Sarah E. Webster

Jacob Stevens-Haas, Sarah E. Webster, Aleksandr Aravkin

We examine acoustic Doppler current profiler (ADCP) measurements from underwater gliders to determine glider position, glider velocity, and subsurface current. ADCPs, however, do not directly observe the quantities of interest; instead, they measure the relative motion of the vehicle and the water column. We examine the lineage of mathematical innovations that have previously been applied to this problem, discovering an unstated but incorrect assumption of independence. We reframe a recent method to form a joint probability model of current and vehicle navigation, which allows us to correct this assumption and extend the classic Kalman smoothing method. Detailed simulations affirm the efficacy of our approach for computing estimates and their uncertainty. The joint model developed here sets the stage for future work to incorporate constraints, range measurements, and robust statistical modeling.

Jonathan Jonker, Aleksandr Y. Aravkin, James V. Burke et al.

State-space models are used in a wide range of time series analysis formulations. Kalman filtering and smoothing are work-horse algorithms in these settings. While classic algorithms assume Gaussian errors to simplify estimation, recent advances use a broader 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 innovations 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 nonsinguar) as constrained convex optimization problems, and develop an efficient algorithm for this reformulation. The convergence rate is {\it 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 {\it 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. We therefore suggest that the proposed approach be the {\it default choice} for estimating state space models outside of the Gaussian context, regardless of whether the error covariances are singular or not.