Shot Noise Process Research Articles

Simulating temporal speckle with prescribed correlation properties in optical lithography

We describe a method of simulating temporal speckle often encountered in optical lithography. The first step of the method is to generate numerically optical fields with prescribed temporal coherence properties using a shot noise process. By properly arranging the data for instantaneous intensities calculated from the generated fields, one can construct virtually temporal speckle patterns that satisfy all the necessary conditions. As some illustrative examples, we examine the variation of the temporal speckle contrast as a function of the exposure time of the detector by means of these speckle patterns. Our method would provide an intuitive understanding of hardly observable temporal speckle and serve as an educational tool for students and professionals in optics.

Sample path moderate deviations for shot noise processes in the high intensity regime

We study the sample-path moderate deviation principle (MDP) for shot noise processes in the high intensity regime. The shot noise processes have a renewal arrival process, non-stationary noises (with arrival-time dependent distributions) and a general shot response function of the noises. The rate function in the MDP exhibits a memory phenomenon in this asymptotic regime, which is in contrast with that in the conventional time–space scaling regime. To prove the sample-path MDP, we first establish that this is equivalent to establishing the sample-path MDP of another process that is easier to study. We prove its finite-dimensional MDP and then establish the exponential tightness under the Skorohod J1 topology. This results in the sample-path MDP in D under the Skorohod J1 topology with a rate function that is derived from the rate function in the finite-dimensional MDP using the tools of reproducing kernel Hilbert space. In the proofs, because of the non-stationarity of shot noise process, we establish a new exponential maximal inequality and use it to prove exponential tightness and the aforementioned equivalence.

Establishing Convergence of Infinite-Server Queues with Batch Arrivals to Shot-Noise Processes

Establishing Convergence of Infinite-Server Queues with Batch Arrivals to Shot-Noise Processes

Real-time image denoising corrupted by shot-noise

In this paper the new model for the image noise is proposed. The video data is modelled as a sequence of images corrupted by the shot-noise. Shot-noise may occur due to the defect in the device’s hardware or in the camera’s sensor. This kind of noise effects not only the current image of the video sequence but also the subsequent images, the effect decays as time passes and disappears totally after a certain time period. The shot-noise process consists of the sequence of jumps that decay gradually as time passes. The times at which the jumps occur are Poisson distributed and the hight of the jumps is normally distributed. It can be proved that under certain conditions shot-noise process tends to Gaussian process. Therefore, it was proposed to apply Kalman filter to remove the noise and to restore the corrupted sequence of images. The numerical experiments that were carried out that show the effectiveness of the proposed approach for denoising the shot-noise corrupted videos.

On the finiteness of the moments of the measure of level sets of random fields

General conditions on smooth real valued random fields are given that ensure the finiteness of the moments of the measure of their level sets. As a by product a new generalized Kac-Rice formula (KRF) for the expectation of the measure of these level sets is obtained when the second moment can be uniformly bounded. The conditions involve (i) the differentiability of the trajectories up to a certain order k, (ii) the finiteness of the moments of the k-th partial derivatives of the field up to another order, (iii) the boundedness of the joint density of the field and some of its derivatives. Particular attention is given to the shot noise processes and fields. Other applications include stationary Gaussian processes, Chi-square processes and regularized diffusion processes. AMS2000 Classifications: Primary 60G60. Secondary 60G15.

A New Upper Bound on Cache Hit Probability for Non-Anticipative Caching Policies

Caching systems have long been crucial for improving the performance of a wide variety of network and web-based online applications. In such systems, end-to-end application performance heavily depends on the fraction of objects transferred from the cache, also known as the cache hit probability . Many caching policies have been proposed and implemented to improve the hit probability. In this work, we propose a new method to compute an upper bound on hit probability for all non-anticipative caching policies and for policies that have no knowledge of future requests. Our key insight is to order the objects according to the ratio of their Hazard Rate (HR) function values to their sizes, and place in the cache the objects with the largest ratios till the cache capacity is exhausted. When object request processes are conditionally independent, we prove that this cache allocation based on the HR-to-size ratio rule guarantees the maximum achievable expected number of object hits across all non-anticipative caching policies. Further, the HR ordering rule serves as an upper bound on cache hit probability when object request processes follow either independent delayed renewal process or a Markov modulated Poisson process. We also derive closed form expressions for the upper bound under some specific object request arrival processes. We provide simulation results to validate its correctness and to compare it to the state-of-the-art upper bounds, such as produced by Bélády’s algorithm. We find it to be tighter than state-of-the-art upper bounds for some specific object request arrival processes such as independent renewal, Markov modulated, and shot noise processes.

Ruin probabilities in a Markovian shot-noise environment

AbstractWe consider a risk model with a counting process whose intensity is a Markovian shot-noise process, to resolve one of the disadvantages of the Cramér–Lundberg model, namely the constant intensity of the Poisson process. Due to this structure, we can apply the theory of piecewise deterministic Markov processes on a multivariate process containing the intensity and the reserve process, which allows us to identify a family of martingales. Eventually, we use change of measure techniques to derive an upper bound for the ruin probability in this model. Exploiting a recurrent structure of the shot-noise process, even the asymptotic behaviour of the ruin probability can be determined.

A question of scale: modeling biomass, gain and mortality distributions of a tropical forest

Abstract. Describing the heterogeneous structure of forests is often challenging. One possibility is to analyze forest biomass in different plots and to derive plot-based frequency distributions. However, these frequency distributions depend on the plot size and thus are scale dependent. This study provides insights about transferring them between scales. Understanding the effects of scale on distributions of biomass is particularly important for comparing information from different sources such as inventories, remote sensing and modeling, all of which can operate at different spatial resolutions. Reliable methods to compare results of vegetation models at a grid scale with field data collected at smaller scales are still missing. The scaling of biomass and variables, which determine the forest biomass, was investigated for a tropical forest in Panama. Based on field inventory data from Barro Colorado Island, spanning 50 ha over 30 years, the distributions of aboveground biomass, biomass gain and mortality were derived at different spatial resolutions, ranging from 10 to 100 m. Methods for fitting parametric distribution functions were compared. Further, it was tested under which assumptions about the distributions a simple stochastic simulation forest model could best reproduce observed biomass distributions at all scales. Also, an analytical forest model for calculating biomass distributions at equilibrium and assuming mortality as a white shot noise process was tested. Scaling exponents of about −0.47 were found for the standard deviations of the biomass and gain distributions, while mortality showed a different scaling relationship with an exponent of −0.3. Lognormal and gamma distribution functions fitted with the moment matching estimation method allowed for consistent parameter transfers between scales. Both forest models (stochastic simulation and analytical solution) were able to reproduce observed biomass distributions across scales, when combined with the derived scaling relationships. The study demonstrates a way of how to approach the scaling problem in model–data comparisons by providing a transfer relationship. Further research is needed for a better understanding of the mechanisms that shape the frequency distributions at the different scales.

Modelling the intensity distribution of underwater optical wave propagation in the presence of turbulence and air bubbles

Objectives:The objective of this paper is to derive a model for the irradiance distribution of underwater optical waves propagation in the combined presence of turbulence and air bubbles. The proposed distribution is based on modelling the medium index of refraction as a shot noise process and using the Rytov solution to the stochastic wave equation. Since the oceanic turbulence spectrum is different than the atmospheric spectrum, recent studies show that the well-known probability density functions poorly describe optical wave propagation underwater in the combined presence of air bubbles and turbulence, and that a mixture of distributions is generally needed. This conclusion, however, does not follow from theoretical investigations. Moreover, the physical aspects of the phenomena, such as the bubble size distribution, are not included in these models, and as such may limit their applicability. The approach introduced in this paper allows for including the environment physical properties directly in to the medium index of refraction. Method:In this paper, intensity distribution model is derived based on Rytov theory of propagation through weak turbulence. The physical processes in effect are directly included by assuming a shot noise index of refraction model. Results:Using the beam propagation method combined with the random phase screen method, the derived distribution is validated by simulation over a large range of turbulence levels and shows excellent agreement with the simulation data. In particular, the qualitative aspect of two peak distribution is modelled by the proposed distribution.

Probabilistic Description of the State of Charge of Batteries Used for Primary Frequency Regulation

Battery participation in the service of power system frequency regulation is universally recognized as a viable means for counteracting the dramatic impact of the increasing utilization of renewable energy sources. One of the most complex aspects, in both the planning and operation stage, is the adequate characterization of the dynamic variation of the state of charge of the battery in view of lifetime preservation as well as the adequate participation in the regulation task. Since the power system frequency, which is the input of the battery regulation service, is inherently of a stochastic nature, it is easy to argue that the most proper methodology for addressing this complex issue is that of the theory of stochastic processes. In the first part of the paper, a preliminary characterization of the power system frequency is presented by showing that with an optimal degree of approximation it can be regarded as an Ornstein–Uhlenbeck process. Some considerations for guaranteeing desirable performances of the control strategy are performed by assuming that the battery-regulating power depending on the frequency can be described by means of a Wiener process. In the second part of the paper, more realistically, the regulating power due to power system changes is described as an Ornstein–Uhlenbeck or an exponential shot noise process driven by a homogeneous Poisson process depending on the frequency response features requested of the battery. Because of that, the battery state of charge is modeled as the output of a dynamic filter having this exponential shot noise process as input and its characterization constitutes the central role for the correct characterization of the battery life. Numerical simulations are carried out for demonstrating the goodness and the applicability of the proposed probabilistic approach.

Matrix calculations for moments of Markov processes

AbstractMatryoshka dolls, the traditional Russian nesting figurines, are known worldwide for each doll’s encapsulation of a sequence of smaller dolls. In this paper, we exploit the structure of a new sequence of nested matrices we callmatryoshkan matricesin order to compute the moments of the one-dimensional polynomial processes, a large class of Markov processes. We characterize the salient properties of matryoshkan matrices that allow us to compute these moments in closed form at a specific time without computing the entire path of the process. This simplifies the computation of the polynomial process moments significantly. Through our method, we derive explicit expressions for both transient and steady-state moments of this class of Markov processes. We demonstrate the applicability of this method through explicit examples such as shot noise processes, growth–collapse processes, ephemerally self-exciting processes, and affine stochastic differential equations from the finance literature. We also show that we can derive explicit expressions for the self-exciting Hawkes process, for which finding closed-form moment expressions has been an open problem since their introduction in 1971. In general, our techniques can be used for any Markov process for which the infinitesimal generator of an arbitrary polynomial is itself a polynomial of equal or lower order.

Variability Signatures of a Burst Process in Flaring Gamma-Ray Blazars

Blazars exhibit stochastic flux variability across the electromagnetic spectrum, often exhibiting heavy-tailed flux distributions, commonly modeled as lognormal. However, Tavecchio et al. and Adams et al. found that the high-energy gamma-ray flux distributions of several of the brightest flaring Fermi-LAT flat-spectrum radio quasars (FSRQs) are well modeled by an even heavier-tailed distribution, which we show is the inverse gamma distribution. We propose an autoregressive inverse gamma variability model in which an inverse gamma flux distribution arises as a consequence of a shot-noise process. In this model, discrete bursts are individually unresolved and averaged over within time bins, as in the analysis of Fermi-LAT data. Stochastic variability on timescales longer than the time-bin duration is modeled using first-order autoregressive structure. The flux distribution becomes approximately lognormal in the limiting case of many weak bursts. The fractional variability is predicted to decrease as the time-bin duration increases. Using simulated light curves, we show that the proposed model is consistent with the typical gamma-ray variability properties of FSRQs and BL Lac objects. The model parameters can be physically interpreted as the average burst rate, the burst fluence, and the timescale of long-term stochastic fluctuations.

Incremental nonlinear stability analysis of stochastic systems perturbed by Lévy noise

AbstractWe present a theoretical framework for characterizing incremental stability of nonlinear stochastic systems perturbed by either compound Poisson shot noise or finite‐measure Lévy noise. For each noise type, we compare trajectories of the perturbed system with distinct noise sample paths against trajectories of the nominal, unperturbed system. We show that for a finite number of jumps arising from the noise process, the mean‐squared error between the trajectories exponentially converge toward a bounded error ball across a finite interval of time under practical boundedness assumptions. The convergence rate for shot noise systems is the same as the exponentially stable nominal system, but with a tradeoff between the parameters of the shot noise process and the size of the error ball. The convergence rate and the error ball for the Lévy noise system are shown to be nearly direct sums of the respective quantities for the shot and white noise systems separately, a result which is analogous to the Lévy–Khintchine theorem. We demonstrate both empirical and analytical computation of the error ball using several numerical examples, and illustrate how varying the parameters of the system affect the tightness of the bound.

Shot-noise cojumps: Exact simulation and option pricing

We consider a parsimonious framework of jump-diffusion models for price dynamics with stochastic price volatilities and stochastic jump intensities in continuous time. They account for conditional heteroscedasticity and also incorporate key features appearing in financial time series of price volatilities and jump intensities, such as persistence of contemporaneous jumps (cojumps), mean reversion and feedback effects. More precisely, the stochastic variance and stochastic intensity are jointly modelled by a generalised bivariate shot-noise process sharing common jump arrivals with any non-negative jump-size distributions. This framework covers many classical and important models in the literature. The main contribution of this paper is that, we develop a very efficient scheme for its exact simulation based on perfect decomposition where neither numerical inversion nor acceptance/rejection scheme is required, which means that it is not only accurate but also the efficiency would not be sensitive to the parameter choice. Extensive numerical implementations and tests are reported to demonstrate the accuracy and effectiveness of this scheme. Our algorithm substantially outperforms the classical discretisation scheme. Moreover, we unbiasedly estimate the prices of discrete-barrier European options to show the applicability and flexibility of our algorithms.

Asymptotic analysis of Poisson shot noise processes, and applications

Poisson shot noise processes are natural generalizations of compound Poisson processes that have been widely applied in insurance, neuroscience, seismology, computer science and epidemiology. In this paper we study sharp deviations, fluctuations and the stable probability approximation of Poisson shot noise processes. Our achievements extend, improve and complement existing results in the literature. We apply the theoretical results to Poisson cluster point processes, including generalized linear Hawkes processes, and risk processes with delayed claims. Many examples are discussed in detail.

Shot noise processes with randomly delayed cluster arrivals and dependent noises in the large-intensity regime

AbstractWe study shot noise processes with cluster arrivals, in which entities in each cluster may experience random delays (possibly correlated), and noises within each cluster may be correlated. We prove functional limit theorems for the process in the large-intensity asymptotic regime, where the arrival rate gets large while the shot shape function, cluster sizes, delays, and noises are unscaled. In the functional central limit theorem, the limit process is a continuous Gaussian process (assuming the arrival process satisfies a functional central limit theorem with a Brownian motion limit). We discuss the impact of the dependence among the random delays and among the noises within each cluster using several examples of dependent structures. We also study infinite-server queues with cluster/batch arrivals where customers in each batch may experience random delays before receiving service, with similar dependence structures.

Ubiquitous Power Law Scaling in Nonlinear Self-Excited Hawkes Processes.

The origin(s) of the ubiquity of probability distribution functions with power law tails is still a matter of fascination and investigation in many scientific fields from linguistic, social, economic, computer sciences to essentially all natural sciences. In parallel, self-excited dynamics is a prevalent characteristic of many systems, from the physics of shot noise and intermittent processes, to seismicity, financial and social systems. Motivated by activation processes of the Arrhenius form, we bring the two threads together by introducing a general class of nonlinear self-excited point processes with fast-accelerating intensities as a function of "tension." Solving the corresponding master equations, we find that a wide class of such nonlinear Hawkes processes have the probability distribution functions of their intensities described by a power law on the condition that (i)the intensity is a fast-accelerating function of tension, (ii)the distribution of marks is two sided with nonpositive mean, and (iii)it has fast-decaying tails. In particular, Zipf's scaling is obtained in the limit where the average mark is vanishing. This unearths a novel mechanism for power laws including Zipf's law, providing a new understanding of their ubiquity.

Failure rate-based models for systems subject to random shocks

Shock process that can increase the failure rate of the systems working under a stochastic environment is known as the shot-noise process. In the existing literature focusing on the shot-noise process, it is usually assumed that the failure rate increments caused by shocks are the same constants or identically distributed variables. However, such an assumption has limitations if the external environment is constantly changing or the shocks can weaken the system gradually. This article proposes an extended shock model in which the failure rate increments of systems caused by shocks need not be identical or identically distributed. The reliability functions of the systems as well as some reliability indices are derived explicitly. Some special cases are discussed. Besides, a preventive maintenance model is also investigated in which the system is replaced when the number of shocks reaches a fixed threshold. Finally, an example on a refrigeration system is provided to illustrate the proposed model.

Shot-noise queueing models

We provide a survey of so-called shot-noise queues: queueing models with the special feature that the server speed is proportional to the amount of work it faces. Several results are derived for the workload in an M/G/1 shot-noise queue and some of its variants. Furthermore, we give some attention to queues with general workload-dependent service speed. We also discuss linear stochastic fluid networks, and queues in which the input process is a shot-noise process.

Functional Limit Theorem Without Centering for General Shot-Noise Processes

Functional Limit Theorem Without Centering for General Shot-Noise Processes