Bollt Erik M
Department of Mathematics and Electrical and Computer Engineering and Clarkson Center for Complex Systems Science (CS), Clarkson University, Potsdam, New York 13699, USA.
Chaos. 2018 Jul;28(7):075309. doi: 10.1063/1.5031109.
A basic systems question concerns the concept of closure, meaning autonomy (closed) in the sense of describing the (sub)system as fully consistent within itself. Alternatively, the system may be nonautonomous (open), meaning it receives influence from an outside subsystem. We assert here that the concept of information flow and the related concept of causation inference are summarized by this simple question of closure as we define herein. We take the forecasting perspective of Weiner-Granger causality that describes a causal relationship exists if a subsystem's forecast quality depends on considering states of another subsystem. Here, we develop a new direct analytic discussion, rather than a data oriented approach. That is, we refer to the underlying Frobenius-Perron (FP) transfer operator that moderates evolution of densities of ensembles of orbits, and two alternative forms of the restricted Frobenius-Perron operator, interpreted as if either closed (deterministic FP) or not closed (the unaccounted outside influence seems stochastic and we show correspondingly requires the stochastic FP operator). Thus follows contrasting the kernels of the variants of the operators, as if densities in their own rights. However, the corresponding differential entropy comparison by Kullback-Leibler divergence, as one would typically use when developing transfer entropy, becomes ill-defined. Instead, we build our Forecastability Quality Metric (FQM) upon the "symmetrized" variant known as Jensen-Shannon divergence, and we are also able to point out several useful resulting properties. We illustrate the FQM by a simple coupled chaotic system. Our analysis represents a new theoretical direction, but we do describe data oriented directions for the future.
一个基本的系统问题涉及封闭性的概念，即（子）系统在自身内部完全一致的意义上的自主性（封闭）。或者，系统可能是非自主的（开放），这意味着它受到外部子系统的影响。我们在此断言，正如我们在此所定义的，信息流的概念和相关的因果推断概念可以通过这个简单的封闭性问题来概括。我们采用维纳 - 格兰杰因果关系的预测视角，即如果一个子系统的预测质量取决于考虑另一个子系统的状态，那么就存在因果关系。在这里，我们进行一种新的直接分析讨论，而不是采用面向数据的方法。也就是说，我们提及底层的弗罗贝尼乌斯 - 佩龙（FP）转移算子，它调节轨道系综密度的演化，以及受限弗罗贝尼乌斯 - 佩龙算子的两种替代形式，其解释为要么是封闭的（确定性FP），要么不是封闭的（未考虑的外部影响似乎是随机的，并且我们表明相应地需要随机FP算子）。因此，要对比这些算子变体的核，就好像它们本身就是密度一样。然而，像在开发转移熵时通常会使用的通过库尔贝克 - 莱布勒散度进行的相应微分熵比较变得定义不明确。相反，我们基于被称为詹森 - 香农散度的“对称化”变体构建我们的可预测性质量度量（FQM），并且我们还能够指出几个有用的结果性质。我们通过一个简单的耦合混沌系统来说明FQM。我们的分析代表了一个新的理论方向，但我们确实也描述了未来面向数据的方向。
