On the mode-change problem for random measures (original) (raw)

Optimal detection of a change-set in a spatial Poisson process

The Annals of Applied Probability, 2010

We generalize the classic change-point problem to a "changeset" framework: a spatial Poisson process changes its intensity on an unobservable random set. Optimal detection of the set is defined by maximizing the expected value of a gain function. In the case that the unknown change-set is defined by a locally finite set of incomparable points, we present a sufficient condition for optimal detection of the set using multiparameter martingale techniques. Two examples are discussed.

Variational Analysis of Poisson Processes

Bocconi & Springer Series, 2016

The expected value of a functional F(η) of a Poisson process η can be considered a function of its intensity measure µ. The paper surveys several results concerning differentiability properties of this functional on the space of signed measures with finite total variation. Then necessary conditions for µ being a local minima of the considered functional are elaborated taking into account possible constraints on µ, most importantly the case of µ with given total mass a. These necessary conditions can be phrased by requiring that the gradient of the functional (being the expected first difference F(η + δ x) − F(η)) is constant on the support of µ. In many important cases, the gradient depends only on the local structure of µ in a neighbourhood of x and so it is possible to work out the asymptotics of the minimising measure with the total mass a growing to infinity. Examples include the optimal approximation of convex functions, clustering problem, optimal search. In non-asymptotic cases, generally it is possible to find the optimal measure using steepest descent algorithms which are based on the obtained explicit form of the gradient.

The use of cumulative sums for detection of changepoints in the rate parameter of a Poisson Process

Computational Statistics & Data Analysis, 2007

This paper studies the problem of multiple changepoints in rate parameter of a Poisson process. We propose a binary segmentation algorithm in conjunction with a cumulative sums statistic for detection of changepoints such that in each step we need only to test the presence of a simple changepoint. We derive the asymptotic distribution of the proposed statistic, prove its consistency and obtain the limiting distribution of the estimate of the changepoint. A Monte Carlo analysis shows the good performance of the proposed procedure, which is illustrated with a real data example.

The Multiple Change-Points Problem for the Spectral Distribution

Bernoulli, 2000

We consider the problem of detecting an unknown number of change-points in the spectrum of a second-order stationary random process. To reach this goal, some maximal inequalities for quadratic forms are ®rst given under very weak assumptions. In a parametric framework, and when the number of changes is known, the change-point instants and the parameter vector are estimated using the Whittle pseudo-likelihood of the observations. A penalized minimum contrast estimate is proposed when the number of changes is unknown. The statistical properties of these estimates hold for strongly mixing and also long-range dependent processes. Estimation in a nonparametric framework is also considered, by using the spectral measure function. We conclude with an application to electroencephalogram analysis.

On Compound Poisson Type Limiting Likelihood Ratio Process Arising in some Change-Point Models

Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. In a previous paper of one of the authors it was established that one of these likelihood ratios, which is an exponential functional of a two-sided Poisson process driven by some parameter, can be approximated (for sufficiently small values of the parameter) by another one, which is an exponential functional of a two-sided Brownian motion. In this paper we consider yet another likelihood ratio, which is the exponent of a two-sided compound Poisson process driven by some parameter. We establish, that similarly to the Poisson type one, the compound Poisson type likelihood ratio can be approximated by the Brownian type one for sufficiently small values of the parameter. We equally discuss the asymptotics for large values of the parameter.

On the classification problem for Poisson point processes

Journal of Multivariate Analysis, 2016

For Poisson processes taking values in any general metric space, we tackle the problem of supervised classification in two different ways: via the classical knearest neighbor rule, by introducing suitable distances between patterns of points and via the Bayes rule, by estimating nonparametricaly the intensity function of the process. In the first approach we prove that, under the separability of the space the rule turns out to be consistent. In the former, we prove the consistency of rule by proving the consistency of the estimated intensities. Both classifiers have shown to have a good behaviour under departures from the Poisson distribution.

Estimating the Change Point of Correlated Poisson Count Processes

Quality Engineering, 2014

Knowing the time of change would narrow the search to find and identify the variables disturbing a process. The knowledge of the change point can greatly aid practitioners in detecting and removing the special cause(s). Count processes with autocorrelation structure are commonly observed in real-world applications and can often be modeled by the firstorder integer-valued autoregressive (INAR) model. The most widely used marginal distribution for count processes is Poisson. In this study, change-point estimators are proposed for the parameters of correlated Poisson count processes. To do this, Newton's method is first used to approximate the parameters of the process. Then, maximum likelihood estimators of the process change point are developed. The performances of these estimators are next evaluated when they are employed in a combined EWMA and c scheme. Finally, for the rate parameter, the proposed estimator is compared with the estimator developed for independent observations.

Bayesian Identification of Multiple Change Points in Poisson Data

Advances in Complex Systems, 2005

The identification of multiple change points is a problem shared by many subject areas, including disease and criminality mapping, medical diagnosis, industrial control, and finance. An algorithm based on the Product Partition Model (PPM) is developed to solve the multiple change point identification problem in Poisson data sequences. In order to address the PPM, a simple and easy way to implement Gibbs sampling scheme is derived. A sensitivity analysis is performed, for different prior specifications. The algorithm is then applied to the analysis of a real data sequence. The results show that the method is quite effective and provides useful inferences.

A maximum likelihood approach to estimate the change point of multistage Poisson count processes

The International Journal of Advanced Manufacturing Technology, 2014

The difference between the signaling time and the real change-point of a process is an important monitoring issue. If the exact time at which the change manifests itself into the process is known, then process engineers can identify and eliminate the root causes of process disturbance efficiently and quickly, resulting in considerable amount of time and cost savings. Multistage count processes that are often observed in production environments must be monitored to assure quality products. In this study, multistage Poisson count processes are first introduced. Then, the process is modeled using a first-order integer-valued autoregressive time-series (INAR(1)). For out-of-control signals obtained by a combined EWMA and c control chart, the Newton's method is next used to approximate the rate and the dependence parameters. Finally, the maximum likelihood method is employed to estimate the out-of-control sample along with the out-of-control stage. Besides, the accuracy and the precision of the proposed estimators are examined through some Monte Carlo simulation experiments. The results show that the estimators are accurate and promising. According to Al-Osh and Alzaid [16] and Weiß [19], the INAR(1) process   , s X s Z  with the range 0 N is defined by means of the difference equation