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High-order parameter approximation for von Mises-Fisher distributions

机译：von Mises-Fisher分布的高阶参数逼近

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This paper concerns the issue of the maximum-likelihood estimation (MLE) for the concentration parameters of the von Mises-Fisher (vMF) distributions, which are crucial to directional data analysis. In particular, we study the numerical approximation approach for solving the implicit nonlinear equation arising from building the MLE of the concentration parameter κ of vMF distributions. In addition, we address the implementation of ~(Is)(x), the modified Bessel function of the first kind, which is the most time-consuming and fundamental ingredient in the proposed approximation scheme of the MLE for κ. The main contribution of this paper is two fold. The first is to present a two-steps Halley based method for exploring a high-order approximation of the MLE for κ, which can significantly contribute to the improvement of estimation accuracy. The second is to develop a novel approach for the implementation of ~(Is)(x), which can make the substantial improvement of computation efficiency for computing the MLE approximation for κ. The numerical experiments were conducted to compare the proposed schemes with those in the existing works by Tanabe et al. [1] and Sra [2]. The experimental results show that, given the same amount of computation as their methods, the proposed high-order scheme can achieve much more accurate approximations while our implementation of ~(Is)(x) is preferable yet desirable for high dimensional applications.
机译：本文涉及von Mises-Fisher（vMF）分布的浓度参数的最大似然估计（MLE）问题，这对定向数据分析至关重要。特别是，我们研究了数值近似方法，用于求解由建立vMF分布的浓度参数κ的MLE引起的隐式非线性方程。另外，我们解决了〜（Is）（x）的实现，这是第一类经过修改的Bessel函数，它是针对κ的MLE拟议近似方案中最耗时和最基本的成分。本文的主要贡献有两个方面。首先是提出一种基于哈雷的两步法，用于探索κ的MLE的高阶近似，这可以极大地有助于估计精度的提高。第二个是开发一种实现〜（Is）（x）的新颖方法，可以大大提高计算κ的MLE近似的计算效率。进行了数值实验，以将所提出的方案与Tanabe等人现有工作中的方案进行比较。 [1]和Sra [2]。实验结果表明，与它们的方法相同的计算量，提出的高阶方案可以实现更精确的逼近，而我们对〜（Is）（x）的实现是可取的，但对于高尺寸应用而言是理想的。

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