A large class of modern probabilistic learning systems assumes symmetric distributions, however, real-world data tend to obey skewed distributions and are thus not always adequately modelled through symmetric distributions. To address this issue, elliptical distributions are increasingly used to generalise symmetric distributions, and further improvements to skewed elliptical distributions have recently attracted much attention. However, existing approaches are either hard to estimate or have complicated and abstract representations. To this end, we propose to employ the von-Mises-Fisher (vMF) distribution to obtain an explicit and simple probability representation of the skewed elliptical distribution. This is shown not only to allow us to deal with non-symmetric learning systems, but also to provide a physically meaningful way of generalising skewed distributions. For rigour, our extension is proved to share important and desirable properties with its symmetric counterpart. We also demonstrate that the proposed vMF distribution is both easy to generate and stable to estimate, both theoretically and through examples.

翻译：大量现代概率学习系统都假定了对称分布,然而,现实世界数据往往服从偏斜分布,因此并不总是通过对称分布进行适当的模拟。为了解决这个问题,正越来越多地利用椭圆分布来概括对称分布,最近人们非常注意对斜椭圆分布的进一步改进。然而,现有的方法要么难以估计,要么具有复杂和抽象的表述方式。为此,我们提议采用正对-米舍-菲舍尔(vMF)分布方法,以获得扭曲的椭圆分布的清晰和简单概率表示。这显示不仅使我们能够处理非对称学习系统,而且还提供了一种具有实际意义的一般偏斜分布方法。为了严谨起见,我们的扩展证明是与其对口方分享重要和可取的属性。我们还表明,拟议的 von-MF分布方法既容易产生,也稳定于估算,无论是从理论上还是通过实例。