A method for identifying causality in the response of nonlinear dynamical systems

TL;DR

The paper addresses identifying the causality between an input signal $x(t)$ and the noisy nonlinear response $y_n(t)$ of a dynamical system under broadband excitation without requiring a complete benchmark model. It introduces a frequency-domain architecture that jointly learns a forward-prediction weight $K(f)$ and a nonlinear predictor by optimally fusing the forward prediction $y_z$ with the observed output $y_n$, enabling estimation of the nonlinear coherence $\gamma^2_{YY_n}$ as a measure of causality. Key contributions include (i) a method to bound and estimate the maximum predictive performance under noise, (ii) a data-driven computation of nonlinear coherence for a broad class of nonlinear ODEs, and (iii) demonstration on three simulations and an experiment showing accurate coherence estimates and actionable insights for resource allocation. The framework supports practical causality assessment and can guide data collection and noise-reduction strategies in applications like active noise control, by indicating how much of the output is truly driven by the input across frequencies.

Abstract

Predicting the response of nonlinear dynamical systems subject to random, broadband excitation is important across a range of scientific disciplines, such as structural dynamics and neuroscience. Building data-driven models requires experimental measurements of the system input and output, but it can be difficult to determine whether inaccuracies in the model stem from modelling errors or noise. This paper presents a novel method to identify the causal component of the input-output data from measurements of a system in the presence of output noise, as a function of frequency, without needing a high fidelity model. An output prediction, calculated using an available model, is optimally combined with noisy measurements of the output to predict the input to the system. The parameters of the algorithm balance the two output signals and are utilised to calculate a nonlinear coherence metric as a measure of causality. This method is applicable to a broad class of nonlinear dynamical systems. There are currently no solutions to this problem in the absence of a complete benchmark model.

Figures (14)

• Figure 1: Architecture. The fast Fourier transform (FFT) of the signals $y_{z}$ and $y_{n}$ is taken, before the two signals are combined at each frequency point using the parameter $K(f)$. The inverse fast Fourier transform (iFFT) is then applied to the resultant signal, which is then mapped to $x$ using a CNN.
• Figure 2: Polynomial stiffness case study.
• Figure 3: Saturating stiffness case study.
• Figure 4: Nonlinear friction case study.
• Figure 5: Diagram of the experimental set up.