Wavelets in Multi-Scale Time Series Analysis: An Application to Seismic Data

Forecasting earthquakes is one of the most important problems in Earth science because of their devastating consequences. Current scientific studies related to earthquake forecasting focus on three key points: when the event will occur, where it will occur, and how large it will be. In this work we investigate the possibility to determine when the earthquake will take place. We formulate the problem as a multiple change-point detection in the time series. In particular, we refer to the multi-scale formulation described in Fryzlewicz (Ann Stat 46(6B): 3390–3421, 2018). In that paper a bottom-up hierarchical structure is defined. At each stage, multiple neighbor regions which are recognized to correspond to locally constant underlying signal are merged. Due to their multi-scale structure, wavelets are suitable as basis functions, since the coefficients of the representation contain local information. The preprocessing stage involves the discrete unbalanced Haar transform, which is a wavelet decomposition of one-dimensional data with respect to an orthonormal Haar-like basis, where jumps in the basis vectors do not necessarily occur in the middle of their support. The algorithm is tested on data from a well-characterized laboratory system described in Rouet-Leduc et al. (Geophys Res Lett 44(18): 9276–9282, 2017).