Predicting extreme avalanches in self-organized critical sandpiles

In a finite-size Abelian sandpile model, extreme avalanches are repelling each other. Taking a time series of the avalanche size and using a decision variable derived from that, we predict the occurrence of a particularly large avalanche in the next time step. The larger the magnitude of these target avalanches, the better is their predictability. The predictability which is based on a finite-size effect, is discussed as a function of the system size.

Figure 1

Recurrence time distribution for $L=64$ and $\eta =10000$ for the observation series and the surrogate data. The distribution was truncated at a time of $10000$ added sand grains to enhance visibility of the depression for small times.

Figure 2

(Color online) Conditional probability $P\left(X=1\mid y\right)$ for the system size $L=64$ as a function of the decision variable $y$, for events exceeding 1%, 2%, 5%, 10%, 15%, 20%, 25%, 30%, and 35% of the maximal avalanche size $s_{\text{max}}$ (from top to bottom).

Figure 3

Illustration of probabilistic predictions (superposition of ten examples): at relative time 0, an extreme event takes place (top panel: avalanche series $s_i$). The decision variable before the event drops to small values (medium panel), whereas the predicted probability for the event to happen is high (bottom panel).

Figure 4

ROC plots for the prediction of extreme avalanches for different minimal event magnitudes $\eta$, comparing $L=64$ and $L=256$ and the surrogate data for $L=64$.

Figure 5

The error sums $ϵ$ for our predictions in a representation which permits the comparison to the results in [9]. The symbols on every line represent the nine different $\eta$ values of Fig. 1.