Non-homogeneous Poisson Model Research Articles

Spatiotemporal nonhomogeneous poisson model with a seasonal component applied to the analysis of extreme rainfall

This paper develops an extension of spatiotemporal models that handle count data using nonhomogeneous Poisson processes. In this new proposal, we incorporate a seasonal cycle component in the definition of the intensity function to control possible effects produced by the occurrence of the event of interest in regular periods. The seasonal cycle can cause problems in estimating the shape parameter of the Weibull and generalized Goel intensity functions. This shape parameter serves to confront the research hypothesis that seeks to identify a trend in the occurrence rate of an event of interest. In the case of the Weibull intensity function, a value significantly equal to one of the shape parameters indicates a constant rate of occurrence, less than one indicates a decreasing rate, and greater than one indicates an increasing rate. In the case of the Goel intensity function, parameter values less than or equal to one indicate a decreasing occurrence rate, and values greater than one indicate the presence of a change point. We also built a spatial model using the Musa-Okumoto intensity function as an alternative to approximate counting processes for which there is a decreasing trend in the occurrence rate of the event of interest. We estimated the parameters of the proposed method from a Bayesian perspective. Finally, we fitted the proposed model and compared it with other approximations to analyze the frequency of extreme rainfall in the northern region of the states of Maranhão and Piauí in northeastern Brazil over ten years. Among the main results, we found that (1) the proposed method has proven superior in terms of fit and prediction performance than the other models, and (2) unlike other approximations, the proposed model does not detect changes in the rate of extreme rainfall occurrences.

An additive hazards cure model with informative interval censoring

The existence of a cured subgroup happens quite often in survival studies and many authors considered this under various situations (Farewell in Biometrics 38:1041-1046, 1982; Kuk and Chen in Biometrika 79:531-541, 1992; Lam and Xue in Biometrika 92:573-586, 2005; Zhou et al. in J Comput Graph Stat 27:48-58, 2018). In this paper, we discuss the situation where only interval-censored data are available and furthermore, the censoring may be informative, for which there does not seem to exist an established estimation procedure. For the analysis, we present a three component model consisting of a logistic model for describing the cure rate, an additive hazards model for the failure time of interest and a nonhomogeneous Poisson model for the observation process. For estimation, we propose a sieve maximum likelihood estimation procedure and the asymptotic properties of the resulting estimators are established. Furthermore, an EM algorithm is developed for the implementation of the proposed estimation approach, and extensive simulation studies are conducted and suggest that the proposed method works well for practical situations. Also the approach is applied to a cardiac allograft vasculopathy study that motivated this investigation.

A non-homogeneous Poisson process geostatistical model with spatial deformation

In this paper, we propose a geostatistical model for the counting process using a non-homogeneous Poisson model. This work aims to model the intensity function as the sum of two components: spatial and temporal. The spatial component is modeled using a Gaussian process in which the covariance structure is assumed to be anisotropic. Anisotropy is incorporated by applying a spatial deformation approach. The temporal component is modeled in such a way that its behavior concerning time has the structure of a Goel process. The inferences for the proposed model are obtained from a Bayesian perspective. The parameter estimation is obtained using Markov Chain Monte Carlo methods. The proposed model is adjusted to a set of real data, referring to the rain precipitation in 29 monitoring stations, distributed in the states of Maranhao and Piaui, in the northeast region of Brazil, in 31 years, from 01/01/1980 to 12/31/2010. The objective is to estimate the frequency of rain that exceeded a certain threshold.

A Stochastic Approach for the Analysis of Long Dry Spells with Different Threshold Values in Southern Italy

A non-homogeneous Poisson model was proposed to analyze the sequences of dry spells below prefixed thresholds as an upgrade of a stochastic procedure previously used to describe long periods of no rainfall. Its application concerned the daily precipitation series in a 60-year time span at four rain gauges (Calabria, southern Italy), aiming at testing the different behaviors of the dry spells below prefixed thresholds in two paired periods (1951–1980 and 1981–2010). A simulation analysis performed through a Monte Carlo approach assessed the statistical significance of the variation of the mean values of dry spells observed at an annual scale in the two 30-year periods. The results evidenced that the dry spells durations increased passing from the first 30-year period to the second one for all the thresholds analyzed. For instance, for the Cassano station, an increase of about 10% of the maximum dry spell duration was detected for a threshold of 5 mm. Moreover, the return periods evaluated for fixed long dry spells through the synthetic data of the period 1981–2010 were lower than the corresponding ones evaluated with the data generated for the previous 30-year period. Specifically, the difference between the two 30-year periods in terms of the return period of long dry spells occurrence increased with the growing thresholds. As an example, for the Cosenza rain gauge with a threshold of 1 mm, the return period for a dry spell length of 70 days decreased from 20 years (in 1951–1980) to about 10 years (in 1981–2010), while for a threshold of 5 mm, the return period for the dry spell lengths of 120 days decreases from 70 years to about 20 years. These results show a higher probability of the occurrence of long dry spells in the more recent period than in the past.

Using a non-homogeneous Poisson model with spatial anisotropy and change-points to study air pollution data

A non-homogeneous Poisson process is used to study the rate at which a pollutant’s concentration exceeds a given threshold of interest. An anisotropic spatial model is imposed on the parameters of the Poisson intensity function. The main contribution here is to allow the presence of change-points in time since the data may behave differently for different time frames in a given observational period. Additionally, spatial anisotropy is also imposed on the vector of change-points in order to account for the possible correlation between different sites. Estimation of the parameters of the model is performed using Bayesian inference via Markov chain Monte Carlo algorithms, in particular, Gibbs sampling and Metropolis-Hastings. The different versions of the model are applied to ozone data from the monitoring network of Mexico City, Mexico. An analysis of the results obtained is also given.

Optimal investment and premium control for insurers with ambiguity

In this paper, we consider the optimal investment and premium control problem for insurers who worry about model ambiguity. Different from previous works, we assume that the insurer’s surplus process is described by a non-homogeneous compound Poisson model and the insurer has ambiguity on both the financial market and the insurance market. Our purpose is to find the impacts of model ambiguity on optimal policies. With the objective of maximizing the expected utility of terminal wealth, the closed-form solutions of the optimal investment and premium policies are obtained by solving HJB equations. Finally, numerical examples are also given to illustrate the results.

Modeling operational risk depending on covariates: an empirical investigation

The importance of operational risk management in financial and commodity markets has increased significantly over the last few decades. This paper demonstrates the application of a nonhomogeneous Poisson model and dynamic extreme value theory (EVT) incorporating covariates on estimating frequency, severity and risk measures for operational risk. Compared with a classical EVT approach, the dynamic EVT gives a better performance with respect to the statistical fit and realism. It is also flexible enough to handle different types of empirical data. In our model, we include firm-specific covariates associated with internal control weaknesses (ICWs) and show empirically that firms with higher incidences of selected ICWs have higher time-varying severities for operational risk. Our methodology provides risk managers and regulators with a tool that uncovers the nonobvious patterns hidden in operational risk data.

EVIDENCE FOR INDUCED SEISMICITY FOLLOWING THE 2001 SKYROS MAINSHOCK

Estimation of the seismicity rate changes caused by a major earthquake is based upon the assumption that the earthquake occurrence can be described by stochastic processes. Three stochastic models are applied to the data, i.e. the homogeneous Poisson model, the non-homogeneous Poisson model with two different rate functions, and the Autoregressive model AR(2). The two latter models seem to be adequate to properly simulate the earthquake production in a given area. The identification of the model which best fits the data, enables the estimations of the seismicity rate changes and the numbers of the earthquakes following a specific main shock.

Pulse Jitter, Delay Spread, and Doppler Shift in Mode-Stirred Reverberation

A time-domain characterization and stochastic model are developed for analyzing the evolution of the early randomization of fields inside mode-stirred reverberation chambers. The chamber is excited by a pulse amplitude modulated RF signal and stirred mechanically by a rotational stirrer. Phase noise (jitter) in the arrival times of a received pulse train is found to contain a systematic contribution that depends on stir speed and stir sense, as well as a random contribution governed by the rate of field depolarization. For the systematic component, a model of the phase delay for a uniformly orbiting scatterer inside a cavity is derived. For the random component, a nonhomogeneous Poisson model for arrivals of wave fronts in a nonstationary stir process exhibits good agreement with the empirical distribution of the measured jitter. At increasing stir velocities, the rate of field mixing increases less rapidly. The excess delay approaches asymptotically an Erlang distribution, in the limit of a homogeneous Poisson process for uniformly distributed late arrival times.

A time series analysis and a non-homogeneous Poisson model with multiple change-points applied to acoustic data

High levels of the so-called community noise may produce hazardous effect on the health of a population exposed to them for large periods of time. Hence, the study of the behaviour of those noise measurements is very important. In this work we analyse that in terms of the probability of exceeding a given threshold level a certain number of times in a time interval of interest. Since the datasets considered contain missing measurements, we use a time series model to estimate the missing values and complete the datasets. Once the data is complete, we use a non-homogeneous Poisson model with multiple change-points to estimate the probability of interest. Estimation of the parameters of the models are made using the usual time series methodology as well as the Bayesian point of view via Markov chain Monte Carlo algorithms. The models are applied to data obtained from two measuring sites in Messina, Italy.

A Bayesian Classifier for X-Ray Pulsars Recognition

Recognition for X-ray pulsars is important for the problem of spacecraft’s attitude determination by X-ray Pulsar Navigation (XPNAV). By using the nonhomogeneous Poisson model of the received photons and the minimum recognition error criterion, a classifier based on the Bayesian theorem is proposed. For X-ray pulsars recognition with unknown Doppler frequency and initial phase, the features of every X-ray pulsar are extracted and the unknown parameters are estimated using the Maximum Likelihood (ML) method. Besides that, a method to recognize unknown X-ray pulsars or X-ray disturbances is proposed. Simulation results certificate the validity of the proposed Bayesian classifier.

Semiparametric regression analysis of panel count data allowing for within-subject correlation

In this paper, a maximum likelihood approach is proposed for analyzing panel count data under the gamma frailty non-homogeneous Poisson process model. The approach allows one to estimate the baseline mean function and the regression parameters jointly while taking the within-subject correlation into account. The within-subject correlation is quantified explicitly by Pearson’s correlation coefficient. Monotone splines are adopted to approximate the unspecified nondecreasing baseline mean function in the model. An expectation–maximization (EM) algorithm is derived to facilitate the computation by exploiting a data augmentation based on Poisson latent variables. The EM algorithm is robust to initial values, easy to implement, converges fast, and provides closed-form variance estimates. It can be also applied to the non-homogeneous Poisson model without frailty. The proposed approach is evaluated through simulations and illustrated by two real-life examples coming from a skin cancer study and a bladder tumor study. A companion R package PCDSpline has been developed and is available on R CRAN for public use.

Non-stationary frequency analysis of extreme precipitation in South Korea using peaks-over-threshold and annual maxima

The conventional approach to the frequency analysis of extreme precipitation is complicated by non-stationarity resulting from climate variability and change. This study utilized a non-stationary frequency analysis to better understand the time-varying behavior of short-duration (1-, 6-, 12-, and 24-h) precipitation extremes at 65 weather stations scattered across South Korea. Trends in precipitation extremes were diagnosed with respect to both annual maximum precipitation (AMP) and peaks-over-threshold (POT) extremes. Non-stationary generalized extreme value (GEV) and generalized Pareto distribution (GPD) models with model parameters made a linear function of time were applied to AMP and POT respectively. Trends detected using the Mann–Kendall test revealed that the stations showing an increasing trend in AMP extremes were concentrated in the mountainous areas (the northeast and southwest regions) of South Korea. Trend tests on POT extremes provided fairly different results, with a significantly reduced number of stations showing an increasing trend and with some stations showing a decreasing trend. For most of stations showing a statistically significant trend, non-stationary GEV and GPD models significantly outperformed their stationary counterparts, particularly for precipitation extremes with shorter durations. Due to a significant-increasing trend in the POT frequency found at a considerable number of stations (about 10 stations for each rainfall duration), the performance of modeling POT extremes was further improved with a non-homogeneous Poisson model. The large differences in design storm estimates between stationary and non-stationary models (design storm estimates from stationary models were significantly lower than the estimates of non-stationary models) demonstrated the challenges in relying on the stationary assumption when planning the design and management of water facilities. This study also highlighted the need of caution when quantifying design storms from POT and AMP extremes by showing a large discrepancy between the estimates from those two approaches.

Stochastic analysis of long dry spells in Calabria (Southern Italy)

A deficit in precipitation may impact greatly on soil moisture, snowpack, streamflow, groundwater and reservoir storage. Among the several approaches available to investigate this phenomenon, one of the most applied is the analysis of dry spells. In this study, a non-homogeneous Poisson model has been applied to a set of high-quality daily rainfall series, recorded in southern Italy (Calabria region) during the period 1981–2010, for the stochastic analysis of dry spells. Firstly, some statistical details of the Poisson models were presented. Then, the proposed model has been applied to the analysis of long dry spells. In particular, a Monte Carlo technique was performed to reproduce the characteristics of the process. As a result, the main characteristics of the long dry spells have shown patterns clearly related to some geographical features of the study area, such as elevation and latitude. The results obtained from the stochastic modelling of the long dry spells proved that the proposed model is useful for the probability evaluation of drought, thus improving environmental planning and management.

An analysis of airport noise data using a non-homogeneous Poisson model with a change-point

In this work, a non-homogeneous Poisson model is considered to study the behaviour of airport noise levels. The model is used to count the number of times that the noise level exceeds a given threshold in a time interval of interest. The rate function of the Poisson process is assumed to be of a Weibull type. Two threshold values are considered. One of them, make it evident the change in the data’s behaviour. Hence, it requires the use of a model with a change-point. The models considered depend on some parameters that need to be estimated. After the estimation of the parameters are made, a way of obtaining the probability that the noise threshold is exceeded a certain number of times in a given time interval is presented. Results related to the mean number of exceedances of the threshold are also provided. Those results can be very useful in the study of the population’s exposure to noise produced by air traffic and other types of noise related to the airport operation. The model is also useful to predict, given the current behaviour of the data, the probability of occurrence of high levels of noise in a near future. An application to the airport noise data from Nice International Airport is given.

A non-homogeneous poisson model with spatial anisotropy applied to ozone data from Mexico City

In this work we consider a non-homogenous Poisson model to study the behaviour of the number of times that a pollutant’s concentration surpasses a given threshold of interest. Spatial dependence is imposed on the parameters of the Poisson intensity function in order to account for the possible correlation between measurements in different sites. An anisotropic model is used due to the nature of the region of interest. Estimation of the parameters of the model is performed using the Bayesian point of view via Markov chain Monte Carlo (MCMC) algorithms. We also consider prediction of the days in which exceedances of the threshold might occur at sites where measurements cannot be taken. This is obtained by spatial interpolation using the information provided by the sites where measurements are available. The prediction procedure allows for estimation of the behaviour of the mean function of the non-homogeneous Poisson process associated with those sites. The models considered here are applied to ozone data obtained from the monitoring network of Mexico City.

Modeling environmental noise exceedances using non-homogeneous Poisson processes.

In this work a non-homogeneous Poisson model is considered to study noise exposure. The Poisson process, counting the number of times that a sound level surpasses a threshold, is used to estimate the probability that a population is exposed to high levels of noise a certain number of times in a given time interval. The rate function of the Poisson process is assumed to be of a Weibull type. The presented model is applied to community noise data from Messina, Sicily (Italy). Four sets of data are used to estimate the parameters involved in the model. After the estimation and tuning are made, a way of estimating the probability that an environmental noise threshold is exceeded a certain number of times in a given time interval is presented. This estimation can be very useful in the study of noise exposure of a population and also to predict, given the current behavior of the data, the probability of occurrence of high levels of noise in the near future. One of the most important features of the model is that it implicitly takes into account different noise sources, which need to be treated separately when using usual models.

A stochastic model for the analysis of the temporal change of dry spells

In the present paper a stochastic approach which considers the arrival of rainfall events as a Poisson process is proposed to analyse the sequences of no rainy days. Particularly, among the different Poisson models, a non-homogeneous Poisson model was selected and then applied to the daily rainfall series registered at the Cosenza rain gauge (Calabria, southern Italy), as test series. The aim was to evaluate the different behaviour of the dry spells observed in two different 30-year periods, i.e. 1951–1980 and 1981–2010. The analyses performed through Monte Carlo simulations assessed the statistical significance of the variation of the mean expected values of dry spells observed at annual scale in the second period with respect to those observed in the first. The model has then been verified by comparing the results of the test series with the ones obtained from other three rain gauges of the same region. Moreover, greater occurrence probabilities for long dry spells in 1981–2010 than in 1951–1980 were detected for the test series. Analogously, the return periods evaluated for fixed long dry spells through the synthetic data of the period 1981–2010 resulted less than half of the corresponding ones evaluated with the data generated for the previous 30-year period.

X-ray pulsar signal detection using photon interarrival time

The distribution probability of the photon interarrival time (PIT) without signal initial phases is derived based on the Poisson model of X-ray pulsar signals, and a pulsar signal detection algorithm employing the PIT sequence is put forward. The joint probability of the PIT sequence is regarded as a function of the distribution probability and used to compare a constant radiation intensity model with the nonhomogeneous Poisson model for the signal detection. The relationship between the number of detected photons and the probabilities of false negative and positive is studied, and the success rate and mean detection time are estimated based on the number of the given photons. For the spacecraft velocity data detection, the changes of time of photon arrival (TOPA) and PIT caused by spacecraft motion are presented first, then the influences on detection are analyzed respectively. By using the analytical pulse profile of PSR B0531+21, the simulation of the X-ray pulsar signal detection is implemented. The simulation results verify the effectiveness of the proposed method, and the contrast tests show that the proposed method is suitable for the spacecraft velocity data detection.

Gastrointestinal Stromal Tumor: A Method for Optimizing the Timing of CT Scans in the Follow-up of Cancer Patients

To develop a mathematical model to adjust the timing of computed tomography (CT) scans with the hazard of cancer recurrence in time to facilitate early detection of cancer recurrence. The clinical data were extracted from the randomized Scandinavian Sarcoma Group (SSG) XVIII/Arbeitsgemeinschaft Internistische Onkologie (AIO) trial database. The SSG XVIII/AIO trial was registered (trial no. NCT00116935) and approved by the national or institutional review boards. In the trial, 1- and 3-year durations of adjuvant imatinib mesylate in the treatment of patients with gastrointestinal stromal tumor (GIST) were compared. A nonhomogeneous Poisson model with a piecewise log-constant hazard in time that accounts for the nonlinear pattern of GIST recurrence was applied to tumor site, mitotic count, and recurrence data. The optimal times to obtain follow-up CT scans were computed by modifying the frequency of CT scans with the hazard of tumor recurrence in time. The hazard-adjusted follow-up schedules were compared with the National Comprehensive Cancer Network (NCCN) guidelines of the United States, which suggest imaging with CT at intervals of 3-6 months for 3-5 years and then annually. Optimized timing of CT scans on the basis of hazard of recurrence resulted in follow-up schedule options where CT is performed more sparsely than in the NCCN guidelines during adjuvant imatinib administration and more frequently, at approximately 3-month intervals, during the first 2 years that follow imatinib discontinuation when the risk of recurrence was the greatest. The number of CT scans could be reduced by a median of 31% (from 13 to nine) compared with the standard schedules within the first 6 years of follow-up without increasing the delay in recurrence detection. Detection of GIST recurrence may be enhanced by adjusting the timing of the CT scans with the hazard of recurrence. The method may be applicable to other human tumor types. Online supplemental material is available for this article.