Randomized multifractal detrended fluctuation analysis of long time series.

A novel general randomized method is proposed to investigate multifractal properties of long time series. Based on multifractal temporally weighted detrended fluctuation analysis (MFTWDFA), we obtain randomized multifractal temporally weighted detrended fluctuation analysis (RMFTWDFA). The innovation of this algorithm is applying a random idea in the process of dividing multiple intervals to find the local trend. To test the performance of the RMFTWDFA algorithm, we apply it, together with the MFTWDFA, to the artificially generated time series and real genomic sequences. For three types of artificially generated time series, consistency tests are performed on the estimated h(q), and all results indicate that there is no significant difference in the estimated h(q) of the two methods. Meanwhile, for different sequence lengths, the running time of RMFTWDFA is reduced by over ten times. We use prokaryote genomic sequences with large scales as real examples, the results obtained by RMFTWDFA demonstrate that these genomic sequences show fractal characteristics, and we leverage estimated exponents to study phylogenetic relationships between species. The final clustering results are consistent with real relationships. All the results reflect that RMFTWDFA is significantly effective and timesaving for long time series, while obtaining an accuracy statistically comparable to other methods.