Distribution Of Recurrence Times Research Articles

Characteristic Sign Renewals of Kardar–Parisi–Zhang Fluctuations

Tracking the sign of fluctuations governed by the $(1+1)$-dimensional Kardar-Parisi-Zhang (KPZ) universality class, we show, both experimentally and numerically, that its evolution has an unexpected link to a simple stochastic model called the renewal process, studied in the context of aging and ergodicity breaking. Although KPZ and the renewal process are fundamentally different in many aspects, we find remarkable agreement in some of the time correlation properties, such as the recurrence time distributions and the persistence probability, while the two systems can be different in other properties. Moreover, we find inequivalence between long-time and ensemble averages in the fraction of time occupied by a specific sign of the KPZ-class fluctuations. The distribution of its long-time average converges to nontrivial broad functions, which are found to differ significantly from that of the renewal process, but instead be characteristic of KPZ. Thus, we obtain a new type of ergodicity breaking for such systems with many-body interactions. Our analysis also detects qualitative differences in time-correlation properties of circular and flat KPZ-class interfaces, which were suggested from previous experiments and simulations but still remain theoretically unexplained.

Word-Length Correlations and Memory in Large Texts: A Visibility Network Analysis

We study the correlation properties of word lengths in large texts from 30 ebooks in the English language from the Gutenberg Project (www.gutenberg.org) using the natural visibility graph method (NVG). NVG converts a time series into a graph and then analyzes its graph properties. First, the original sequence of words is transformed into a sequence of values containing the length of each word, and then, it is integrated. Next, we apply the NVG to the integrated word-length series and construct the network. We show that the degree distribution of that network follows a power law, P ( k ) ∼ k - γ , with two regimes, which are characterized by the exponents γ s ≈ 1 . 7 (at short degree scales) and γ l ≈ 1 . 3 (at large degree scales). This suggests that word lengths are much more strongly correlated at large distances between words than at short distances between words. That finding is also supported by the detrended fluctuation analysis (DFA) and recurrence time distribution. These results provide new information about the universal characteristics of the structure of written texts beyond that given by word frequencies.

Correlations of correlations: Secondary autocorrelations in finite harmonic systems.

The momentum or velocity autocorrelation function C(t) for a tagged oscillator in a finite harmonic system decays like that of an infinite system for short times, but exhibits erratic behavior at longer time scales. We introduce the autocorrelation function of the long-time noisy tail of C(t) ("a correlation of the correlation"), which characterizes the distribution of recurrence times. Remarkably, for harmonic systems with same-mass particles this secondary correlation may coincide with the primary correlation C(t) (when both functions are normalized) either exactly, or over a significant initial time interval. When the tagged particle is heavier than the rest, the equality does not hold, correlations show nonrandom long-time scale pattern, and higher-order correlations converge to the lowest normal mode.

On Mechanistic Explanation of the Shape of the Universal Curve of Earthquake Recurrence Time Distributions

This paper outlines an idea for an explanation of a mechanism underlying the shape of the universal curve of the Earthquake Recurrence Time Distributions. The proposed simple stochastic cellular automaton model is reproducing the gamma distribution fit with the proper value of parameter γ characterizing the Earth’s seismicity and also imitates a deviation from the fit at short interevent times, as observed in real data. Thus the model suggests an explanation of the universal pattern of rescaled Earthquake Recurrence Time Distributions in terms of combinatorial rules for accumulation and abrupt release of seismic energy.

Characterizing weak chaos using time series of Lyapunov exponents.

We investigate chaos in mixed-phase-space Hamiltonian systems using time series of the finite-time Lyapunov exponents. The methodology we propose uses the number of Lyapunov exponents close to zero to define regimes of ordered (stickiness), semiordered (or semichaotic), and strongly chaotic motion. The dynamics is then investigated looking at the consecutive time spent in each regime, the transition between different regimes, and the regions in the phase space associated to them. Applying our methodology to a chain of coupled standard maps we obtain (i) that it allows for an improved numerical characterization of stickiness in high-dimensional Hamiltonian systems, when compared to the previous analyses based on the distribution of recurrence times; (ii) that the transition probabilities between different regimes are determined by the phase-space volume associated to the corresponding regions; and (iii) the dependence of the Lyapunov exponents with the coupling strength.

A population study of type II bursts in the Rapid Burster

Type II bursts are thought to arise from instabilities in the accretion flow onto a neutron star in an X-ray binary. Despite having been known for almost 40 years, no model can yet satisfactorily account for all their properties. To shed light on the nature of this phenomenon and provide a reference for future theoretical work, we study the entire sample of Rossi X-ray Timing Explorer data of type II bursts from the Rapid Burster (MXB 1730-335). We find that type II bursts are Eddington-limited in flux, that a larger amount of energy goes in the bursts than in the persistent emission, that type II bursts can be as short as 0.130 s, and that the distribution of recurrence times drops abruptly below 15-18 s. We highlight the complicated feedback between type II bursts and the NS surface thermonuclear explosions known as type I bursts, and between type II bursts and the persistent emission. We review a number of models for type II bursts. While no model can reproduce all the observed burst properties and explain the source uniqueness, models involving a gating role for the magnetic field come closest to matching the properties of our sample. The uniqueness of the source may be explained by a special combination of magnetic field strength, stellar spin period and alignment between the magnetic field and the spin axis.

Statistics of the island-around-island hierarchy in Hamiltonian phase space.

The phase space of a typical Hamiltonian system contains both chaotic and regular orbits, mixed in a complex, fractal pattern. One oft-studied phenomenon is the algebraic decay of correlations and recurrence time distributions. For area-preserving maps, this has been attributed to the stickiness of boundary circles, which separate chaotic and regular components. Though such dynamics has been extensively studied, a full understanding depends on many fine details that typically are beyond experimental and numerical resolution. This calls for a statistical approach, the subject of the present work. We calculate the statistics of the boundary circle winding numbers, contrasting the distribution of the elements of their continued fractions to that for uniformly selected irrationals. Since phase space transport is of great interest for dynamics, we compute the distributions of fluxes through island chains. Analytical fits show that the "level" and "class" distributions are distinct, and evidence for their universality is given.

Dynamics of an open quantum system interacting with a quantum environment

We examine the dynamics of subsystems of bipartite and tripartite quantum systems with nonlinear Hamiltonians. We consider two models which capture the generic features of open quantum systems: a three-level atom interacting with a single-mode radiation field, and a three-level atom interacting with two field modes which do not directly interact with each other. The entanglement of specific initially unentangled states of the atom–field system is examined through the time-varying subsystem von Neumann entropy (SVNE). The counterparts of near-revivals and fractional revivals of the initial state are clearly identifiable in the SVNE in all cases where revival phenomena occur. The Mandel Q parameter corresponding to the photon number of a radiation field is obtained as a function of time in both models. In those cases where revivals are absent, a time series analysis of the mean photon number reveals a variety of ergodicity properties (as manifested in return maps, recurrence-time distributions and Lyapunov exponents), depending on the strength of the nonlinearity and the degree of coherence of the initial state of the radiation field(s).

Estimation of a recurrent event gap time distribution: an application to morbidity outcomes following lower extremity fracture in Veterans with spinal cord injury

Recurrent event data is frequently encountered in biomedical research. Estimators describing the time to the next event occurrence could be useful information in healthcare settings to deliver targeted, preemptive, patient-specific care. Distribution estimation of recurrence times is varied based on both the underlying assumptions of the estimator (e.g., dependence vs. independence of recurrence times within individual) and estimator construct. In an example among Veterans with spinal cord injury, distributional changes in recurrence times of health complications subsequent lower extremity (LE) fracture are explored. Veterans with LE fracture (cases) were matched to Veterans without LE fracture (controls) on demographic (age, race), level of injury (paraplegia vs. tetraplegia), extent of injury, Veterans Affairs connected service status, and comorbidities using Mahalanobis metric matching. Stratified distributional estimates of recurrence times between successive morbidity outcomes are compared between Veterans with/without LE fracture using three estimators: independent identically distributed product limit estimator, Wang–Chang product limit estimator, and a gamma frailty maximum likelihood estimator. It can be seen that the estimator selection can provide a very different showcasing of a recurrence time distribution. There is a change in the time-to-recurrence of recurrent urinary tract infections and pressure ulcers for fracture cases directly following LE fracture, however testing if this difference is statistically significant remains unclear. Causal inference of gap time analyses in observational data with recurrent events is considered and a call for methods in this area is much warranted.

New Insights on the Strongest Historical Earthquake in the Pollino Region (Southern Italy)

The Italian seismic catalog (Rovida et al., 2011) portrays a significant gap of seismicity in the Pollino Range area, the southernmost segment of the southern Apennines at the boundary with the Calabrian arc. In this region, the only significant seismic event of the instrumental era occurred in 1998 north of the Pollino Range (Mw 5.6, Fig. 1). No seismic event with intensity greater than VIII on the Mercalli–Cancani–Sieberg (MCS) scale is present in the historical records, whereas strong earthquakes occurred immediately north (1836 M 6.0; 1857 M 7.0) and south (1184 M 6.7; 1836 M 6.2) of the area (Fig. 1). However, geologic data (Bousquet, 1973; Russo and Schiattarella, 1992) have shown clear evidence of Quaternary faulting along two major normal seismogenic faults, the Pollino and the Castrovillari faults. Paleoseismological data (Michetti et al., 1997; Cinti et al., 1997, 2002) recognized the occurrence of large magnitude events during medieval times along those two faults, but no trace is left of these major events in the historical record. In the case of the Pollino area, the discrepancy between true seismic history and recorded seismic history is due to a combination of a documentary gap of the historical sources (Scionti et al., 2006) and to the low population and scarcity of settlements in the epicentral area (D’Addezio et al., 1995; Cinti et al., 1997). Furthermore, there is the sound possibility that such documentary limitation also might affect moderate (M 5–6) earthquakes, thus providing the opportunity for new assessments that could re-estimate their sizes. Noticeable recent examples of this come from Scionti et al. (2006), with the reestimation to Mw 6.2 of an earthquake that occurred 100 km southeast of the Pollino region, from Tertulliani et al. (2012) for the 1762 earthquake close to L’Aquila, and from Azzaro et al. (2007) for northeastern Sicily. We want to emphasize that there is recent growing interest in the question of reestimating the sizes of historical earthquakes (see Rong et al., 2011; Hough, 2013). Therefore, reducing the uncertainties in the locations and sizes of seismic events for areas like Pollino can be an important contribution toward a clearer picture of the seismic potential and the distribution of earthquake recurrence times that can affect the seismic hazard of a territory. The first attempts at in-depth analyses of the historical seismicity in the Pollino seismic gap date back to the first half of the 1990s (Storia Geofisica Ambiente [SGA], 1994; Valensise et al., 1994; Guidoboni and Mariotti, 1997), whereas more recent developments are by SGA (2000), Camassi and Castelli (2004), and Castelli and Camassi (2005). The seismic histories (the list of earthquakes affecting a site during historic time) of towns and villages of the area (Locati et al., 2011) do not contain earthquakes that occurred before A.D. 1600, neither local nor remote, which could have affected such localities. The oldest traces of earthquakes are associated with the 1638 M 7 event that occurred in central Calabria ∼100 km south of the Pollino area and produced damage in Castrovillari. The more ancient event documented in the study area is the 1693 earthquake, which was listed for the first time in a catalog in 2007 (Guidoboni et al., 2007). In this study, we focus on this latter event, which also appears as the strongest shock in the region and is characterized by a relative scarcity of information and few intensity datapoints. We gathered all potential primary sources of information available for the 1693 earthquake in the attempt to improve its intensity map. We also address two further issues in the reappraisal of the 1693 earthquake: (1) the temporal coincidence between 8 January 1693 Pollino earthquake and its aftershocks and 9–11 January 1693 catastrophic earthquakes of eastern Sicily, along with the consequent quest for a correct attribution of the information regarding the two seismic sequences; and (2) the occurrence in October 2012 (following a sequence of small earthquakes lasting three years) of an Mw 5.3 mainshock, which is by far the strongest event in the instrumental era in the area.

Recurrence time distribution and temporal clustering properties of a cellular automaton modelling landslide events

Abstract. Reasonable prediction of landslide occurrences in a given area requires the choice of an appropriate probability distribution of recurrence time intervals. Although landslides are widespread and frequent in many parts of the world, complete databases of landslide occurrences over large periods are missing and often such natural disasters are treated as processes uncorrelated in time and, therefore, Poisson distributed. In this paper, we examine the recurrence time statistics of landslide events simulated by a cellular automaton model that reproduces well the actual frequency-size statistics of landslide catalogues. The complex time series are analysed by varying both the threshold above which the time between events is recorded and the values of the key model parameters. The synthetic recurrence time probability distribution is shown to be strongly dependent on the rate at which instability is approached, providing a smooth crossover from a power-law regime to a Weibull regime. Moreover, a Fano factor analysis shows a clear indication of different degrees of correlation in landslide time series. Such a finding supports, at least in part, a recent analysis performed for the first time of an historical landslide time series over a time window of fifty years.

Discovering focal regions of slightly-aggregated sparse signals

The characteristic aspects of dynamic distortions on a lengthy time series of i.i.d. pure noise when embedded with slightly-aggregating sparse signals are summarized into a significantly shorter recurrence time process of a chosen extreme event. We first employ the Kolmogorov---Smirnov statistic to compare the empirical recurrence time distribution with the null geometry distribution when no signal being present in the original time series. The power of such a hypothesis testing depends on varying degrees of aggregation of sparse signals: from a completely random distribution of singletons to batches of various sizes on the entire temporal span. We demonstrate the Kolmogorov---Smirnov statistic capturing the dynamic distortions due to slightly-aggregating sparse signals better than does Tukey's Higher Criticism statistic even when the batch size is as small as five. Secondly, after confirming the presence of signals in the pure noise time series, we apply the hierarchical factor segmentation (HFS) algorithm again based on the recurrence time process to compute focal segments that contain a significantly higher intensity of signals than do the rest of the temporal regions. In a computer experiment with a given fixed number of signals, the focal segments identified by the HFS algorithm afford many folds of signal intensity which also critically depend on the degree of aggregation of sparse signals. This ratio information can facilitate better sensitivity, equivalent to a smaller false discovery rate, if the signal-discovering protocol implemented within the computed focal regions is different from that used outside of the focal regions. We also numerically compute the specificity as the total number of signals contained in the computed collection of focal regions, which indicates the inherent difficulty in the task of sparse signal discovery.

Fertility heterogeneity as a mechanism for power law distributions of recurrence times

We study the statistical properties of recurrence times in the self-excited Hawkes conditional Poisson process, the simplest extension of the Poisson process that takes into account how the past events influence the occurrence of future events. Specifically, we analyze the impact of the power law distribution of fertilities with exponent α, where the fertility of an event is the number of triggered events of first generation, on the probability distribution function (PDF) f(τ) of the recurrence times τ between successive events. The other input of the model is an exponential law quantifying the PDF of waiting times between an event and its first generation triggered events, whose characteristic time scale is taken as our time unit. At short-time scales, we discover two intermediate power law asymptotics, f(τ)~τ(-(2-α)) for τ<<τ(c) and f(τ)~τ(-α) for τ(c)<<τ<<1, where τ(c) is associated with the self-excited cascades of triggered events. For 1<<τ<<1/ν, we find a constant plateau f(τ)=/~const, while at long times, 1/ν</~τ, f(τ)=/~e(-ντ) has an exponential tail controlled by the arrival rate ν of exogenous events. These results demonstrate a novel mechanism for the generation of power laws in the distribution of recurrence times, which results from a power law distribution of fertilities in the presence of self-excitation and cascades of triggering.

The 6 October 1762 Middle Aterno Valley (L'Aquila, Central Italy) Earthquake: New Constraints and New Insights

The effort for reducing the uncertainties in the location and size of historical earthquakes, even moderate-size ones, is not a peripheral issue, as it plays a major role in the distribution of earthquake recurrence times that can affect the maps of seismic hazard of a territory. The L’Aquila area (Abruzzo, Central Italy) struck by the 6 April 2009 Mw 6.3 seismic event is a typical example of an earthquake-prone region, the location of which historical seismicity needs to be better located. Thanks to a large body of geological, seismological, and geodetic evidence, the deep source of the 2009 mainshock has been imaged as a ~15-km-long, northwest-trending, southwest-dipping, almost-pure normal fault coinciding with the Paganica fault system at the surface (Fig. 1). Conversely, so far, very few papers have addressed the issue of reconstructing the seismic history of the Paganica fault, charting the relationships between this fault and the adjacent ones, and finding out the causative source for each of the earthquakes listed in the Italian earthquake catalog (CPTI Working Group, 2004; Rovida et al., 2011). In particular, the 1461 M 6.4 and the 1762 M 6.0 earthquakes (Fig. 1) took place in the same general area and display a comparable magnitude with that of the 2009 earthquake. According to recently provided paleoseismological data (Cinti et al., 2011), the shock of the 1461 earthquake was the penultimate event that ruptured the Paganica fault. The case of the 1762 earthquake was more open to debate: Tertulliani et al. (2009) suggested that it could have had the same fault source as that of the 2009 shock (west of the catalog epicenter) on account of some similarities in the distribution of the maximum intensities and in the temporal evolution of the 1762 and 2009 sequences. At the same time, however, Tertulliani et al. (2009) were aware of, and cautioned against, the very short recurrence interval implied by their hypothesis. The issue of the too-short recurrence interval, along with some observations on the too-low throw rate measured across the Paganica fault, induced Roberts et al. (2010) to invoke the role of other neighboring sources and to propose for the 1762 shock an alternative location some 10 km east-southeast of the epicenter adopted by the catalog. Such a variance of opinions concerning the epicentral location of the 1762 earthquake is explained by the scantiness of the data set on which investigations must rely and which consists of the shaking intensities at three localities only. As Di Bucci et al. (2011) pointed out, such a data set made it impossible to associate the 1762 event to any fault “beyond mere speculation”. We are left from this stark and somewhat frustrating consideration to systematically gather all potential primary sources of information available for the 1762 earthquake in the area to attempt to improve its intensity map and to compare it with the 2009 one. In the following, we will (1) set out the evidence newly collected from archives in L’Aquila and Naples and from other repositories of contemporary sources; (2) explain the rationale that allowed us to turn demographic and historical information into data sensitive to macroseismic analyses; (3) show the new intensity map for the 1762 earthquake and discuss its main differences from the previous intensity data sets (Boschi et al., 2000 and the most recent versions of the Italian macroseismic database, i.e., Stucchi et al., 2007; Locati and DBMI Working Group, 2009; Locati et al., 2011); (4) provide a more accurate location and propose new source parameters for the event; and (5) discuss the constraints on its epicenter location and source parameters with the possible seismotectonic implications.

Phase coherence and attractor geometry of chaotic electrochemical oscillators

Chaotic attractors are known to often exhibit not only complex dynamics but also a complex geometry in phase space. In this work, we provide a detailed characterization of chaotic electrochemical oscillations obtained experimentally as well as numerically from a corresponding mathematical model. Power spectral density and recurrence time distributions reveal a considerable increase of dynamic complexity with increasing temperature of the system, resulting in a larger relative spread of the attractor in phase space. By allowing for feasible coordinate transformations, we demonstrate that the system, however, remains phase-coherent over the whole considered parameter range. This finding motivates a critical review of existing definitions of phase coherence that are exclusively based on dynamical characteristics and are thus potentially sensitive to projection effects in phase space. In contrast, referring to the attractor geometry, the gradual changes in some fundamental properties of the system commonly related to its phase coherence can be alternatively studied from a purely structural point of view. As a prospective example for a corresponding framework, recurrence network analysis widely avoids undesired projection effects that otherwise can lead to ambiguous results of some existing approaches to studying phase coherence. Our corresponding results demonstrate that since temperature increase induces more complex chaotic chemical reactions, the recurrence network properties describing attractor geometry also change gradually: the bimodality of the distribution of local clustering coefficients due to the attractor's band structure disappears, and the corresponding asymmetry of the distribution as well as the average path length increase.

The negative binomial–beta Weibull regression model to predict the cure of prostate cancer

In this article, for the first time, we propose the negative binomial–beta Weibull (BW) regression model for studying the recurrence of prostate cancer and to predict the cure fraction for patients with clinically localized prostate cancer treated by open radical prostatectomy. The cure model considers that a fraction of the survivors are cured of the disease. The survival function for the population of patients can be modeled by a cure parametric model using the BW distribution. We derive an explicit expansion for the moments of the recurrence time distribution for the uncured individuals. The proposed distribution can be used to model survival data when the hazard rate function is increasing, decreasing, unimodal and bathtub shaped. Another advantage is that the proposed model includes as special sub-models some of the well-known cure rate models discussed in the literature. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes. We analyze a real data set for localized prostate cancer patients after open radical prostatectomy.

Recurrence of particles in static and time varying oval billiards

Dynamical properties are studied for escaping particles, injected through a hole in an oval billiard. The dynamics is considered for both static and periodically moving boundaries. For the static boundary, two different decays for the recurrence time distribution were observed after exponential decay for short times: A changeover to: (i) power law or; (ii) stretched exponential. Both slower decays are due to sticky orbits trapped near KAM islands, with the stretched exponential apparently associated with a single group of large islands. For time dependent case, survival probability leads to the conclusion that sticky orbits are less evident compared with the static case.

Testing a scaling law for the earthquake recurrence time distributions

The earthquake recurrence time distribution in a given space-time window is being studied, using earthquake catalogues from different seismic regions (Southern California, Canada, and Central Asia). The quality of the available catalogues, taking into account the completeness of the magnitude, is examined. Based on the analysis of the catalogues, it was determined that the probability densities of the earthquake recurrence times can be described by a universal gamma distribution, in which the time is normalized with the mean rate of occurrence. The results show a deviation from the gamma distribution at the short interevent times, suggesting the existence of clustering. This holds from worldwide to local scales and for quite different tectonic environments.

Distinguishing dynamics using recurrence-time statistics

The probability densities of the mean recurrence time, which is the average time needed for a system to recur to a previously visited neighborhood, are investigated in various dynamical regimes and are found to be in agreement with those of the finite-time Lyapunov exponents. The important advantages of the former ones are that they are easy to estimate and that comparable short time series are sufficient. Asymmetric distributions with exponential tails are observed for intermittency and crisis-induced intermittency, while for typical chaos, the distribution has a Gaussian shape. Further, the shape of the distribution distinguishes intermittent strange nonchaotic attractors from those appearing through fractalization and tori collision mechanisms. Furthermore, statistics performed on the peaks in the frequency distribution of recurrence times unveil scaling behavior in agreement with that obtained from the spectral distribution function defined as the number of peaks in the Fourier spectrum greater than a predefined value. The results of the present recurrence statistics are of relevance in classifying different dynamics and providing important insights into the dynamics of a system when only one realization of this system is available. The practical use of this approach for experimental data is shown on experimental electrochemical time series.

Re-evaluation of conditional probability of rupture of the Wellington-Hutt Valley segment of the Wellington Fault

New information on the activity of the Wellington-Hutt Valley segment of the Wellington Fault, New Zealand, has become available from geological and modelling studies undertaken in the last several years as part of the “It’s Our Fault” project. There are now revised estimates of: 1) the timing of the most recent rupture, and the previous four older ruptures; 2) the size of single-event displacements; 3) the Holocene dextral slip rate; and 4) rupture statistics of the Wellington-Wairarapa fault-pair, as deduced from synthetic seismicity modelling. The conditional probability of rupture of this segment over the next 100 years is re-evaluated in light of this new information, assuming a renewal process framework. Four recurrence-time distributions (exponential, lognormal, Weibull and Brownian passage-time) are explored. The probability estimates take account of both data and parameter uncertainties. A sensitivity analysis is conducted, entertaining different bounds and shapes of the probability distributions of important fault rupture data and parameters. Important findings and conclusions include:&#x0D; &#x0D; The estimated probability of rupture of the Wellington-Hutt Valley segment of the Wellington Fault in the next 100 years is ~11% (with sensitivity results ranging from 4% to 15%), and the probability of rupture in the next 50 years is about half of that (~5%).&#x0D; In all cases, the inclusion of the new data has reduced the estimated probability of rupture of the Wellington Fault by ~50%, or more, compared to previous estimates.&#x0D;