Data-driven prediction of thresholded time series of rainfall and self-organized criticality models

We study the occurrence of events, subject to threshold, in a representative self-organized criticality (SOC) sandpile model and in high-resolution rainfall data. The predictability in both systems is analyzed by means of a decision variable sensitive to event clustering, and the quality of the predictions is evaluated by the receiver operating characteristic (ROC) method. In the case of the SOC sandpile model, the scaling of quiet-time distributions with increasing threshold leads to increased predictability of extreme events. A scaling theory allows us to understand all the details of the prediction procedure and to extrapolate the shape of the ROC curves for the most extreme events. For rainfall data, the quiet-time distributions do not scale for high thresholds, which means that the corresponding ROC curves cannot be straightforwardly related to those for lower thresholds. In this way, ROC curves are useful for highlighting differences in predictability of extreme events between toy models and real-world phenomena.

Figure 1

(a) Quiet-time PDFs for the Manna model $\left(L=1024\right)$ for different activity thresholds $n_c$ characterized by their quantiles $q$. Statistics are collected over $1\times {10}^7$ particle additions. PDFs for different $q$ are displaced vertically for clarity. The solid black lines have slope $-1.67$. (b) Data collapse of the suitably rescaled quiet-time PDFs. (c) Alternative data collapse in order to visualize the scaling function $f_0$. In all plots, quantile thresholds $q$ correspond to activity thresholds $n_c=51$, 123, 255, 584, and 806.

Figure 2

(a) Quiet-time PDFs for different rain rate thresholds, as measured on Manus island for the period 15 February 2005–18 March 2012. PDFs for different $q$ are displaced vertically for clarity. A power-law decay with exponent $\beta =1.14$ has been fitted to the $q=0.00$ data to serve as a visual guide. (b) Data collapse of the suitably rescaled quiet-time PDFs. (c) Alternative data collapse in order to visualize the scaling function $f_0$. Units in the ordinate are ${\text{min}}^{0.14}$. Quantile thresholds $q$ correspond to rate thresholds $r_c=0.20$, 0.67, 1.75, 4.74, 22.02, and 42.76 mm/h.

Figure 3

Pictorial definition of the hazard function. Event 2 falls between $t_w$ and $d{t}_w$, having exceeded event 1 by $t_w$.

Figure 4

(a) Hazard functions for quiet times in the Manna model for different activity thresholds (same data as in Fig. 1). (b) Hazard functions for quiet times of Manus island rain data for different rain rate thresholds (same data as in Fig. 2).

Figure 5

ROC curves for different thresholds in the activity or rate for (a) the Manna model with $L=1024$ and (b) rainfall data from Manus island. Same data as in Figs. 1 and 2.

Figure 6

Data collapse after rescaling the ROC curves for different activity thresholds in the Manna model with $L=1024$. Same data as in Fig. 1.