Impulse Response Functions for Nonlinear, Nonstationary, and Heterogeneous Systems, Estimated by Deconvolution and Demixing of Noisy Time Series

Abstract

Impulse response functions (IRFs) are useful for characterizing systems’ dynamic behavior and gaining insight into their underlying processes, based on sensor data streams of their inputs and outputs. However, current IRF estimation methods typically require restrictive assumptions that are rarely met in practice, including that the underlying system is homogeneous, linear, and stationary, and that any noise is well behaved. Here, I present data-driven, model-independent, nonparametric IRF estimation methods that relax these assumptions, and thus expand the applicability of IRFs in real-world systems. These methods can accurately and efficiently deconvolve IRFs from signals that are substantially contaminated by autoregressive moving average (ARMA) noise or nonstationary ARIMA noise. They can also simultaneously deconvolve and demix the impulse responses of individual components of heterogeneous systems, based on their combined output (without needing to know the outputs of the individual components). This deconvolution–demixing approach can be extended to characterize nonstationary coupling between inputs and outputs, even if the system’s impulse response changes so rapidly that different impulse responses overlap one another. These techniques can also be extended to estimate IRFs for nonlinear systems in which different input intensities yield impulse responses with different shapes and amplitudes, which are then overprinted on one another in the output. I further show how one can efficiently quantify multiscale impulse responses using piecewise linear IRFs defined at unevenly spaced lags. All of these methods are implemented in an R script that can efficiently estimate IRFs over hundreds of lags, from noisy time series of thousands or even millions of time steps.