Real-time update of multi-fractal analysis on dynamic time series using incremental discrete wavelet transforms

Abstract

An algorithm is presented to update the multi-fractal spectrum of a time series in constant time when new data arrives. The discrete wavelet transform (DWT) of the time series is first updated for the new data value. This is done optimally in terms of sharing previous computations, in O(L) constant time, with L the number of levels of decomposition. The multi-fractal spectrum is then updated also in constant-time. New pre-computation techniques are presented to further accelerate this process. All possible 2^L data alignments are taken into account in the course of the incremental updates. The resulting spectrum estimate is more stable, compared to the current DWT method using only one dyadic frame, as precise, and more efficient. It is adapted for real-time on-line updates of the time series.