Robust and Trend-following Kalman Smoothers using Student's t

TL;DR

This paper introduces two nonlinear Kalman smoothers based on Student's t distributions, enhancing robustness to outliers and ability to track sudden changes in the process, with proven convergence and practical implementation methods.

Contribution

The paper presents novel Student's t-based Kalman smoothers that improve robustness to outliers and enable trend-following, with specialized algorithms and convergence analysis.

Findings

T-Robust smoother outperforms l1-Laplace in extreme outlier scenarios.

T-Trend smoother effectively tracks sudden process changes.

Both smoothers are easy to implement with minor modifications.

Abstract

We propose two nonlinear Kalman smoothers that rely on Student's t distributions. The T-Robust smoother finds the maximum a posteriori likelihood (MAP) solution for Gaussian process noise and Student's t observation noise, and is extremely robust against outliers, outperforming the recently proposed l1-Laplace smoother in extreme situations (e.g. 50% or more outliers). The second estimator, which we call the T-Trend smoother, is able to follow sudden changes in the process model, and is derived as a MAP solver for a model with Student's t-process noise and Gaussian observation noise. We design specialized methods to solve both problems which exploit the special structure of the Student's t-distribution, and provide a convergence theory. Both smoothers can be implemented with only minor modifications to an existing L2 smoother implementation. Numerical results for linear and nonlinear models illustrating both robust and fast tracking applications are presented.