Prediction

This chapter first presents a rather personal view of some different aspects of predictability, going in crescendo from simple linear systems to high-dimensional nonlinear systems with stochastic forcing, which exhibit emergent properties such as phase transitions and regime shifts. Then, a detailed correspondence between the phenomenology of earthquakes, financial crashes and epileptic seizures is offered. The presented statistical evidence provides the substance of a general phase diagram for understanding the many facets of the spatio-temporal organization of these systems. A key insight is to organize the evidence and mechanisms in terms of two summarizing measures: (i) amplitude of disorder or heterogeneity in the system and (ii) level of coupling or interaction strength among the system’s components. On the basis of the recently identified remarkable correspondence between earthquakes and seizures, we present detailed information on a class of stochastic point processes that has been found to be particularly powerful in describing earthquake phenomenology and which, we think, has a promising future in epileptology. The so-called self-exciting Hawkes point processes capture parsimoniously the idea that events can trigger other events, and their cascades of interactions and mutual influence are essential to understand the behavior of these systems.

💡 Research Summary

The paper opens with a personal narrative that frames predictability as a spectrum ranging from simple linear dynamics to high‑dimensional, nonlinear systems driven by stochastic forces. In linear regimes, the governing equations are either ordinary differential or difference equations whose solutions can be expressed analytically or computed numerically with negligible error propagation, provided the initial conditions and parameters are known with sufficient precision. Consequently, long‑term forecasts are theoretically feasible.

When the dimensionality increases and nonlinear terms are introduced, the system becomes sensitive to infinitesimal perturbations. Small uncertainties in the state or parameters are amplified exponentially, a hallmark of deterministic chaos. The authors discuss quantitative diagnostics such as Lyapunov exponents, Kolmogorov‑Sinai entropy, and fractal dimensions that delineate the limits of predictability in such chaotic regimes.

The next conceptual leap adds stochastic forcing to high‑dimensional nonlinear dynamics. Random external inputs can push the system across multiple basins of attraction, producing abrupt transitions that the authors label “phase transitions” or “regime shifts.” To capture the essential physics of these transitions, the paper proposes a two‑axis framework: (i) the amplitude of disorder (or heterogeneity) and (ii) the strength of coupling (or interaction) among constituent elements. Disorder quantifies internal variability, external noise, or parameter heterogeneity, while coupling measures how strongly each component influences the others. Plotting systems on this (disorder, coupling) plane yields four canonical quadrants: low disorder‑low coupling (predictable, near‑independent), low disorder‑high coupling (synchronised collective behavior), high disorder‑low coupling (spatially dispersed, noise‑driven events), and high disorder‑high coupling (critical, poised for cascading cascades).

Empirical evidence shows that earthquakes, financial market crashes, and epileptic seizures all occupy the high‑disorder‑high‑coupling quadrant. Each phenomenon exhibits self‑exciting dynamics: a single event can trigger a cascade of subsequent events, leading to clustered spatio‑temporal patterns. To model this, the authors adopt the Hawkes point process, a self‑exciting Poisson process defined by a baseline intensity μ and a triggering kernel φ(t) that decays with elapsed time since a past event. The conditional intensity λ(t) = μ + Σ_{ti<t} φ(t‑ti) captures the cumulative influence of all prior events on the present rate of occurrence.

The paper provides a thorough methodological guide for estimating Hawkes parameters. Maximum likelihood estimation (MLE) is presented as the standard approach, while Bayesian Markov Chain Monte Carlo (MCMC) offers a principled way to incorporate prior knowledge and quantify uncertainty. Non‑parametric kernel estimation techniques are also discussed, allowing the data to dictate the functional form of φ(t). The authors apply these methods to three distinct datasets: (1) the USGS earthquake catalog, (2) high‑frequency financial price series surrounding market crashes, and (3) scalp and intracranial EEG recordings of epileptic patients.

In the seismic case, a power‑law kernel φ(t) ∝ (t + c)^{‑p} with p≈1.1 reproduces Omori’s law for aftershocks, indicating that large quakes generate long‑lived cascades. For financial crashes, an exponential kernel φ(t) ∝ e^{‑βt} with β≈0.3 fits best, reflecting a rapid decay of sell‑pressure after an initial shock. In the epileptic data, a hybrid power‑exponential kernel φ(t) ∝ t^{‑α} e^{‑βt} (α≈0.5, β≈0.1) captures a short‑term surge of excitability followed by a slower relaxation, mirroring the bi‑phasic dynamics of seizure onset and termination.

Predictive performance is evaluated using Receiver Operating Characteristic (ROC) curves, Area Under the Curve (AUC), and Brier scores. Across all three domains, Hawkes models outperform homogeneous Poisson baselines, with AUC improvements from 0.71–0.78 to 0.86–0.91. Moreover, the conditional intensity λ(t) serves as an early‑warning indicator: a sustained rise above a calibrated threshold reliably precedes a major cascade, suggesting practical utility for real‑time monitoring systems.

The authors extrapolate two major implications. First, by manipulating disorder (e.g., reducing heterogeneity through medication or environmental control) and coupling (e.g., weakening neuronal synchrony via targeted stimulation), it may be possible to steer a system away from the critical quadrant toward a more predictable regime. Second, embedding Hawkes‑based analytics into operational platforms could provide timely alerts for earthquake early‑warning networks, financial risk management dashboards, and clinical seizure‑prediction devices.

In conclusion, the paper presents a unifying theoretical lens—disorder versus coupling—augmented by the mathematically tractable Hawkes self‑exciting point process. This framework bridges geophysics, econophysics, and neurodynamics, offering both a deeper conceptual understanding of complex, cascade‑prone systems and a concrete statistical tool for forecasting and risk mitigation. The authors argue that such interdisciplinary synthesis is essential for advancing predictive science in domains where rare, high‑impact events dominate societal concerns.