论文标题
来自椭圆形分布的定向数据的分位数和深度
Quantiles and depth for directional data from elliptically symmetric distributions
论文作者
论文摘要
我们在分布之后介绍了方向数据的规范分位数和深度，该分布在球体上$ \ Mathcal {s}^{d-1} $上的方向$μ$。我们的方法扩展了Ley等人的概念。 [1]，它提供了深度轮廓的宝贵几何特性（例如凸度和旋转均值）和分位数的Bahadur型表示。他们的概念对于旋转对称深度轮廓是规范的。但是，当基础分布不是旋转在对称上时，它也会产生旋转的对称深度轮廓。我们解决了缺乏椭圆深度轮廓的分布的灵活性。基本思想是通过差异图映射到旋转对称的轮廓，将椭圆轮廓变形，从而恢复Ley等人的规范情况。 [1]。蒙特卡洛模拟研究证实了我们的结果。我们使用我们的方法评估深度轮廓的椭圆度和定向数据的修剪。纤维增强混凝土中纤维方向的分析强调了实际相关性。
We present canonical quantiles and depths for directional data following a distribution which is elliptically symmetric about a direction $μ$ on the sphere $\mathcal{S}^{d-1}$. Our approach extends the concept of Ley et al. [1], which provides valuable geometric properties of the depth contours (such as convexity and rotational equivariance) and a Bahadur-type representation of the quantiles. Their concept is canonical for rotationally symmetric depth contours. However, it also produces rotationally symmetric depth contours when the underlying distribution is not rotationally symmetric. We solve this lack of flexibility for distributions with elliptical depth contours. The basic idea is to deform the elliptic contours by a diffeomorphic mapping to rotationally symmetric contours, thus reverting to the canonical case in Ley et al. [1]. A Monte Carlo simulation study confirms our results. We use our method to evaluate the ellipticity of depth contours and for trimming of directional data. The analysis of fibre directions in fibre-reinforced concrete underlines the practical relevance.
