Robust and Trend-Following Student's t Kalman Smoothers

Two nonlinear Kalman smoothers are proposed using the Student’s t distribution. The first, which we call the T-Robust smoother, finds the maximum a posteriori (MAP) solution for Gaussian process noise and Student’s t observation noise. It is extremely robust against outliers, outperforming the recently proposed `1-Laplace smoother in extreme situations with data containing 20% or more outliers. The second, which we call the T-Trend smoother, is a MAP solver for a model with Student’s t-process noise and Gaussian observation noise. This smoother tracks sudden changes in the process model. A key ingredient of our approach is a novel technique to overcome the non-convexity of the Student’s t loss function. By exploiting the structure of the underlying dynamics, the computational effort per iteration grows linearly with the length of the time series. Convergence analysis is provided for both smoothers. Numerical results for linear and nonlinear models illustrate the performance of the new smoothers for both robust and fast tracking applications, including an underwater tracking application with real data.