Understanding dynamics in complex systems is challenging because there are many degrees of freedom, and those that are most important for describing events of interest are often not obvious. The leading eigenfunctions of the transition operator are useful for visualization, and they can provide an efficient basis for computing statistics such as the likelihood and average time of events (predictions). Here we develop inexact iterative linear algebra methods for computing these eigenfunctions (spectral estimation) and making predictions from a data set of short trajectories sampled at finite intervals. We demonstrate the methods on a low-dimensional model that facilitates visualization and a high-dimensional model of a biomolecular system. Implications for the prediction problem in reinforcement learning are discussed.

翻译：理解复杂系统中的动力学行为具有挑战性，因为系统存在大量自由度，而描述感兴趣事件最关键的变量往往并不显见。转移算子的主导本征函数可用于可视化，并能为计算事件似然度与平均发生时间（预测）等统计量提供高效基函数。本文针对基于有限间隔采样短轨迹数据集的场景，提出用于计算这些本征函数（谱估计）并实现预测的非精确迭代线性代数方法。我们分别在便于可视化的低维模型与生物分子系统的高维模型上验证了该方法，并讨论了其对强化学习中预测问题的启示。