Abstract
Robustness is a major problem in Kalman filtering and smoothing
that can be solved using heavy tailed distributions; e.g.,
l1-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
l1-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 l1-Laplace.
It is also tested on actual data with a nonlinear measurement model for
an underwater tracking experiment.
The l1-Laplace smoother is able to construct a smoothed fit,
without data removal, from data with very large outliers.