发表机构
University of Chicago; New York University（芝加哥大学; 纽约大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明确定性平方根集合卡尔曼滤波器在线性高斯系统中具有时间一致精度，集合大小取决于有效秩和不稳定子空间维数；对局地化滤波器，集合大小取决于局部维数且随空间块数对数增长。
AI 中文摘要
本文在线性-高斯状态空间模型且具有完美模型动力学的条件下，为确定性平方根集合卡尔曼滤波器建立了时间一致精度保证。对于双曲、可检测系统，我们证明集合均值和协方差在时间上一致地逼近其卡尔曼滤波器对应值，且具有高概率，无需膨胀或重采样。所需的集合大小取决于初始协方差的有效秩和不稳定子空间的维数，而非环境状态维数。我们还分析了一种用于弱耦合空间系统的局地化平方根集合卡尔曼滤波器。我们证明了稳定的卡尔曼协方差从动力学中继承空间衰减，并导出了将采样误差与局地化偏差分离的精度界限。对于局地化滤波器，所需集合大小取决于局部固有维数，并且仅随空间块数的对数增长，而局地化偏差则随相互作用强度缩放。我们的分析结合了可能奇异的Riccati递推的稳定性与遗忘性、非渐近协方差集中性以及局地化协方差动力学的扰动估计。
英文摘要
This paper establishes time-uniform accuracy guarantees for deterministic square-root ensemble Kalman filters in linear--Gaussian state-space models under perfect-model dynamics. For hyperbolic, detectable systems, we prove that the ensemble means and covariances approximate their Kalman filter counterparts uniformly in time, with high probability and without inflation or resampling. The required ensemble size depends on the effective rank of the initial covariance and the dimension of the unstable subspace, rather than on the ambient state dimension. We also analyze a localized square-root ensemble Kalman filter for weakly coupled spatial systems. We show that the stabilizing Kalman covariance inherits spatial decay from the dynamics and derive accuracy bounds that separate sampling error from localization bias. For the localized filter, the required ensemble size depends on the local intrinsic dimension and only logarithmically on the number of spatial blocks, while the localization bias scales with the interaction strength. Our analysis combines stability and forgetting for possibly singular Riccati recursions, nonasymptotic covariance concentration, and perturbation estimates for localized covariance dynamics.
Comments62 pages
