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.
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.
Consider a simple dynamical system with the following linear auto-regressive dynamics r(t) = βr(t -1) + ǫ(t) ,
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 σ 2 ǫ . The dependence structure between successive values of r(t) is entirely captured by the correlation function which is non-zero only for the time lag of one unit step (in addition to the zero-time lag of course). Indeed, the correlation coefficient between the random variable at some time and its realization at the following time step is nothing but β. Correspondingly, the covariance of r(t -1) and r(t) is β × σ 2 r , where σ 2 r = σ 2 ǫ /(1-β 2 ) is the variance of r(t). More generally, consider an extension of expression (1) into a linear auto-regressive process of larger order, so that we can consider an arbitrary covariance matrix C(t, t ′ ) between r(t) and r(t ′ ) for all possible instant pairs t and t ′ . A simple mathematical calculation shows that the best linear predictor m t for r(t) at time t, knowing the past history r t-1 , r t-2 , …, r i , … is given by
where B(i, t) is the coefficient (i, t) of the inverse matrix of the covariance matrix C(t, t ′ ). This formula (2) expresses that each past values r i impacts on the future r t in proportion to its value with a coefficient B(i, t)/B(t, t) which is non-zero only if there is non-zero correlation between the realization of the variable at time i and time t. This formula (2) provides the best linear predictor in the sense that it minimizes the errors in a variance sense. Armed with this prediction, useful operational strategies can be developed, depending on the context. For instance, if the set {r(t)} denotes the returns of a financial asset, then, one could use this prediction (2) to invest as follows: buy if m t > 0 (expected future price increase) and sell if m t < 0 (expected future price decrease). Such predictor can be applied to general moving average and auto-regressive processes with long memory, whose general expression reads [41] 1 -
where L is the lag operator defined by Lr(t) = r(t -1) and p, q and d can be arbitrary integers. Such predictors are optimal or close to optimal as long as there is no change of regime, that is, if the process is stationary and the coefficients {φ i } and {θ j } and the orders of moving average q, of auto-regression p and of fractional derivation d do not change during the course of the dynamics. Otherwise, other methods, including Monte Carlo Markov Chains, are needed [41]. In the case where the initial conditions or observations during the course of the dynamics are obtained with noise or uncertainty, Kalman filtering and more generally data assimilation methods [58] provide significant improvements in predicting the dynamics of the system.
There is an enormous amount of literature on this subject since the last 1970s (see for instance [6,74,72,89,125] and references therein). The idea of how to develop predictors for lowdimensional deterministic chaotic systems is very natural: because of determinism, and provided that the dynamics is in some sense sufficiently regular, the short-time evolution remembers the initial conditions, so that two trajectories that are found in a neighborhood of each other remain close to each other for a time t f roughly given by the inverse of the largest Lyapunov exponent. Thus, if one monitors past evolution, however complicated, a future path which comes in the vicinity of a previously visited point will then evolve along a trajectory shadowing the previous one over a time of the order of t f [31,127,103]. The previously recorded dynamical evolution of a domain over some short-time horizon can thus provide in principle a short-term prediction through the knowledge of the transformation of this domain.
However, in practice, there are many caveats to this idealized situation. Model errors and noise, additive and/or multiplicative (also called “parametric”), complicate and limit predictability. Model errors refer to the generic problem that the used model is at best only an approximation of the true dynamics, and more generally neglects some possibly important ingredients, making prediction questionable.
In the simplest case of additive noise decorating deterministic chaotic dynamics, it turns out that the standard statistic methods for the estimation of the parameters of the model break down. For instance, the application of the maximum likelihood method to unstable nonlinear systems distorted by noise has no mathematical ground so far [98]. There are inherent difficulties in the statistical analysis of deterministically chaotic time
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