Discriminant analysis for a folded Watson distribution

When directional data fall in the positive orthant of the unit hypersphere, a folded directional distribution is preferred over a simple directional distribution for modeling the data. Since directional data, especially axial data, can be modeled using a Watson distribution, this paper considers a folded Watson distribution for such cases. We first address the parameter estimation of this distribution using maximum likelihood, which requires a numerical algorithm to solve the likelihood equations. We use the Expectation-Maximization (EM) algorithm to obtain these estimates and to analyze the properties of the concentration estimator through simulation. Next, we propose the Bayes rule for a folded Watson distribution and evaluate its performance through simulation in various scenarios, comparing it with the Bayes rule for the Watson distribution. Finally, we present examples using both simulated and real data available in the literature.

当方向数据位于单位超球面的正卦限（positive orthant）时，折叠方向分布（folded directional distribution）相较于简单方向分布更适合用于数据建模。由于方向数据（尤其是轴数据（axial data））可采用沃森分布（Watson distribution）建模，本文针对此类情况考虑折叠沃森分布（folded Watson distribution）。我们首先采用最大似然法（maximum likelihood）解决该分布的参数估计问题，这需要通过数值算法求解似然方程；为此，我们使用期望最大化（Expectation-Maximization，EM）算法获取这些估计值，并通过模拟分析浓度估计量（concentration estimator）的性质。接下来，我们提出折叠沃森分布的贝叶斯法则（Bayes rule），并通过多种场景下的模拟评估其性能，同时与沃森分布的贝叶斯法则进行对比。最后，我们利用文献中可得的模拟数据与真实数据呈现实例分析。