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

Comparing prediction efficiency in the BTW and Manna sandpiles

The state-of-the-art in the theory of self-organized criticality reveals that a certain inactivity precedes extreme events, which are located on the tail of the event probability distribution with respect to their sizes. The existence of the inactivity allows for the prediction of these events in advance. In this work, we explore the predictability of the Bak-Tang-Wiesenfeld (BTW) and Manna models on the square lattice as a function of the lattice length. For both models, we use an algorithm that forecasts the occurrence of large events after a fall in activity. The efficiency of the prediction can be universally described in terms of the event size divided by an appropriate power-law function of the lattice length. The power-law exponents are projected to be 2.75 and 3 for the Manna and BTW models respectively. The scaling with the exponent 2.75 is known for collapsing of the entire size-frequency relationship in the Manna model. However, the correspondence between events on different lattices in the BTW model requires a variety of exponents where 3 is the largest. This indicates that in thermodynamic limit, prediction does exist in the Manna but not in the BTW model, at least based on inactivity. The difference in the universality classes may underline the difference in the prediction.

Signatures of self-organized dynamics in rapidly driven critical sandpiles

We study two prototypical models of self-organized criticality, namely sandpile automata with deterministic (Bak-Tang-Wiesenfeld) and probabilistic (Manna model) dynamical rules, focusing on the nature of stress fluctuations induced by driving-adding grains during avalanche propagation, and dissipation through avalanches that hit the system boundary. Our analysis of stress evolution time series reveals robust cyclical trends modulated by collective fluctuations with dissipative avalanches. These modulated cycles attain higher harmonics, characterized by multifractal measures within a broad range of timescales. The features of the associated singularity spectra capture the differences in the dynamic rules behind the self-organized critical states at adiabatic driving and their pertinent response to the increased driving rate, which alters the process of stochasticity and causes a loss of avalanche scaling. In sequences of outflow current carried by dissipative avalanches, the first return distributions follow the q-Gaussian law in the adiabatic limit. They appear to follow different laws at an intermediate scale with an increased driving rate, describing different pathways to the gradual loss of cooperative behavior in these two models. The robust appearance of cyclical trends and their multifractal modulation thus represents another remarkable feature of self-organized dynamics beyond the scaling of avalanches. It can also help identify the prominence of self-organizational phenomenology in an empirical time series when underlying interactions and driving modes remain hidden.

Explanation of flicker noise with the Bak-Tang-Wiesenfeld model of self-organized criticality

With the original Bak-Tang-Wisenefeld (BTW) sandpile we uncover the 1/φ noise in the mechanism maintaining self-organized criticality (SOC)-the question raised together with the concept of SOC. The BTW sandpile and the phenomenon of SOC in general are built on the slow time scale at which the system is loaded and the fast time scale at which the stress is transported outward from overloaded locations. Exploring the dynamics of stress in the slow time in the BTW sandpile, we posit that it follows cycles of gradual stress accumulation that end up with an abrupt stress release and the drop of the system to subcritical state. As the system size grows, the intracycle dynamics exhibits the 1/φ-like spectrum that extends boundlessly and corresponds to the stress release within the critical state.

Eigen microstates in self-organized criticality

We employ the eigen microstates approach to explore the self-organized criticality (SOC) in two celebrated sandpile models, namely the BTW model and the Manna model. In both models, phase transitions from the absorbing state to the critical state can be understood by the emergence of dominant eigen microstates with significantly increased weights. Spatial eigen microstates of avalanches can be uniformly characterized by a linear system size rescaling. The first temporal eigen microstates reveal scaling relations in both models. Furthermore, by finite-size scaling analysis of the first eigen microstates, we numerically estimate critical exponents, i.e., sqrt[σ_{0}w_{1}]/v[over ̃]_{1}∝L^{D} and v[over ̃]_{1}∝L^{D(1-τ_{s})/2}. Our findings could provide profound insights into eigen microstates of the universality and phase transition in nonequilibrium complex systems governed by self-organized criticality.

Increased extinction probability of the Madden-Julian oscillation after about 27 days

The Madden-Julian oscillation (MJO) is a tropical weather system that has an important influence in the tropics and beyond; however, many of its characteristics are poorly understood, including their initiation and termination. Here we define Madden-Julian events as contiguous time periods with an active MJO, and we show that both the durations and the sizes of these events are well described by a double power-law distribution. Thus, small events have no characteristic scale, and the same for large events; nevertheless, both types of events are separated by a characteristic duration of about 27 days (this corresponds to half a cycle, roughly). Thus, after 27 days, there is a sharp increase in the probability that an event becomes extinct. We find that this effect is independent of the starting and ending phases of the events, which seems to point to an internal mechanism of exhaustion rather than to the effect of an external barrier. Our results would imply an important limitation of the MJO as a driver of subseasonal predictability.

Universal predictability of large avalanches in the Manna sandpile model

Substantiated explanations of the unpredictability regarding sandpile models of self-organized criticality (SOC) gave way to efficient forecasts of extremes in a few models. The appearance of extremes requires a preparation phase that ends with general overloading of the system and spatial clustering of the local stress. Here, we relate the predictability of large events to the system volume in the Manna and Bak-Tang-Wiesenfeld sandpiles, which are basic models of SOC. We establish that in the Manna model, the events located to the right of the power-law segment of the size-frequency relationship are predictable and the prediction efficiency is described by the universal linear dependence on the event size scaled by a power-law function of the lattice volume. Our scaling-based approach to predictability contributes to the theory of SOC and may elucidate the forecast of extremes in the dynamics of such systems with SOC as neuronal networks and earthquakes.

Self-Organized Criticality of Precipitation in the Rainy Season in East China

Based on daily precipitation data from 1960 to 2017 in the rainy season in east China, to a given percentile threshold of one observation station, the time that precipitation spends below threshold is defined as quiet time τ. The probability density functions τ in different thresholds follow power-law distributions with exponent β of approximately 1.2 in the day, pentad and ten-day period time scales, respectively. The probability density functions τ in different regions follow the same rules, too. Compared with sandpile model, Γ function describing the collapse behavior can effectively scale the quiet time distribution of precipitation events. These results confirm the assumption that for observation station data and low-resolution precipitation data, even in China, affected by complex weather and climate systems, precipitation is still a real world example of self-organized criticality in synoptic. Moreover, exponent β of the probability density function τ, mean quiet time τ¯q and hazard function Hq of quiet times can give sensitive regions of precipitation events in China. Usual intensity precipitation events (UPEs) easily occur and cluster mainly in the middle Yangtze River basin, east of the Sichuan Province and north of the Gansu Province. Extreme intensity precipitation events (EPEs) more easily occur in northern China in the rainy season. UPEs in the Hubei Province and the Hunan Province are more likely to occur in the future. EPEs in the eastern Sichuan Province, the Guizhou Province, the Guangxi Province and Northeast China are more likely to occur.

Interlayer impacts to deep-coupling dynamical networks: A snapshot of equilibrium stability

This paper defines deep-coupling networks with two categories of interlayer structures and investigates the equilibrium stability, when every node in the network is governed by a differential dynamical system. Several stability criteria of the equilibrium of deep-coupling networks are arrived at when the equilibrium of each node system is stable or unstable in two categories of interlayer couplings, which clarify interlayer impacts with the dependence of network topology and intralayer intensity as well. Several two-layer and three-layer numerical examples are provided to verify the analytical results.

Probability estimation of a Carrington-like geomagnetic storm

Intense geomagnetic storms can cause severe damage to electrical systems and communications. This work proposes a counting process with Weibull inter-occurrence times in order to estimate the probability of extreme geomagnetic events. It is found that the scale parameter of the inter-occurrence time distribution grows exponentially with the absolute value of the intensity threshold defining the storm, whereas the shape parameter keeps rather constant. The model is able to forecast the probability of occurrence of an event for a given intensity threshold; in particular, the probability of occurrence on the next decade of an extreme event of a magnitude comparable or larger than the well-known Carrington event of 1859 is explored, and estimated to be between 0.46% and 1.88% (with a 95% confidence), a much lower value than those reported in the existing literature.

Forecasting extreme atmospheric events with a recurrence-interval-analysis-based autoregressive conditional duration model

With most city dwellers in China subjected to air pollution, forecasting extreme air pollution spells is of paramount significance in both scheduling outdoor activities and ameliorating air pollution. In this paper, we integrate the autoregressive conditional duration model (ACD) with the recurrence interval analysis (RIA) and also extend the ACD model to a spatially autoregressive conditional duration (SACD) model by adding a spatially reviewed term to quantitatively explain and predict extreme air pollution recurrence intervals. Using the hourly data of six pollutants and the air quality index (AQI) during 2013–2016 collected from 12 national air quality monitoring stations in Beijing as our test samples, we attest that the spatially reviewed recurrence intervals have some general explanatory power over the recurrence intervals in the neighbouring air quality monitoring stations. We also conduct a one-step forecast using the RIA-ACD(1,1) and RIA-SACD(1,1,1) models and find that 90% of the predicted recurrence intervals are smaller than 72 hours, which justifies the predictive power of the proposed models. When applied to more time lags and neighbouring stations, the models are found to yield results that are consistent with reality, which evinces the feasibility of predicting extreme air pollution events through a recurrence-interval-analysis-based autoregressive conditional duration model. Moreover, the addition of a spatial term has proved effective in enhancing the predictive power.

Increasing power-law range in avalanche amplitude and energy distributions

Power-law-type probability density functions spanning several orders of magnitude are found for different avalanche properties. We propose a methodology to overcome empirical constraints that limit the range of truncated power-law distributions. By considering catalogs of events that cover different observation windows, the maximum likelihood estimation of a global power-law exponent is computed. This methodology is applied to amplitude and energy distributions of acoustic emission avalanches in failure-under-compression experiments of a nanoporous silica glass, finding in some cases global exponents in an unprecedented broad range: 4.5 decades for amplitudes and 9.5 decades for energies. In the latter case, however, strict statistical analysis suggests experimental limitations might alter the power-law behavior.

Universal avalanche statistics and triggering close to failure in a mean-field model of rheological fracture

The hypothesis of critical failure relates the presence of an ultimate stability point in the structural constitutive equation of materials to a divergence of characteristic scales in the microscopic dynamics responsible for deformation. Avalanche models involving critical failure have determined common universality classes for stick-slip processes and fracture. However, not all empirical failure processes exhibit the trademarks of criticality. The rheological properties of materials introduce dissipation, usually reproduced in conceptual models as a hardening of the coarse grained elements of the system. Here, we investigate the effects of transient hardening on (i) the activity rate and (ii) the statistical properties of avalanches. We find the explicit representation of transient hardening in the presence of generalized viscoelasticity and solve the corresponding mean-field model of fracture. In the quasistatic limit, the accelerated energy release is invariant with respect to rheology and the avalanche propagation can be reinterpreted in terms of a stochastic counting process. A single universality class can be defined from such analogy, and all statistical properties depend only on the distance to criticality. We also prove that interevent correlations emerge due to the hardening-even in the quasistatic limit-that can be interpreted as "aftershocks" and "foreshocks."

Testing universality in critical exponents: The case of rainfall

One of the key clues to consider rainfall as a self-organized critical phenomenon is the existence of power-law distributions for rain-event sizes. We have studied the problem of universality in the exponents of these distributions by means of a suitable statistic whose distribution is inferred by several variations of a permutational test. In contrast to more common approaches, our procedure does not suffer from the difficulties of multiple testing and does not require the precise knowledge of the uncertainties associated to the power-law exponents. When applied to seven sites monitored by the Atmospheric Radiation Measurement Program the tests lead to the rejection of the universality hypothesis, despite the fact that the exponents are rather close to each other. We discuss the reasons of the rejection.