Sparse Estimation Techniques for l1 Mean and Trend Filtering

Sammanfattning: It is often desirable to find the underlying trends in time series data. This is a wellknown signal processing problem that has many applications in areas such as financial dataanalysis, climatology, biological and medical sciences etc. Mean filtering finds a piece-wiseconstant trend in the data while trend filtering finds a piece-wise linear trend. When thesignal is noisy, the main difficulty is finding the changing points in the data. These are thepoints where the mean or the trend changes. We focus on a quadratic cost function with apenalty term on the number of changing points. We use the `1 norm for the penalty termas it leads to a sparse solution. This is attractive because the problem is convex in theunknown parameters and well known optimization algorithms exist for this problem. Weinvestigate the Alternating Direction Method of Multipliers (ADMM) algorithm and twofast taut string methods in terms of computational speed and performance. A well knownproblem is the occurrence of false changing point detection. We incorporate a techniqueto remove these false changing points to the fast mean filtering algorithm resulting in anefficient method with fewer false detections. We also propose an extension of the fast meanfiltering technique to the trend filtering problem. This is an approximate solution that workswell for signals with low noise levels.