299 research outputs found
The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics
We prove that the distributional limit of the normalised number of returns to
small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical
systems is compound Poisson. The returns to small balls around a fixed point in
the phase space correspond to the occurrence of rare events, or exceedances of
high thresholds, so that there is a connection between the laws of Return Times
Statistics and Extreme Value Laws. The fact that the fixed point in the phase
space is a repelling periodic point implies that there is a tendency for the
exceedances to appear in clusters whose average sizes is given by the Extremal
Index, which depends on the expansion of the system at the periodic point.
We recall that for generic points, the exceedances, in the limit, are
singular and occur at Poisson times. However, around periodic points, the
picture is different: the respective point processes of exceedances converge to
a compound Poisson process, so instead of single exceedances, we have entire
clusters of exceedances occurring at Poisson times with a geometric
distribution ruling its multiplicity.
The systems to which our results apply include: general piecewise expanding
maps of the interval (Rychlik maps), maps with indifferent fixed points
(Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.Comment: To appear in Communications in Mathematical Physic
Customer contributions and roles in service delivery
Focuses on the roles of customers in creating quality and productivity in service experiences. Presents two conceptual frameworks to aid managerial understanding and focus research efforts on customer participation. The first framework captures levels of customer participation across different types of services. The second discusses three major roles of customers in the service delivery process. Two examples of the concepts are presented ‐ one in a weight loss context and the other in a mammography screening setting. Both are based on empirical research and illustrate specific applications of customers’ roles in creating the service experience
Extreme value statistics for dynamical systems with noise
We study the distribution of maxima ( extreme value statistics ) for sequences of observables computed along orbits generated by random transformations. The underlying, deterministic, dynamical system can be regular or chaotic. In the former case, we show that, by perturbing rational or irrational rotations with additive noise, an extreme value law appears, regardless of the intensity of the noise, while unperturbed rotations do not admit such limiting distributions. In the case of deterministic chaotic dynamics, we will consider observables specially designed to study the recurrence properties in the neighbourhood of periodic points. Hence, the exponential limiting law for the distribution of maxima is modified by the presence of the extremal index , a positive parameter not larger than one, whose inverse gives the average size of the clusters of extreme events. The theory predicts that such a parameter is unitary when the system is perturbed randomly. We perform sophisticated numerical tests to assess how strong the impact of noise level is when finite time series are considered. We find agreement with the asymptotic theoretical results but also non-trivial behaviour in the finite range. In particular, our results suggest that, in many applications where finite datasets can be produced or analysed, one must be careful in assuming that the smoothing nature of noise prevails over the underlying deterministic dynamics
Modelling and analysis of turbulent datasets using Auto Regressive Moving Average processes
International audienceWe introduce a novel way to extract information from turbulent datasets by applying an Auto Regressive Moving Average (ARMA) statistical analysis. Such analysis goes well beyond the analysis of the mean flow and of the fluctuations and links the behavior of the recorded time series to a discrete version of a stochastic differential equation which is able to describe the correlation structure in the dataset. We introduce a new index ϒ that measures the difference between the resulting analysis and the Obukhov model of turbulence, the simplest stochastic model reproducing both Richardson law and the Kolmogorov spectrum. We test the method on datasets measured in a von Kármán swirling flow experiment. We found that the ARMA analysis is well correlated with spatial structures of the flow, and can discriminate between two different flows with comparable mean velocities, obtained by changing the forcing. Moreover, we show that the ϒ is highest in regions where shear layer vortices are present, thereby establishing a link between deviations from the Kolmogorov model and coherent structures. These deviations are consistent with the ones observed by computing the Hurst exponents for the same time series. We show that some salient features of the analysis are preserved when considering global instead of local observables. Finally, we analyze flow configurations with multistability features where the ARMA technique is efficient in discriminating different stability branches of the system
Universal behavior of extreme value statistics for selected observables of dynamical systems
The main results of the extreme value theory developed for the investigation
of the observables of dynamical systems rely, up to now, on the Gnedenko
approach. In this framework, extremes are basically identified with the block
maxima of the time series of the chosen observable, in the limit of infinitely
long blocks. It has been proved that, assuming suitable mixing conditions for
the underlying dynamical systems, the extremes of a specific class of
observables are distributed according to the so called Generalized Extreme
Value (GEV) distribution. Direct calculations show that in the case of
quasi-periodic dynamics the block maxima are not distributed according to the
GEV distribution. In this paper we show that, in order to obtain a universal
behaviour of the extremes, the requirement of a mixing dynamics can be relaxed
if the Pareto approach is used, based upon considering the exceedances over a
given threshold. Requiring that the invariant measure locally scales with a
well defined exponent - the local dimension -, we show that the limiting
distribution for the exceedances of the observables previously studied with the
Gnedenko approach is a Generalized Pareto distribution where the parameters
depends only on the local dimensions and the value of the threshold. This
result allows to extend the extreme value theory for dynamical systems to the
case of regular motions. We also provide connections with the results obtained
with the Gnedenko approach. In order to provide further support to our
findings, we present the results of numerical experiments carried out
considering the well-known Chirikov standard map.Comment: 7 pages, 1 figur
Numerical convergence of the block-maxima approach to the Generalized Extreme Value distribution
In this paper we perform an analytical and numerical study of Extreme Value
distributions in discrete dynamical systems. In this setting, recent works have
shown how to get a statistics of extremes in agreement with the classical
Extreme Value Theory. We pursue these investigations by giving analytical
expressions of Extreme Value distribution parameters for maps that have an
absolutely continuous invariant measure. We compare these analytical results
with numerical experiments in which we study the convergence to limiting
distributions using the so called block-maxima approach, pointing out in which
cases we obtain robust estimation of parameters. In regular maps for which
mixing properties do not hold, we show that the fitting procedure to the
classical Extreme Value Distribution fails, as expected. However, we obtain an
empirical distribution that can be explained starting from a different
observable function for which Nicolis et al. [2006] have found analytical
results.Comment: 34 pages, 7 figures; Journal of Statistical Physics 201
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