Dynamical behavior of combined detrended cross-correlation analysis methods in random walks and Lévy flights

In this paper, the utility and the correlation technique of the detrended cross-correlation analysis and the combined detrended cross-correlation analysis methods are treated for the stochastic properties of the random walk and the Lévy flight via nucleotide sequence data. Data are extracted definitely from twelve genomic DNA sequences (HUMHBB 1-6, Amazonmolly, Bushbaby, Cow, Dolphin, Elephant, and Tarsier) of the GenBank databank. We mainly simulate and analyze scaling exponents for three methods in 1D and 2D random walks and Lévy flights. To compare each other between one gene and the other gene in the twelve genomic DNA sequences, the value of the detrended cross-correlation coefficient ρ_DCCA in {HUMHBB3} and {Dolphin} genes is shown the largest value 0.214 in the 1D Lévy flight with μ=1.5 compared to other sets. The detrended cross-correlation coefficient ρ_DCCA in {HUMHBB4} and {HUMHBB6} genes is shown the smallest value -0.231 in the 1D Lévy flight with μ=1.5. We particularly find that the combined detrended cross-correlation coefficient ρ_CDCCA in the two cases of {HUMHBB3}+{Bushbaby} and {HUMHBB6}+{Dolphin} has the largest value 0.331 in the 1D random walk compared to other sets, while that in {HUMHBB3}+{Tarsier} and {HUMHBB4}+{Elephant} has the smallest value -0.295 in the 1D Lévy flight with μ=1.5.