Grain boundary (GB) energy is a fundamental property that affects the form of grain boundary and plays an important role to unveil the behavior of polycrystalline materials. With a better understanding of grain boundary energy distribution (GBED), we can produce more durable and efficient materials that will further improve productivity and reduce loss. The lack of robust GB structure-property relationships still remains one of the biggest obstacles towards developing true bottom-up models for the behavior of polycrystalline materials. Progress has been slow because of the inherent complexity associated with the structure of interfaces and the vast five-dimensional configurational space in which they reside. Estimating the GBED is challenging from a statistical perspective because there are not direct measurements on the grain boundary energy. We only have indirect information in the form of an unidentifiable homogeneous set of linear equations. In this paper, we propose a new statistical model to determine the GBED from the microstructures of polycrystalline materials. We apply spline-based regression with constraints to successfully recover the GB energy surface. Hamiltonian Monte Carlo and Gibbs sampling are used for computation and model fitting. Compared with conventional methods, our method not only gives more accurate predictions but also provides prediction uncertainties.
Bayesian bandwidth estimation for a nonparametric functional regression model with mixed types of regressors and unknown error density
在误差密度未知的框架下，研究了具有函数值、连续实值和离散值回归变量的非参数函数回归模型的带宽估计问题.从Shang（2013，Computational Statistics & Data Analysis）的最近工作扩展，我们通过残差的核密度估计来近似未知误差密度，其中回归函数由允许混合类型回归变量的函数Nadaraya-Watson估计器估计。在核形式误差密度下，推导了带宽参数的核似然和后验密度，提出了一种同时估计带宽的贝叶斯带宽估计方法。仿真研究表明，所提出的贝叶斯方法的回归函数和误差密度的估计精度。以食品质量控制中的光谱数据集为例，我们应用贝叶斯方法来选择具有混合类型回归变量的非参数函数回归模型中的最佳带宽。
