Conditional Poisson Distribution Research Articles

Classification of multivariate count data with multivariate log-linear conditional Poisson distribution

Classification of multivariate count data with multivariate log-linear conditional Poisson distribution

SOME RESULTS ON r-TRUNCATED DEGENERATE POISSON RANDOM VARIABLES

The zero-truncated Poisson distributions are certain discrete probability distributions whose supports are the set of positive integers, which are also known as the conditional Poisson distributions or the positive Poisson distributions. Recently, as a natural extension of those distributions, Kim–Kim studied the zero-truncated degenerate Poisson distributions. In this paper, we introduce the [Formula: see text]-truncated degenerate Poisson random variable with parameter [Formula: see text], whose probability mass function is given by [Formula: see text], and investigate various properties of this random variable.

Spectrum Sensing for Optical Wireless Scattering Communications Over Málaga Fading—A Cooperative Approach With Hard Decision Fusion

This paper presents collaborative spectrum sensing for optical wireless scattering communications. The received signal is characterized by the conditional Poisson distribution, given the mean photon count following Málaga distributed correlated fading. We consider multi-scattered non-coplanar links and use the stochastic analytical method to model the channel. A system model is developed, utilizing the cooperative diversity drawn from the fact that the users are randomly distributed. A novel centralized cooperative spectrum sensing technique is developed that all the raw data available at each collaborative user are combined at the fusion center to make a decision. Another novel decentralized technique is also proposed where only one-bit information is required to be sent to the fusion center. For this decentralized technique, we propose a unique voting rule that leverages the location flexibility of randomly distributed users with each user experiencing correlated fading. Results demonstrate their effectiveness and provide insights into the proposed cooperative spectrum sensing system. Interestingly, it is also found that the use of the proposed voting rule requires fewer than the total number of collaborative users while satisfying a given error rate, and yields the Bernoulli distributed output of the fusion center.

Degenerate Zero-Truncated Poisson Random Variables

Recently, the degenerate Poisson random variable with parameter $$\alpha > 0$$ , whose probability mass function is given by $$P_{\lambda}(i) = e_{\lambda}^{-1} (\alpha) \frac{\alpha^{i}}{i!} (1)_{i,\lambda}$$ $$(i = 0,1,2,\dots)$$ , was studied. In probability theory, the zero-truncated Poisson distributions are certain discrete probability distributions whose supports are the set of positive integers. These distributions are also known as the conditional Poisson distributions or the positive Poisson distributions. In this paper, we introduce the degenerate zero-truncated Poisson random variables whose probability mass functions are a natural extension of the zero-truncated Poisson distributions, and investigate various properties of those random variables.

Abstract OT2-07-07: Image-guided de-escalation of neoadjuvant chemotherapy in HER2-positive breast cancer: The TRAIN-3 study

Abstract Background The addition of pertuzumab to trastuzumab containing chemotherapy has boosted pathologic complete response (pCR) rates after neoadjuvant chemotherapy for HER2-positive breast cancer. PCR rates over 80% have been described and achieving a pCR is associated with a favorable long-term outcome. In addition, achieving a radiologic complete response (rCR) is predictive of the pathologic response in HER2-positive tumors. Therefore it is hypothesized that image-guided evaluation based on the early occurrence of rCR can be used to tailor the number of chemotherapy cycles. Trial design This is a single arm, multicenter study evaluating the efficacy of image-guided de-escalation of neoadjuvant treatment with paclitaxel, Herceptin®, carboplatin, and pertuzumab (PTC-ptz). Radiologic evaluation with contrast-enhanced breast MRI and ultrasound of the axilla (in cN+ patients) is performed at baseline and after 3, 6, and 9 cycles of treatment. In case of rCR of the breast (and axilla) after 3 or 6 cycles, early surgery will be performed. If residual tumor is present after 3 and 6 cycles, patients will continue the PTC-ptz regimen to complete a total of 9 cycles. All patients will receive adjuvant Herceptin® and pertuzumab to complete 1 year of anti-HER2 blockade and endocrine treatment according to local guidelines if HR-positive. The study will be performed in the Netherlands in approximately 35 centers. Eligibility criteria Eligible patients have histologically proven stage II/III HER2-positive primary breast cancer with known hormone-receptor status. Patients must have a measurable breast tumor on baseline MRI and can be either node negative or node positive. Specific aims The aim is to evaluate the efficacy of image-guided de-escalation of neoadjuvant chemotherapy in HER2-positive breast cancer on event-free survival (EFS) at 3 years as primary endpoint. Secondary endpoints are overall survival, rCR, concordance between rCR and pCR (ypT0/is, ypN0), differences in EFS and OS following pCR between patients who received 3, 6, or 9 cycles, and toxicity. Statistical methods This is a single-arm, two stage study with one interim-analysis and a final analysis. Statistics will be performed for each hormone receptor subgroup separately. Stopping rules are based on 3-year EFS-rates described in literature (88% for HR-negative tumors and 90% for HR-positive tumors) and calculated using the exact conditional Poisson distribution. The study is successful with ≤34 EFS-events in the HR-negative subgroup and ≤38 events in the HR-positive subgroup after 700 patient-years of follow-up. The three-year EFS-estimate will be calculated using Kaplan-Meier statistics. Present accrual and target accrual Target accrual is 231 patients for the HR-negative group and 231 patients for the HR-positive group. Present accrual will follow. Funding Investigator initiated trial sponsored by the Dutch Breast Cancer Research Group (BOOG), funded by Roche. Contact information for people with a specific interest in the trial Study coordinator: A van der Voort, MD The Netherlands Cancer Institute 1006 BE Amsterdam E: a.vd.voort@nki.nl, P:+31 20 512 2951 Citation Format: van der Voort A, Dezentjé VO, van der Steeg WA, Winter-Warnars GA, Schipper R-J, Scholten AN, Wesseling J, van Werkhoven ED, van Duijnhoven FH, Vrancken Peeters M-JT, Sonke GS. Image-guided de-escalation of neoadjuvant chemotherapy in HER2-positive breast cancer: The TRAIN-3 study [abstract]. In: Proceedings of the 2018 San Antonio Breast Cancer Symposium; 2018 Dec 4-8; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2019;79(4 Suppl):Abstract nr OT2-07-07.

Poisson Dependency Networks: Gradient Boosted Models for Multivariate Count Data

Although count data are increasingly ubiquitous, surprisingly little work has employed probabilistic graphical models for modeling count data. Indeed the univariate case has been well studied, however, in many situations counts influence each other and should not be considered independently. Standard graphical models such as multinomial or Gaussian ones are also often ill-suited, too, since they disregard either the infinite range over the natural numbers or the potentially asymmetric shape of the distribution of count variables. Existing classes of Poisson graphical models can only model negative conditional dependencies or neglect the prediction of counts or do not scale well. To ease the modeling of multivariate count data, we therefore introduce a novel family of Poisson graphical models, called Poisson Dependency Networks (PDNs). A PDN consists of a set of local conditional Poisson distributions, each representing the probability of a single count variable given the others, that naturally facilitates a simple Gibbs sampling inference. In contrast to existing Poisson graphical models, PDNs are non-parametric and trained using functional gradient ascent, i.e., boosting. The particularly simple form of the Poisson distribution allows us to develop the first multiplicative boosting approach: starting from an initial constant value, alternatively a log-linear Poisson model, or a Poisson regression tree, a PDN is represented as products of regression models grown in a stage-wise optimization. We demonstrate on several real world datasets that PDNs can model positive and negative dependencies and scale well while often outperforming state-of-the-art, in particular when using multiplicative updates.

Detecting overdispersion in INARCH(1) processes

A statistical test for the degree of overdispersion of count data time series based on the empirical version of the (Poisson) index of dispersion is considered. The test design relies on asymptotic properties of this index of dispersion, which in turn have been analyzed for time series stemming from a compound Poisson (Poisson‐stopped sum) INAR(1) model. This approach is extended to the popular Poisson INARCH(1) model, which exhibits unconditional overdispersion but has an (equidispersed) conditional Poisson distribution. The asymptotic distribution of the index of dispersion if applied to time series stemming from such a model is derived. These results allow us to investigate the ability of the dispersion test to discriminate between Poisson INAR(1) and INARCH(1) models. Furthermore, the question is considered if the index of dispersion could be used to test the null of a Poisson INARCH(1) model against the alternative of an INARCH(1) model with additional conditional overdispersion.

A likelihood approach to estimate the number of co-infections.

The number of co-infections of a pathogen (multiplicity of infection or MOI) is a relevant parameter in epidemiology as it relates to transmission intensity. Notably, such quantities can be built into a metric in the context of disease control and prevention. Having applications to malaria in mind, we develop here a maximum-likelihood (ML) framework to estimate the quantities of interest at low computational and no additional costs to study designs or data collection. We show how the ML estimate for the quantities of interest and corresponding confidence-regions are obtained from multiple genetic loci. Assuming specifically that infections are rare and independent events, the number of infections per host follows a conditional Poisson distribution. Under this assumption, we show that a unique ML estimate for the parameter () describing MOI exists which is found by a simple recursion. Moreover, we provide explicit formulas for asymptotic confidence intervals, and show that profile-likelihood-based confidence intervals exist, which are found by a simple two-dimensional recursion. Based on the confidence intervals we provide alternative statistical tests for the MOI parameter. Finally, we illustrate the methods on three malaria data sets. The statistical framework however is not limited to malaria.

North Atlantic Tropical Storm Frequency Response to Anthropogenic Forcing: Projections and Sources of Uncertainty

Abstract The impact of future anthropogenic forcing on the frequency of tropical storms in the North Atlantic basin has been the subject of intensive investigation. However, whether the number of North Atlantic tropical storms will increase or decrease in a warmer climate is still heavily debated and a consensus has yet to be reached. To shed light on this issue, the authors use a recently developed statistical model, in which the frequency of North Atlantic tropical storms is modeled by a conditional Poisson distribution with rate of occurrence parameter that is a function of tropical Atlantic and mean tropical sea surface temperatures (SSTs). It is shown how the disagreement among dynamical modeling projections of late-twenty-first-century tropical storm frequency can be largely explained by differences in large-scale SST patterns from the different climate model projections used in these studies. The results do not support the notion of large (~200%) increases in tropical storm frequency in the North Atlantic basin over the twenty-first century in response to increasing greenhouse gases (GHGs). Because the statistical model is computationally inexpensive, it is used to examine the impact of different climate models and climate change scenarios on the frequency of North Atlantic tropical storms. The authors estimate that the dominant drivers of uncertainty in projections of tropical storm frequency over the twenty-first century are internal climate variations and systematic intermodel differences in the response of SST patterns to increasing GHGs. Relative to them, uncertainties in total GHG emissions or other climate forcings, within the scenarios explored here, represent a minor source of uncertainty in tropical storm frequency projections. These results suggest that reducing uncertainty in future projections of North Atlantic tropical storm frequency may depend as critically on reducing the uncertainty in the sensitivity of tropical Atlantic warming relative to the tropical mean, in response to GHG increase, as on improving dynamical or statistical downscaling techniques. Moreover, the large uncertainties on century-scale trends that are due to internal climate variability are likely to remain irreducible for the foreseeable future. As a further illustration of the statistical model’s utility, the authors model projected changes in U.S. landfalling tropical storm activity under a variety of different climate change scenarios and climate models. These results are similar to those for the overall number of North Atlantic tropical storms, and do not point to a large increase in U.S. landfalling tropical storms over the twenty-first century in response to increasing GHGs.

Optimal designs for radiation retention with Poisson correlated response

In this paper we describe a non-linear model with correlated observations which accounts for the elimination rate of radiation in the lung of individuals who have been exposed to an accidental intake at some time. The response is then modelled as a conditional Poisson distribution. When the leak is moderate or the size of the particles is large a theoretical justification of this assumption is given and D-optimal designs are computed.

Time series of count data: modeling, estimation and diagnostics

Various models for time series of counts which can account for discreteness, overdispersion and serial correlation are compared. Besides observation- and parameter-driven models based upon corresponding conditional Poisson distributions, a dynamic ordered probit model as a flexible specification to capture the salient features of time series of counts is also considered. For all models, appropriate efficient estimation procedures are presented. For the parameter-driven specification this requires Monte-Carlo procedures like simulated maximum likelihood or Markov chain Monte Carlo. The methods, including corresponding diagnostic tests, are illustrated using data on daily admissions for asthma to a single hospital. Estimation results turn out to be remarkably similar across the different models.

Hierarchical Bayesian spatial models for alcohol availability, drug "hot spots" and violent crime

BackgroundEcologic studies have shown a relationship between alcohol outlet densities, illicit drug use and violence. The present study examined this relationship in the City of Houston, Texas, using a sample of 439 census tracts. Neighborhood sociostructural covariates, alcohol outlet density, drug crime density and violent crime data were collected for the year 2000, and analyzed using hierarchical Bayesian models. Model selection was accomplished by applying the Deviance Information Criterion.ResultsThe counts of violent crime in each census tract were modelled as having a conditional Poisson distribution. Four neighbourhood explanatory variables were identified using principal component analysis. The best fitted model was selected as the one considering both unstructured and spatial dependence random effects. The results showed that drug-law violation explained a greater amount of variance in violent crime rates than alcohol outlet densities. The relative risk for drug-law violation was 2.49 and that for alcohol outlet density was 1.16. Of the neighbourhood sociostructural covariates, males of age 15 to 24 showed an effect on violence, with a 16% decrease in relative risk for each increase the size of its standard deviation. Both unstructured heterogeneity random effect and spatial dependence need to be included in the model.ConclusionThe analysis presented suggests that activity around illicit drug markets is more strongly associated with violent crime than is alcohol outlet density. Unique among the ecological studies in this field, the present study not only shows the direction and magnitude of impact of neighbourhood sociostructural covariates as well as alcohol and illicit drug activities in a neighbourhood, it also reveals the importance of applying hierarchical Bayesian models in this research field as both spatial dependence and heterogeneity random effects need to be considered simultaneously.

Large deviations and the maximum entropy principle for marked point random fields

We establish large deviation principles for the stationary and the individual empirical fields of Poisson, and certain interacting, random fields of marked point particles in ℝd. The underlying topologies are induced by a class of not necessarily bounded local functions, and thus finer than the usual weak topologies. Our methods yield further that the limiting behaviour of conditional Poisson distributions, as well as certain distributions of Gibbsian type, is governed by the maximum entropy principle. We also discuss various applications and examples.

Minimum chi-square estimator and test for incomplete multidimensional contingency tables

A modified transformed chi-square statistic is defined for testing hypotheses of quasi-independence in the incomplete multi-dimensional contingency table and a simple method for determining degrees of freedom is given. A modified transformed chi-squareestimator of the expected cell frequencies is given in closed form for a general class of exact linear constraints. The co-variance matrix of estimated cell frequencies is derived under the assumption of a conditional Poisson distribution.

The Asymptotic Covariance Matrix of the Maximum Likelihood Parameter Estimator in Conditional Poisson Log‐linear Models

AbstractThe asymptotic covariance matrix of the maximum likelihood estimator for the log‐linear model is given for a general class of conditional Poisson distributions which include the unconditional Poisson, multinomial and product‐multinomial, as special cases. The general conditions are given under which the maximum likelihood covariance matrix is equal to the covariance matrix of an equivalent closed‐form weighted least squares estimator.

Some Extensions of a Linear Model for Categorical Variables

The Grizzle-Starmer-Koch (GSK) model is extended to include the traditional log-linear model and a general class of Poisson and conditional Poisson distributions. Estimators of the model parameters are defined under general exact and stochastic linear constraints.