Probabilities for large events in driven threshold systems

Many driven threshold systems display a spectrum of avalanche event sizes, often characterized by power-law scaling. An important problem is to compute probabilities of the largest events (“Black Swans”). We develop a data-driven approach to the problem by transforming to the event index frame, and relating this to Shannon information. For earthquakes, we find the 12-month probability for magnitude $m>6$ earthquakes in California increases from about 30$\%$ after the last event, to 40$\%$–50$\%$ prior to the next one.

Figure 1

Optimized time series for earthquake probabilities in the California-Nevada region during 1997 to the present, where date quality is best. Top: Chance of an earthquake having magnitude $m_L>6$ during the next 12 months as a function of time. Bottom: Magnitudes of all events in the region as a function of time. Blue dots and vertical dashed lines are earthquakes in the region having $m_L>6$. $\times$ represents current 12-month probability of $\tilde{5}1\%$.

Figure 2

Reliability (scatter plot) diagram for earthquake probabilities corresponding to the time series shown in Fig. 1, and to the data in Table . In the figure, there are seven bins having forecast probability, and these are plotted against observed frequencies. A perfectly reliable forecast would have zero reliability error, and the blue dots would lie along the black dashed line having $\text{slope}=1$. The red dashed line is the no-skill line, and the large red dot is the “climatology point,' or average forecast. The light turquoise crosses are the reliability errors constructed from superposing the reliability diagrams for the 500 random forecasts associated with the bootstrap error analysis. The inset shows the histogram of forecast probability bins.

Figure 3

Temporal receiver operating characteristic diagram for the forecast of Fig. 1. The solid blue line is the ROC diagram for 1997 to present, and the light turquoise crosses represent a superposition of the ROC diagrams for the 500 random forecasts.