Temporal Point Process Research Articles

Investigating the time-scaling behavior of the 2004–2010 seismicity of Aswan area (Egypt) by means of the Allan factor statistics and the detrended fluctuation analysis

Abstract. The time dynamics of seismicity of Aswan area (Egypt) from 2004 to 2010 was investigated by means of the (i) Allan Factor, which is a powerful tool allowing the capture of time-clusterized properties of temporal point processes; and the (ii) detrended fluctuation analysis, which is capable of detecting scaling in nonstationary time series. The analysis was performed varying the depth and the magnitude thresholds. The 2004–2010 Aswan seismicity is characterized by significant three-fold time-clustering behaviors with scaling exponents ~0.77 for timescales between 104.16 s and 105.14 s, ~0.34 for timescales between 105.14 s and 106.53 s, and ~1 for higher timescales. The seismic interevent times and distances are characterized by persistent temporal fluctuations for most of the magnitude and depth thresholds.

Noise characteristics of heterodyne/homodyne frequency-domain measurements

We theoretically develop and experimentally validate the noise characteristics of heterodyne and/or homodyne measurements that are widely used in frequency-domain diffusive imaging. The mean and covariance of the modulated heterodyne output are derived by adapting the random amplification of a temporal point process. A multinomial selection rule is applied to the result of the temporal noise analysis to additionally model the spatial distribution of intensified photons measured by a charge-coupled device (CCD), which shows that the photon detection efficiency of CCD pixels plays an important role in the noise property of detected photons. The approach of using a multinomial probability law is validated from experimental results. Also, experimentally measured characteristics of means and variances of homodyne outputs are in agreement with the developed theory. The developed noise model can be applied to all photon amplification processes.

Time-clustering analysis of the 1978–2008 sub-crustal seismicity of Vrancea region

Abstract. The analysis of time-clustering behaviour of the sub-crustal seismicity (depth larger than 60 km) of the Vrancea region has been performed. The time span of the analyzed catalogue is from 1978 to 2008, and only the events with a magnitude of Mw ≥ 3 have been considered. The analysis, carried out on the full and aftershock-depleted catalogues, was performed using the Allan Factor (AF) that allows the identificatiion and quantification of correlated temporal structures in temporal point processes. Our results, whose significance was analysed by means of two methods of generation of surrogate series, reveal the presence of time-clustering behaviour in the temporal distribution of seismicity data of the full catalogue. The analysis performed on the aftershock-depleted catalogue indicates that the time-clustering is associated mainly to the aftershocks generated by the two largest events occurred on 30 August 1986 (Mw = 7.1) and 30 May 1990 (Mw = 6.9).

Gamma Kernel Intensity Estimation in Temporal Point Processes

In this article, we propose a nonparametric approach for estimating the intensity function of temporal point processes based on kernel estimators. In particular, we use asymmetric kernel estimators characterized by the gamma distribution, in order to describe features of observed point patterns adequately. Some characteristics of these estimators are analyzed and discussed both through simulated results and applications to real data from different seismic catalogs.

PtProcess: AnRPackage for Modelling Marked Point Processes Indexed by Time

This paper describes the package <b>PtProcess</b> which uses the <b>R</b> statistical language. The package provides a unified approach to fitting and simulating a wide variety of temporal point process or temporal marked point process models. The models are specified by an intensity function which is conditional on the history of the process. The user needs to provide routines for calculating the conditional intensity function. Then the package enables one to carry out maximum likelihood fitting, goodness of fit testing, simulation and comparison of models. The package includes the routines for the conditional intensity functions for a variety of standard point process models. The package is intended to simplify the fitting of point process models indexed by time in much the same way as generalized linear model programs have simplified the fitting of various linear models. The primary examples used in this paper are earthquake sequences but the package is intended to have a much wider applicability.

Output Dead-Time in Point Processes

When events of temporal point processes are too close to each other they can be erased by dead-time effects. Among various possible mechanisms of dead-time, the output dead-time is the most important. Dead-time effects modify the statistical properties of point processes and some of these modifications are analyzed in this article. To do so, we note that a point process is defined by the distance between its successive points called life-time which constitutes a discrete time positive signal. The dead-time mechanism is a system which transforms such a signal into another discrete time positive signal. Except in very specific cases this transformation cannot be expressed in closed form. We show, however, that it can be written in a recursive form analogous to the state representation of systems. By using this recursion, various statistical properties of point processes with dead-time are analyzed in computer experiments. In this study, we focus on the probability distribution of the intervals between points and the coincidence function which describes the second-order properties of the point process. For the rare processes where theoretical calculations are possible there is an excellent agreement between experiment and theory.

Spatial—temporal marked point processes: a spectrum of stochastic models

AbstractMany processes that develop through space and time do so in response not only to their own individual growth mechanisms but also in response to interactive pressures induced by their neighbours. The growth of trees in a forest which compete for light and nutrient resources, for example, provides a classic illustration of this general spatial—temporal growth‐interaction process. Not only has its mathematical representation proved to be a powerful tool in the study and analysis of marked point patterns since it may easily be simulated, but it has also been shown to be highly flexible in terms of its application since it is robust with respect to incorrect choice of model selection. Moreover, it is highly amenable to maximum likelihood and least squares parameter estimation techniques. Currently the algorithm comprises deterministic growth and interaction coupled with a stochastic arrival and departure mechanism. So for systems with a fixed number of particles there is an inherent lack of randomness. A variety of different stochastic approaches are therefore presented, from the exact event—time model through to the associated stochastic differential equation, taking in time‐increment and Tau‐ and Langevin‐Leaping approximations en route. The main algorithm is illustrated through application to forest management and high‐intensity packing of hard particle systems, and comparisons are made with the established force biased approach. Copyright © 2009 John Wiley &amp; Sons, Ltd.

A Hypothesis‐Free Multiple Scan Statistic with Variable Window

In this article we propose a new technique for identifying clusters in temporal point processes. This relies on the comparision between all the m -order spacings and it is totally independent of any alternative hypothesis. A recursive procedure is introduced and allows to identify multiple clusters independently. This new scan statistic seems to be more efficient than the classical scan statistic for detecting and recovering cluster alternatives. These results have applications in epidemiological studies of rare diseases.

A spatial–temporal point process model of rainfall for the Thames catchment, UK

A spatial–temporal point process model of rainfall is fitted to data taken from 23 sites across the Thames catchment, London. For the period January 1970 to December 1988 the sites have daily data, whilst for the period January 1989 to November 2003 the sites have hourly data. In addition, the records contain missing values. The fitted model is used to infill the missing values, to disaggregate the daily data to hourly data and to generate spatially consistent series at a further 25 sites across the catchment. The model is validated by considering statistical properties that are not used in the fitting procedure, which include extreme values, the proportion of dry intervals, and annual totals. Overall the performance of the model is good, with the exception of an over-estimation in cross-correlation between pairs of sites for the disaggregated series.

Searching for time-scaling features in rainfall sequences

Temporal point processes can be used to model rain event sequences. Time-scaling analysis of rainfall data has been performed by means of the statistical approaches of Fano Factor analysis (FF) and Allan Factor analysis (AF). The obtained results suggest the presence of clustering in the time distribution of the sequence of rain events, with scaling exponents α FF ∼ 0.33–0.47 and α AF ∼ 0.41–0.48.

Consistent parametric estimation of the intensity of a spatial–temporal point process

We consider conditions under which parametric estimates of the intensity of a spatial–temporal point process are consistent. Although the actual point process being estimated may not be Poisson, an estimate involving maximizing a function that corresponds exactly to the log-likelihood if the process is Poisson is consistent under certain simple conditions. A second estimate based on weighted least squares is also shown to be consistent under quite similar assumptions. The conditions for consistency are simple and easily verified, and examples are provided to illustrate the extent to which consistent estimation may be achieved. An important special case is when the point processes being estimated are in fact Poisson, though other important examples are explored as well.

Transforming echoes into pseudo-action potentials for classifying plants.

Animals perceive their environment by converting sensory stimuli into action potentials, or temporal point processes, that are interpreted by the brain. This paper investigates the information content of point processes extracted from echoes from in situ plants in an effort to understand how bats recognize landmarks in the field. A mobile sonar converts echoes into biologically similar temporal point processes. termed pseudo-action potentials (PAPs), whose inter-PAP interval relates to echo amplitude. The sonar forms a sector scan of an object to produce a spatial-temporal PAP field. Classifier neurons apply delays and coincidence detection to the PAP field to identify three distinct echo types, glints, blobs, and fuzz, which characterize plant features. Glints are large amplitude echoes exhibiting coherence over successive echoes in the sector scan, typically produced by favorably oriented isolated specular reflectors. Blobs are large echoes lacking coherence, typically bordering glints or formed by collections of interfering reflectors. Fuzz represents weak echoes, typically produced by collection of weak scatterers or by reflectors on the beam periphery. A small mirror reflector models a flat leaf surface and motivates the glint criteria. Classifiers are applied to experimental data from two types of tree trunks, a glint-producing sycamore (Platanus occidenatalis) and a glint-absent Norway maple (Acer platanoides) and two plants, a glint-producing rhododendron (Rhododendron maximus) and a glint-absent yew (Taxus media). We speculate that our narrow-band sonar models the activity of a single frequency bin in the frequency-modulated (FM) sweep emitted by bats, and that one function of the frequency bins in the FM sweep is to form a sector scan of the environment.

Scaling laws and forecasting in athletic world records

In this study, we analysed running world records and found that the mean speed of the race, μ, as a function of the record time, τ, can be described asymptotically by two well-defined scaling laws of the form u ˜s τ - β. There is a break in the scaling laws (˜1000m) between the shorter and the longer races at a characteristic time of around 150-170 s, after which a new scaling regime emerges. This is the first occasion that this characteristic time has been clearly found in physical terms; we interpreted it as the transition time between the anaerobic and the aerobic energy expenditure of athletes. This phenomenon is independent of the athletes' sex and is also found in swimming races with similar values of the characteristic time. We also investigated the forecasting of world records using historical data. Using an approach based on the identification of non-Poissonian events for a sequence of temporal point processes, we found that the sequence of improvements in all athletic records from 1900 to the present day cannot be considered as a sequence of completely random events.