Heavy tails is a common feature of filtering distributions that results from the nonlinear dynamical and observation processes as well as the uncertainty from physical sensors. In these settings, the Kalman filter and its ensemble version - the ensemble Kalman filter (EnKF) - that have been designed under Gaussian assumptions result in degraded performance. t-distributions are a parametric family of distributions whose tail-heaviness is modulated by a degree of freedom $\nu$. Interestingly, Cauchy and Gaussian distributions correspond to the extreme cases of a t-distribution for $\nu = 1$ and $\nu = \infty$, respectively. Leveraging tools from measure transport (Spantini et al., SIAM Review, 2022), we present a generalization of the EnKF whose prior-to-posterior update leads to exact inference for t-distributions. We demonstrate that this filter is less sensitive to outlying synthetic observations generated by the observation model for small $\nu$. Moreover, it recovers the Kalman filter for $\nu = \infty$. For nonlinear state-space models with heavy-tailed noise, we propose an algorithm to estimate the prior-to-posterior update from samples of joint forecast distribution of the states and observations. We rely on a regularized expectation-maximization (EM) algorithm to estimate the mean, scale matrix, and degree of freedom of heavy-tailed \textit{t}-distributions from limited samples (Finegold and Drton, arXiv preprint, 2014). Leveraging the conditional independence of the joint forecast distribution, we regularize the scale matrix with an $l1$ sparsity-promoting penalization of the log-likelihood at each iteration of the EM algorithm. By sequentially estimating the degree of freedom at each analysis step, our filter can adapt its prior-to-posterior update to the tail-heaviness of the data. We demonstrate the benefits of this new ensemble filter on challenging filtering problems.

翻译：重尾性是滤波分布的一个常见特征，源于非线性动力学和观测过程以及物理传感器的不确定性。在这种场景下，基于高斯假设设计的卡尔曼滤波器及其集成版本——集成卡尔曼滤波器（EnKF）——会导致性能下降。t-分布是一类参数化分布族，其尾部厚度由自由度$\nu$调节。有趣的是，柯西分布和高斯分布分别对应t-分布在$\nu=1$和$\nu=\infty$时的极端情况。利用测度传输工具（Spantini等，《SIAM Review》，2022），我们提出了一种EnKF的推广形式，其先验-后验更新能够精确推断t-分布。实验表明，该滤波器对小$\nu$值下观测模型生成的异常合成观测值具有较低的敏感性。此外，当$\nu=\infty$时，它退化为标准卡尔曼滤波器。针对具有重尾噪声的非线性状态空间模型，我们提出了一种算法，通过从状态与观测的联合预报分布样本中估计先验-后验更新。我们采用正则化期望最大化（EM）算法，从有限样本中估计重尾t-分布的均值、尺度矩阵和自由度（Finegold与Drton，arXiv预印本，2014）。利用联合预报分布的条件独立性，我们在EM算法的每次迭代中对尺度矩阵施加$l1$稀疏性促进惩罚项来正则化对数似然函数。通过在每个分析步骤中顺序估计自由度，我们的滤波器能够根据数据的尾部厚度自适应调整其先验-后验更新。最后，我们在具有挑战性的滤波问题上验证了这一新型集成滤波器的优势。