SphericalDistributionsRand
The random sample generators for the two spherical distributions: von Mises-Fisher, Watson.
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This paper considers statistical estimation problems where the probability distribution of the observed random variable is invariant with respect to actions of a finite topological group. It is shown that any such distribution must satisfy a restricted finite mixture representation. When specialized to the case of distributions over the sphere that are invariant to the actions of a finite spherical symmetry group G, a group-invariant extension of the Von Mises Fisher (VMF) distribution is obtained. The G-invariant VMF is parameterized by location and scale parameters that specify the distribution's mean orientation and its concentration about the mean, respectively. Using the restricted finite mixture representation these parameters can be estimated using an Expectation Maximization (EM) maximum likelihood (ML) estimation algorithm. This is illustrated for the problem of mean crystal orientation estimation under the spherically symmetric group associated with the crystal form, e.g., cubic or octahedral or hexahedral. Simulations and experiments establish the advantages of the extended VMF EM-ML estimator for data acquired by Electron Backscatter Diffraction (EBSD) microscopy of a polycrystalline Nickel alloy sample.
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The random sample generators for the two spherical distributions: von Mises-Fisher, Watson.
This paper considers estimation of parameters of distributions whose domain is a particular non-Euclidean geometry: a topological space divided into equivalence classes by actions of a finite spherical symmetry group. A well known example of a finite spherical symmetry group is the point group in 3 dimensions describing the soccer ball, or football, with truncated icosahedral symmetry that also corresponds to the symmetry of the Carbon-60 molecule. This paper formulates a general approach to parameter estimation in distributions defined over such domains. First we establish a restricted finite mixture representation for probability distributions that are invariant to actions of any topological group. This representation has the property that the number of mixture components is equal to the order of the group, the distributions in the mixture are all parameterized by the same parameters, and the mixture coefficients are all equal. This is practically significant since many reliable algorithms have been developed for parameter estimation when samples come from finite mixture distributions.
We illustrate the power of the representation for an important problem in materials science: analysis of mean orientation in polycrystals. Crystal orientation characterizes properties of materials including electrical conductivity and thermal conductivity. Mechanical properties, such as, stiffness, elasticity, and deformability, can also depend on the distribution of crystal orientations over the material. Thus accurate estimation of crystal orientation is useful for materials evaluation, testing and prediction.
The mean orientation of the crystal, characterized by its Euler angles, can only be specified modulo a set of angular rotations determined by the symmetry group associated with the specific type of crystal. This multiplicity of equivalent Euler angles complicates the development of reliable mean orientation estimators. By extending the Von Mises Fisher (VMF) model under the proposed finite mixture representation, and applying the expectation maximization (EM) maximum likelihood (ML) algorithm for mixtures, we obtain an accurate iterative estimator of the mean Euler angle parameter and angular concentration parameter of the extended VMF distribution. Specifically, the VMF extension is accomplished as follows. We start with the standard VMF model, which is a density parameterized by location (angle mean) and scale (angle concentration) defined over the -dimensional sphere [1]
. In this model, a point on the sphere is specified by its direction vector, and the angle between two vectors is the arc-cosine of the normalized inner product between them. The spherical symmetry group extension is accomplished by applying the mixture representation to the standard VMF distribution using the group of quaternion rotation matrices.
The performance of the proposed EM-ML orientation estimator is evaluated by simulation and compared to two other angle estimators. The ML orientation estimator is then illustrated on EBSD data collected from a Nickel alloy whose crystal form induces the cubic point symmetry group. We establish that the ML orientation estimator results in significantly improved estimates of the mean direction in addition to providing an accurate estimate of concentration about the mean.
The paper is organized as follows. Section II describes group invariant random variables and gives the mixture representation for their densities. Section III specializes to random variables invariant relative to actions of the spherical symmetry group and develops the -invariant VMF distribution along with EM-ML parameter estimator. The crystallography application is presented in section IV along with experimental comparisons. Section V has concluding remarks.
Consider a finite topological group of distinct actions on a topological space , and a binary operation ”*” defining the action composition , denoted . has the properties that composition of multiple actions is associative, for every action there exists an inverse action, and there exists an identity action [2]. A real valued function on is said to be invariant under if: for . Let be a random variable defined on . We have the following theorem for the probability density of .
The density function is invariant under if and only if
(1) |
Proof: If (1) holds then . Since is a group so that and . On the other hand, if then .
Theorem II.1 says that any density that is invariant under group can be represented as a finite mixture of its translates under the group’s actions . This simple result has important implications on
-invariant density estimation and parameter estimation. In particular it can be used to construct maximum likelihood estimators for parametric densities and kernel density estimators of non-parametric
-invariant densities with finite sample guaranteed performance.To illustrate the non-parametric case, assume that has topological dimension with Lebesgue -invariant density . Define the symmetric non-negative second order kernel function , i.e., , , , and . For the finite group , define the -invariant kernel function . Given a realization of i.i.d. samples from define the kernel density estimator . Assume that is a sequence of kernel widths that satisfies while . Then, if is smooth, using Thm II.1 and concentration results from [3], it can be shown that as goes to infinity
where is the norm difference.
For the parametric case, let be a density on that is parameterized by a parameter in a parameter space . We extend to a -invariant density by using Thm. II.1, obtaining:
(2) |
where . This density is of the form of a finite mixture of densities of known parametric form where the mixture coefficients are all identical and equal to . Maximum likelihood (ML) estimation of the parameter from an i.i.d. sample from any -invariant density can now be performed using finite mixture model methods [4] such as the Expectation-Maximization (EM) algorithm [5]
or the restricted Boltzman machine (RBM)
[6].In this section we specialize to estimation of parameters for the case that the probability density is on a sphere and is invariant to actions in a spherical symmetry group. In Sec. IV this will be applied to a crystallography example under a Von-Mises-Fisher likelihood model for the mean crystal orientation. The measured and mean orientations can be represented in three equivalent ways.
The orientation is defined by a set of three successive rotations of a reference unit vector about the specified axes [7]. Denote the Euler angles as , where and .
The quaternion representation describes the orientation as a 4D vector on the 3D sphere [8]: , where . The main advantage of this representation is that any rotation of is simply accomplished via left multiplication by a orthogonal matrix called a quaternion matrix.
The Rodrigues vector describes the orientation by rotating a reference vector along one direction by angle according to the right hand rule [9]. It is denoted as , where and .
Any of the aforementioned orientation representations have inherent ambiguity due to crystal symmetries. For example, if the crystal has cubic symmetry, its orientation is only uniquely defined up to a 48-fold set of rotations, reflections and inversions of the cube about its symmetry axes. These rotations reflections, and inversions can be represented as a point symmetry group , called , of quaternionic matrices operating on the 3D sphere. Two orientations, e.g., represented in Euler angle, Quaternion or Rodrigues forms, are called symmetry-equivalent to each other if they are mapped to an equivalent orientation under . A fundamental zone (FZ), also called the fundamental domain, is a conic solid that can be specified to disambiguate any particular orientation . However, as will be seen in Sec. IV, reduction of the entire data sample to a FZ destroys information necessary for maximum likelihood estimation: the entire -invariant density (2) must be used.
The von Mises-Fisher (VMF) distribution arises in directional statistics [1]
as a natural generalization of the multivariate Gaussian distribution to the
-dimensional sphere , where . The VMF distribution is parameterized by the mean direction and the concentration parameter :(3) |
where and is the modified Bessel function of the first kind of order . Given an i.i.d sample from the VMF distribution the ML estimator has the closed form expressions [1]
(4) |
where and .
Let be a group of symmetric actions acting on the quaternionic representation of orientation on the dimensional sphere . This group is called a spherical symmetry group. We extend the the VMF distribution (3) using the mixture representation Thm II.1:
(5) | |||||
(6) |
where in going from (5) to (6) we used the inner product form in (3) and the symmetry of . The expression (6) for the extended VMF distribution is in the form of a finite mixture of standard VMF distributions on the same random variable having different mean parameters but having the same concentration parameters .
The finite mixture (6) for the -invariant density is in a form for which an EM algorithm [5] can be implemented to compute the ML estimates of and . Denoting the parameter pair as the EM algorithm generates a sequence of estimates that monotonically increase the likelihood and are given by , where is a latent variable assigning to a particular mixture component in (6) and is the likelihood function of given the complete data . Specifically,
where . The EM algorithm takes the form:
E-step:
(8) |
M-step:
(9) |
where .
Polycrystalline materials are composed of grains, of varying size and orientation, where each grain contains crystal forms with similar orientations. The quality of the material is mainly determined by the grain structure i.e. the arrangement of the grains and their sizes, as well as the distribution of the precipitates. Analyzing the crystal orientation of the grains helps us predict how materials fail and what modes of failure are more likely to occur [10].
Electron backscatter diffraction (EBSD) microscopy acquires crystal orientation at multiple locations within a grain by capturing the Kikuchi diffraction patterns of the backscatter electrons [11]. A Kikuchi pattern can be translated to crystal orientation through Hough Transformation analysis [12] or Dictionary-Based indexing [13]. The process of assigning mean orientation values to each grain is known as indexing. Crystal forms possess point symmetries, e.g. triclinic, tetragonal, or cubic, leading to a probability density of measured orientations that is invariant over an associated spherical symmetry group . Therefore, when the type of material has known symmetries, e.g., cubic-type symmetry for nickel or gold, the extended VMF model introduced in the previous section can be applied to estimate the mean orientation and the concentration associated with each grain.
A set of i.i.d. samples were simulated from the -invariant VMF distribution with given for the point symmetry group associated with the symmetries of cubic crystal lattice planes. The number of samples for each simulation was set to and was swept from to while, for each simulation run, was selected uniformly at random. The experiment was repeated times and the average values of and the inner product are shown in Fig. 2 and 1. In the figures we compare performance for the following methods: (1) the naive ML estimator for the standard VMF model that does not account for the point group structure (4) (labeled ”ML for VMF”). (2) Mapping the samples to a single fundamental zone of on the sphere followed by performing ML for the standard VMF distribution over this FZ (labeled ”Modified ML for VMF”). (3) Applying our proposed exact EM-ML algorithm directly to the samples using the the mixture of VMF distribution (8)-(9) (labeled ”EM-ML for mVMF”).
Figure 1 shows the inner product values . The proposed EM-ML estimator achieves perfect recovery of the mean orientation () much faster than the other methods as the concentration parameter increases (lower dispersion of the samples about the mean). Notice that when , none of the methods can accurately estimate the mean orientation. The reason is that when
is small the samples become nearly uniformly distributed over the sphere. The threshold
value at which performance starts to degrade depends on the point symmetry group. In Fig. 2 it is seen that the bias of the proposed EM-ML estimator is significantly lower than that of the other methods compared. While the modified ML for VMF performs better than the naive ML estimator for VMF, its bias is significantly worse bias than the proposed EM-ML approach.We next illustrate the proposed EM-ML orientation estimator on a real IN100 sample acquired from US Air Force Research Laboratory (AFRL) [13]. The IN100 sample is a polycrystalline Ni superalloy which has cubic symmetry in the point symmetry group. EBSD orientation measurements were aquired on a pixel grid, corresponding to spatial resolution of nm. The Kikuchi diffraction patterns were recorded on a photosensitive detector for each of the pixels.
Figure 3 (a) shows a sub-region of the full EBSD sample where the orientations are shown in the inverse pole figure (IPF) coloring obtained from the OEM EBSD imaging software and (c) is the back-scattered electron (BSE) image. Note that the OEM-estimated orientations in some grain regions of the IPF image are very inhomogeneous, which is likely due to a fundamental zone wrap-around problem. Figure 3 (b) shows the estimates of the mean orientations of each grain using the proposed EM-ML algorithm. Figures 3 (d) show the estimated concentration parameter for the grains using the proposed EM-ML algorithm.
We have obtained a general finite mixture representation for densities on domains whose topologies have group invariances. This representation was used to extend the Von-Mises-Fisher distribution to a mixture VMF distributions that possess spherically symmetric group invariances. An efficient EM algorithm was derived for estimation of parameters of this extended VMF model. The extended VMF model was applied to the problem of estimation of mean grain orientation parameters in polycrystalline materials whose orientations lie in the point symmetry group. Application of the finite mixture representation to other types of groups would be worthwhile future work.
The authors are grateful for inputs from Megna Shah, Mike Jackson and Mike Groeber.
In [14] a set of conditions defining the fundamental zone are given in terms of Rodrigues space:
(10) |
Here and are the rotation angle and the unit vector of the rotation axis of the -th element of the rotation symmetry group. is the order of the group and corresponds to the identity operator.
For cubic symmetry corresponding to the point symmetry group, there are Rodrigues symmetry operators. The transformation equations between the Rodrigues vector and the quaternion are as follows:
(11) |
The Expectation-Maximization algorithm is an iterative approach for obtaining maximum-likelihood parameter estimates in models where there are hidden latent variables and, in particular, finite mixture models. The algorithm alternates between performing two steps: ”Expectation step (E-step)” and ”Maximization step (M-step)” in each iteration. In the E-step, one calculates the expectation of the complete data log-likelihood function assuming the parameters of the model are fixed. In the M-step, the updated parameters are estimated by maximizing the expectation function. The process is repeated until the objective function converges.
Here we assume that all the observed quaternions are realizations from the -fold finite mixture of Von Mises-Fisher (VMF) distributions which has the PDF (13).
(13) |
Notice that The only parameters we need to estimate are the mean parameter and the concentration parameter . The latent variable indicates the index of the distribution a sample , , belongs to. Based on the model, the log-likelihood function given the data has the following form:
where is the set of parameters. The EM objective function (called the function) to be maximized is:
(14) |
where
is the posterior probability that
. For the E-step, denote the parameters to be estimated as at -th iteration, can be calculated by:(15) |
In the M-step the parameters are updated by maximizing the function. Taking the partial derivative w.r.t of .(14)) subject to the constraint we have:
(16) |
where is a Lagrange multiplier. By taking the partial derivative w.r.t , the ML estimator of is:
(17) |
where .
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