Department of Environmental Systems Science, ETH Zurich, CH-8092 Zürich, Switzerland.
Swiss Federal Research Institute WSL, CH-8903 Birmensdorf, Switzerland.
Sensors (Basel). 2022 Apr 25;22(9):3291. doi: 10.3390/s22093291.
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.
脉冲响应函数（IRF）基于系统输入和输出的传感器数据流，有助于描述系统的动态行为并深入了解其潜在过程。然而，当前的 IRF 估计方法通常需要满足很少在实践中遇到的严格假设，包括基础系统是均匀的、线性的和稳定的，以及任何噪声都是可控的。在这里，我提出了数据驱动、模型独立、非参数的 IRF 估计方法，这些方法放宽了这些假设，从而扩大了 IRF 在实际系统中的适用性。这些方法可以从受自回归滑动平均（ARMA）噪声或非平稳 ARIMA 噪声严重污染的信号中准确有效地反卷积 IRF。它们还可以根据组合输出同时反卷积和解混异质系统中各个组件的脉冲响应（而无需知道各个组件的输出）。这种去卷积-解混方法可以扩展到描述输入和输出之间的非平稳耦合，即使系统的脉冲响应变化如此之快，以至于不同的脉冲响应相互重叠。这些技术还可以扩展到估计其中不同输入强度产生具有不同形状和幅度的脉冲响应的非线性系统，然后在输出中相互叠加。我进一步展示了如何使用在不均匀间隔的滞后处定义的分段线性 IRF 来有效地量化多尺度脉冲响应。所有这些方法都在一个 R 脚本中实现，可以从数千甚至数百万个时间步长的嘈杂时间序列中高效地估计数百个滞后的 IRF。
