Optimal Estimate of Monotonic Trend with Sparse Jumps

This paper discusses a problem for recovering an underlying trend from noisy data. The key assumption is that the trend is monotonic, e.g., reflects accumulation of irreversible system deterioration. The trend is obtained as a maximum a posteriori probability estimate. The overall problem setup is related to alpha-beta filter and Hodrick-Prescott filter. The main difference is that instead of a Gaussian process noise, a one-sided exponentially distributed noise is assumed. The batch estimate is a solution to a Quadratic Programming problem. The approach works exceptionally well for piece-wise linear trends that have a small number of jumps in the trended variable or its increase rate. Theoretical analysis justifies the sparsity properties for the jumps in the solution.