Peak Persistence Diagrams for Shape-Based Signal Estimation

Signal estimation from noisy data is a fundamental problem in signal processing and data analysis. Existing literature offers various estimators based on different model choices and estimation criteria. This paper uses an innovative framework that leverages topological and geometric features of the data for signal estimation. The proposed approach builds on penalized elastic signal alignment (PESA) framework and introduces a topological tool - peak-persistence diagram (PPD) - to analyze shapes of signals over the range of potential alignments. In the first step, the PPD estimates the unknown signal shape, defined as the number of internal peaks and valleys. In the second step, we estimate the underlying signal using shape-constrained optimization. This PESA approach strikes a balance between two extreme solutions: signal averaging without any alignment and signal averaging with full elastic alignment. Importantly, the proposed framework provides an estimator under a statistical model where the signal is affected by both additive and warping noise. A computationally efficient procedure for implementing this solution is presented, and its effectiveness is demonstrated through simulations and real-world examples. The latter include COVID rate curves and household electricity consumption curves. The results showcase superior performance of the proposed approach compared to several current state-of-the-art techniques.