263 research outputs found
Bath's law Derived from the Gutenberg-Richter law and from Aftershock Properties
The empirical Bath's law states that the average difference in magnitude
between a mainshock and its largest aftershock is 1.2, regardless of the
mainshock magnitude. Following Vere-Jones [1969] and Console et al. [2003], we
show that the origin of Bath's law is to be found in the selection procedure
used to define mainshocks and aftershocks rather than in any difference in the
mechanisms controlling the magnitude of the mainshock and of the aftershocks.
We use the ETAS model of seismicity, which provides a more realistic model of
aftershocks, based on (i) a universal Gutenberg-Richter (GR) law for all
earthquakes, and on (ii) the increase of the number of aftershocks with the
mainshock magnitude. Using numerical simulations of the ETAS model, we show
that this model is in good agreement with Bath's law in a certain range of the
model parameters.Comment: major revisions, in press in Geophys. Res. Let
Grobner Bases for Finite-temperature Quantum Computing and their Complexity
Following the recent approach of using order domains to construct Grobner
bases from general projective varieties, we examine the parity and
time-reversal arguments relating de Witt and Lyman's assertion that all path
weights associated with homotopy in dimensions d <= 2 form a faithful
representation of the fundamental group of a quantum system. We then show how
the most general polynomial ring obtained for a fermionic quantum system does
not, in fact, admit a faithful representation, and so give a general
prescription for calcluating Grobner bases for finite temperature many-body
quantum system and show that their complexity class is BQP
Properties of Foreshocks and Aftershocks of the Non-Conservative SOC Olami-Feder-Christensen Model: Triggered or Critical Earthquakes?
Following Hergarten and Neugebauer [2002] who discovered aftershock and
foreshock sequences in the Olami-Feder-Christensen (OFC) discrete block-spring
earthquake model, we investigate to what degree the simple toppling mechanism
of this model is sufficient to account for the properties of earthquake
clustering in time and space. Our main finding is that synthetic catalogs
generated by the OFC model share practically all properties of real seismicity
at a qualitative level, with however significant quantitative differences. We
find that OFC catalogs can be in large part described by the concept of
triggered seismicity but the properties of foreshocks depend on the mainshock
magnitude, in qualitative agreement with the critical earthquake model and in
disagreement with simple models of triggered seismicity such as the Epidemic
Type Aftershock Sequence (ETAS) model [Ogata, 1988]. Many other features of OFC
catalogs can be reproduced with the ETAS model with a weaker clustering than
real seismicity, i.e. for a very small average number of triggered earthquakes
of first generation per mother-earthquake.Comment: revtex, 19 pages, 8 eps figure
Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes
We present the first exact analysis of some of the temporal properties of
multivariate self-excited Hawkes conditional Poisson processes, which
constitute powerful representations of a large variety of systems with bursty
events, for which past activity triggers future activity. The term
"multivariate" refers to the property that events come in different types, with
possibly different intra- and inter-triggering abilities. We develop the
general formalism of the multivariate generating moment function for the
cumulative number of first-generation and of all generation events triggered by
a given mother event (the "shock") as a function of the current time . This
corresponds to studying the response function of the process. A variety of
different systems have been analyzed. In particular, for systems in which
triggering between events of different types proceeds through a one-dimension
directed or symmetric chain of influence in type space, we report a novel
hierarchy of intermediate asymptotic power law decays of the rate of triggered events as a function of the
distance of the events to the initial shock in the type space, where for the relevant long-memory processes characterizing many natural
and social systems. The richness of the generated time dynamics comes from the
cascades of intermediate events of possibly different kinds, unfolding via a
kind of inter-breeding genealogy.Comment: 40 pages, 8 figure
Prediction of extreme events in the OFC model on a small world network
We investigate the predictability of extreme events in a dissipative
Olami-Feder-Christensen model on a small world topology. Due to the mechanism
of self-organized criticality, it is impossible to predict the magnitude of the
next event knowing previous ones, if the system has an infinite size. However,
by exploiting the finite size effects, we show that probabilistic predictions
of the occurrence of extreme events in the next time step are possible in a
finite system. In particular, the finiteness of the system unavoidably leads to
repulsive temporal correlations of extreme events. The predictability of those
is higher for larger magnitudes and for larger complex network sizes. Finally,
we show that our prediction analysis is also robust by remarkably reducing the
accessible number of events used to construct the optimal predictor.Comment: 5 pages, 4 figure
Scale free networks of earthquakes and aftershocks
We propose a new metric to quantify the correlation between any two
earthquakes. The metric consists of a product involving the time interval and
spatial distance between two events, as well as the magnitude of the first one.
According to this metric, events typically are strongly correlated to only one
or a few preceding ones. Thus a classification of events as foreshocks, main
shocks or aftershocks emerges automatically without imposing predefined
space-time windows. To construct a network, each earthquake receives an
incoming link from its most correlated predecessor. The number of aftershocks
for any event, identified by its outgoing links, is found to be scale free with
exponent . The original Omori law with emerges as a
robust feature of seismicity, holding up to years even for aftershock sequences
initiated by intermediate magnitude events. The measured fat-tailed
distribution of distances between earthquakes and their aftershocks suggests
that aftershock collection with fixed space windows is not appropriate.Comment: 7 pages and 7 figures. Submitte
Generation-by-Generation Dissection of the Response Function in Long Memory Epidemic Processes
In a number of natural and social systems, the response to an exogenous shock
relaxes back to the average level according to a long-memory kernel with . In the presence of an epidemic-like
process of triggered shocks developing in a cascade of generations at or close
to criticality, this "bare" kernel is renormalized into an even slower decaying
response function . Surprisingly, this means that the
shorter the memory of the bare kernel (the larger ), the longer the
memory of the response function (the smaller ). Here, we present a
detailed investigation of this paradoxical behavior based on a
generation-by-generation decomposition of the total response function, the use
of Laplace transforms and of "anomalous" scaling arguments. The paradox is
explained by the fact that the number of triggered generations grows
anomalously with time at so that the contributions of active
generations up to time more than compensate the shorter memory associated
with a larger exponent . This anomalous scaling results fundamentally
from the property that the expected waiting time is infinite for . The techniques developed here are also applied to the case
and we find in this case that the total renormalized response is a {\bf
constant} for followed by a cross-over to
for .Comment: 27 pages, 4 figure
Universal features of correlated bursty behaviour
Inhomogeneous temporal processes, like those appearing in human
communications, neuron spike trains, and seismic signals, consist of
high-activity bursty intervals alternating with long low-activity periods. In
recent studies such bursty behavior has been characterized by a fat-tailed
inter-event time distribution, while temporal correlations were measured by the
autocorrelation function. However, these characteristic functions are not
capable to fully characterize temporally correlated heterogenous behavior. Here
we show that the distribution of the number of events in a bursty period serves
as a good indicator of the dependencies, leading to the universal observation
of power-law distribution in a broad class of phenomena. We find that the
correlations in these quite different systems can be commonly interpreted by
memory effects and described by a simple phenomenological model, which displays
temporal behavior qualitatively similar to that in real systems
Dragon-kings: mechanisms, statistical methods and empirical evidence
This introductory article presents the special Discussion and Debate volume
"From black swans to dragon-kings, is there life beyond power laws?" published
in Eur. Phys. J. Special Topics in May 2012. We summarize and put in
perspective the contributions into three main themes: (i) mechanisms for
dragon-kings, (ii) detection of dragon-kings and statistical tests and (iii)
empirical evidence in a large variety of natural and social systems. Overall,
we are pleased to witness significant advances both in the introduction and
clarification of underlying mechanisms and in the development of novel
efficient tests that demonstrate clear evidence for the presence of
dragon-kings in many systems. However, this positive view should be balanced by
the fact that this remains a very delicate and difficult field, if only due to
the scarcity of data as well as the extraordinary important implications with
respect to hazard assessment, risk control and predictability.Comment: 20 page
Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency financial data
We define a numerical method that provides a non-parametric estimation of the
kernel shape in symmetric multivariate Hawkes processes. This method relies on
second order statistical properties of Hawkes processes that relate the
covariance matrix of the process to the kernel matrix. The square root of the
correlation function is computed using a minimal phase recovering method. We
illustrate our method on some examples and provide an empirical study of the
estimation errors. Within this framework, we analyze high frequency financial
price data modeled as 1D or 2D Hawkes processes. We find slowly decaying
(power-law) kernel shapes suggesting a long memory nature of self-excitation
phenomena at the microstructure level of price dynamics.Comment: 6 figure
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