This work introduces a method to select linear functional measurements of a vector-valued time series optimized for forecasting distant time-horizons. By formulating and solving the problem of sequential linear measurement design as an infinite-horizon problem with the time-averaged trace of the Cram\'{e}r-Rao lower bound (CRLB) for forecasting as the cost, the most informative data can be collected irrespective of the eventual forecasting algorithm. By introducing theoretical results regarding measurements under additive noise from natural exponential families, we construct an equivalent problem from which a local dimensionality reduction can be derived. This alternative formulation is based on the future collapse of dimensionality inherent in the limiting behavior of many differential equations and can be directly observed in the low-rank structure of the CRLB for forecasting. Implementations of both an approximate dynamic programming formulation and the proposed alternative are illustrated using an extended Kalman filter for state estimation, with results on simulated systems with limit cycles and chaotic behavior demonstrating a linear improvement in the CRLB as a function of the number of collapsing dimensions of the system.

翻译：本文提出一种方法，用于选取面向远期时域预测的向量值时间序列线性泛函测量。通过将顺序线性测量设计问题建模为以时间平均的克拉美-罗下界（CRLB）预测追踪为代价函数的无限时域问题，可收集最具信息量的数据，而无需依赖最终的预测算法。基于自然指数族加性噪声下测量的理论成果，我们构造了一个等效问题，并由此推导出局部降维方案。该替代形式源于许多微分方程极限行为中固有的未来维度坍缩现象，可直接从预测CRLB的低秩结构观测到。采用扩展卡尔曼滤波器进行状态估计，分别实现了近似动态规划公式及其替代方案，在具有极限环和混沌行为的仿真系统上，结果表明CRLB随系统坍缩维度数量呈线性改善。