The Multivariate Generalised von Mises Distribution: Inference and Applications
The Multivariate Generalised von Mises Distribution: Inference and Applications
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DOI：
10.1609/aaai.v31i1.10943
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发表时间：
2016-02
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影响因子：
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通讯作者：
Alexandre K. W. Navarro;J. Frellsen;Richard E. Turner
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文献类型：
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作者：
Alexandre K. W. Navarro;J. Frellsen;Richard E. Turner
循环变量出现在从机器人到社会科学的大量数据建模环境中，但它们在很大程度上被机器学习社区所忽视。本文通过将一些标准的概率建模工具扩展到圆形域来部分纠正这种不平衡。首先，我们介绍一个新的多元分布的圆形变量，称为多元广义冯米塞斯（mGvM）分布。该分布可以通过将一般多元高斯分布限制和重正化到单位超环面来构造。先前提出的多元圆形分布被证明是这种结构的特殊情况。其次，我们介绍了一个新的概率模型的启发高斯过程的循环回归，和概率主成分分析方法与循环隐藏变量。这些模型可以利用标准建模工具（例如核函数和自动相关性确定）。第三，我们证明了这些模型中的后验分布是一个mGvM分布，它可以开发一个有效的变分自由能方案来进行近似推理和近似最大似然学习。
Circular variables arise in a multitude of data-modelling contexts ranging from robotics to the social sciences, but they have been largely overlooked by the machine learning community. This paper partially redresses this imbalance by extending some standard probabilistic modelling tools to the circular domain. First we introduce a new multivariate distribution over circular variables, called the multivariate Generalised von Mises (mGvM) distribution. This distribution can be constructed by restricting and renormalising a general multivariate Gaussian distribution to the unit hyper-torus. Previously proposed multivariate circular distributions are shown to be special cases of this construction. Second, we introduce a new probabilistic model for circular regression inspired by Gaussian Processes, and a method for probabilistic Principal Component Analysis with circular hidden variables. These models can leverage standard modelling tools (e.g. kernel functions and automatic relevance determination). Third, we show that the posterior distribution in these models is a mGvM distribution which enables development of an efficient variational free-energy scheme for performing approximate inference and approximate maximum-likelihood learning.