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. chapter in “Epilepsy: The Intersection of Neurosciences, Mathematics, and Engineering” ,Taylor & Francis Group, Ivan Osorio, Mark G. Frei, Hitten Zaveri, Susan Arthurs, eds (2010) 1 A brief classification of predictability Characterizations of the predictability (or unpredictability) of a system provide useful theoretical and practical measure of its complexity [9, 72]. It is also a grail in epileptology, as advanced warnings by a few minutes may drastically improve the quality of life of these patients. 1.1 Predictability of linear stochastic systems Consider a simple dynamical system with the following linear auto-regressive dynamics r(t) = βr(t− 1) + ǫ(t) , (1) where 0 < β < 1 is a constant and ǫ(t) is a i.i.d. (independently identically distributed) random variable, i.e., a noise, with variance σ2 ǫ . The dependence structure between successive values of