Department of Bioengineering, School of Engineering and Applied Sciences, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA.
Chaos. 2020 Feb;30(2):021101. doi: 10.1063/1.5142462.
Whether listening to overlapping conversations in a crowded room or recording the simultaneous electrical activity of millions of neurons, the natural world abounds with sparse measurements of complex overlapping signals that arise from dynamical processes. While tools that separate mixed signals into linear sources have proven necessary and useful, the underlying equational forms of most natural signals are unknown and nonlinear. Hence, there is a need for a framework that is general enough to extract sources without knowledge of their generating equations and flexible enough to accommodate nonlinear, even chaotic, sources. Here, we provide such a framework, where the sources are chaotic trajectories from independently evolving dynamical systems. We consider the mixture signal as the sum of two chaotic trajectories and propose a supervised learning scheme that extracts the chaotic trajectories from their mixture. Specifically, we recruit a complex dynamical system as an intermediate processor that is constantly driven by the mixture. We then obtain the separated chaotic trajectories based on this intermediate system by training the proper output functions. To demonstrate the generalizability of this framework in silico, we employ a tank of water as the intermediate system and show its success in separating two-part mixtures of various chaotic trajectories. Finally, we relate the underlying mechanism of this method to the state-observer problem. This relation provides a quantitative theory that explains the performance of our method, and why separation is difficult when two source signals are trajectories from the same chaotic system.
无论是在嘈杂的房间里聆听重叠的对话，还是记录数百万个神经元的同时电活动，自然界中都充满了复杂重叠信号的稀疏测量，这些信号源于动态过程。虽然已经证明将混合信号分离为线性源的工具是必要且有用的，但大多数自然信号的基础等式形式是未知的，而且是非线性的。因此，需要有一种框架，该框架足够通用，可以在不了解其生成方程的情况下提取源，并且足够灵活，可以适应非线性甚至混沌源。在这里，我们提供了这样一个框架，其中源是来自独立演化动力系统的混沌轨迹。我们将混合信号视为两个混沌轨迹的和，并提出了一种监督学习方案，从混合物中提取混沌轨迹。具体来说，我们招募一个复杂的动力系统作为中间处理器，该处理器不断受到混合物的驱动。然后，我们通过训练适当的输出函数，根据这个中间系统获得分离的混沌轨迹。为了在计算机上证明这个框架的通用性，我们将水箱作为中间系统，并展示它在分离各种混沌轨迹的两部分混合物方面的成功。最后，我们将这种方法的基本机制与状态观测器问题联系起来。这种关系提供了一种定量理论，解释了我们方法的性能，以及为什么当两个源信号是来自同一混沌系统的轨迹时，分离会很困难。
