Efficient filtering using monotonic walk model

This paper proposes a nonlinear filter for estimating monotonic underlying trend from noisy observations. The filter computes maximum aposteriori probability (MAP) estimate using a monotonic walk model instead of the random walk model in standard linear filtering. The batch estimate is a solution of quadratic programming (QP) problem. This paper shows that the QP has a form of isotonic regression (IR) and has a linear computational complexity. The filter is implemented in a moving horizon estimation (MHE) setting. The data beyond the estimation horizon are replaced by the initial condition parameters (arrival cost). The MHE for IR is nonsmooth, so the existing nonlinear MHE theory is not applicable. By exploiting properties of the IR solution, we develop an update of the MHE arrival cost, which is provably close to the full information MAP solution and stable. The analysis is complemented by a Monte Carlo simulation study of the proposed nonlinear filtering algorithm. The simulation results confirm improved performance of the proposed filter compared with a linear filter and the earlier version of the MHE update.