Estimation of Granger causality through Artificial Neural Networks: applications to physiological systems and chaotic electronic oscillators

Abstract

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.

Time-varying information measures: an adaptive estimation of information storage with application to brain-heart interactions

“…Future developments will aim at testing the proposed novel on different biosignals in the context of network physiology, to provide new tools to analyze over-time the information stored in physiological ( Antonacci et al, 2021a ; Koutlis et al, 2021 ) and non-physiological ( Antonacci et al, 2021b ) complex systems. Moreover, a complete time-varying estimation of the information processing could be provided in a network of multiple interacting dynamical systems in the framework of information dynamics ( Faes et al, 2016 ).…”

A Network approach to find poor orthostatic tolerance by simple tilt maneuvers

“…The new approach, suggested by Network Physiology to put complex biological problems into network descriptions (Ivanov, 2021), does not yet have a mathematical technique of its own. A number of proposals and examples have been published: for instance, when the particular properties of the dataset clearly required application of non-Normal statistics (West, 2014), Granger Causality estimation (Antonacci et al, 2021), time delay stability testing (Bashan et al, 2012) or a combination of Wavelet Phase Coherence and Conditional Mutual Information (Clemson et al,FIGURE 3 Percentage cerebral blood flow velocity (FV) from supine as function of % end tidal pCO2 (ETCO2). Red triangles: + 70 °tilt, black squares + 30 °.…”

Optimal architecture for artificial neural networks as pressure estimator

“…However, the high number of parameters and training iteration can provide overfitting and/or overtraining problems and ANN accuracy becomes poorly (Tetko et al, 1997;Lin & Wu, 2021). In some ANNs where the overfitting and/or overtraining can be a problem, an additional algorithm, such as the one implemented by Antonacci et al (2021) is necessary. Therefore, the problem of finding an optimal architecture, its weights and biases is hard and a field for application of several optimization techniques.…”