Compound Poisson-gamma Research Articles

Bias correction and ensemble techniques in statistical downscaling model for rainfall prediction using Tweedie-LASSO in West Java, Indonesia

Rainfall is a climate element with high variations in space and time scales, so it is not easy to predict. One way to predict rainfall is statistical downscaling (SD). SD can predict local rainfall based on Global Circulation Model (GCM) data. The Decadal Climate Prediction Project (DCPP), one of the GCMs, originates from adjacent grids and experiences multicollinearity problems. Rainfall as a response variable is Tweedie Compound Poisson Gamma (TCPG) distribution data because it has a discrete component (rainfall events) and a continuous component (rainfall intensity), so SD modelling will be carried out using Tweedie-LASSO. This research aims to compare the performance of bias correction and ensemble methods in SD in predicting rainfall in West Java, Indonesia. Bias correction uses Empirical Quantile Mapping (EQM) with CHIRPS data, and the ensemble method uses a stacking technique with Random Forest (Stacking-RF) due to the varied characteristics of DCPP model sources. Evaluation results using Root Mean Square Error Prediction (RMSEP) and correlation coefficient show that bias correction improves single-model performance but not ensemble models. Besides that, ensemble models outperform single models both before and after bias correction. The combination of bias correction and ensemble modelling can be recommended when conducting SD to enhance the prediction capability of rainfall at stations and other areas.

Bonus-Malus Scale premiums for Tweedie’s compound Poisson models

Abstract Based on the recent papers, two distributions for the total claims amount (loss cost) are considered: compound Poisson-gamma and Tweedie. Each is used as an underlying distribution in the Bonus-Malus Scale (BMS) model. The BMS model links the premium of an insurance contract to a function of the insurance experience of the related policy. In other words, the idea is to model the increase and the decrease in premiums for insureds who do or do not file claims. We applied our approach to a sample of data from a major insurance company in Canada. Data fit and predictability were analyzed. We showed that the studied models are exciting alternatives to consider from a practical point of view, and that predictive ratemaking models can address some important practical considerations.

Genome-wide association study of a semicontinuous trait: illustration of the impact of the modeling strategy through the study of Neutrophil Extracellular Traps levels.

Over the last years, there has been a considerable expansion of genome-wide association studies (GWAS) for discovering biological pathways underlying pathological conditions or disease biomarkers. These GWAS are often limited to binary or quantitative traits analyzed through linear or logistic models, respectively. In some situations, the distribution of the outcome may require more complex modeling, such as when the outcome exhibits a semicontinuous distribution characterized by an excess of zero values followed by a non-negative and right-skewed distribution. We here investigate three different modeling for semicontinuous data: Tobit, Negative Binomial and Compound Poisson-Gamma. Using both simulated data and a real GWAS on Neutrophil Extracellular Traps (NETs), an emerging biomarker in immuno-thrombosis, we demonstrate that Compound Poisson-Gamma was the most robust model with respect to low allele frequencies and outliers. This model further identified the MIR155HG locus as significantly (P=1.4×10-8) associated with NETs plasma levels in a sample of 657 participants, a locus recently highlighted to be involved in NETs formation in mice. This work highlights the importance of the modeling strategy for GWAS of a semicontinuous outcome and suggests Compound Poisson-Gamma as an elegant but neglected alternative to Negative Binomial for modeling semicontinuous outcome in the context of genomic investigations.

Bayesian Variable Selection in Double Generalized Linear Tweedie Spatial Process Models

Double generalized linear models provide a flexible framework for modeling data by allowing the mean and the dispersion to vary across observations. Common members of the exponential dispersion family including the Gaussian, Poisson, compound Poisson-gamma (CP-g), Gamma and inverse-Gaussian are known to admit such models. The lack of their use can be attributed to ambiguities that exist in model specification under a large number of covariates and complications that arise when data display complex spatial dependence. In this work we consider a hierarchical specification for the CP-g model with a spatial random effect. The spatial effect is targeted at performing uncertainty quantification by modeling dependence within the data arising from location based indexing of the response. We focus on a Gaussian process specification for the spatial effect. Simultaneously, we tackle the problem of model specification for such models using Bayesian variable selection. It is effected through a continuous spike and slab prior on the model parameters, specifically the fixed effects. The novelty of our contribution lies in the Bayesian frameworks developed for such models. We perform various synthetic experiments to showcase the accuracy of our frameworks. They are then applied to analyze automobile insurance premiums in Connecticut, for the year of 2008.

Scaling behaviour of braided active channels: a Taylor’s power law approach

At a channel (reach) scale, braided channels are fluvial, geomorphological, complex systems that are characterized by a shift of bars during flood events. In such events water flows are channeled in multiple and mobile channels across a gravel floodplain that remain in unmodified conditions. From a geometrical point of view, braided patterns of the active hydraulic channels are characterized by multicursal nature with structures that are spatially developed by either simple- and multi-scaling behavior. Since current studies do not take into account a general procedure concerning scale measurements, the latter behavior is still not well understood. The aim of our investigation is to analyze directly, through a general procedure, the scaling behavior of hydraulically active channels per transect and per reach analyzed. Our generalized stochastic approach is based on Taylor’s law, and the theory of exponential dispersion distributions. In particular, we make use of a power law, based on the variance and mean of the active channel fluctuations. In this way we demonstrate that the number of such fluctuations with respect to the unicursal behavior of the braided rivers, follows a jump-process of Poisson and compound Poisson–Gamma distributions. Furthermore, a correlation is also provided between the scaling fractal exponents obtained by Taylor’s law and the Hurst exponents.

The spatio-temporal model for the Tweedie compound Poisson gamma response in statistical downscaling

This research aims to develop a Spatio-temporal generalized linear mixed model with the h-likelihood estimation method for Statistical Downscaling modeling with the Tweedie compound Poisson Gamma distribution which can produce estimates for fixed effects, random effects, and variance components simultaneously. The results showed that the proposed model has a good performance characterized by the lowest root mean square error prediction and able to reduce the variety of random effects caused by spatial and temporal dependencies.

Bayesian Processing of Data on Bursts of Pressure Vessels

Two alternative Bayesian approaches are proposed for the prediction of fragmentation of pressure vessels triggered off by accidental explosions (bursts) of these containment structures. It is shown how to carry out this prediction with post-mortem data on fragment numbers counted after past explosion accidents. Results of the prediction are estimates of probabilities of individual fragment numbers. These estimates are expressed by means of Bayesian prior or posterior distributions. It is demonstrated how to elicit the prior distributions from relatively scarce post-mortem data on vessel fragmentations. Specifically, it is suggested to develop priors with two Bayesian models known as compound Poisson-gamma and multinomial-Dirichlet probability distributions. The available data is used to specify non-informative prior for Poisson parameter that is subsequently transformed into priors of individual fragment number probabilities. Alternatively, the data is applied to a specification of Dirichlet concentration parameters. The latter priors directly express epistemic uncertainty in the fragment number probabilities. Example calculations presented in the study demonstrate that the suggested non-informative prior distributions are responsive to updates with scarce data on vessel explosions. It is shown that priors specified with Poisson-gamma and multinomial-Dirichlet models differ tangibly; however, this difference decreases with increasing amount of new data. For the sake of brevity and concreteness, the study was limited to fire induced vessel bursts known as boiling liquid expanding vapour explosions (BLEVEs).

Degree distributions in AB random geometric graphs

In this paper, we provide degree distributions for AB random geometric graphs, in which points of type A connect to the closest k points of type B. The motivating example to derive such degree distributions is in 5G wireless networks with multi-connectivity, where users connect to their closest k base stations. In this setting, it is important to know how many users a particular base station serves, which gives the degree of that base station. To obtain these degree distributions, we investigate the distribution of area sizes of the kth order Voronoi cells of B-points. Assuming that the A-points are Poisson distributed, we investigate the amount of users connected to a certain B-point, which is equal to the degree of this point. In the simple case where the B-points are placed in an hexagonal grid, we show that all kth order Voronoi areas are equal and thus all degrees follow a Poisson distribution. However, this observation does not hold for Poisson distributed B-points, for which we show that the degree distribution follows a compound Poisson–Erlang distribution in the 1-dimensional case. We then approximate the degree distribution in the 2-dimensional case with a compound Poisson-Gamma degree distribution and show that this one-parameter fit performs well for different values of k. Moreover, we show that for increasing k, these degree distributions become more concentrated around the mean. This means that k-connected AB random graphs balance the loads of B-type nodes more evenly as k increases. Finally, we provide a case study on real data of base stations. We show that with little shadowing in the distances between users and base stations, the Poisson distribution does not capture the degree distribution of these data, especially for k>1. However, under strong shadowing, our degree approximations perform quite good even for these non-Poissonian location data.

A new approach to statistical downscaling using Tweedie compound Poisson gamma response and lasso regularization

Statistical Downscaling (SDS) is a technique in climatology to analyze the functional relationship of global scale GCM (Global Circulation Model) output data as predictor variables and local scale rainfall data as a response variable. Rainfall contains continuous and discrete components. The continuous component is related to the intensity of rainfall more than zero which can be assumed to be Gamma distribution while the discrete component is related to the occurrence of rain including zero which can be assumed to be Poisson distribution. A combination of both distributions is called Tweedie compound poisson gamma (TCPG). SDS modeling with TCPG response can be used to predict the occurrence of rain and also the rainfall intensity simultaneously. GCM output data generally contain multicollinearity problems which can be overcome by Lasso regularization. This study discusses SDS modeling which assumes TCPG distributed response and uses Lasso to predict some characteristics of rainfall such as the average number of daily rainfall events per month (, shape parameter , the average intensity of daily rainfall per month , probability of no rain event per month , the number of no rain per month . Based on the smallest RMSEP and the high correlation of the actual and predicted data, the TCPG model with Lasso regulation is more reliable and needs to be considered for modeling rainfall than TCPG-generalized linear models and TCPG-principal component analysis.

Poisson-Gamma mixture processes and applications to premium calculation

In the paper, Poisson-Gamma mixture process is first brought forward, which is dynamically expanded from the well-known Poisson-Gamma mixture model. Some properties on Poisson-Gamma mixture process are presented, including the distribution of increment, Markov property, infinitesimal generator, joint density function of jump/waiting times, and the limit distribution of compound Poisson-Gamma mixture process, etc., which provide a thorough grounding in application of Poisson-Gamma mixture process. At last, some premium calculation principles are presented to show the application of Poisson-Gamma mixture process, which include expected value premium, stop-loss premium, mean-variance premium, and exponential premium.

Compound Poisson models for weighted networks with applications in finance

We develop a modelling framework for estimating and predicting weighted network data. The edge weights in weighted networks often arise from aggregating some individual relationships between the nodes. Motivated by this, we introduce a modelling framework for weighted networks based on the compound Poisson distribution. To allow for heterogeneity between the nodes, we use a regression approach for the model parameters. We test the new modelling framework on two types of financial networks: a network of financial institutions in which the edge weights represent exposures from trading Credit Default Swaps and a network of countries in which the edge weights represent cross-border lending. The compound Poisson Gamma distributions with regression fit the data well in both situations. We illustrate how this modelling framework can be used for predicting unobserved edges and their weights in an only partially observed network. This is for example relevant for assessing systemic risk in financial networks.

Compound Poisson Models for Financial Networks

We develop a modelling framework for estimating and predicting weighted network data. The edge weights in weighted networks often arise from aggregating some individual relationships between the nodes. Motivated by this, we introduce a modelling framework for weighted networks based on the compound Poisson distribution. To allow for heterogeneity between the nodes, we use a regression approach for the model parameters. We test the new modelling framework on two types of financial networks: a network of financial institutions in which the edge weights represent exposures from trading Credit Default Swaps and a network of countries in which the edge weights represent cross-border lending. The compound Poisson Gamma distributions with regression fit the data well in both situations. We illustrate how this modelling framework can be used for predicting unobserved edges and their weights in an only partially observed network. This is for example relevant for assessing systemic risk in financial networks.

Predicting cyanobacterial biovolume from water temperature and conductivity using a Bayesian compound Poisson-Gamma model

Eutrophication and climate change scenarios engender the need to develop good predictive models for harmful cyanobacterial blooms (CyanoHABs). Nevertheless, modeling cyanobacterial biomass is a challenging task due to strongly skewed distributions that include many absences as well as extreme values (dense blooms). Most modeling approaches alter the natural distribution of the data by splitting them into zeros (absences) and positive values, assuming that different processes underlie these two components. Our objectives were (1) to develop a probabilistic model relating cyanobacterial biovolume to environmental variables in the Río de la Plata Estuary (35°S, 56°W, n = 205 observations) considering all biovolume values (zeros and positive biomass) as part of the same process; and (2) to use the model to predict cyanobacterial biovolume under different risk level scenarios using water temperature and conductivity as explanatory variables. We developed a compound Poisson-Gamma (CPG) regression model, an approach that has not previously been used for modeling phytoplankton biovolume, within a Bayesian hierarchical framework. Posterior predictive checks showed that the fitted model had a good overall fit to the observed cyanobacterial biovolume and to more specific features of the data, such as the proportion of samples crossing three threshold risk levels (0.2, 1 and 2 mm³ L−1) at different water temperatures and conductivities. The CPG model highlights the strong control of cyanobacterial biovolume by nonlinear and interactive effects of water temperature and conductivity. The highest probability of crossing the three biovolume levels occurred at 22.2 °C and at the lowest observed conductivity (∼0.1 mS cm−1). Cross-validation of the fitted model using out-of-sample observations (n = 72) showed the model’s potential to be used in situ, as it enabled prediction of cyanobacterial biomass based on two readily measured variables (temperature and conductivity), making it an interesting tool for early alert systems and management strategies. Furthermore, this novel application demonstrates the potential of the Bayesian CPG approach for predicting cyanobacterial dynamics in response to environmental change.

Developing and assessing a density surface model in a Bayesian hierarchical framework with a focus on uncertainty: insights from simulations and an application to fin whales (Balaenoptera physalus).

Density surface models (DSMs) are an important tool in the conservation and management of cetaceans. Most previous applications of DSMs have adopted a two-step approach to model fitting (hereafter referred to as the Two-Stage Method), whereby detection probabilities are first estimated using distance sampling detection functions and subsequently used as an offset when fitting a density-habitat model. Although variance propagation techniques have recently become available for the Two-Stage Method, most previous applications have not propagated detection probability uncertainty into final density estimates. In this paper, we describe an alternative approach for fitting DSMs based on Bayesian hierarchical inference (hereafter referred to as the Bayesian Method), which is a natural framework for simultaneously propagating multiple sources of uncertainty into final estimates. Our framework includes (1) a mark-recapture distance sampling observation model that can accommodate two team line transect data, (2) an informed prior for the probability a group of animals is at the surface and available for detection (i.e. surface availability) (3) a density-habitat model incorporating spatial smoothers and (4) a flexible compound Poisson-gamma model for count data that incorporates overdispersion and zero-inflation. We evaluate our method and compare its performance to the Two-Stage Method with simulations and an application to line transect data of fin whales (Balaenoptera physalus) off the east coast of the USA. Simulations showed that both methods had low bias (<1.5%) and confidence interval coverage close to the nominal 95% rate when variance was propagated from the first step. Results from the fin whale analysis showed that density estimates and predicted distribution patterns were largely similar among methods; however, the coefficient of variation of the final abundance estimate more than doubled (0.14 vs 0.31) when detection variance was correctly propagated into final estimates. An analysis of the variance components demonstrated that overall detectability as well as surface availability contributed substantial amounts of variance in the final abundance estimates whereas uncertainty in mean group size contributed a negligible amount. Our method provides a Bayesian alternative to DSMs that incorporates much of the flexibility available in the Two-Stage Method. In addition, these results demonstrate the degree to which uncertainty can be underestimated if certain components of a DSM are assumed fixed.

Applying the Tweedie model for improved microinsurance pricing

Microinsurance usually involves basic rate making due to the scarcity of data and the lack of actuarial skills, causing microinsurers to add significant charges to premiums to compensate for unexpected variations in risk assessment. The situation could be improved by applying predictive modelling techniques to enable better pricing in the microinsurance market. This paper proposes the application of a generalised linear model with the Tweedie compound Poisson–Gamma distribution for a bundled microinsurance in the Philippines. The risk factors considered are mainly derived from the available database and are related to the features of the insured. The results of the predictive analysis point out that there are three predictors that best fit the model: age of the insured, policy age, and population density. The application of the Tweedie model provides fair and accurate risk premiums that outperform the model currently applied by the company and enhance its risk control.

Developing spatio-temporal models using multiple data types for evaluating population trends and habitat usage

AbstractSpatio-temporal models have become key tools for evaluating population trends and habitat usage. We developed a spatio-temporal modelling framework employing a combination of encounter/non-encounter, count, and biomass data collected by different monitoring programs (“combined data”). The three data types are predicted using a computationally efficient approximation to a compound Poisson-gamma process. We fitted spatio-temporal models to combined data for Gulf of Mexico (GOM) red snapper (Lutjanus campechanus) for 2006–2014. These spatio-temporal models provided insights into GOM red snapper spatial distribution patterns, which we corroborated by comparing to past predictions generated using only encounter/non-encounter data. However, relying on biomass and count data in addition to encounter/non-encounter data also allowed us to reconstruct biomass trends for GOM red snapper and to examine patterns of distribution shifts and range expansion/contraction for this population for the first time. Moreover, combining multiple data types improved the precision of reconstructed population trends and some variables quantifying habitat usage. Finally, scenarios and simulation experiments conditioned upon red snapper data showed that the improvement in fitting to combined data is greater when biomass data for the study population are lacking for an entire subregion and, to a lesser extent, for an entire time period (e.g. in early years).

Three problems with the conventional delta-model for biomass sampling data, and a computationally efficient alternative

Ecologists often analyse biomass sampling data that result in many zeros, where remaining samples can take any positive real number. Samples are often analysed using a “delta-model” that combines two separate generalized linear models, GLMs (for encounter probability and positive catch rates), or less often using a compound Poisson-gamma (CPG) distribution that is computationally expensive. I discuss three theoretical problems with the conventional delta-model: difficulty interpreting covariates for encounter probability, the assumed independence of the two GLMs, and the biologically implausible form when eliminating covariates for either GLM. I then derive an alternative “Poisson-link model” that solves these problems. To illustrate, I use biomass samples for 113 fish populations to show that the Poisson-link model improves fit (and decreases residual spatial variation) for &gt;80% of populations relative to the conventional delta-model. A simulation experiment illustrates that CPG and Poisson-link models estimate covariate effects that are similar and biologically interpretable. I therefore recommend the Poisson-link model as a useful alternative to the conventional delta-model with similar properties to the CPG distribution.

The Gerber-Shiu function in the perturbed compound Poisson Gamma Omega model with a dividend barrier

In this paper, the perturbed compound Poisson Gamma Omega model with a barrier dividend strategy is studied. Using the strong Markov property and Taylor formula, the integro-differential equations for the Gerber-Shiu expected discounted penalty functions are derived. The explicit solutions of the Gerber-Shiu expected discounted penalty functions are also obtained when the claim size is exponentially distributed. Furthermore, a numerical example is presented to illustrate some properties of the function.

The Gerber-Shiu function in the perturbed compound Poisson Gamma Omega model with a dividend barrier

In this paper, the perturbed compound Poisson Gamma Omega model with a barrier dividend strategy is studied. Using the strong Markov property and Taylor formula, the integro-differential equations for the Gerber-Shiu expected discounted penalty functions are derived. The explicit solutions of the Gerber-Shiu expected discounted penalty functions are also obtained when the claim size is exponentially distributed. Furthermore, a numerical example is presented to illustrate some properties of the function.

Properties of Poisson processes directed by compound Poisson-Gamma subordinators

In the paper we consider time-changed Poisson processes where the time is expressed by compound Poisson-Gamma subordinators $G(N(t))$ and derive the expressions for their hitting times. We also study the time-changed Poisson processes where the role of time is played by the processes of the form $G(N(t)+at)$ and by the iteration of such processes.