Understanding volatility correlation behavior with a magnitude cross-correlation function

We propose an approach for analyzing the basic relation between correlation properties of the original signal and its magnitude fluctuations by decomposing the original signal into its positive and negative fluctuation components. We use this relation to understand the following phenomenon found in many naturally occurring time series: the magnitude of the signal exhibits long-range correlation, whereas the original signal is short-range correlated. The applications of our approach to heart rate variability signals and high-frequency foreign exchange rates reveal that the difference between the correlation properties of the original signal and its magnitude fluctuations is induced by the time organization structure of the correlation function between the magnitude fluctuations of positive and negative components. We show that this correlation function can be described well by a stretched-exponential function and is related to the nonlinearity and the multifractal structure of the signals.

Figure 1

(a) The fluctuation $x\left(t\right)$ for the out-of-phase correlated state in a segment $\upsilon$ of time length $n=400$. The solid (dotted) line corresponds to $x_p\left(t\right)$ $\left[x_n\left(t\right)\right]$ of $x\left(t\right)$. (b) The in-phase correlated state. (c) The vectors ${\vec{V}}_p^{\upsilon }$ (solid) of $x_p\left(t\right)$ and ${\vec{V}}_n^{\upsilon }$ (dotted) of $x_n\left(t\right)$ for $x\left(t\right)$ in (a). (d) ${\vec{V}}_p^{\upsilon }$ and ${\vec{V}}_n^{\upsilon }$ for $x\left(t\right)$ in (b). The data are from the FX exchange rate of lira (Italy) for the U.S. dollar in 1996 [8].

Figure 2

(a) The scale exponents $H$, $H_{\Vert }$ (•), and $H_{pn}$ (엯) of the interbeat interval signals $x\left(t\right)$ for 36 HRV data. (b) The scale exponents $H$ and $H_{\Vert }$ of price change signals $x\left(t\right)$ of nine foreign exchange rates for the U.S. dollar. Data indices: $i=1$ (Germany, DEM), 2 (Japan, JPY), 3 (Italy, ITL), 4 (France, FRF), 5 (Switzerland, CHF), 6 (Belgium, BEF), 7 (Denmark, DKK), 8 (Spain, ESP), and 9 (Finland, FIM).

Figure 3

(a) The scaling behaviors of $F_{\Vert }\left(n\right)$ (∎) and $f_{pn}\left(n\right)$ (●) for the HRV signal of a healthy subject [8]. (b) The scaling behaviors of $F_{\Vert }\left(n\right)$ and $f_{pn}\left(n\right)$ for the JPY-USD FX data in Fig. 2b.

Figure 4

(Color online) (a) The magnitude cross-correlation function $C\left(n\right)$ for five people randomly selected from the 36 HRV data. The function $C\left(n\right)$ is fitted by a stretched-exponential function (solid curves). (b) The function $C\left(n\right)$ (square dotted) of HRV data in Fig. 4a is plotted for comparison with $C\left(n\right)$ of FX data.

Figure 5

(a) The rescaled functions $R\left(n\right)$ in Eq. (9) for the HRV (●) and FX (×) data in Figs. 4a, 4b, respectively. (b) The distribution of $\gamma$ and $T$ for the magnitude cross-correlation $C\left(n\right)$ of 36 HRV data (●) and nine FX data (×).

Figure 6

(a) The functions $C\left(n\right)$ (●) for $x\left(t\right)$’s of five HRV data in Fig. 3a and (×) for $x\left(t\right)$’s of their surrogate data. (b) The function $C\left(n\right)$ (●) for $x\left(t\right)$’s of five FX data in Fig. 3b and (×) for $x\left(t\right)$’s of their surrogate data.

Figure 7

(a) The multifractal spectrum for the typical HRV (●) and FX (엯) data. (b) The minimum $\alpha$ (dotted line) and the maximum $\alpha$ (solid line) in the $\alpha \text{−}f\left(\alpha \right)$ curve for FX $(i=1–5)$ and HRV $(i=6–10)$ data.