Simulation analysis of multifractal detrended methods based on the ARFIMA process

The two-component autoregressive fractionally integrated moving average (ARFIMA) process and mix-correlated ARFIMA(MC-ARFIMA) are applied in this paper to generate artificial sequence with Hxy = 1/2(Hx + Hy) and Hxy < 1/2(Hx + Hy) respectively and simulate the results of multifractal detrended cross-correlation analysis (MFXDFA), multifractal detrending moving average cross-correlation analysis (MFXDMA), MFDCCA based on maximum overlap wavelet transform (MFDCCA-MODWT), and multifractal detrended partial cross-correlation analysis(MF-DPXA). The advantages and disadvantages of MFXDFA, MFXDMA(θ = 0,0.5,1), and MFDCCA-MODWT are compared to the long-memory of sequences. In the case of Hxy < 1/2(Hx + Hy), these three methods keep around the theoretical value with small fluctuations in a variety of sequence lengths. In the case of Hxy = 1/2(Hx + Hy), the curves are significantly stable and are slightly smaller than the theoretical value. The precision of these estimators may be influenced by the relationship between Hxy and 1/2(Hx + Hy). Multifractal features is detected and the result shows that MFXDMA-0 and MFXDMA-1 is optimal to detect the multifractality. An interesting finding is that MFDCCA-MODWOT performs best in both case of Hxy = 1/2(Hx + Hy) and Hxy < 1/2(Hx + Hy), but it performs worst to detect the multifractality. When Gaussian noise is added to the sequences with different long-memory levels, MFDPXA can eliminate the noise interference compared with MFXDFA, thereby verifying the effectiveness of this method.