一种基于Koopman算子的预测算法及其在COVID-19大流行中的应用
A Koopman Operator-Based Prediction Algorithm and its Application to COVID-19 Pandemic
AI总结：
本文提出基于Koopman算子的动态置信度切换预测算法，以数据驱动方式应对动力系统突变，并验证其在流感和COVID-19病例预测中的鲁棒性。
摘要：
20世纪下半叶，随着人们发现在演化规律不随时间改变的简单物理动力系统中也可能存在混沌动力学，动力系统行为预测问题经历了一次范式转变。该范式的本质是轨迹的长期指数发散。然而，该范式没有解释另一种不可预测性：“黑天鹅”事件。它也没有说明这样一个事实：即使在具有指数发散的系统中，短期预测通常也是可能的。在我们的框架中，当底层动力系统在不同类型的动力学之间突然转换时，就会出现黑天鹅型动力学。学习和预测系统应当能够识别以“置信度丧失”为代表的行为转变。在这一范式中，预测能力会被动态评估，并利用置信度水平在长期预测与局部时间预测之间切换。本文研究了表现出此类行为的系统中的预测问题。我们的理论和算法的数学基础基于一种算子论方法，该方法将系统动力学嵌入到一个无限维空间中。我们将该算法应用于若干案例研究，包括流感病例预测和COVID-19大流行。结果表明，该预测算法对可用数据的扰动具有鲁棒性，例如由报告延迟或检测能力提高导致的病例突然增加所引发的扰动。这完全是以数据驱动方式实现的，不需要底层的疾病数学模型。
英文摘要：
The problem of prediction of behavior of dynamical systems has undergone a paradigm shift in the second half of the 20th century with the discovery of the possibility of chaotic dynamics in simple, physical, dynamical systems for which the laws of evolution do not change in time. The essence of the paradigm is the long term exponential divergence of trajectories. However, that paradigm does not account for another type of unpredictability: the ``Black Swan" event. It also does not account for the fact that short-term prediction is often possible even in systems with exponential divergence. In our framework, the Black Swan type dynamics occurs when an underlying dynamical system suddenly shifts between dynamics of different types. A learning and prediction system should be capable of recognizing the shift in behavior, exemplified by ``confidence loss". In this paradigm, the predictive power is assessed dynamically and confidence level is used to switch between long term prediction and local-in-time prediction. Here we explore the problem of prediction in systems that exhibit such behavior. The mathematical underpinnings of our theory and algorithms are based on an operator-theoretic approach in which the dynamics of the system are embedded into an infinite-dimensional space. We apply the algorithm to a number of case studies including prediction of influenza cases and the COVID-19 pandemic. The results show that the predictive algorithm is robust to perturbations of the available data, induced for example by delays in reporting or sudden increase in cases due to increase in testing capability. This is achieved in an entirely data-driven fashion, with no underlying mathematical model of the disease.
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