Robust enhanced trend filtering with unknown noise

One important step in time series analysis is the extraction of an underlying trend. However, the true trend is often submerged by complex background noise, especially non-Gaussian noise or outliers. Accurate trend extraction against outliers from a raw signal is a challenging task. To address this challenge, this paper extends l₁ trend filtering to a robust enhanced trend filtering called RobustETF by combining mix of Gaussian (MoG) and non-convex sparsity-inducing functions. We first model the noise as a MoG distribution to allow RobustETF to be robust in the presence of any type of non-Gaussian noise or outliers. After that, to handle the biased estimation of the l₁ norm, we use the Gibbs distribution embedding smoothed and non-convex sparsity-inducing functions to faithfully preserve the amplitude of the trend. Furthermore, we design an extended EM algorithm to solve the resulting non-convex optimization problem. Finally, we show the results of experiments on both real-world and synthetic data to compare the performance of the proposed algorithm against other state-of-the-art methods. Finally, the corresponding Matlab codes are available at https://github.com/ZhaoZhibin/RobustETF.