Seasonal Signal Extraction from GPS Coordinate Time Series Using Low-Rank Matrix Approximation Based on Nonconvex Log-Sum Function Minimization

Inspired by an earlier work, which showed that the Hankel matrix constructed by a global positioning system (GPS) coordinate time series has a low rank, this study proposes a low-rank matrix approximation based on a nonconvex log-sum function minimization for extracting seasonal signals from GPS coordinate time series. Compared to a convex nuclear norm minimization, the nonconvex log-sum function is more approximate for the rank function. Unlike a nuclear norm minimization, which can be solved by singular value thresholding, obtaining the solution of a nonconvex log-sum function minimization is challenging. By using the iterative reweighted nuclear norm algorithm, the nonconvex low-rank matrix approximation problem is addressed in this paper by iteratively updating the weighted nuclear norm minimization problem, which has a globally optimal solution under certain conditions. Correspondingly, the residuals were analyzed using a maximum likelihood estimation. Both the proposed method and the weighted nuclear norm minimization approach were compared with classical methods in the presence of time-correlated noise. Compared to the classical seasonally extracted methods, the proposed method provides a guaranteed root mean square error performance in the presence of time-correlated noise. The experimental analysis results of both the simulated time series and real station data demonstrated that the proposed method outperformed the weighted nuclear norm minimization approach, which outperformed the classical approaches under different noise levels.