A local fitting based multifractal detrend fluctuation analysis method

Abstract

In this study, we propose a one-dimensional multifractal local fitting detrending fluctuation analysis (MF-LF-DFA) algorithm using traditional multifractal detrended fluctuation analysis (MF-DFA). The classical MF-DFA uses the least squares technique to fit the local trend of the time series, and the fitting order is fixed. However, the proposed MF-LF-DFA algorithm sets the fitting order reasonably for different local intervals according to the fluctuation characteristics of the sequence. We examine the performances of MF-DFA and MF-LF-DFA using the time series made using p-model multiplication cascade. By comparing the numerical results and analytical values of the generalized Hurst exponent H(q), the Scaling exponent τ(q) and multifractal spectrum f(α), the empirical tests demonstrate that H(q), τ(q) and f(α) calculated by MF-LF-DFA algorithm are closer to the analytical values. Furthermore, we compute the relative errors Herr and τerr of the two methods, and the relative errors calculated by MF-LF-DFA are smaller. Besides, we examine the monofractal structure of MF-LF-DFA by constructing a random sequence and calculating the value of H(2). In addition, we generate and test shuffled and phase Fourier randomized sequences of the original random time series and multiplicative cascade time series. We also study different length sequences within the same model, and verify the stability of the results. Finally, we change the range of segmentation window s and the fitting order of MF-DFA model to check the robust property of the proposed MF-LF-DFA algorithm. The computational results indicate that MF-LF-DFA outperforms the traditional MF-DFA in all situations.