Polycrystalline materials, such as metals, are comprised of heterogeneously oriented crystals. Observed crystal orientations are modelled as a sample from an orientation distribution function (ODF), which determines a variety of material properties and is therefore of great interest to practitioners. Observations consist of quaternions, 4-dimensional unit vectors reflecting both orientation and rotation of a single crystal. Thus, an ODF must account for known crystal symmetries as well as satisfy the unit length constraint. A popular method for estimating ODFs non-parametrically is symmetrized kernel density estimation. However, disadvantages of this approach include difficulty in interpreting results quantitatively, as well as in quantifying uncertainty in the ODF. We propose to use a mixture of symmetric Bingham distributions as a flexible parametric ODF model, inferring the number of mixture components, the mixture weights, and scale and location parameters based on crystal orientation data. Furthermore, our Bayesian approach allows for structured uncertainty quantification of the parameters of interest. We discuss details of the sampling methodology and conclude with analyses of various orientation datasets, interpretations of parameters of interest, and comparison with kernel density estimation methods.
翻译:观测由四维、四维单位矢量组成,反映单一晶体的方向和旋转。因此,ODF必须说明已知的晶体对称性,并满足单位长度限制。一种非对称性估计ODFs的流行方法是平衡内核密度估计。然而,这一方法的缺点包括难以从数量上解释结果,以及难以量化ODF的不确定性。我们提议使用一种对称Bingham分布的混合物混合物,作为灵活的ODF模型,推断混合物成分的数量、混合物重量以及基于晶体定向数据的尺度和位置参数。此外,我们采用Bayesian方法,可以对兴趣参数进行结构上的不确定性量化。我们讨论取样方法的细节,并在分析各种定向数据组时作出结论。