Non-homogeneous Poisson Models Research Articles

Non-homogeneous Poisson and linear regression models as approaches to study time series with change-points

In this study we consider some stochastic models and a Bayesian approach to analyze count time series data in the presence of one or more change-points. When the observed count data are all different of zero, it is possible to perform an analysis using linear regression models with normal distribution errors for the log-transformed data. When we have the presence of zero counts at different times, the statistical model based on the log-transformation is not suitable. Hence, in this case, it is possible to consider non-homogeneous Poisson processes (NHPP) models with a suitable rate function. In the present work we consider both models (linear regression and Poisson) to analyze three data sets. These sets are data recording the monthly visitors to New Zealand with the purpose of education, yearly tuberculosis incidence data from New York City, and monthly measles incidence data from Brazil. When the NHPP model is used a PLP (power law process) model is assumed for the intensity function. Additionally, in both models, the presence of change-points will be allowed.

Agricultural drought periods analysis by using nonhomogeneous poisson models and regionalization of appropriate model parameters

Precipitation has a dominant role in agriculture, and a regular rain pattern is usually vital to agriculture; excessive or inadequate rainfall can be harmful. In this paper, an agricultural drought index is utilized to study the agricultural drought periods and analyze them with their intensities at various locations. Some nonhomogeneous Poisson models are also used to calculate the probability of agricultural droughts (number of times occurred) in a time interval of interest. It is to be assumed that the number of agricultural drought event occurrences is a Nonhomogeneous Poisson Process (NHPP) has a rate function, which depends on some parameters that must be estimated. Two cases of these functions are considered: the Weibull and linear intensity function. The Bayesian approach with Gibbs sampling under the Markov Chain Monte Carlo (MCMC) algorithm is used to estimate the parameters of these functions. The most appropriate fitted model is selected by using Deviance Information Criteria (DIC) and use that appropriately fitted model to calculate the accumulated events of agricultural drought in a time interval of interest at each location. Ordinary Kriging (OK) is used to regionalize the parameters and present its spatial behavior. The results based on the DIC indicate that the Power Law Process (PLP) performs better than the linear intensity function, NHPP model. The interpolated parameter values for the appropriate model, their patterns, and fluctuations for the study area are efficiently presented using contour maps. It is a novel and straightforward approach to assess the selected model parameter values used to predict the accumulated drought events at un-sampled locations. The proposed framework might also help to analyze other spatial variables of interest and can be used for climate-change study, ecosystem modeling, etc. The findings can also help to make decisions for sustainable environmental management in Pakistan.

Analysis of agricultural and hydrological drought periods by using non-homogeneous Poisson models: Linear intensity function

The abnormal behavior of the precipitation in recent years has caused shrinkage of water supplies, floods and rising seas and rivers in all over the world. Pakistan is one of those countries which are highly effected from this abnormal behavior. In this paper, we utilized the concept of drought indices for the calculation and exploratory analysis of the agricultural and hydrological droughts with its intensities at multiple locations. The concept of point processes to analyze the drought periods at different locations was also investigated. The 3-months and 12-months Standardized Precipitation Index (SPI) were used to calculate the agricultural and hydrological drought events. The accumulated number of agricultural and hydrological drought events are calculated by considering the threshold of −1.0, and analyzed the drought periods by using Non-Homogeneous Poisson Process (NHPP). The parameters of the model were estimated by using Bayesian approach with Gibbs sampling under Markov Chain Monte Carlo (MCMC) algorithm. The results revealed that SPI provides the better and detailed structure of the agricultural and hydrological droughts with different intensity levels at multiple locations. The study showed that every region of Pakistan was significantly influenced by the agricultural and hydrological droughts with different intensity levels and magnitudes. The NHPP was found to be better in the assessment and analysis of drought events occur during a specific period. The findings of this study can be an important reference for the policymakers and government to take decisions in advance for sustainable environmental management in Pakistan.

Comparative evaluation of various frequentist and Bayesian non-homogeneous Poisson counting models

In this paper a comparative evaluation study on popular non-homogeneous Poisson models for count data is performed. For the study the standard homogeneous Poisson model (HOM) and three non-homogeneous variants, namely a Poisson changepoint model (CPS), a Poisson free mixture model (MIX), and a Poisson hidden Markov model (HMM) are implemented in both conceptual frameworks: a frequentist and a Bayesian framework. This yields eight models in total, and the goal of the presented study is to shed some light onto their relative merits and shortcomings. The first major objective is to cross-compare the performances of the four models (HOM, CPS, MIX and HMM) independently for both modelling frameworks (Bayesian and frequentist). Subsequently, a pairwise comparison between the four Bayesian and the four frequentist models is performed to elucidate to which extent the results of the two paradigms (‘Bayesian vs. frequentist’) differ. The evaluation study is performed on various synthetic Poisson data sets as well as on real-world taxi pick-up counts, extracted from the recently published New York City Taxi database.

Analysis of Ozone Behaviour in the City of Puebla-Mexico Using Non-Homogeneous Poisson Models with Multiple Change-Points

In this work, some non-homogeneous Poisson models are considered to study the behaviour of ozone in the city of Puebla, Mexico. Several functions are used as the rate function for the non-homogeneous Poisson process. In addition to their dependence on time, these rate functions also depend on some parameters that need to be estimated. In order to estimate them, a Bayesian approach will be taken. The expressions for the distributions of the parameters involved in the models are very complex. Therefore, Markov chain Monte Carlo algorithms are used to estimate them. The methodology is applied to the ozone data from the city of Puebla, Mexico.

Comparing the Adequacy of Some Non-Homogeneous Poisson Models to Estimate Ozone Exceedances in Mexico City

We consider some non-homogeneous Poisson models to estimate the mean number of times that a given environmental threshold of interest is surpassed by a given pollutant. Seven different rate functions for the Poisson processes describing the models are taken into account. The rate functions considered are the Weibull, exponentiated-Weibull, and their generalisation the Beta-Weibull rate function. We also use the Musa-Okumoto, the Goel-Okumoto, a generalised Goel- Okumoto and the Weibull-geometric rate functions. Whenever thought justifiable, the model allowing the presence of change-points is also going to be considered. The different models are applied to the daily maximum ozone measurements data provided by the monitoring network of the Metropolitan Area of Mexico City. The aim is to compare the adjustment of different rate functions to the data. Even though, some of the rate functions have been considered before, now we are applying them to the same data set. In previous works they were used in different data sets and therefore a comparison of the adequacy of those models were not possible. The measurements considered here were obtained after a series of environmental measures were implemented in Mexico City. Hence, the data present a different behaviour from that of earlier studies.

Using non-homogeneous Poisson models with multiple change-points to estimate the number of ozone exceedances in Mexico City

In this paper, we consider some non-homogeneous Poisson models to estimate the probability that an air quality standard is exceeded a given number of times in a time interval of interest. We assume that the number of exceedances occurs according to a non-homogeneous Poisson process (NHPP). This Poisson process has rate function , , which depends on some parameters that must be estimated. We take into account two cases of rate functions: the Weibull and the Goel—Okumoto. We consider models with and without change-points. When the presence of change-points is assumed, we may have the presence of either one, two or three change-points, depending of the data set. The parameters of the rate functions are estimated using a Gibbs sampling algorithm. Results are applied to ozone data provided by the Mexico City monitoring network. In a first instance, we assume that there are no change-points present. Depending on the adjustment of the model, we assume the presence of either one, two or three change-points. Copyright © 2009 John Wiley & Sons, Ltd.

Non-homogeneous Poisson models with a change-point: an application to ozone peaks in Mexico city

In this paper, we use some non-homogeneous Poisson models in order to study the behavior of ozone measurements in Mexico City. We assume that the number of ozone peaks follows a non-homogeneous Poisson process. We consider four types of rate function for the Poisson process: power law, Musa–Okumoto, Goel–Okumoto, and a generalized Goel–Okumoto rate function. We also assume that a change-point may or may not be present. The analysis of the problem is performed by using a Bayesian approach via Markov chain Monte Carlo methods. The best model is chosen using the DIC criterion as well as graphical approach.

RENEWAL AGING IN NON-HOMOGENEOUS POISSON PROCESSES WITH PERIODIC RATE MODULATION

The statistical properties of sequences of Waiting Times generated by Non-Homogeneous Poisson Processes are investigated by means of Renewal Aging, i.e. a statistical analysis being able to detect the presence of genuine renewal events with non-Poisson statistics. The Renewal Aging features of two non-homogeneous Poisson models with different prescriptions of the rate modulation are compared. Both prescriptions are essentially periodic. In the first model, the rate is a totally deterministic and smooth harmonic function. In the second model, the rate modulation is deterministic and smooth almost everywhere, except in some singular points occurring with a periodicity affected by weak random fluctuations. The main parameter of both models is the modulation speed, defined as the ratio between the modulation period and the average Waiting Time. This allows to distinguish between slow, intermediate and fast modulation of the rate. As expected, in the very slow modulation case both models have a strongly reduced Renewal Aging, practically zero, because the analysis is affected by local clusters of Waiting Times with the same Poisson rate. In the intermediate range of modulation speeds, both models show non-zero Renewal Aging and the behavior of the two models tends to be more different as the modulation speed decreases. Going towards the fast modulation regime, the behavior of the two models becomes completely different from one another. Model I does not show a definite global Renewal Aging. On the time scale of the modulation period, local oscillations arise in the Probability Density Function and an apparent oscillating aging is also observed. However, this form of aging is not the manifestation of genuine renewal non-Poisson events, but it is related to the local oscillations. On the contrary, Model II displays Renewal Aging features in agreement with a homogeneous Renewal non-Poisson Process. It is argued that this is related to the singular points in the rate modulation of Model II, which, due to their random occurrence, could be identified with genuine critical non-Poisson events. The approach presented here could have interesting applications in problems involving dichotomous noise and, in particular, in the statistical characterization of ON–OFF fluorescence intermittency experimentally observed in complex systems, such as nano-crystals and single bio-molecules stimulated by laser fields.

Optimal testing/maintenance design in a software development project

In this paper, we consider the problem of determining the optimal testing period and the optimal maintenance limit for software. Software debugging in the testing phase and the operation stage is described by a software reliability model based on a nonhomogeneous Poisson process and models the difference between the software execution environments in both stages as the operational profile. Furthermore, the time when continued maintenance by the project team ends is defined as the planned maintenance limit. Each of the optimal testing period (release time) and the optimal planned maintenance limit that minimize the total expected software cost is determined. Finally, we present an algorithm that determines the joint optimal policies for both, and perform the optimal testing/maintenance design through numerical examples. © 2006 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 89(8): 1–9, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.20224

Doubly periodic non-homogeneous Poisson models for hurricane data

Non-homogeneous Poisson processes with periodic claim intensity rate have been proposed as claim counts in risk theory. Here a doubly periodic Poisson model with short- and long-term trends is studied. Beta-type intensity functions are presented as illustrations. The likelihood function and the maximum likelihood estimates of the model parameters are derived. Doubly periodic Poisson models are appropriate when the seasonality does not repeat exactly the same short-term pattern every year, but has a peak intensity that varies over a longer period. This reflects periodic environments like those forming hurricanes, in alternating El Niño/La Niña years. An application of the model to the data set of Atlantic hurricanes affecting the United States (1899–2000) is discussed in detail.

Analysis of Data from a Series of Events by a Geometric Process Model

Geometric process was first introduced by Lam[10,11]. A stochastic process {X i , i = 1, 2, · · ·} is called a geometric process (GP) if, for some a > 0, {a i -1X i , i = 1, 2, · · ·} forms a renewal process. In this paper, the GP is used to analyze the data from a series of events. A nonparametric method is introduced for the estimation of the three parameters in the GP. The limiting distributions of the three estimators are studied. Through the analysis of some real data sets, the GP model is compared with other three homogeneous and nonhomogeneous Poisson models. It seems that on average the GP model is the best model among these four models in analyzing the data from a series of events.

Three nonhomogeneous Poisson models for the probability of basaltic volcanism: Application to the Yucca Mountain region, Nevada

The distribution and timing of areal basaltic volcanism are modeled using three nonhomogeneous methods: spatio‐temporal nearest neighbor, kernel, and nearest‐neighbor kernel. These models give nonparametric estimates of spatial or spatio‐temporal recurrence rate based on the positions and ages of cinder cones and related vent structures and can account for migration and shifts in locus, volcano clustering, and development of regional vent alignments. The three methods are advantageous because (1) recurrence rate and probability maps can be made, facilitating comparison with other geological information; (2) the need to define areas or zones of volcanic activity, required in homogeneous approaches, is eliminated; and (3) the impact of uncertainty in the timing and distribution of individual events is particularly easy to assess. The models are applied to the Yucca Mountain region (YMR), Nevada, the site of a proposed high‐level radioactive waste repository. Application of the Hopkins F test, Clark‐Evans test, and K function indicates volcanoes cluster in the YMR at the >95% confidence level. Weighted‐centroid cluster analysis indicates that Plio‐Quaternary volcanoes are distributed in four clusters: three of these clusters include cinder cones formed <1 Ma. Probability of disruption within the 8 km2 area of the proposed repository by formation of a new basaltic vent is calculated to be between 1 × 10−4 and 5 × 10−4 in 104 years (the kernel and nearest‐neighbor kernel methods give a maximum probability of 5 × 10−4 in 104 years), assuming regional recurrence rates of 5–10 volcanoes/m.y. An additional finding, illustrating the strength of nonhomogeneous methods, is that maps of the probability of volcanic eruption for the YMR indicate the proposed repository lies on a steep probability gradient: volcanism recurrence rate varies by more than 2 orders of magnitude within 20 km. Insight into this spatial scale of probability variation is a distinct benefit of application of these methods to hazard analysis in areal volcanic fields.