Stochastic Point Process Research Articles

How the ETAS models were created, used, and evolved -- Personal views and perspectives

Statistical seismology originated in Japan 130 years ago. Through the marriage with stochastic point processes in about half a century, we can now provide on-line estimation, real-time forecasting and direct diagnosis of seismic activity. This manuscript describes the origin and recent development of the ETAS models, their relationship to forecasting problems, and their diagnostic studies of the physical phenomena of seismic activity. I will take this opportunity to focus on my research experiences and views.

A doubly stochastic point process approach for spatio-temporal dynamics of crime data

Crime, in general, is at the base of crucial issues for many societies living in large cities worldwide. Indeed, crime and neighbourhood disorder may negatively impact the health of urban residents. Thus, the crime rate reduction is at the core of many local policies driven by active plans supported by police action and local authorities. Considering crime reports as a spatio-temporal point pattern, we propose spatio-temporal log-Gaussian Cox processes as a modelling framework for crimes in space and time. We model the spatial and temporal variation through generalized parametric additive and linear models, and a Gaussian space-time process approximates the residual variation. The inference is performed via Markov chain Monte Carlo through MALA algorithms. We provide short-term forecasts of future crimes and suggest a surveillance system that operates by reporting predictive probabilities. Our data come from the reported crimes in the locality of Kennedy (Bogota) over several years and several types of crimes. The police department may use our method to help allocate police resources and design crime prevention strategies and policies, such as surveillance, in specific zonal planning units.

Predicting the Probability of Abrupt Changes to Wave‐Generated Seafloor Sand Ripples

AbstractA new, non‐dimensional ripple reset parameter and a stochastic point process model is used to estimate the likelihood of propagating ocean waves to form ripples on sandy seabeds. The ripple reset parameter is a function only of water depth, significant wave height, and mean grain size. Ripple formation is estimated by the magnitude of an intensity function based on a time series of the ripple reset parameter. The point process model is trained with a time series of observed waves and ripple change, and is then applied to predict the probability that a ripple field with a different geometry will form within a given time interval from another time series of wave data. The model is trained and tested with four field deployments at three field sites to determine its skill in predicting the ripple formation (a) at one field site over one time period after being trained with observations from the same site over a different time period, and (b) at one field site after being trained with observations from another field site. Results show that while the model is sufficient at predicting ripple formation in both scenarios, it is sensitive to the quality and quantity of the training data. Increasing the amount of training data greatly improves model performance. Employing a stochastic model based on a simple ripple reset parameter reduces tunable model parameters and provides a prediction of the probability for ripple formation given only a water depth, grain size, and time series of wave heights.

Spatio-Temporal Hawkes Point Processes: A Review

AbstractHawkes processes are a particularly interesting class of stochastic point processes that were introduced in the early seventies by Alan Hawkes, notably to model the occurrence of seismic events. They are also called self-exciting point processes, in which the occurrence of an event increases the probability of occurrence of another event. The Hawkes process is characterized by a stochastic intensity, which represents the conditional probability density of the occurrence of an event in the immediate future, given the observations in the past. In this paper, we present some background and all major aspects of Hawkes processes, with a particular focus on simulation methods, and estimation techniques widely used in practical modeling aspects. We aim to provide a rich and self-contained overview of these stochastic processes as a way to have an overall vision of Hawkes processes in only one piece of paper. We also discuss possibilities for future research in the area of self-exciting processes.

Simulation of doubly stochastic Poisson point processes and application to nucleation of nanocrystals and evaluation of exciton fluxes

Abstract In this study we solve the following problem: Simulate random 2D Poisson point processes with a desired correlation function. To solve this problem we suggest the following algorithm: (1) simulate a positive valued random process with the desired correlation function, (2) use this process as an intensity of the doubly stochastic Poisson random point process. We apply this algorithm to simulate random distribution of nanocrystals on a plane. Then we apply the developed methods to calculate excitonic fluxes to the family of generated nanocrystals.

Vapor bubble nucleation in flowing liquids

In this work, a stochastic diffuse interface model coupled with the Navier–Stokes equations has been exploited to numerically investigate vapor nucleation in a non-equilibrium system (a flowing liquid). Both homogeneous and heterogeneous nucleation is addressed and the influence of macroscopic flows on nucleation observables is discussed. The extended mesoscale simulations allow us to infer the spatial distributions of the nucleated bubbles via Voronoi tessellation analysis and to represent the nucleation phenomenon as a stochastic Random Poisson Point process. The findings open the way to design multiscale fluid simulations experiencing phase change.

On Gaussian Markov processes and Polya processes

In previous work we characterized Gaussian Markov processes with stationary increments and showed that they arise as asymptotic approximations for stochastic point processes with a random rate such as Polya processes, which can be useful to model over-dispersion and path-dependent behavior in service system arrival processes. Here we provide additional insight into these stochastic processes.

Continuous‐time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations

AbstractWe develop a new continuous‐time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model's parameters using an estimate for the gradient of the stationary distribution. The gradient estimate is simultaneously updated using forward propagation of the SDE state derivatives, asymptotically converging to the direction of steepest descent. We rigorously prove convergence of the online forward propagation algorithm for linear SDE models (i.e., the multidimensional Ornstein–Uhlenbeck process) and present its numerical results for nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the direction of steepest descent. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds for the solutions of a new class of Poisson partial differential equations (PDEs), which are then used to analyze the parameter fluctuations in the algorithm. Our algorithm is applicable to a range of mathematical finance applications involving statistical calibration of SDE models and stochastic optimal control for long time horizons where ergodicity of the data and stochastic process is a suitable modeling framework. Numerical examples explore these potential applications, including learning a neural network control for high‐dimensional optimal control of SDEs and training stochastic point process models of limit order book events.

On spare parts demand and the installed base concept: A theoretical approach

Original Equipment Manufacturers (OEMs) aim to design their service supply chain before the introduction of their products to maximize their aftersales business revenues, reduce waste and achieve sustainability. In this study, we develop a stochastic model that unifies the installed base, i.e., the number of products in use, spare parts demand, and the number of discarded products within a single modeling framework based on three product characteristics: sales rate, usage time, and failure rate. Our model describes the installed base and spare part demand evolution over the entire life cycle of a parent product using stochastic point processes. At the same time we propose under very general assumptions on the cdf of the usage time and the mean arrival functions of the sales and failure processes an easy bisection procedure to compute the time at which the expected installed base and rate of the expected demand for spare parts is maximal. Our numerical experiments show that the volume of aftersales services increases in the expected usage time if the products face an increasing failure rate. The same experiments also reveal a 20 percent shift of the time at which the expected installed base is maximal in case the expected usage time is increased threefold. At the same time, we observe a boosting effect of the intensity of the sales process on this point in time.

Modeling scenarios for mitigating outbreaks in congregate settings.

The explosive outbreaks of COVID-19 seen in congregate settings such as prisons and nursing homes, has highlighted a critical need for effective outbreak prevention and mitigation strategies for these settings. Here we consider how different types of control interventions impact the expected number of symptomatic infections due to outbreaks. Introduction of disease into the resident population from the community is modeled as a stochastic point process coupled to a branching process, while spread between residents is modeled via a deterministic compartmental model that accounts for depletion of susceptible individuals. Control is modeled as a proportional decrease in the number of susceptible residents, the reproduction number, and/or the proportion of symptomatic infections. This permits a range of assumptions about the density dependence of transmission and modes of protection by vaccination, depopulation and other types of control. We find that vaccination or depopulation can have a greater than linear effect on the expected number of cases. For example, assuming a reproduction number of 3.0 with density-dependent transmission, we find that preemptively reducing the size of the susceptible population by 20% reduced overall disease burden by 47%. In some circumstances, it may be possible to reduce the risk and burden of disease outbreaks by optimizing the way a group of residents are apportioned into distinct residential units. The optimal apportionment may be different depending on whether the goal is to reduce the probability of an outbreak occurring, or the expected number of cases from outbreak dynamics. In other circumstances there may be an opportunity to implement reactive disease control measures in which the number of susceptible individuals is rapidly reduced once an outbreak has been detected to occur. Reactive control is most effective when the reproduction number is not too high, and there is minimal delay in implementing control. We highlight the California state prison system as an example for how these findings provide a quantitative framework for understanding disease transmission in congregate settings. Our approach and accompanying interactive website (https://phoebelu.shinyapps.io/DepopulationModels/) provides a quantitative framework to evaluate the potential impact of policy decisions governing infection control in outbreak settings.

Analyzing the Correlations and the Statistical Distribution of Moderate to Large Earthquakes Interevent Times in Greece

Seismic temporal properties constitute a fundamental component in developing probabilistic models for earthquake occurrence in a specific area. Earthquake occurrence is neither periodic nor completely random but often accrues into bursts in both short- and long-term time scales, and involves a complex summation of triggered and independent events (ΔT). This behavior underlines the impact of the correlations on many potential applications such as the stochastic point process for the earthquake interevent times. In this respect, we intend firstly to determine the appropriate magnitude thresholds, Mthr, indicating the temporal crossover between correlated and statistically independent earthquakes in each 1 of the 10 distinctive sub-areas of the Aegean region. The second goal is the investigation of the statistical distribution that optimally fits the interevent times’ data for earthquakes with M≥Mthr after evaluating the Gamma, Weibull, Lognormal and Exponential distributions performance. Results concerning the correlations analysis evidenced that the temporal crossover of the earthquake interevent time data ranges from Mthr≥ 4.7 up to Mthr≥ 5.1 among the 10 sub-areas. The distribution fitting and comparison reveals that the Gamma distribution outperforms the other three distributions for all the data sets. This finding indicates a burst or clustering behavior in the earthquake interevent times, in which each earthquake occurrence depends upon only the occurrence time of the last one and not from the full seismic history.

Stimulus presentation can enhance spiking irregularity across subcortical and cortical regions.

Stimulus presentation is believed to quench neural response variability as measured by fano-factor (FF). However, the relative contributions of within-trial spike irregularity and trial-to-trial rate variability to FF fluctuations have remained elusive. Here, we introduce a principled approach for accurate estimation of spiking irregularity and rate variability in time for doubly stochastic point processes. Consistent with previous evidence, analysis showed stimulus-induced reduction in rate variability across multiple cortical and subcortical areas. However, unlike what was previously thought, spiking irregularity, was not constant in time but could be enhanced due to factors such as bursting abating the quench in the post-stimulus FF. Simulations confirmed plausibility of a time varying spiking irregularity arising from within and between pool correlations of excitatory and inhibitory neural inputs. By accurate parsing of neural variability, our approach reveals previously unnoticed changes in neural response variability and constrains candidate mechanisms that give rise to observed rate variability and spiking irregularity within brain regions.

Inhomogeneous spatio-temporal point processes on linear networks for visitors’ stops data

We analyse the spatio-temporal distribution of visitors’ stops by touristic attractions in Palermo (Italy), using theory of stochastic point processes living on linear networks. We first propose an inhomogeneous Poisson point process model with a separable parametric spatio-temporal first-order intensity. We account for the spatial interaction among points on the given network, fitting a Gibbs point process model with mixed effects for the purely spatial component. This allows us to study first-order and second-order properties of the point pattern, accounting both for the spatio-temporal clustering and interaction and for the spatio-temporal scale at which they operate. Due to the strong degree of clustering in the data, we then formulate a more complex model, fitting a spatio-temporal log-Gaussian Cox process to the point process on the linear network, addressing the problem of the choice of the most appropriate distance metric.

Reducing Bias in Event Time Simulations via Measure Changes

Stochastic point process models of event timing are common in many areas, including finance, insurance, and reliability. Monte Carlo simulation is often used to perform computations for these models. The standard sampling algorithm, which is based on a time-change argument, is widely applicable but generates biased simulation estimators. This article develops and analyzes a change of probability measure that can reduce or even eliminate the bias without restricting the scope of the algorithm. A result of independent interest offers new conditions that guarantee the existence of a broad class of point process martingales inducing changes of measure. Numerical results illustrate our approach.

Performance evaluation of IoT networks: A product density approach

Mobile cellular-based Internet of Things (IoT) networking is set to be enhanced with the addition of two important pillars of 5G — massive Machine Type Communications and ultra-Reliable Low Latency Communications. Temporal traffic uncertainty from diverse applications in IoT networks mandates novel modeling approaches for performance evaluation. This paper introduces a novel stochastic point process approach that can be used to evaluate the time-dependent performance of IoT base stations. Special correlation functions called Product Densities are used to (a) evaluate time-dependent offered traffic, and (b) analyze delay performance. These performance measures are evaluated at IoT base station for Poisson as well as non-Homogeneous Poisson (Beta distributed) traffic arrival processes suggested by 3rd Generation Partnership Project(3GPP). The Product Density estimates of offered traffic are found to be more accurate than the point-wise stationary approximation (PSA) under non-stationary traffic arrival rates. Results from the proposed analytical model are compared with results from a simulation of two queuing models of the base station; infinite server model for offered traffic and multi-server model for delay performance. Analytical results from Product Density functions also correlate with the simulation outcomes suggesting that the proposed Product Density technique is effective when modeling the time-dependent performance of IoT networks subjected to non-stationary traffic conditions, which reflects the real-world scenario.

Characteristic and Necessary Minutiae in Fingerprints

AbstractFingerprints feature a ridge pattern with moderately varying ridge frequency (RF), following an orientation field (OF), which usually features some singularities. Additionally at some points, called minutiae, ridge lines end or fork and this point pattern is usually used for fingerprint identification and authentication. Whenever the OF features divergent ridge lines (e.g., near singularities), a nearly constant RF necessitates the generation of more ridge lines, originating at minutiae. We call these the necessary minutiae. It turns out that fingerprints feature additional minutiae which occur at rather arbitrary locations. We call these the random minutiae or, since they may convey fingerprint individuality beyond the OF, the characteristic minutiae. In consequence, the minutiae point pattern is assumed to be a realization of the superposition of two stochastic point processes: a Strauss point process (whose activity function is given by the divergence field) with an additional hard core, and a homogeneous Poisson point process, modelling the necessary and the characteristic minutiae, respectively. We perform Bayesian inference using an Markov-Chain-Monte-Carlo (MCMC)-based minutiae separating algorithm (MiSeal). In simulations, it provides good mixing and good estimation of underlying parameters. In application to fingerprints, we can separate the two minutiae patterns and verify by example of two different prints with similar OF that characteristic minutiae convey fingerprint individuality.

A stochastic process based modular tool-box for simulating COVID-19 infection spread

BackgroundThe novel coronavirus disease (COVID-19) culminated in a pandemic with many countries affected in varying stages. We aimed to develop a simulation environment for COVID-19 spread, taking environmental and social factors into account. MethodsThe program was written in R language. A stochastic point process simulation model for simulating epidemics, a maximum-likelihood estimation model, an exponential growth rate model for calculating the basic reproduction number (R0), and functions for generating graphical representations of the simulations were utilized.Geographical area definition, population size, the number of initial infected individuals, period of simulation, parameters accounting for the radius of spread like masks usage, mobility level, intrinsic viral virulence, average infectious period, fraction of population vaccinated, time of vaccination, the efficacy of the vaccine, presence or absence of quarantine centers, time of establishment of quarantine centers, the efficacy of case detection and average time to quarantine from the detection of the infection were considered. ResultsWhen the defined parameters were input, the model performed successfully producing the epidemic curve, R0 and an animation of infection spread. It was found that when parameters of known epidemics such as COVID-19 in California, Texas and, Florida were input, the epidemic curve generated was comparable to the epidemic curve in reality. ConclusionThis model can be utilized by many countries to visualize the effects of various mitigation strategies applied in their stage of disease and for policy makers to make informed decisions. It is applicable to many infectious diseases and hence can be used for research and educational purposes.

Statistical dependence analysis of Erbium doped fiber ring lasers (EDFRL) chaos

Utilizing ergodic nature of chaotic lasers, the probability density function (PDF) estimation is carried out for peak amplitudes of pulsed chaos in Erbium Doped Fiber Ring Laser (EDFRL) in relation to varying the four key cavity parameters i.e. cavity gain, pump power, modulation index and modulation frequency. Kernel based non-parametric probability distributions are found more suitable to fit the chaos peaks data instead of parametric Gaussian, Rayleigh or Poisson Probability Density Functions. The resulting non-parametric PDFs’ shape, modes, skew and variance is found controllable by changing cavity parameters which is beneficial for random number generation with desired properties. The non-parametric distribution changed its shape between uni-modal and bimodal Gaussian like PDFs alternately and the variance also changes. The PDF shape corresponds to density of points in chaotic data bifurcation plots. This statistical in depth analysis is important while employing the EDFRL for random number generation and in chaotic secure communication applications. EDFRL chaos is found here to be a non-Poisson doubly stochastic point process implying it does depend on history of the chaos generation process, which is hidden in the slowly varying population inversion density of the laser. The bimodality skew keeps altering between left and right while becoming uni-modal every time between switching with modulation index and pump power changes. At high gain the variance of PDF decreases and it approaches uni-modal Gaussian. At high modulation frequency the PDF is left skewed with higher number of shorter pulses bursting.

Age replacement with Markovian opportunity process

This paper focuses on the age replacement problems under the assumption that the arrival of opportunities follows a stochastic point process. The opportunity is a chance to make the replacement, and one can replace a unit with a low cost at the opportunity. In this paper, the opportunity-based age replacement models by Dekker and Dijkstra (1992) and Zhao and Nakagawa (2012) are extended in the environment where the opportunity process is in accordance with a Markovian arrival process (MAP). The MAP is one of the general point processes, and involves a Poisson process and some renewal process classes. This work introduces the generalization of the arrival process of the preventive replacement problems with opportunities. Furthermore, a comprehensive evaluation of the opportunity-based age replacement policies by Dekker and Dijkstra (1992) and Zhao and Nakagawa (2012) in the generalized modeling framework is carried out.

Detecting Change between Urban Road Environments along a Route Based on Static Road Object Occurrences

For over a decade, urban road environment detection has been a target of intensive research. The topic is relevant for the design and implementation of advanced driver assistance systems. Typically, embedded systems are deployed in these for the operation. The environments can be categorized into road environment-types. Abrupt transitions between these pose a traffic safety risk. Road environment-type transitions along a route manifest themselves also in changes in the distribution of traffic signs and other road objects. Can the placement and the detection of traffic signs be modelled jointly with an easy-to-handle stochastic point process, e.g., an inhomogeneous marked Poisson process? Does this model lend itself for real-time application, e.g., via analysis of a log generated by a traffic sign detection and recognition system? How can the chosen change detector help in mitigating the traffic safety risk? A change detection method frequently used for Poisson processes is the cumulative sum (CUSUM) method. Herein, this method is tailored to the specific stochastic model and tested on realistic logs. The use of several change detectors is also considered. Results indicate that a traffic sign-based road environment-type change detection is feasible, though it is not suitable for an immediate intervention.