Generalized Poisson Process Research Articles

Distinguishing between stochastic models of heterogeneity and contagion

Historically, the Poisson process has been the “benchmark” model for many stochastic processes. When event counts from a particular process fail to be described adequately by a Poisson distribution, a researcher may turn to generalized Poisson processes to model the empirical data more accurately. Two common generalizations are the (1) heterogeneous Poisson process, in which the process rate is a Gamma random variable, and (2) contagious Poisson process, in which the process rate depends linearly on the current state of the process. Paradoxically, both the heterogeneous and contagious processes yield the same theoretical distribution for the number of events that occur in a single interval of time. Consequently, distinguishing between these two using event counts can be difficult. This situation is discussed, first reviewing the models and distribution, then giving strategies for choosing between them using event count data, and demonstrating these techniques on episodes of hospitalization for schizophrenia.

Some statistical properties of a sequence of historical Calabro‐Peloritan Earthquakes

A statistical analysis is applied to the sequence of earthquakes, with epicentral intensity I0 ≥ VIII MSK, which occurred in the Calabro‐Peloritan Arc region (southern Italy) during the period 1600–1974. The study reveals, by the application of the Poisson test, an evident level of interdependence of the shocks which appear grouped in temporal clusters satisfying the simple Poisson process. In fact, the equations which characterize the generalized Poisson process give a good fit of the observed data, the distribution of the events in the clusters being of the Pareto type (discrete). The values 0.9 and 2.8 have been found for k′ and E parameters, which are the average number of clusters occurring in unit time and the parameter of the Pareto distribution, respectively. A subsequent elaboration of these results makes the calculation of some quantities, like the clustering length, possible. Finally, a statistical interpretation of the sequence, with the characterization of the clusters, is proposed.

Nonlinear Estimation of Generalized Vector Shot Processes

A class of nonlinear filters for dynamical systems driven by generalized Poisson processes is developed. One of the filters, the maximum a posteriori (MAP) filter, is shown by a numerical example to be superior to other known predictors in getting the highest target hit probabilities, and it is relatively simple to implement. This filter has applications in both fire control and air traffic control of maneuvering piloted vehicles.

On the Size of the First Jump for a Process with Independent Increments

Previous article Next article On the Size of the First Jump for a Process with Independent IncrementsA. A. Mogul’skiiA. A. Mogul’skiihttps://doi.org/10.1137/1121060PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] B. A. Rogozin, On the distribution of functionals connected with boundary problems for processes with independent increments, Theory Prob. Applications, 11 (1966), 580–591 10.1137/1111062 LinkGoogle Scholar[2] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, New York, 1968 0598.60002 Google Scholar[3] K. Yushida, Functional Analysis, Mir, Moscow, 1967, (In Russian.) Google Scholar[4] A. A. Borovkov, Probabilistic methods in the theory of mass service (queueing “Nauka”), Moscow, 1972, (In Russian.) 0984.60050 Google Scholar[5] E. A. Pecherskii and , B. A. Rogozin, On joint distributions of random variables associated with fluctuations of a process with independent increments, Theory Prob. Applications, 14 (1969), 410–423 10.1137/1114054 LinkGoogle Scholar[6] Yu. V. Prokhorov, Convergence of random processes and limit theorems in probability theory, Theory Prob. Applications, 1 (1956), 158–214 10.1137/1101016 LinkGoogle Scholar[7] A. I. Fokht, On the distribution of the first jump for a generalized Poisson process with drift, Theory Prob. Applications, 19 (1974), 159–163 10.1137/1119014 0318.60062 LinkGoogle Scholar[8] A. A. Mogul'skii, Absolute estimates for moments of certain boundary functionals, Theory Prob. Applications, 18 (1973), 340–347 10.1137/1118035 0298.60031 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Inequalities in a Two-Sided Boundary Crossing Problem for Stochastic Processes27 May 2021 | Siberian Mathematical Journal, Vol. 62, No. 3 Cross Ref Generalized parking problems for levy processesSequential Analysis, Vol. 17, No. 2 Cross Ref Asymptotic Solution of the Kiefer–Weiss Problem for Processes with Independent IncrementsV. P. Dragalin and A. A. Novikov17 July 2006 | Theory of Probability & Its Applications, Vol. 32, No. 4AbstractPDF (865 KB)Asymptotic expansions for 2-SPRT19 September 2006 Cross Ref Volume 21, Issue 3| 1977Theory of Probability & Its Applications History Submitted:15 January 1976Published online:28 July 2006 InformationCopyright © 1977 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1121060Article page range:pp. 470-481ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Optimum Design for Infrequent Disturbances

Sporadic disturbances such as earthquakes and tornadoes are idealized as renewal processes. Uncertainties are classified according to their time correlation into disturbance, structure, and analyst random variables. Only the latter admit Bayesian updating. In this light a result due to Hasofer is revised. Structures are idealized as having a single degree-of-freedom and as having potential limit states in cascade, i.e., limit states can be entered only in a fixed order. Equivalent second-moment (beta method) criteria are developed. Treatment is then specialized to earthquake resistant design, in which disturbances are idealized as generalized Poisson processes. Explicit optimal-design formulas are given for this case. Appendices include novel second-moment approximate treatment of uncertainties not requiring calculation of derivatives. Those variables whose distributions are evidently close to Gaussian or lognormal are given a treatment that is exact for these types of distribution.

Estimation of linear functionals by maximum likelihood*

SUMMARYA classical problem of stereology is estimation of linear functionals of particle size distributions from intersections on planar probes. The maximum likelihood method of estimation is outlined and compared with classical linear methods and with numerical interpolation methods.A population of spherical particles is distributed in an infinite matrix with position defined by a Poisson field. Diameters of circular intersections on a planar probe constitute a generalized Poisson process. Maximum likelihood theory fits a parametric family of planar density functions to observed diameter frequency data. The integral of an appropriate kernel with respect to the fitted density function is the linear functional estimate.The approach can handle composite data involving observation at several magnifications. Estimation of complementary cumulative density of pores in uranium oxide fuel pellets illustrates the maximum likelihood method of data collected with two microscopes at four magnifications.

On the Distribution of the First Jump for a Generalized Poisson Process with Drift

On the Distribution of the First Jump for a Generalized Poisson Process with Drift

Reliability Optimization in Isostatic Structures

Reliability optimization is formulated here in a way that permits dealing explicitly with the benefit derived from the existence of the structure, its initial cost, and the expected losses due to failure. This formulation is adapted to the policies in which the structure is either not rebuilt or systematically rebuilt if it fails. Introduction of the time variable in one-component structures leads to expressions that can be put in the form of those for a single load application. Attention is given to disturbances which are generalized Poisson processes and to criteria for deciding on the advisability of providing defense plateaus in specific cases. The treatment is extended to statically determinate structures having uncorrelated component resistance functions or a certain type of correlation of these variables. An iteration procedure is developed for these cases.

On the convergence of the sums of multidimensional stochastic step processes of random number to generalized Poisson processes

The abstracts (in two languages) can be found in the pdf file of the article.
 Original author name(s) and title in Russian and Lithuanian:
 Р. Т. Банис. О сходимости сумм случайного числа многомерных ступенчатых случайных процессов к обобщенным пуассоновским
 R. Banys. Apie atsitiktinio skaičiaus daugiamačių laiptuotų atsitiktinių procesų sumų konvergenciją į apibendrintus Puasono procesus

On the distribution of the sequence of discards of a generalized Poisson process on a phase straight line

On the distribution of the sequence of discards of a generalized Poisson process on a phase straight line

A clustering model for earthquake occurrences

A statistical analysis of the time occurrence of the earthquakes listed in the USCGS catalog was performed. Except for the deep earthquakes, goodness of fit tests have rejected the simple Poisson process as a model of earthquake occurrences at a very low significance level. It was inferred that another parameter related to the Poisson index of dispersion is required to describe earthquake occurrences. Attempts to fit the distribution of the number of days with n earthquakes with standard two-parameter distributions like the lognormal and the γ distribution were not very successful. The data were tested with respect to the generalized Poisson process, a model in which there are finite probabilities for more than one event occurring at the same time instant. It was deduced that the distribution of the number of earthquakes generated in an aftershock sequence is reasonably approximated by an inverse power law distribution with exponent E being between 2.5 and 4.0. Good fits were obtained with respect to the generalized Poisson distribution, and regional variations in the parameter E could be detected. Estimates of the autocovariance function and power spectra failed to detect definite periodicities of earthquake occurrences in the range of 2 to 256 days. Positive correlations of earthquake activity were observed for time intervals as long as 10 days.

Extension of the collective risk theory

Abstract In its original form the collective risk theory is based upon the assumption that the r.v. Y(t), the total amount of claims up to the (operational) time t, is a generalized Poisson process and thus has a d.f. of the form a c.f. of the form where is the generalized c.f. of the claim distribution P(y).

Reduced density matrices of the anisotropic Heisenberg model

The reduced density matrices of the anisotropic Heisenberg model are studied by means of a functional integral representation based on a generalized Poisson process. Integral equations, which are analogous to the classical Kirkwood-Salzburg equations, are then used to prove the existence of the infinite volume limit of the reduced density matrices, analyticity properties with respect to the fugacity (or magnetic field) and the potentials, and a cluster property, in the low fugacity (high magnetic field) region.

Optimal Control of Systems with Random Gain of Plant †

ABSTRACT A theoretical study is made of optimizing an nth order system which has a random variation of gain of the plant of (a) Gaussian white noise, or (b) generalized Poisson process. The problem is formulated by dynamic programming.

Optimal Control of Systems With Generalized Poisson Inputs

This paper discusses the optimal control of systems having known dynamics with command and disturbance inputs which may be described as generalized Poisson processes. An engineering interpretation of the physical nature of Poisson processes is given, with brief comments on their use as mathematical models of practical situations. Representative examples of partial differential equations of optimal control are quoted. Explicit solutions of the control equations are given for the following systems with quadratic performance indexes: (a) The general linear system with linear Poisson inputs; (b) a linear regulator with a quadratically nonlinear Poisson disturbance; (c) a linear regulator with a random step-wave disturbance.