标题： 实数和复数值多变量数据的总相关性/对偶相关性/相干性，冗余/协同性，复杂性，以及O-信息 显示英文标题

标题： Total/dual correlation/coherence, redundancy/synergy, complexity, and O-information for real and complex valued multivariate data

摘要： 首先，假设高斯性，给出了以下信息理论度量的方程：总相关性/相干性（TC）、双总相关性/相干性（DTC）、O-信息、TSE复杂性以及冗余-协同指数（RSI）。 由于这些度量是协方差矩阵"S"及其逆矩阵"S^-1"的函数，因此相关的Wishart分布和逆Wishart分布值得关注。 DTC被证明是逆Wishart对"S^-1"和其对角矩阵"D=diag(S^-1)"之间的Kullback-Leibler（KL）散度，这有助于将其解释为“总部分相关性”的度量，-lndetP，检验假设H0: P=I，其中"P"是标准化的逆协方差（即P=(D^-1/2)(S^-1)(D^-1/2)）。 本文的第二个目标是将所有这些度量推广到结构化的变量组。 例如，考虑三个或更多组，每组包含三个或更多变量，每组内主要存在冗余，但组间存在协同作用。 O-信息将忽略组间的协同作用（因为系统中冗余更常见）。 相反，这里提出的结构化O-信息度量将正确报告组间的主导协同作用。 这是向结构化多变量信息度量的一个相关推广。 第三个目标是提出一个框架，用于量化变量之间的“连接”对系统TC、DTC、O-信息和TSE复杂性的贡献。 第四个目标是提出冗余-协同指数的推广，用于量化一组变量对系统冗余-协同平衡的贡献。 最后，表明此处推导的表达式可以直接应用于来自几种其他椭圆分布的数据。 所有程序代码、数据文件和可执行文件均可获得（https://osf.io/jd37g/）。 显示英文摘要

摘要： Firstly, assuming Gaussianity, equations for the following information theory measures are presented: total correlation/coherence (TC), dual total correlation/coherence (DTC), O-information, TSE complexity, and redundancy-synergy index (RSI). Since these measures are functions of the covariance matrix "S" and its inverse "S^-1", the associated Wishart and inverse-Wishart distributions are of note. DTC is shown to be the Kullback-Leibler (KL) divergence for the inverse-Wishart pair "(S^-1)" and its diagonal matrix "D=diag(S^-1)", shedding light on its interpretation as a measure of "total partial correlation", -lndetP, with test hypothesis H0: P=I, where "P" is the standardized inverse covariance (i.e. P=(D^-1/2)(S^-1)(D^-1/2). The second aim of this paper introduces a generalization of all these measures for structured groups of variables. For instance, consider three or more groups, each consisting of three or more variables, with predominant redundancy within each group, but with synergistic interactions between groups. O-information will miss the between group synergy (since redundancy occurs more often in the system). In contrast, the structured O-information measure presented here will correctly report predominant synergy between groups. This is a relevant generalization towards structured multivariate information measures. A third aim is the presentation of a framework for quantifying the contribution of "connections" between variables, to the system's TC, DTC, O-information, and TSE complexity. A fourth aim is to present a generalization of the redundancy-synergy index for quantifying the contribution of a group of variables to the system's redundancy-synergy balance. Finally, it is shown that the expressions derived here directly apply to data from several other elliptical distributions. All program codes, data files, and executables are available (https://osf.io/jd37g/).