Koopman Kalman Filter (KKF): An asymptotically optimal nonlinear filtering algorithm with error bounds and its application to parameter estimation

TL;DR

This work addresses nonlinear state estimation by reframing nonlinear filtering in an infinite-dimensional linear framework via the Koopman operator. It introduces the Koopman Kalman Filter (KKF), which combines RKHS-based kernel methods with Extended Dynamic Mode Decomposition (kEDMD) to obtain finite-dimensional approximations of the Koopman operators and derive a Kalman-like recursion. A key contribution is a rigorous $O(N^{-1/2})$ error bound for the finite-dimensional KKF relative to the infinite-dimensional reference, together with a KKF-based parameter estimation procedure that competes with MCMC methods in speed. Numerical experiments on linear and nonlinear dynamics (e.g., SIR/SIRS/SEIRS models) validate the error bounds, show KKF achieving competitive accuracy with substantially lower runtime, and demonstrate practical applicability to data assimilation in epidemiology.

Abstract

In this article, we propose a new filtering algorithm based in the Koopman operator, showing that a nonlinear filtering problem can be seen as an equivalent problem where the dynamics is infinite dimensional, but linear. Using Extended Dynamic Mode Decomposition (EDMD), we create a finite dimensional approximation of the filtering problem of dimension $N$, in state and error covariance matrix, that accomplishes an error bound of order \(O(N^{-1/2})\) in both where $N$ denotes the number of points used in the Koopman approximation. The algorithm is denominated Koopman Kalman Filter (KKF), and has computational complexity \(O(T\cdot N^3)\) in time, and \(O(T \cdot N^2)\) in space, where \(T\) is the number of iterations of the filtering problem. We test the algorithm in linear and nonlinear dynamics cases, showing and effective error bound with respect to the Kalman filter, that corresponds to the optimal solution in the linear case, and equals the error performance of other methods in the state of the art, but with a much lower execution time. Also, we propose a parameter estimation algorithm based in KKF, comparing it with Markov Chain Monte Carlo techniques, showing similar performance with lower execution time.

Figures (5)

• Figure 1: Errors of KKF trajectories with respect to Kalman filter trajectories. The dotted red line represents the best fit curve of the form $C\cdot N^{\alpha}$ that fits the empirical errors.
• Figure 2: Realization of KKF in a SIR model.
• Figure 3: SIRS system parameter evolution in the KKF estimation. The first iterations used for warm-up are shown in red. The later iterations for the distribution are shown in orange. The constant true parameter is shown as a blue line. In this case, we can see the successful estimation as the final iterations accumulate around the true parameters.
• Figure 4: Mean parameter densities for the three parameters of the SIRS system. The mean is taken across all cores that are running a process in parallel. The dashed vertical line indicates the true parameter, which should correspond to, or be near, the mode of the distribution.
• Figure 5: Samples of the trajectories from the parameter distribution generated by each parameter estimation algorithm for the SIRS model.

Theorems & Definitions (33)

• Proposition 1: Kalman1960AProblems
• Proposition 2: Kalman1960AProblems
• Definition 1: Transition measures
• Definition 2: Invariant spaces
• Definition 3: Reproducing Kernel Hilbert Space (RKHS) Mercer1909XVI.Equations
• Theorem 1
• Definition 4: Kronecker Product
• Definition 5: Covariance operators
• Definition 6: Conditional Embedding Operator
• Theorem 2