Detrended partial cross-correlation analysis of two nonstationary time series influenced by common external forces

When common factors strongly influence two power-law cross-correlated time series recorded in complex natural or social systems, using detrended cross-correlation analysis (DCCA) without considering these common factors will bias the results. We use detrended partial cross-correlation analysis (DPXA) to uncover the intrinsic power-law cross correlations between two simultaneously recorded time series in the presence of nonstationarity after removing the effects of other time series acting as common forces. The DPXA method is a generalization of the detrended cross-correlation analysis that takes into account partial correlation analysis. We demonstrate the method by using bivariate fractional Brownian motions contaminated with a fractional Brownian motion. We find that the DPXA is able to recover the analytical cross Hurst indices, and thus the multiscale DPXA coefficients are a viable alternative to the conventional cross-correlation coefficient. We demonstrate the advantage of the DPXA coefficients over the DCCA coefficients by analyzing contaminated bivariate fractional Brownian motions. We calculate the DPXA coefficients and use them to extract the intrinsic cross correlation between crude oil and gold futures by taking into consideration the impact of the U.S. dollar index. We develop the multifractal DPXA (MF-DPXA) method in order to generalize the DPXA method and investigate multifractal time series. We analyze multifractal binomial measures masked with strong white noises and find that the MF-DPXA method quantifies the hidden multifractal nature while the multifractal DCCA method fails.

Figure 1

Detrended partial cross-correlation exponents. (a, b) Dependence of the average Hurst indices $h_{r_x}$ and $h_{r_x}$ of the two components of the generated bivariate fractional Brownian motions on the input Hurst indices $H_{r_x}$ and $H_{r_x}$. (c) Dependence of the average Hurst index $h_z$ of the generated univariate fractional Brownian motions on the input Hurst index $H_z$. (d) Dependence of $h_x$ on $H_z$ for different $h_{r_x}$ values. (e) Relative error $\mathrm{\Delta }{h}_{r_x{r}_y}=h_{r_x{r}_y}-H_{r_x{r}_y}$. (f) Relative error $\mathrm{\Delta }{h}_{xy:z}$ between the DPXA estimate ${\langle h_{xy:z}\rangle }_z$ and the true value $h_{r_x{r}_y}$ as a function of $h_{r_x}$ and $h_{r_y}$.

Figure 2

Detrended partial cross-correlation coefficients. (a) Performance of different methods by comparing three cross-correlation coefficients ${\rho }_{x,y}$, ${\rho }_{r_x,r_y}$, and ${\rho }_{x,y:z}$ of the mathematical model in Eq. (8). (b) Estimation and comparison of the cross-correlation levels between the two return time series ($\circ$) and two volatility time series ($\square$) of crude oil and gold when including and excluding the influence of the U.S. dollar (USD) index.

Figure 3

Multifractal detrended partial cross-correlation analysis of two binomial measures contaminated by Gaussian noise with very low signal-to-noise ratio. (a)–(c) The segments of the binomial signal $r_x\left(t\right)$ with $p_x=0.3$, the Gaussian noise $z\left(t\right)$, and the “observed” signal $x\left(t\right)$. (d)–(f) Power-law dependence of the fluctuations $F_{xy}(q,s)$, $F_{xy:z}(q,s)$, $F_{r_x{r}_y}(q,s)$ on the scale $s$ for $q=-4$, 0, and 4 (from top to bottom). The values of $F_{xy:z}$ and $F_{r_x{r}_y}$ have been multiplied by $1\times {10}^5$. (g) Multifractal mass exponents ${\tau }_{xy}\left(q\right)$, ${\tau }_{xy:z}\left(q\right)$, and ${\tau }_{r_x{r}_y}\left(q\right)$, with the theoretical curve ${\mathcal{T}}_{r_x{r}_y}$ shown as a continuous line. (d) Multifractal spectra $f_{xy}\left(\alpha \right)$, $f_{xy:z}\left(\alpha \right)$, and $f_{r_x{r}_y}\left(\alpha \right)$ of the singularity strength $\alpha$. The continuous curve is the theoretical spectrum ${\mathcal{F}}_{r_x{r}_y}\left(\alpha \right)$.