Zero-inflated Poisson Distribution Research Articles

A double generally weighted moving average control chart for monitoring zero-inflated Poisson processes

. The zero-inflated Poisson (ZIP) distribution is used to model count data with an excessive number of zeros. In this article, we present a double generally weighted moving average control chart for monitoring ZIP processes (regarded as ZIP-DGWMA chart). The run-length distribution of the proposed chart is evaluated by performing Monte Carlo simulations. A comparison study with the ZIP-GWMA chart indicates that the ZIP-DGWMA chart is more effective for shifts in the zero-inflated parameter p and for small shifts in both of the zero-inflated and Poisson parameters. The proposed chart is also applied on an example to demonstrate its application.

A beta-Poisson model for infectious disease transmission.

Outbreaks of emerging and zoonotic infections represent a substantial threat to human health and well-being. These outbreaks tend to be characterised by highly stochastic transmission dynamics with intense variation in transmission potential between cases. The negative binomial distribution is commonly used as a model for transmission in the early stages of an epidemic as it has a natural interpretation as the convolution of a Poisson contact process and a gamma-distributed infectivity. In this study we expand upon the negative binomial model by introducing a beta-Poisson mixture model in which infectious individuals make contacts at the points of a Poisson process and then transmit infection along these contacts with a beta-distributed probability. We show that the negative binomial distribution is a limit case of this model, as is the zero-inflated Poisson distribution obtained by combining a Poisson-distributed contact process with an additional failure probability. We assess the beta-Poisson model's applicability by fitting it to secondary case distributions (the distribution of the number of subsequent cases generated by a single case) estimated from outbreaks covering a range of pathogens and geographical settings. We find that while the beta-Poisson mixture can achieve a closer to fit to data than the negative binomial distribution, it is consistently outperformed by the negative binomial in terms of Akaike Information Criterion, making it a suboptimal choice on parsimonious grounds. The beta-Poisson performs similarly to the negative binomial model in its ability to capture features of the secondary case distribution such as overdispersion, prevalence of superspreaders, and the probability of a case generating zero subsequent cases. Despite this possible shortcoming, the beta-Poisson distribution may still be of interest in the context of intervention modelling since its structure allows for the simulation of measures which change contact structures while leaving individual-level infectivity unchanged, and vice-versa.

The Applications of Generalized Poisson Regression Models to Insurance Claim Data

Predictive modeling has been widely used for insurance rate making. In this paper, we focus on insurance claim count data and address their common issues with more flexible modeling techniques. In particular, we study the zero-inflated and hurdle-generalized Poisson and negative binomial distributions in a functional form for modeling insurance claim count data. It is shown that these models are useful in addressing the problem of excess zeros and over-dispersion of the claim count variable. In addition, we show that including the exposure as a covariate in both the zero and the count part of the model is an effective approach to incorporating exposure information in zero-inflated and hurdle models. We illustrate the effectiveness and versatility of the introduced models using three real datasets. The results suggest their promising applications in insurance risk classification and beyond.

Bayesian CART models for insurance claims frequency

The accuracy and interpretability of a (non-life) insurance pricing model are essential qualities to ensure fair and transparent premiums for policy-holders, that reflect their risk. In recent years, classification and regression trees (CARTs) and their ensembles have gained popularity in the actuarial literature, since they offer good prediction performance and are relatively easy to interpret. In this paper, we introduce Bayesian CART models for insurance pricing, with a particular focus on claims frequency modelling. In addition to the common Poisson and negative binomial (NB) distributions used for claims frequency, we implement Bayesian CART for the zero-inflated Poisson (ZIP) distribution to address the difficulty arising from the imbalanced insurance claims data. To this end, we introduce a general MCMC algorithm using data augmentation methods for posterior tree exploration. We also introduce the deviance information criterion (DIC) for tree model selection. The proposed models are able to identify trees which can better classify the policy-holders into risk groups. Simulations and real insurance data will be used to illustrate the applicability of these models.

ZERO-INFLATED POISSON TRANSMUTED WEIGHTED EXPONENTIAL DISTRIBUTION: PROPERTIES AND APPLICATIONS

Count observations with high frequencies of zero counts abound in diverse fields. In actuary science, for example, insurance claims are often underreported, leading to a higher frequency of zero counts. This invariably reduces the mean and leads to over-dispersion. Different techniques to model such occurrences exist. This study uses the cubic rank transmutation map to compound the weighted exponential distribution and obtain a new count distribution in the mixed Poisson paradigm. The zero-inflated form of the proposition is obtained along with some mathematical properties. Simulated skewness, kurtosis, and dispersion index for the new distributions reveal they are suitable for model dispersed observation with positive skewness. Five datasets with high frequencies of zero counts are used to assess the performance of the new distribution along with some popular count distributions by using the maximum likelihood for parameter estimation. Results show that the natural form of the new proposition performs creditably better than its zero-inflated forms, even when there is a higher proportion of zero counts. The classical negative binomial distribution is also observed to outperform its zero-inflated form in most cases. In contrast, the zero-inflated Poisson distribution better fits the datasets than the classical Poisson distribution.

Inferring the distribution of norovirus in individual oysters below the limit of quantification by pooled sampling.

Norovirus (NoV) in oysters is a food safety risk of much concern. In order to assess the risk of the exposure, the distribution of the number of NoV copies contained in each oyster should be acquired first for comprehensively quantifying the associated risks. However, the part of the distribution below the limit of quantification cannot be obtained directly by laboratory detecting methods, which hampers accurate assessment. To tackle this challenging problem, a systematic method (Distribution Inference Method by Pooled Sampling) is proposed to infer the unobservable part of distribution based upon all measurements of the pooled samples with n=2. Using convolutional integrals and real-coded genetic algorithm for inferring, this method has neither requirements for the type or properties of the original distribution, nor requirements for historical data, even nor requirements for the relationship between observable and unobservable parts of the distribution. A series of experiments were conducted on simulated datasets of a variety of types, including normal distribution, uniform distribution, gamma distribution, lognormal distribution, zero-inflated Poisson distribution, their combinations, and even their splicing, covering common distribution types in oyster NoV scenario and more general scenarios. The results show that almost all inferred simulation data and their original counterparts passed Kolmogorov-Smirnov tests, which implies that they are essential of the same distribution. Based on this method, a ready-to-use web system was developed for researchers to infer their original distribution with pooled-sampling measurements from the detection of NoV or even other substances.

The multivariate component zero‐inflated Poisson model for correlated count data analysis

SummaryMultivariate zero‐inflated Poisson (ZIP) distributions are important tools for modelling and analysing correlated count data with extra zeros. Unfortunately, existing multivariate ZIP distributions consider only the overall zero‐inflation while the component zero‐inflation is not well addressed. This paper proposes a flexible multivariate ZIP distribution, called the multivariate component ZIP distribution, in which both the overall and component zero‐inflations are taken into account. Likelihood‐based inference procedures including the calculation of maximum likelihood estimates of parameters in the model without and with covariates are provided. Simulation studies indicate that the performance of the proposed methods on the multivariate component ZIP model is satisfactory. The Australia health care utilisation data set is analysed to demonstrate that the new distribution is more appropriate than the existing multivariate ZIP distributions.

Assessing Area under the Curve as an Alternative to Latent Growth Curve Modeling for Repeated Measures Zero-Inflated Poisson Data: A Simulation Study

Researchers interested in the assessment of substance use trajectories, and predictors of change, have several data analysis options. These include, among others, generalized estimating equations and latent growth curve modeling. One difficulty in the assessment of substance use, however, is the nature of the variables studied. Although counting instances of use (e.g., the number of cigarettes smoked per day) would seem to be the best option, such data present difficulties in that the distribution of these variables is not likely normal. Count variables often follow a Poisson distribution, and when dealing with substance use in the general population, there is a preponderance of zeros (representing not using). As such, substance use counts may approximate a zero-inflated Poisson distribution. Unfortunately, analyses with zero-inflated Poisson random variables are not easily accommodated in many types of software and may be beyond access to most researchers. As such, an easier method would benefit researchers interested in assessing substance use change. The purpose of this study is to assess the area under the curve as an option when dealing with repeated measures data and contrast it to one popular method of longitudinal data analysis, latent growth curve modeling. Using a Monte Carlo simulation study with varying sample sizes, we found that the area under the curve performed well with different sample sizes and compared favorably to the performance of latent growth curve modeling, particularly when dealing with smaller sample sizes. The area under the curve may be a simpler alternative for researchers, especially when dealing with smaller sample sizes.

An index of capability for bivariate zero-inflated processes

Rapid technological advancement and implementation of automation and computerization in today's manufacturing set up resulted in many high quality processes, where defects are rarely observed. There are many high quality manufacturing processes where two or more types of defects may be generated from different types of equipment/process problems. The zeroinflated defects data containing two types of defects are commonly modeled by bivariate zero-inflated (BZI) Poisson distribution. Pal and Gauri (2022a) proposed a methodology for measuring capability of a BZI Poisson process. However, they ignored the count of zero defect (ZD) products produced in a BZI process. Because of that, Pal and Gauri (2022a) proposed approach fails to discriminate the BZI processes which produces different proportions of ZD units but having almost the same proportion of nonconforming items with respect to the USL of combined number of defects or USLs of individual defect types. In this paper, a new measure of process capability for BZI processes is proposed that can truly discriminate different BZI processes taking into account the USL of combined number of defects (or USLs of individual defect types) as well as the proportion of ZD units produced in these processes. The proposed methodology is illustrated using two case studies. The results of the case studies show that the proposed index well represents the true capability of BZI processes.

Inferring predator-prey interactions from camera traps: ABayesian co-abundance modeling approach.

Predator-prey dynamics are a fundamental part of ecology, but directly studying interactions has proven difficult. The proliferation of camera trapping has enabled the collection of large datasets on wildlife, but researchers face hurdles inferring interactions from observational data. Recent advances in hierarchical co-abundance models infer species interactions while accounting for two species' detection probabilities, shared responses to environmental covariates, and propagate uncertainty throughout the entire modeling process. However, current approaches remain unsuitable for interacting species whose natural densities differ by an order of magnitude and have contrasting detection probabilities, such as predator-prey interactions, which introduce zero inflation and overdispersion in count histories. Here, we developed a Bayesian hierarchical N-mixture co-abundance model that is suitable for inferring predator-prey interactions. We accounted for excessive zeros in count histories using an informed zero-inflated Poisson distribution in the abundance formula and accounted for overdispersion in count histories by including a random effect per sampling unit and sampling occasion in the detection probability formula. We demonstrate that models with these modifications outperform alternative approaches, improve model goodness-of-fit, and overcome parameter convergence failures. We highlight its utility using 20 camera trapping datasets from 10 tropical forest landscapes in Southeast Asia and estimate four predator-prey relationships between tigers, clouded leopards, and muntjac and sambar deer. Tigers had a negative effect on muntjac abundance, providing support for top-down regulation, while clouded leopards had a positive effect on muntjac and sambar deer, likely driven by shared responses to unmodelled covariates like hunting. This Bayesian co-abundance modeling approach to quantify predator-prey relationships is widely applicable across species, ecosystems, and sampling approaches and may be useful in forecasting cascading impacts following widespread predator declines. Taken together, this approach facilitates a nuanced and mechanistic understanding of food-web ecology.

IccCounts: An R Package to Estimate the Intraclass Correlation Coefficient for Assessing Agreement with Count Data

The intraclass correlation coefficient (ICC) is a widely used index to assess agreement with continuous data. The common approach for estimating the ICC involves estimating the variance components of a linear mixed model under assumptions such as linearity and normality of effects. However, if the outcomes are counts these assumptions are not met and the ICC estimates are biased and inefficient. In this situation, it is necessary to use alternative approaches that are better suited for count data. Here, the iccCounts R package is introduced for estimating the ICC under the Poisson, Negative Binomial, Zero-Inflated Poisson and Zero-Inflated Negative Binomial distributions. The utility of the iccCounts package is illustrated by three examples that involve the assessment of repeatability and concordance with count data.

Zero-inflated count distributions for capture-mark-reencounter data.

The estimation of demographic parameters is a key component of evolutionary demography and conservation biology. Capture–mark–recapture methods have served as a fundamental tool for estimating demographic parameters. The accurate estimation of demographic parameters in capture–mark–recapture studies depends on accurate modeling of the observation process. Classic capture–mark–recapture models typically model the observation process as a Bernoulli or categorical trial with detection probability conditional on a marked individual's availability for detection (e.g., alive, or alive and present in a study area). Alternatives to this approach are underused, but may have great utility in capture–recapture studies. In this paper, we explore a simple concept: in the same way that counts contain more information about abundance than simple detection/non‐detection data, the number of encounters of individuals during observation occasions contains more information about the observation process than detection/non‐detection data for individuals during the same occasion. Rather than using Bernoulli or categorical distributions to estimate detection probability, we demonstrate the application of zero‐inflated Poisson and gamma‐Poisson distributions. The use of count distributions allows for inference on availability for encounter, as well as a wide variety of parameterizations for heterogeneity in the observation process. We demonstrate that this approach can accurately recover demographic and observation parameters in the presence of individual heterogeneity in detection probability and discuss some potential future extensions of this method.

Evaluating capability of a bivariate zero-inflated poisson process

A zero-inflated Poisson (ZIP) distribution is commonly used for modelling zero-inflated process data with single type of defect, and for developing appropriate tools for instituting statistical process control of manufacturing processes. However, in reality, such manufacturing scenarios are very common where more than one type of defect can occur. For example, occurrences of defects like solder short circuits (shorts) and absence of solder (skips) are very common on printed circuit boards. In literature, different forms of bivariate zero-inflated Poisson (BZIP) distributions are proposed, which can be used for modelling the manufacturing scenarios where two types of defects can occur. Control charts are designed for monitoring for such processes using BZIP models. Although evaluation of capability is an integral part of statistical process control of a manufacturing process, researchers have given very little effort on this aspect of zero-inflated processes. Only a few articles attempted to evaluate the capability of a univariate zero-inflated process and no work is reported on evbaluating capability of a bivariate zero-inflated process. In this paper, a methodology for measuring capability of a bivariate zero-inflated process is presented. The proposed methodology is illustrated using two case studies.

Primed for positive perceptions? Applying the acquired preparedness model to explain college students' e-cigarette use and dependence.

College students use electronic nicotine delivery systems (ENDS) more often than any other US demographic group. In a novel application of the acquired preparedness model, we examined how proximal (e.g., cognitions) and distal (e.g., dispositional) influences accounted for ENDS use and dependence. Undergraduates (N = 1075; 72% female, 74% White) from seven US campuses completed an online survey between October 2019-March 2020. We modeled ENDS use and dependence, respectively, as zero-inflated Poisson distributions with impulsivity as an independent variable and perceived risks and benefits of e-cigarettes as mediators. Students higher in impulsivity perceived more benefits and, in turn, reported greater ENDS use and dependence. Curiosity and friends' use motivated ENDS initiation; stress management and nicotine motivated continued use. ENDS interventions should be tailored to students higher in impulsivity, as they hold more favorable perceptions of ENDS, and should enhance skills to manage stress and nicotine cravings.

Smoothed Quantiles for Measuring Discrete Risks

Many risk measures can be defined through the quantile function of the underlying loss variable (e.g., a class of distortion risk measures). When the loss variable is discrete or mixed, however, the definition of risk measures has to be broadened, which makes statistical inference trickier. To facilitate a straightforward transition from the risk measurement literature of continuous loss variables to that of discrete, in this article we study smoothing of quantiles for discrete variables. Smoothed quantiles are defined using the theory of fractional or imaginary order statistics, which was originated by Stigler (1977). To prove consistency and asymptotic normality of sample estimators of smoothed quantiles, we utilize the results of Wang and Hutson (2011) and generalize them to vectors of smoothed quantiles. Further, we thoroughly investigate extensions of this methodology to discrete populations with infinite support (e.g., Poisson and zero-inflated Poisson distributions). Furthermore, large- and small-sample properties of the newly designed estimators are investigated theoretically and through Monte Carlo simulations. Finally, applications of smoothed quantiles to risk measurement (e.g., estimation of distortion risk measures such as Value at Risk, conditional tail expectation, and proportional hazards transform) are discussed and illustrated using automobile accident data. Comparisons between the classical (and linearly interpolated) quantiles and smoothed quantiles are performed as well.

Designing and Selection of Bayesian Skip Lot Sampling Plan Type BSkSP-2 with Single Sampling Plan Under Zero-Inflated Poisson Distribution

Sampling plans are effective statistical technique for testing and deciding whether to accept or not a lot based on quality inspection studied under highway construction and materials. This article presents a designing methodology for the selection of Bayesian Skip lot Sampling Plan type 2 (BSkSP-2) based on Gamma-Zero Inflated Poisson (GZIP) distribution. A Gamma distribution is a baseline distribution that is to be considered as a conjugate prior for Zero-Inflated Poisson (ZIP) distribution. Generally, ZIP distribution is used for count data with an excessive number of zeros. In this article, ZIP distribution is mainly focused upon to study the occurrence of defective units in a well-designed manufacturing environment. In such case, one can develop an attribute sampling plan under the SkSP-2 through a G-ZIP model, to protect both the producers and consumers with the minimum inspection efforts rather than the conventional Poisson distribution. Necessary tables are developed for the proposed sampling plan and numerical illustrations are made for the applications on highway construction and materials are given for the execution of the plan.

N-mixture models provide informative crocodile (Crocodylus moreletii) abundance estimates in dynamic environments.

Estimates of animal abundance provide essential information for population ecological studies. However, the recording of individuals in the field can be challenging, and accurate estimates require analytical techniques which account for imperfect detection. Here, we quantify local abundances and overall population size of Morelet’s crocodiles (Crocodylus moreletii) in the region of Calakmul (Campeche, Mexico), comparing traditional approaches for crocodylians (Minimum Population Size—MPS; King’s Visible Fraction Method—VFM) with binomial N-mixture models based on Poisson, zero-inflated Poisson (ZIP) and negative binomial (NB) distributions. A total of 191 nocturnal spotlight surveys were conducted across 40 representative locations (hydrologically highly dynamic aquatic sites locally known as aguadas) over a period of 3 years (2017–2019). Local abundance estimates revealed a median of 1 both through MPS (min–max: 0–89; first and third quartiles, Q1–Q3: 0–7) and VFM (0–112; Q1–Q3: 0–9) non-hatchling C. moreletii for each aguada, respectively. The ZIP based N-mixture approach shown overall superior confidence over Poisson and NB, and revealed a median of 6 ± 3 individuals (min = 0; max = 120 ± 18; Q1 = 0; Q3 = 18 ± 4) jointly with higher detectabilities in drying aguadas with low and intermediate vegetation cover. Extrapolating these inferences across all waterbodies in the study area yielded an estimated ~10,000 (7,000–11,000) C. moreletii present, highlighting Calakmul as an important region for this species. Because covariates enable insights into population responses to local environmental conditions, N-mixture models applied to spotlight count data result in particularly insightful estimates of crocodylian detection and abundance.

Phase II monitoring of zero inflated Poisson regression profiles

In many quality control applications, the quality of some products or processes is best characterized by a functional relationship (profile) between a response variable and one or more explanatory variables. Profile monitoring involves the use of control charts to monitor the stability of this type of quality control processes. Several studies have discussed the problem of monitoring normal response profiles. More recently, researchers started studying the case where the response variable follows a discrete distribution such as the Poisson or the Bernoulli distributions. Due to recent technological advancement and the high level of automation used in almost all manufacturing processes, there exist near-zero defect manufacturing processes, hence, the zero-inflated Poisson distribution is expected to be more appropriate than the ordinary Poisson distribution for monitoring such processes. This study aims at extending three of the existing methods for phase II monitoring of profiles namely; MEWMA, Hotelling’s T2, and EWMA-R to the case of zero-inflated Poisson profiles. A simulation study is used to compare the performance of the competing approaches in terms of the average run length (ARL) and the standard deviation run length (SDRL). The results revealed that the EWMA-R chart is generally superior to the other competing methods.

Approximation of Zero-Inflated Poisson Credibility Premium via Variational Bayes Approach

While both zero-inflation and the unobserved heterogeneity in risks are prevalent issues in modeling insurance claim counts, determination of Bayesian credibility premium of the claim counts with these features are often demanding due to high computational costs associated with a use of MCMC. This article explores a way to approximate credibility premium for claims frequency that follows a zero-inflated Poisson distribution via variational Bayes approach. Unlike many existing industry benchmarks, the proposed method enables insurance companies to capture both zero-inflation and unobserved heterogeneity of policyholders simultaneously with modest computation costs. A simulation study and an empirical analysis using the LGPIF dataset were conducted and it turned out that the proposed method outperforms many industry benchmarks in terms of prediction performances and computation time. Such results support the applicability of the proposed method in the posterior ratemaking practices.

The heritability of Nematodirus battus fecal egg counts.

Although Nematodirus battus is a serious threat to the health and survival of young lambs, there are few options to control this parasite. Bayesian Monte Carlo Markov Chain modelling with a zero-inflated Poisson distribution was used to estimate the heritability of egg counts in both June and July for each of five consecutive cohorts of 200 Scottish Blackface lambs. In one of the 10 analyses, the results failed the diagnostic tests. In seven of the analyses, there was no convincing evidence that the variation in egg counts was heritable. In the 2 years of high infection, the heritability was approximately 0.4 in June but the estimates lacked precision and the 95% highest posterior density credible intervals ranged from just above zero to 0.7. Selective breeding for resistance to N. battus will be difficult because genetically resistant or susceptible lambs cannot be consistently identified by phenotypic markers.