Characterization of intermittency in renewal processes: application to earthquakes

Complexity-calibrated benchmarks for machine learning reveal when prediction algorithms succeed and mislead

Recurrent neural networks are used to forecast time series in finance, climate, language, and from many other domains. Reservoir computers are a particularly easily trainable form of recurrent neural network. Recently, a "next-generation" reservoir computer was introduced in which the memory trace involves only a finite number of previous symbols. We explore the inherent limitations of finite-past memory traces in this intriguing proposal. A lower bound from Fano's inequality shows that, on highly non-Markovian processes generated by large probabilistic state machines, next-generation reservoir computers with reasonably long memory traces have an error probability that is at least ∼ 60 % higher than the minimal attainable error probability in predicting the next observation. More generally, it appears that popular recurrent neural networks fall far short of optimally predicting such complex processes. These results highlight the need for a new generation of optimized recurrent neural network architectures. Alongside this finding, we present concentration-of-measure results for randomly-generated but complex processes. One conclusion is that large probabilistic state machines-specifically, large ϵ -machines-are key to generating challenging and structurally-unbiased stimuli for ground-truthing recurrent neural network architectures.

Statistics of the number of renewals, occupation times, and correlation in ordinary, equilibrium, and aging alternating renewal processes

The renewal process is a point process where an interevent time between successive renewals is an independent and identically distributed random variable. Alternating renewal process is a dichotomous process and a slight generalization of the renewal process, where the interevent time distribution alternates between two distributions. We investigate statistical properties of the number of renewals and occupation times for one of the two states in alternating renewal processes. When both means of the interevent times are finite, the alternating renewal process can reach an equilibrium. On the other hand, an alternating renewal process shows aging when one of the means diverges. We provide analytical calculations for the moments of the number of renewals, occupation time statistics, and the correlation function for several case studies in the interevent-time distributions. We show anomalous fluctuations for the number of renewals and occupation times when the second moment of interevent time diverges. When the mean interevent time diverges, distributional limit theorems for the number of events and occupation times are shown analytically. These are known as the Mittag-Leffler distribution and the generalized arcsine law in probability theory.

Spectral decomposition of nonlinear systems with memory

We present an alternative approach to the analysis of nonlinear systems with long-term memory that is based on the Koopman operator and a Lévy transformation in time. Memory effects are considered to be the result of interactions between a system and its surrounding environment. The analysis leads to the decomposition of a nonlinear system with memory into modes whose temporal behavior is anomalous and lacks a characteristic scale. On average, the time evolution of a mode follows a Mittag-Leffler function, and the system can be described using the fractional calculus. The general theory is demonstrated on the fractional linear harmonic oscillator and the fractional nonlinear logistic equation. When analyzing data from an ill-defined (black-box) system, the spectral decomposition in terms of Mittag-Leffler functions that we propose may uncover inherent memory effects through identification of a small set of dynamically relevant structures that would otherwise be obscured by conventional spectral methods. Consequently, the theoretical concepts we present may be useful for developing more general methods for numerical modeling that are able to determine whether observables of a dynamical system are better represented by memoryless operators, or operators with long-term memory in time, when model details are unknown.

Origin of 1/f noise in hydration dynamics on lipid membrane surfaces

Water molecules on lipid membrane surfaces are known to contribute to membrane stability by connecting lipid molecules and acting as a water bridge. Although water structures and diffusivities near the membrane surfaces have been extensively studied, hydration dynamics on the surfaces has remained an open question. Here we investigate residence time statistics of water molecules on the surface of lipid membranes using all-atom molecular dynamics simulations. We show that hydration dynamics on the lipid membranes exhibits 1/f noise. Constructing a dichotomous process for the hydration dynamics, we find that residence times in each state follow a power-law with exponential cutoff and that the process can be regarded as a correlated renewal process where interoccurrence times are correlated. The results imply that the origin of the 1/f noise in hydration dynamics on the membrane surfaces is a combination of a power-law distribution with cutoff of interoccurrence times of switching events and a long-term correlation between the interoccurrence times. These results suggest that the 1/f noise attributed to the correlated renewal process may contribute to the stability of the hydration layers and lipid membranes.

Generalization of the Einstein relation for single trajectories in deterministic subdiffusion

Diffusion coefficients are intrinsically random in subdiffusion attributable to power-law trapping. Using deterministic biased and unbiased diffusion models, we investigate the Einstein relation for single trajectories in subdiffusion. The difference in the generalized Lyapunov exponent between biased and unbiased deterministic diffusions is related to the velocity under a bias. By Hopf's ergodic theorem, the ratios between the velocities and the Lyapunov exponents for single trajectories converge to a universal constant, which is proportional to the strength of the bias. Based on a certain transport coefficient obtained from a single trajectory, we provide a relation for the transport coefficients divided by the Lyapunov exponent and generalize the Einstein relation for single trajectories.

Optimal observation time window for forecasting the next earthquake

We report that the accuracy of predicting the occurrence time of the next earthquake is significantly enhanced by observing the latest rate of earthquake occurrences. The observation period that minimizes the temporal uncertainty of the next occurrence is on the order of 10 hours. This result is independent of the threshold magnitude and is consistent across different geographic areas. This time scale is much shorter than the months or years that have previously been considered characteristic of seismic activities.