Non-homogeneous Poisson Research Articles

A mixture spatial point pattern model for the quasar luminosity function

Consider the problem of estimating the Quasar Luminosity Function (QLF). In a 2007 Ph.D. dissertation, Hugeback considers the QLF as a nonhomogeneous poisson process and estimates the intensity function under SDSS DR3 data (The University of Chicago, AAT 3273021). The present study follows Hugeback’s approach but introduces a mixture component which improves Hugeback’s model in several respects. Namely, the database is partitioned into two groups according to redshift: z < 2.75 and z ⩾ 2.75. Next, a mixture model for the QLF was derived using the concept of pseudolikelihood, the addition of a K function to allow for inter-point interaction, and evaluation of residuals diagnostic plots. This mixture model (i) improves the deviance of Hugeback’s model, and (ii) satisfies residual assumptions that are violated under Hugeback’s model. Moreover, this study confirms Hugeback’s finding of inhomogeneity in the QLF, and provides stronger evidence for the existence of an interaction between redshift and absolute magnitude.

Reliability prediction using support vector regression

Reliability prediction of machinery is crucial to schedule overhauls, spare parts replacement and maintenance interventions. Many AI tools have been used in order to provide these predictions for the industry. Support vector regression (SVR) is a nonlinear regression technique extended from support vector machine. SVR can fit data flexibly and it has a wide variety of applications. This paper utilizes SVR combining time series to predict the next failure time based on historical failure data. To solve the parameter selection problem a method has been proposed. This method approximates the widely used leave-one-out method. To bound the prediction error, a confidence interval is proposed based on the non-homogeneous poisson process. A numerical case from the mining industry is presented to demonstrate the developed approach.

A point process model for the parabigeminal nucleus as a recursive estimator

Event Abstract Back to Event A point process model for the parabigeminal nucleus as a recursive estimator The parabigemial nucleus (PBN) is a satellite of the superior colliculus (SC), the midbrain saccadic oculomotor control center. PBN neurons fire action potentials at a rate proportional to the retinal position error (RPE) of attended visual targets, whether stationary or moving. The tuning functions of PBN neurons are usually not Gaussian, but are better described by an exponentially decaying sigmoid function. Recently, using periodically blinking moving targets, it has been discovered that PBN activity continues to predict the unseen (virtual) target?s trajectory, as inferred from kinetic information before the target disappears. The sigmoidal tuning function is maintained during target-off phases, except that its dynamic range shrinks somewhat. Such predictive neural events are critical for behavior, but are not well understood, either from the physiological or computational standpoint. This is study is focused on the encoding problem, attempting to understand the computational mechanism for the predictive neuronal activity in the PBN by building statistical models that can be tested against real spike train data. The point process theoretical framework that we used treats the spike train as a non-homogeneous poisson process, enabling us to relate various possible factors, such as auto-regressive self-firing history and predicted virtual target positions, to the logarithm of conditional intensity of spiking probability at each discrete time bin via a linear model. We assume that the PBN has full knowledge of the dynamics of the visual target and the eye movements made by the animal. As a consequence, when the target is momentarily turned off, the animal will assume its continual existence and predict the virtual target?s position using a Kalman filter. The mean of the virtual target positions predicted by the Kalman filter lies very close to the ?true? target position, whether visible or not. The variance of the prediction is near zero when the target is on, grows linearly with time after the target is turned off, and suddenly drops back to near zero when the target is turned back on. We have also incorporated the temporal evolution of the sigmoid tuning function dynamic range during a complete duty cycle by modulating its parameters as functions of the variance of the Kalman filter prediction. The results of this modeling were evaluated with a K-S plot constructed according to the time rescaling theorem, as well as an auto-correlation scatter plot of the consecutive rescaled inter-spike intervals. The curve falls mainly within the 95% confidence intervals, and the auto-correlation scatter plot is spread basically evenly on the unit square area. These observations suggest the model provides a satisfactory fit for the data presented. Our model has successfully captured the characteristics of PBN spike trains related to the encoding of extrapolated positions of virtual targets, and thus provided a plausible explanation for the origin of predictive activity, and how this internal representation of the external world is reflected in neural spike trains. Conference: Computational and systems neuroscience 2009, Salt Lake City, UT, United States, 26 Feb - 3 Mar, 2009. Presentation Type: Poster Presentation Topic: Poster Presentations Citation: (2009). A point process model for the parabigeminal nucleus as a recursive estimator. Front. Syst. Neurosci. Conference Abstract: Computational and systems neuroscience 2009. doi: 10.3389/conf.neuro.06.2009.03.014 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 29 Jan 2009; Published Online: 29 Jan 2009. Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Google Google Scholar PubMed Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

A software reliability growth model addressing learning

Goel proposed generalization of the Goel–Okumoto (G–O) software reliability growth model (SRGM), in order to model the failure intensity function, i.e. the rate of occurrence of failures (ROCOF) that initially increases and then decreases (I/D), which occurs in many projects due to the learning phenomenon of the testing team and a few other causes. The ROCOF of the generalized non-homogenous poisson process (NHPP) model can be expressed in the same mathematical form as that of a two-parameter Weibull function. However, this SRGM is susceptible to wide fluctuations in time between failures and sometimes it seems unable to recognize the I/D pattern of ROCOF present in the datasets and hence does not adequately describe such data. The authors therefore propose a shifted Weibull function ROCOF instead for the generalized NHPP model. This modification to the Goel-generalized NHPP model results in an SRGM that seems to perform better consistently, as confirmed by the goodness of fit statistic and predictive validity metrics, when applied to failure datasets of 11 software projects with widely varying characteristics. A case study on software release time determination using the proposed SRGM is also given.

Model for the Distribution of Aftershock Interoccurrence Times

In this work the distribution of interoccurrence times between earthquakes in aftershock sequences is analyzed and a model based on a nonhomogeneous Poisson (NHP) process is proposed to quantify the observed scaling. In this model the generalized Omori's law for the decay of aftershocks is used as a time-dependent rate in the NHP process. The analytically derived distribution of interoccurrence times is applied to several major aftershock sequences in California to confirm the validity of the proposed hypothesis.

A unified scheme of some nonhomogenous poisson process models for software reliability estimation

In this paper, we describe how several existing software reliability growth models based on Nonhomogeneous Poisson processes (NHPPs) can be comprehensively derived by applying the concept of weighted arithmetic, weighted geometric, or weighted harmonic mean. Furthermore, based on these three weighted means, we thus propose a more general NHPP model from the quasi arithmetic viewpoint. In addition to the above three means, we formulate a more general transformation that includes a parametric family of power transformations. Under this general framework, we verify the existing NHPP models and derive several new NHPP models. We show that these approaches cover a number of well-known models under different conditions.

NHPP Models for Reliability of Software with Imperfect Debugging and Testing Effort

Two Non-homogeneous poisson process (NHPP) models are introduced which incorporate the impact of test effort and imperfection in debugging. In model I it is assumed that every time an error is rectified there is a probability p that the rectification is imperfect and the error may precipitate at a later stage. This p is assumed to be same for all the errors. In model II we classify errors into two types. For type I error, which is suspected to have imperfect rectification, this probability is p*. For Type II error we are sure to have perfect rectification. Also, we present the problem of estimation of parameters and method of finding optimal stopping time that minimises the total cost subject to the specified reliability.

Bayesian computation for the superposition of nonhomogeneous poisson processes

AbstractBayesian inference for the superposition of nonhomogeneous Poisson processes is studied. A Markov‐chain Monte Carlo method with data augmentation is developed to compute the features of the posterior distribution. For each observed failure epoch, a latent variable is introduced that indicates which component of the superposition model gives rise to the failure. This data‐augmentation approach facilitates specification of the transitional kernel in the Markov chain. Moreover, new Bayesian tests are developed for the full superposition model against simpler submodels. Model determination by a predictive likelihood approach is studied. A numerical example based on a real data set is given.

The infinite server queue with arrivals generated by a non-homogeneous compound poisson process and heterogeneous customers

We consider the infinite server queue where customers arrive in groups of various sizes according to a non-homogeneous Poisson process. The customers in each group may be of different types and also may have dependent service times. Assuming that the system is initially empty, we obtain the probability generating function of the transient distribution of the numbers of customers of various types present in the system, thus extending recent relevant results. At the same time the probability generating function of the corresponding departure counting process is derived as well.

A natural conjugate prior for the nonhomogeneous poisson process with an exponential intensity function

This article presents a natural conjugate prior for the nonhomogeneous Poisson process (NHPP) with an exponential intensity function, for modeling the failure rate of repairable systems. The behavior of the conjugate prior distribution with respect to its parameters is studied, and the use of this prior in Bayesian estimation is compared to two other estimation approaches (the use of independent prior distributions, and the bivariate normal distribution). The use of the conjugate prior proposed here facilitates Bayesian statistical analysis of aging. In particular, the proposed prior allows us to explicitly account for dependence between the initial failure rate and the aging rate. This is a significant improvement over the assumptions made in most prior work (either the assumption that the aging rate is known, or the assumption that the initial failure rate and the aging rate are independent). Monte Carlo simulation shows that Bayesian estimation using the proposed prior generally performs at least as well as Bayesian estimation using independent priors for the initial failure rate and the aging rate,except in the case where the prior distribution underestimates both the initial failure rate and the aging rate.

Higher-order approximation techniques for estimating stochastic parameter of a sediment transport model

Higher-order approximation techniques for estimating stochastic parameter of the non-homogeneous Poisson (NHP) model are presented. The NHP model is characterized by a two-parameter cumulative probability distribution function (CDF) of sediment displacement. Those two parameters are the temporal and spatial intensity functions, physically representing the inverse of the average rest period and step length of sediment particles, respectively. Difficulty of estimating the parameters has, however, restricted the applications of the NHP model. The approximation techniques are proposed to address such problem. The basic idea of the method is to approximate a model involving stochastic parameters by Taylor series expansion. The expansion preserves certain higher-order terms of interest. Using the experimental (laboratory or field) data, one can determine the model parameters through a system of equations that are simplified by the approximation technique. The parameters so determined are used to predict the cumulative distribution of sediment displacement. The second-order approximation leads to a significant reduction of the CDF error (of the order of 47%) compared to the first-order approximation. Error analysis is performed to evaluate the accuracy of the first- and second-order approximations with respect to the experimental data. The higher-order approximations provide better estimations of the sediment transport and deposition that are critical factors for such environment as spawning gravel-bed.

A natural conjugate prior for the non-homogeneous poisson process with a power law intensity function

This paper develops a natural conjugate prior for the non-homogeneous Poisson process (NHPP) with a power law intensity function. This prior allows for dependence between the scale factor and the aging rate of the NHPP. The proposed prior has relatively simple closed-form expressions for its moments, facilitating the assessment of prior parameters. The use of this prior in Bayesian estimation is compared to other estimation approaches using Monte Carlo simulation. The results show that Bayesian estimation using the proposed prior generally performs at least as well as either maximum likelihood estimation or Bayesian estimation using independent prior

Preservation of some partial orderings under non-homogeneous poisson shock model and laplace transform

In this paper, we study the preservation of some partial orderings under non-homogeneous Poisson shock model and Laplace transform. These partial orderings are: likelihood ratio ordering (LR); failure rate ordering (FR); stochastic ordering (ST); variable ordering (V); mean residual life ordering (MR); variance residual life ordering (VR); concave ordering (CV); harmonic average mean residual life ordering (HAMR).

Nonhomogeneous poisson processes as overhaul models

AbstractHollander and Proschan (1974) studied nonhomogeneous Poisson processes as models for systems subject to overhauls. They did not postulate a functional form for the intensity, but showed that certain basic assumptions about the deterioration of the system implied that the mean function is superadditive. They studied tests of the null hypothesis that the intensity is constant with the alternative restricted to superadditive mean functions. For estimation purposes, the class of superadditive mean functions is too broad. We assume that the intensity is nondecreasing between overhauls and that at an overhaul it does not fall below its average prior to the overhaul. These two assumptions imply that the mean function is star‐shaped. We obtain the restricted maximum‐likelihood estimates under these two assumptions and under the star‐shaped restriction. The two estimates are compared on a data set.

Simulating nonhomogeneous poisson processes with proportional intensities

Simulating nonhomogeneous poisson processes with proportional intensities

Multivariate survival distributions of age and residual lifetime processes in nonhomogenous poisson process

Let {N(t), t≥0} be the nonhomogenous Poisson process with cumulative intensity parameter Ν (t), {δt,t ⩾ 0} the age process, and {γt,t ⩾ 0} the residual lifetime process. In the present paper the expression of n-dimensional survival distribution functions of the processes {δt} and {γt}, and their Lebesgue decompositions are derived.

Variate generation for a nonhomogeneous poisson process with time dependent covariates

Algorithms are developed for generating a sequence of event times from a nonhomogeneous Poisson process that is influenced by the values of covariates that vary with time. Closed form expressions for random variate generation are shown for several baseline intensity and link functions. Two specific models linking the baseline process to the general model are considered: the accelerated time model and the proportional intensity model. In the accelerated time model, the cumulative intensity function of a nonhomogeneous Poisson process under covariate effects is , where z is a covariate vector, ⋀0(t) is the baseline cumulative intensity function and ψ(z) is the link function. In the proportional intensity model, the cumulative intensity function of a nonhomogeneous Poisson process under covariate effects is , where λ0(t) is the baseline intensity function.

Bayesian inference for poisson process models with censored data.

Bayesian statistical inference for a nonhomogeneous Poisson point process model with censored data is considered. Weighted gamma process priors are founded to be the natural conjugate priors for the cumulative intensity of this model. Upon deflating the prior, the posterior mean of the cumulative intensity becomes a relative of the well-known Nelson-Aalen estimator. Small- and large-sample approximations to the posterior distribution using the Bayesian bootstrap methods are discussed.

An analysis of the trend in ground-level ozone using non-homogeneous poisson processes

This paper provides a method for measuring the long-term trend in the frequency with which ground-level ozone present in the ambient air exceeds the U.S. Environmental Protection Agency's National Ambient Air Quality Standard (NAAQS) for ozone. A major weakness of previous studies that estimate the long-term trend in the very high values of ozone, and therefore the long-term trend in the probability of satisfying the NAAQS for ozone, is their failure to account for the confounding effects of meterological conditions on ozone levels. Meteorological variables such as temperature, wind speed, and frontal passage play an important role in the formation of ground-level ozone. A non-homogenous Poisson process is used to account for the relationship between very high values of ozone and meteorological conditions. This model provides an estimate of the trend in the ozone values after allowing for the effects of meteorological conditions. Therefore, this model provides a means to measure the effectiveness of pollution control programs after accounting for the effects of changing weather conditions. When our approach is applied to data collected at two sites in Houston, TX, we find evidence of a gradual long-term downward trend in the frequency of high values of ozone. The empirical results indicate how possibly misleading results can be obtained if the analysis does not account for changing weather conditions.

On reliability calculations when stresses are generated by a non-homogeneous poisson process

Abstract Reliability of a structure with random strength is investigated assuming that the occurrences of random stresses are governed by a non-homogeneous Poisson process. Here calculations are carried out based on the observation that the failure corresponds to the occurence of the first record stress (new high in the sequence of stresses) relative to the random strength of the component. Explicit formulae for two practically important cases are derived and two examples using real data for wind and earthquake are included.