An $\ell _{1}$-Laplace Robust Kalman Smoother

Abstract

Robustness is a major problem in Kalman filtering and smoothing that can be solved using heavy tailed distributions; e.g., ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -Laplace. This paper describes an algorithm for finding the maximum a posteriori (MAP) estimate of the Kalman smoother for a nonlinear model with Gaussian process noise and ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -Laplace observation noise. The algorithm uses the convex composite extension of the Gauss-Newton method. This yields convex programming subproblems to which an interior point path-following method is applied. The number of arithmetic operations required by the algorithm grows linearly with the number of time points because the algorithm preserves the underlying block tridiagonal structure of the Kalman smoother problem. Excellent fits are obtained with and without outliers, even though the outliers are simulated from distributions that are not ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -Laplace. It is also tested on actual data with a nonlinear measurement model for an underwater tracking experiment. The ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -Laplace smoother is able to construct a smoothed fit, without data removal, from data with very large outliers.