共形化卡尔曼滤波器用于具有可信置信区域的状态估计
Conformalized Kalman Filters for State Estimation with Trustworthy Confidence Regions
- Ben-Gurion University of the Negev（内盖夫本-古里安大学）
- Tel-Aviv University（特拉维夫大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出共形预测框架为卡尔曼滤波器提供统计可靠的置信区域，通过三种构造方法（高斯校准、分位数回归、高斯混合）实现有限样本覆盖保证，实验验证其有效性和优势。
AI中文摘要：
卡尔曼型滤波器广泛用于跟踪动态系统，然而，由其估计协方差导出的置信区域在非线性、非高斯扰动和模型失配情况下可能变得不可靠。在本工作中，我们开发了一个共形预测（CP）框架，为卡尔曼型滤波器配备统计上可靠的置信区域。与应用于黑箱估计器的CP不同，我们的方法利用了状态后验的递归估计的一阶和二阶矩，这些矩捕捉了由底层动态引起的时间变化的不确定性。基于这些统计特征，我们提出了三种互补的构造方法。第一种对由滤波器矩诱导的高斯置信区域进行共形校准。第二种采用分位数回归将估计的矩映射到自适应凸区域的边界，随后对这些区域进行校准。第三种学习后验的高斯混合表示，并对由此产生的基于密度的区域进行共形化，从而能够表征多模态和非凸的不确定性集合。我们为样本-wise置信和轨迹-wise置信建立了有限样本覆盖保证，前者控制单个时间实例的误覆盖，后者在预定时间范围内联合覆盖状态序列。在多种线性和非线性动态系统上的数值实验表明，所提出的方法达到了规定的覆盖率，同时产生紧凑且信息丰富的置信区域，并突出了三种构造在不同后验特征下的相对优势。
英文摘要：
Kalman-type filters are widely used for tracking dynamic systems, yet the confidence regions commonly derived from their estimated covariances can become unreliable under nonlinearities, non-Gaussian disturbances, and model mismatch. In this work, we develop a conformal prediction (CP) framework for equipping Kalman-type filters with statistically reliable confidence regions. Unlike CP applied to black-box estimators, our approach exploits the recursively estimated first- and second-order moments of the state posterior, which capture the time-varying uncertainty induced by the underlying dynamics. Based on these statistical features, we propose three complementary constructions. The first conformally calibrates Gaussian confidence regions induced by the filter moments. The second employs quantile regression to map the estimated moments into the boundaries of adaptive convex regions, which are subsequently calibrated. The third learns a Gaussian-mixture representation of the posterior and conformalizes the resulting density-based regions, enabling the characterization of multimodal and non-convex uncertainty sets. We establish finite- sample coverage guarantees for both sample-wise confidence, which controls miscoverage at individual time instances, and trajectory- wise confidence, which jointly covers the state sequence over a prescribed horizon. Numerical experiments across diverse linear and nonlinear dynamic systems demonstrate that the proposed methods attain the prescribed coverage while producing tight and informative confidence regions, and highlight the relative merits of the three constructions under different posterior characteristics.
