Activity-dependent branching ratios in stocks, solar x-ray flux, and the Bak-Tang-Wiesenfeld sandpile model

We define an activity-dependent branching ratio that allows comparison of different time series $X_t$. The branching ratio $b_x$ is defined as $b_x=E\left[{\xi }_x/x\right]$. The random variable ${\xi }_x$ is the value of the next signal given that the previous one is equal to $x$, so ${\xi }_x=\left\{X_{t+1}\mid X_t=x\right\}$. If $b_x>1$, the process is on average supercritical when the signal is equal to $x$, while if $b_x<1$, it is subcritical. For stock prices we find $b_x=1$ within statistical uncertainty, for all $x$, consistent with an “efficient market hypothesis.” For stock volumes, solar x-ray flux intensities, and the Bak-Tang-Wiesenfeld (BTW) sandpile model, $b_x$ is supercritical for small values of activity and subcritical for the largest ones, indicating a tendency to return to a typical value. For stock volumes this tendency has an approximate power-law behavior. For solar x-ray flux and the BTW model, there is a broad regime of activity where $b_x\simeq 1$, which we interpret as an indicator of critical behavior. This is true despite different underlying probability distributions for $X_t$ and for ${\xi }_x$. For the BTW model the distribution of ${\xi }_x$ is Gaussian, for $x$ sufficiently larger than 1, and its variance grows linearly with $x$. Hence, the activity in the BTW model obeys a central limit theorem when sampling over past histories. The broad region of activity where $b_x$ is close to one disappears once bulk dissipation is introduced in the BTW model—supporting our hypothesis that it is an indicator of criticality.

Figure 1

(Color online) The branching ratio $b_p$ vs price for different stocks. The $x$ axis has been scaled by the mean price. All data shown have a resolution of one minute, where the price used is that at the start of the minute. Error bars in this and subsequent figures indicate one standard deviation. The data for qqqq were taken over the period 09:30 23/05/08–13:37 20/06/08, the data for csco are from 09:30 23/05/08–13:37 20/06/08, the data for aapl are from 9:30 27/05/08–14:08 23/06/08, and the data for F are from 09:30 27/05/08–14:03 23/06/08. These results are consistent with the weak EMH.

Figure 2

(Color online) Same as Fig. 1 for the Dow Jones industrial average. The data are from 01/10/1928–23/05/2008, and has a resolution of 1 day using the opening price. The behavior is also consistent with the weak EMH.

Figure 3

(Color online) Activity-dependent branching ratios for stock volumes during the same time period as in Fig. 1. A line with slope $m=-0.69$ is included as a guide for the eye. The behavior of the Apple stock (aapl) appears to deviate from the other three.

Figure 4

(Color online) Comparison of cumulative distribution function (CDF) $P\left({\xi }_x/x\ge b\right)$ for stock volumes and solar intensities at a given value of activity $x$. For stock volumes, (f) Ford was used, and the CDFs calculated at $x=V/⟨V⟩=0.2\pm 0.05$ (supercritical region) and $x=V/⟨V⟩=2\pm 1$ (subcritical). For solar intensities the entire time series, “All,” was used, and the CDFs calculated at $x=I/⟨I⟩=0.1\pm 0.005$ (supercritical) and $x=I/⟨I⟩=100\pm 10$ (subcritical). The Ford CDF at $x=0.2\pm 0.05$ is compared with a Gaussian having the same mean and variance. The distributions are broader than Gaussian in all cases.

Figure 5

(Color online) Activity-dependent branching ratios for solar flux intensity at solar minimum, solar maximum, and for the entire data set, “All,” as defined in the text. This shows a weak tendency to return to a typical value although $b_I\approx 1$ for a broad range of intensities, $I$. This behavior is comparable to that shown for the SOC BTW model in Fig. 7.

Figure 6

(Color online) The probability distribution function of solar x-ray intensities, $P\left(I\right)$. The straight line indicates a power-law fit with exponent 2.3 for the “All” time series. This is different from the approximately exponential behavior seen in the SOC BTW model as shown in Fig. 8.

Figure 7

(Color online) Activity-dependent branching ratio for the SOC BTW model for different system sizes, $L$. Error bars are smaller than symbol size. As the system size increases, the region where $b_n\approx 1$ broadens. The entries beginning with D denote the dissipative BTW model. The dissipative “D-BTW” model does not show a broad region where $b_n\approx 1$.

Figure 8

(Color online) The probability distribution for activity in the SOC BTW model, $P\left(n\right)$. Error bars are smaller than symbol size. The decay is approximately exponential. The best data collapse of the tail of the distribution is obtained by rescaling with $L^{1.2}$. Since the SOC BTW model does not exhibit finite size scaling this rescaling is only for the purpose of plotting all the data together.

Figure 9

(Color online) Probability distribution function $P\left({\xi }_n/n\right)$ for the SOC BTW model. Error bars are smaller than symbol size. The distribution is shown for $n=15$, 150, and 450 with $L=500$. The system for $n=15$ lies in the supercritical region, while for $n=150$ and $n=450$ it lies in the subcritical region. In all cases the distributions are indistinguishable from Gaussian.

Figure 10

(Color online) The variance of the random variable ${\xi }_n$, ${\sigma }^2\left({\xi }_n\right)/n$ vs $n$ for three different system sizes. The ratio goes to a constant for large $n$. Hence the variance of ${\xi }_n$ grows as $n$.

Figure 11

(Color online) Time series of activity $n_t$ for different initial conditions for the BTW model with periodic boundary conditions. This displays different oscillatory behaviors for different initial conditions. In each case a system of size $L=500$ was used.