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极具挑战性，因为系统存在众多自由度，且描述感兴趣事件时起关键作用的自由度往往并不明确。转移算子的主导本征函数不仅可用于可视化分析，还可为计算事件发生概率与平均时间（预测）等统计量提供高效基础。本文提出了一种基于不精确迭代线性代数方法，通过有限间隔采样的短轨迹数据集计算这些本征函数（谱估计）并进行预测。我们分别在易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，并讨论了该方法对强化学习中预测问题的启示。