Statistical accuracy of the ensemble Kalman filter in the near-linear setting
E. Calvello, J. A. Carrillo, F. Hoffmann, P. Monmarché, A. M. Stuart, U. Vaes
TL;DR
This work analyzes state estimation from partial, noisy observations, where particle filters struggle in high dimensions due to weight collapse and the ensemble Kalman filter (EKF) provides equal-weight particles as an alternative. It adopts a mean-field perspective to derive a nonlinear, nonautonomous update on the law of the state, linking finite-$J$ EKF dynamics to a mean-field map $\mathsf T$ that is exact on Gaussian measures via $\mathsf T(\pi, y^{\dagger})=\mathsf B(\pi, y^{\dagger})$ for $\pi$ Gaussian. The main results include a stability bound $\mathrm d_g(\mu^{\rm EK}_N, \mu_N) \le C\max_n \mathrm d_g(\mathsf Q\mathsf P\mu_n, \mathsf G\mathsf Q\mathsf P\mu_n)$ and a finite-$J$ error bound $(\mathbb E|\mu^{\rm EK,J}_N[\varphi]-\mu_N[\varphi]|^2)^{1/2} \le C(1/\sqrt{J}+\varepsilon)$ under affine baselines and small perturbations, with the constants growing with the horizon $N$. The results extend accuracy analysis beyond the linear Gaussian setting and suggest directions for long-time behavior, continuous-time variants, and inverse problems.
Abstract
Estimating the state of a dynamical system from partial and noisy observations is a ubiquitous problem in a large number of applications, such as probabilistic weather forecasting and prediction of epidemics. Particle filters are a widely adopted approach to the problem and provide provably accurate approximations of the statistics of the state, but they perform poorly in high dimensions because of weight collapse. The ensemble Kalman filter does not suffer from this issue, as it relies on an interacting particle system with equal weights. Despite its wide adoption in the geophysical sciences, mathematical analysis of the accuracy of this filter is predominantly confined to the setting of linear dynamical models and linear observations operators, and analysis beyond the linear Gaussian setting is still in its infancy. In this short note, we provide an accessible overview of recent work in which the authors take first steps to analyze the accuracy of the filter beyond the linear Gaussian setting.