Many time series in natural and social sciences can be seen as resulting from an interplay between exogenous influences and an endogenous organization. We use a simple (ETAS) model of events occurring sequentially, in which future events are influenced (partially triggered) by past events to ask the question of how well can one disentangle the exogenous events from the endogenous ones. We apply both model-dependant and model-independent stochastic declustering methods to reconstruct the tree of ancestry and estimate key parameters. In contrast with previously reported positive results, we have to conclude that declustered catalogs are rather unreliable for the synthetic catalogs that we have investigated, which contains of the order of thousands of events, typical of realistic applications. The estimated rates of exogenous events suffer from large errors. The key branching ratio $n$, quantifying the fraction of events that have been triggered by previous events, is also badly estimated in general from declustered catalogs. We find however that the errors tend to be smaller and perhaps acceptable in some cases for small triggering efficiency and branching ratios. The high level of randomness together with the very long memory makes the stochastic reconstruction of trees of ancestry and the estimation of the key parameters perhaps intrinsically unreliable for long memory processes. For shorter memories (larger "bare" Omori exponent), the results improve significantly.
Deep Dive into Limits of Declustering Methods for Disentangling Exogenous from Endogenous Events in Time Series with Foreshocks, Main shocks and Aftershocks.
Many time series in natural and social sciences can be seen as resulting from an interplay between exogenous influences and an endogenous organization. We use a simple (ETAS) model of events occurring sequentially, in which future events are influenced (partially triggered) by past events to ask the question of how well can one disentangle the exogenous events from the endogenous ones. We apply both model-dependant and model-independent stochastic declustering methods to reconstruct the tree of ancestry and estimate key parameters. In contrast with previously reported positive results, we have to conclude that declustered catalogs are rather unreliable for the synthetic catalogs that we have investigated, which contains of the order of thousands of events, typical of realistic applications. The estimated rates of exogenous events suffer from large errors. The key branching ratio $n$, quantifying the fraction of events that have been triggered by previous events, is also badly estimated
A large variety of natural and social systems are characterized by a stochastic intermittent flow of sudden events: landslides, earthquakes, storms, floods, volcanic eruptions, biological extinctions, traffic gridlocks, power blackouts, breaking news, commercial blockbusters, financial crashes, economic crises, terrorist acts, geopolitical events, and so on. Sequences of such sudden events constitute often the most crucial features of the evolutionary dynamics of complex systems, both in terms of their description, characterization and understanding.
Accordingly, a useful class of models of complex systems views their dynamics as a sequence of intermittent discrete short-lived events. In the limit where the time scales, over which the change of regimes associated with the occurrence of the events occur, are small compared with the inter-event intervals, the catalog of events can be modeled using the mathematics of point-processes [2,6]. This modeling strategy emphasizes that the system is active during short-lived events and inactive otherwise. This amounts to separating a more or less incoherent background activity (such as small undetectable earthquakes) from the occurrence of structured events (large earthquakes), which are the focus of interest. We note that the class of stochastic point processes is fundamentally different from that of discrete and continuous stochastic processes, for which the activity is non-zero most of the time.
Having a time series or catalog of discrete events, we are interested in understanding the generating process that led to the observed sequence. The difficulty in deciphering the underlying mechanisms stems from the fact that the above systems of interest are on the one hand subjected to external forcing which on the other hand provide them the stimuli to self-organize via negative and positive feedback mechanisms. Most natural and social systems are indeed continuously subjected to external stimulations, noises, shocks, solicitations, and forcing, which can widely vary in amplitude. It is thus not clear a priori if the observed activity is due to a strong exogenous shock, to the internal dynamics of the system organizing in response to the continuous flow of information and perturbations, or maybe to a combination of both. In general, a combination of external inputs and internal organization is at work and it seems hopeless to disentangle the different contributions to the observed collective human response. Determining the chain of causality for such questions requires disentangling interwoven exogenous and endogenous contributions with either no clear or too many signatures. How can one assert with confidence that a given event or characteristics is really due to an endogenous self-organization of the system, rather than to the response to an external shock?
It turns out that a significant understanding of the complex flow of observed events can be achieved by precisely framing the problem in terms of a classification of two limited classes of events: (i) those that are the response of the system to exogenous shocks to the system and (ii) those that appear endogenously without any obvious external causes. This can be done by looking as the specific endogenous and exogenous signatures and their mutual relations, which are reminiscent of the fluctuation-susceptibility theorem in statistical physics [26,32].
This approach provides a useful framework for understanding many complex systems and has been successfully applied in several contexts: commercial book successes [7,33], social crises [25], financial volatility [35], financial bubbles and crashes [16,31], earthquakes [13,15], diseases in complex biological organisms [39], epileptic seizures [23] and so on.
A common feature observed in these different systems is the fact that events are not independent as they would be if generated by a Poisson process. Instead, they exhibit pronounced inter-dependencies, characterized by “self-excitation”, i.e., past events are found to often promote or trigger (in part) future events, leading to epidemic-like cascades of events. The analogy with triggering and cascade processes occurring in viral epidemics is so vivid in some instances that the name “Epidemic-type aftershock model” (ETAS) has been given to one of the most popular model of earthquake aftershock processes [12,14,20].
The ETAS model belongs to the class of self-excited conditional point processes introduced in mathematics by Hawkes [9,10,11]. It constitutes an excellent first-order approximation to describe the spatio-temporal organization of earthquakes [30] and is now taken as a standard benchmark. This class of “self-excited” point processes also provides quantitative predictions on the different decay rates after exogenous peaks of activity on the one hand and endogenous peaks of activity on the other hand. These predictions have been verified in a unique data set of almost 5,000,000 time series of human activi
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
