Multiscale trend analysis: a new approach to studying complex time series

Practically every time series emerging in science involves interplay of trends at different scales. Obvious examples are alternations of seismic activation and quiescence, intermittence of warming and cooling of atmosphere and ocean, upward and downward slopes of a landscape, etc. Information on these trends is very important in many problems ranging from classical statistical interpolation of time series to prediction of natural disasters. Powerful Fourier and wavelet analyses are not always convenient since they reflect information on trends only indirectly. We introduce a new technique for decomposition of time series into a hierarchy of trends at different scales. As a result, a time series is represented by a tree whose nodes correspond to single trends. The larger is the scale at which the trend is observed, the higher in the tree is the corresponding node. Various applications of the multiscale trend analysis are demonstrated using fractal Brownian motion, and synthetic and observed seismicity.