One of the most challenging problems in the study of complex dynamical systems is to find the statistical interdependencies among the system components. Granger causality (GC) represents one of the most employed approaches, based on modeling the system dynamics with a linear vector autoregressive (VAR) model and on evaluating the information flow between two processes in terms of prediction error variances. In its most advanced setting, GC analysis is performed through a state-space (SS) representation of the VAR model that allows to compute both conditional and unconditional forms of GC by solving only one regression problem. While this problem is typically solved through Ordinary Least Square (OLS) estimation, a viable alternative is to use Artificial Neural Networks (ANNs) implemented in a simple structure with one input and one output layer and trained in a way such that the weights matrix corresponds to the matrix of VAR parameters. In this work, we introduce an ANN combined with SS models for the computation of GC. The ANN is trained through the Stochastic Gradient Descent L1 (SGD-L1) algorithm, and a cumulative penalty inspired from penalized regression is applied to the network weights to encourage sparsity. Simulating networks of coupled Gaussian systems, we show how the combination of ANNs and SGD-L1 allows to mitigate the strong reduction in accuracy of OLS identification in settings of low ratio between number of time series points and of VAR parameters. We also report how the performances in GC estimation are influenced by the number of iterations of gradient descent and by the learning rate used for training the ANN. We recommend using some specific combinations for these parameters to optimize the performance of GC estimation. Then, the performances of ANN and OLS are compared in terms of GC magnitude and statistical significance to highlight the potential of the new approach to reconstruct causal coupling strength and network topology even in challenging conditions of data paucity. The results highlight the importance of of a proper selection of regularization parameter which determines the degree of sparsity in the estimated network. Furthermore, we apply the two approaches to real data scenarios, to study the physiological network of brain and peripheral interactions in humans under different conditions of rest and mental stress, and the effects of the newly emerged concept of remote synchronization on the information exchanged in a ring of electronic oscillators. The results highlight how ANNs provide a mesoscopic description of the information exchanged in networks of multiple interacting physiological systems, preserving the most active causal interactions between cardiovascular, respiratory and brain systems. Moreover, ANNs can reconstruct the flow of directed information in a ring of oscillators whose statistical properties can be related to those of physiological networks.

中文翻译：

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通过人工神经网络估计格兰杰因果关系：在生理系统和混沌电子振荡器中的应用


复杂动力系统研究中最具挑战性的问题之一是找到系统组件之间的统计相互依赖性。格兰杰因果关系 (GC) 是最常用的方法之一，它基于使用线性向量自回归 (VAR) 模型对系统动力学进行建模，并根据预测误差方差评估两个过程之间的信息流。在最高级的设置中，GC 分析是通过 VAR 模型的状态空间 (SS) 表示来执行的，该模型允许通过仅解决一个回归问题来计算有条件和无条件形式的 GC。虽然这个问题通常通过普通最小二乘 (OLS) 估计来解决，但一种可行的替代方案是使用人工神经网络 (ANN)，该网络以具有一个输入和一个输出层的简单结构实现，并以权重矩阵对应的方式进行训练到 VAR 参数矩阵。在这项工作中，我们引入了结合 SS 模型的 ANN 来计算 GC。人工神经网络通过随机梯度下降 L1 (SGD-L1) 算法进行训练，并将受惩罚回归启发的累积惩罚应用于网络权重以鼓励稀疏性。通过模拟耦合高斯系统的网络，我们展示了 ANN 和 SGD-L1 的组合如何在时间序列点数量与 VAR 参数之间比率较低的情况下减轻 OLS 识别精度的大幅降低。我们还报告了梯度下降迭代次数和用于训练 ANN 的学习率如何影响 GC 估计的性能。我们建议对这些参数使用一些特定的组合来优化 GC 估计的性能。 然后，从 GC 幅度和统计显着性方面比较 ANN 和 OLS 的性能，以突显新方法即使在数据匮乏的挑战性条件下重建因果耦合强度和网络拓扑的潜力。结果强调了正确选择正则化参数的重要性，该参数决定了估计网络的稀疏程度。此外，我们将这两种方法应用于真实的数据场景，研究人类在不同的休息和精神压力条件下大脑和外周相互作用的生理网络，以及新出现的远程同步概念对信息交换的影响。电子振荡器环。结果强调了人工神经网络如何对多个相互作用的生理系统网络中交换的信息提供介观描述，从而保留心血管、呼吸和大脑系统之间最活跃的因果相互作用。此外，人工神经网络可以重建振荡器环中的定向信息流，其统计特性可以与生理网络的统计特性相关。