在这项研究中,我们提出了一种量化多变量数据集中依赖关系的新方法,基于通过最小生成树(MST)估计Rényi互信息。随机变量的依赖程度是一个重要的问题,例如,用于不确定性量化和敏感性分析。后者与以下问题密切相关:例如,计算机模拟,是在单个随机输入变量上。为了从数据中估计伦尼互信息,我们使用一种由于英雄等人的方法。依赖于计算数据的最小生成树(MST),并在熵的估计器中使用MST的长度。为了降低为大型数据集构建精确MST的计算成本,我们探索计算精确MST近似的方法,并找到了钟等人最近提出的多层次方法。(2015)是最准确的。因为MST计算不需要分布的知识(或估计),我们的方法非常适合只有数据可用的情况。此外,我们证明,在只需要几个依赖关系的排名而不是它们的确切值的情况下,没有必要计算雷尼分歧,但只能从中得出一个估计器。本文的主要贡献是引入了这种依赖量词,以及使用MST的近似方法与通过MST估计Rényi互信息的新颖组合。我们将我们提出的方法应用于基于Ishigami函数的人工测试用例,以及涉及厄尔尼诺现象数据集的现实测试案例。
In this study, we present a novel method for quantifying dependencies in multivariate datasets, based on estimating the Rényi mutual information by minimum spanning trees (MSTs). The extent to which random variables are dependent is an important question, e.g., for uncertainty quantification and sensitivity analysis. The latter is closely related to the question how strongly dependent the output of, e.g., a computer simulation, is on the individual random input variables. To estimate the Rényi mutual information from data, we use a method due to Hero et al. that relies on computing minimum spanning trees (MSTs) of the data and uses the length of the MST in an estimator for the entropy. To reduce the computational cost of constructing the exact MST for large datasets, we explore methods to compute approximations to the exact MST, and find the multilevel approach introduced recently by Zhong et al. (2015) to be the most accurate. Because the MST computation does not require knowledge (or estimation) of the distributions, our methodology is well-suited for situations where only data are available. Furthermore, we show that, in the case where only the ranking of several dependencies is required rather than their exact value, it is not necessary to compute the Rényi divergence, but only an estimator derived from it. The main contributions of this paper are the introduction of this quantifier of dependency, as well as the novel combination of using approximate methods for MSTs with estimating the Rényi mutual information via MSTs. We applied our proposed method to an artificial test case based on the Ishigami function, as well as to a real-world test case involving an El Nino dataset.