðt 22Þ p r I p r 2¼ Xp ðt i t Þ 2 þ 2 X t 2 ij þ Oðn =2 Þ; i¼rþ rþi < jp ð93þ where t ;...; t p denote the eigenvalues of T and t ¼ trðt 22 Þ=ðp rþ. It follows from equation (89) and the Proposition at the beginning of this appendix that T 2c 22 trðt 2 22Þ rþ;rþ2 p r I p r : 2 ðp r Þðp rþ2þ=2; n!: ð94þ From equation (87), a suitable estimate of c rþ;rþ2 is ^c rþ;rþ2 ¼ 2 P r i¼ i þ P r P r i¼ j¼ ^c ij ; ðp rþðp r þ 2Þ ð95þ where ^ i ¼ t i ; i ¼ ;...; r; P ^c i j ¼ n ðx T kt i Þ 2 ðx T kt j Þ 2 ; i; j ¼ ;...; r; ð96þ k¼ with t ;...; t p being the unit eigenvectors of T. A3.. The case r =. This gives Bingham s (974) test of uniformity (see p. 232 of Mardia & Jupp, 2). Uniformity is rejected for large values of pðp þ 2Þ nkt p I 2 p k 2 ¼! pðp þ 2Þ n Xp t 2 i ; ð97þ 2 p i¼ where t ;...; t p denote the eigenvalues of T. Under uniformity,! pðp þ 2Þ n Xp t 2 i : 2 2 p ðpþ2þðp Þ=2; n!: ð98þ i¼ A3.2. The case r =. This gives Prentice s (984) test of symmetry about an axis (see p. 235 of Mardia & Jupp, 2). Symmetry about the major axis is rejected for large values of T 2^c 22 trðt 2 22Þ 23 p I p ¼ nðp 2 Þ½ P p i¼2 t 2 i ð t Þ 2 =ðp ÞŠ ; ð99þ 2ð 2t þ ^c Þ where t ;...; t p (with t t p ) denote the eigenvalues of T corresponding to eigenvectors t ;...; t p and ^c ¼ n i¼ 4: x T i t ðþ Under symmetry, nðp 2 Þ½ P p i¼2 t 2 i ð t Þ 2 =ðp ÞŠ : 2 2ð 2t þ ^c Þ ðpþþðp 2Þ=2; n!: ðþ Similarly, symmetry about the minor axis p is rejected for large values of T 2^c 22 trðt 2 22Þ 23 p I p ¼ nðp 2 Þ½ P p i¼ t 2 i ð t p Þ 2 =ðp ÞŠ ; ð2þ 2ð 2t p þ ^c pp Þ where ^c pp ¼ n i¼ ðx T i t p Þ 4 : ð3þ Under symmetry, nðp 2 Þ½ P p i¼ t 2 i ð t p Þ 2 =ðp ÞŠ : 2 2ð 2t p þ ^c pp Þ ðpþþðp 2Þ=2; n!: ð4þ A3.3. The case r =2. We consider the prolate and oblate cases in turn. (a) Prolate case Oðp 2Þ symmetry of the distribution about the plane spanned by and 2 is rejected for large values of T 2^c 22 trðt 2 22Þ 34 p 2 I p 2 ¼ npðp 2Þ½ P p i¼3 t 2 i ð t t 2 Þ 2 =ðp 2ÞŠ ; ð5þ 2½ 2ðt þ t 2 Þþ^c þ 2^c 2 þ ^c 22 Š where t ;...; t p (with t t p ) denote the eigenvalues of T corresponding to eigenvectors t ;...; t p and ^c ij ¼ n k¼ x Tkt i 2 x T kt j 2: ð6þ Under symmetry, npðp 2Þ½ P p i¼3 t 2 i ð t t 2 Þ 2 =ðp 2ÞŠ : 2 2½ 2ðt þ t 2 Þþ^c þ 2^c 2 þ ^c 22 Š pðp 3Þ=2; n!; ð7þ and this statistic is asymptotically equal to w prolate of equation (69). (b) Oblate case Similarly, Oðp 2Þ symmetry of the distribution about the plane spanned by and p is rejected for large values of 352 Florian Bachmann et al. Inferential statistics of EBSD data J. Appl. Cryst. (2). 43,