Fast robust methods for singular state-space models

Abstract

State-space models are used in a wide range of time series analysis applications. Kalman filtering and smoothing are work-horse algorithms in these settings. While classic algorithms assume Gaussian errors to simplify estimation, recent advances use a broad range of optimization formulations to allow outlier-robust estimation, as well as constraints to capture prior information.Here we develop methods on state-space models where either transition or error covariances may be singular. These models frequently arise in navigation (e.g. for 'colored noise' models or deterministic integrals) and are ubiquitous in auto-correlated time series models such as ARMA. We reformulate all state-space models (singular as well as nonsingluar) as constrained convex optimization problems, and develop an efficient algorithm for this reformulation. The convergence rate is locally linear, with constants that do not depend on the conditioning of the problem.Numerical comparisons show that the new approach outperforms competing approaches for nonsingular models, including state of the art interior point (IP) methods. IP methods converge at superlinear rates; we expect them to dominate. However, the steep rate of the proposed approach (independent of problem conditioning) combined with cheap iterations wins against IP in a run-time comparison. This suggests that the proposed approach can be a default choice for estimating state space models outside of the Gaussian context for singular and nonsingular models. To highlight the capabilities of the new framework, we focus on navigation applications that use singular process covariance models, and analyze data from a drifting mooring as a proxy for an autonomous underwater vehicle.

Robust Singular Smoothers for Tracking Using Low-Fidelity Data

“…In our implementation, we need only compute a single block bidiagonal factorization once, which can then be used to solve (8) in O(n 2 N ) operations in each iteration, no more expensive than a single matrix-vector multiply. For piecewise-linear quadratic ρ [26,15], DRS converges to an optimal solution at a local linear rate [16], which does not depend on the condition number of A. A good initialization makes DRS competitive with the fastest available solvers, even second order methods with quadratic local rates [15].…”

“…In this paper, we build on the recently proposed framework of [16] for singular models, and systematically develop complementary modeling elements: robust penalties, informative constraints, and singular models. The resulting approach exploits the structure of singular covariances head on rather than using workarounds such as pseudo-inverses or variance boosting that either do not work in the general setting or introduce unnecessary changes to the fundamental model (see discussion in [16]). A simple The red dash-dot shows a'robust' Huberized approach implemented using a pseudo-inverse; green dash shows the proposed singular 2 estimate; blue solid shows the proposed singular Huber estimate, which clearly tracks the true state.…”

“…None of these techniques generalize to singular models in the setting of (2), and some naive generalizations fail dramatically ( Figure 1). This paper builds on the reformulation of [16] for singular models. We develop a systematic approach and test it using a navigation model with a real-world mooring dataset.…”

Efficient Robust Parameter Identification in Generalized Kalman Smoothing Models

“…rr (r m )R −1/2 (Hx) θ We now have, fully and explicitly, first and second derivatives of the value function v(θ) in (7) for the nonsingular case. Though these results are straightforward, they do not appear in any smoothing literature we are aware of in this compact form, even for least squares losses.…”

“…A singular covariance matrix precludes any errors and innovations that are not in its range. We follow [7] in formulating this problem:…”