Modeling earthquake numbers by Negative Binomial Hidden Markov models

The question of how to characterize the bacterial density in a body of water when data are available as counts from a number of small-volume samples was examined for cases where either the Poisson or negative binomial probability distributions could be used to describe the bacteriological data. The suitability of the Poisson distribution when replicate analyses were performed under carefully controlled conditions and of the negative binomial distribution for samples collected from different locations and over time were illustrated by two examples. In cases where the negative binomial distribution was appropriate, a procedure was given for characterizing the variability by dividing the bacterial counts into homogeneous groups. The usefulness of this procedure was illustrated for the second example based on survey data for Lake Erie. A further illustration of the difference between results based on the Poisson and negative binomial distributions was given by calculating the probability of obtaining all samples sterile, assuming various bacterial densities and sample sizes.

If X 1, X 2, … is a sequence of independent Bernoulli random variables, the number of successes in the first n trials has a binomial distribution and the number of failures before the rth success has a negative binomial distribution. From both the binomial and the negative binomial distributions, the Poisson distribution is obtainable as a limit. Moreover, gamma distributions (integer shape parameters) are limits of negative binomial distributions, and the normal distribution is a limit of negative binomial, Poisson, and gamma distributions. These basic facts from elementary probability have natural extensions to two dimensions because there is a unique natural bivariate Bernoulli distribution. In this article, such extensions yielding a family of bivariate distributions are obtained and studied.

A seismic hazard and risk analysis is one of the requirements under the Dutch Mining Law when applying for a license for an activity in the deep subsurface. Current methods vary from qualitative to fully probabilistic methods reflecting the level of knowledge and expertise. This paper illustrates the methods used and their treatment of uncertainty.

Methodologic Challenges in the Analysis of Count Data in Radiology Health Services Research

Estimating Retail Demand with Poisson Mixtures and Out-of-Sample Likelihood

The foliage in a fully-leafed deciduous forest canopy is clumped. Consequently, theory indicates that the probability of beam penetration will be estimated more accurately with a model based on the negative binomial distribution than with a model based on the Poisson distribution, incorporating an assumption of a spherical leaf inclination angle distribution. Flux densities of photosynthetically active radiation (PAR) were measured in and above a deciduous forest canopy and were computed with the canopy radiative transfer models based on the negative binomial and Poisson distributions. These radiation values were used to compute canopy photosynthesis and stomatal conductance. Canopy photosynthesis and stomatal conductance, based on the negative binomial model, overestimated values computed from measured PAR profiles by 8 and 9%. respectively. The canopy photosynthesis and stomatal conductance values computed with the spherical Poisson model under-estimated measured values by 17 and 10%, respectively. Thus, the negative binomial radiative transfer model improves estimates of canopy photosynthesis and, to a lesser extent, stomatal conductance, inside a deciduous forest.

BackgroundRNA sequencing (RNA-seq) is a powerful tool for genome-wide expression profiling of biological samples with the advantage of high-throughput and high resolution. There are many existing algorithms nowadays for quantifying expression levels and detecting differential gene expression, but none of them takes the misaligned reads that are mapped to non-exonic regions into account. We developed a novel algorithm, XBSeq, where a statistical model was established based on the assumption that observed signals are the convolution of true expression signals and sequencing noises. The mapped reads in non-exonic regions are considered as sequencing noises, which follows a Poisson distribution. Given measureable observed and noise signals from RNA-seq data, true expression signals, assuming governed by the negative binomial distribution, can be delineated and thus the accurate detection of differential expressed genes.ResultsWe implemented our novel XBSeq algorithm and evaluated it by using a set of simulated expression datasets under different conditions, using a combination of negative binomial and Poisson distributions with parameters derived from real RNA-seq data. We compared the performance of our method with other commonly used differential expression analysis algorithms. We also evaluated the changes in true and false positive rates with variations in biological replicates, differential fold changes, and expression levels in non-exonic regions. We also tested the algorithm on a set of real RNA-seq data where the common and different detection results from different algorithms were reported.ConclusionsIn this paper, we proposed a novel XBSeq, a differential expression analysis algorithm for RNA-seq data that takes non-exonic mapped reads into consideration. When background noise is at baseline level, the performance of XBSeq and DESeq are mostly equivalent. However, our method surpasses DESeq and other algorithms with the increase of non-exonic mapped reads. Only in very low read count condition XBSeq had a slightly higher false discovery rate, which may be improved by adjusting the background noise effect in this situation. Taken together, by considering non-exonic mapped reads, XBSeq can provide accurate expression measurement and thus detect differential expressed genes even in noisy conditions.

Negative binomial maximum likelihood expectation maximization (NB-MLEM) algorithm for reconstruction of pre-corrected PET data

Effectiveness of sampling plans by attributes based on mixture distributions characterising microbial clustering in food

This paper introduced a new mixed distribution, namely the negative binomial-generalized exponential (NB-GE) distribution, which it obtained by mixing the negative binomial (NB) distribution with a generalized exponential (GE) distribution. The NB-GE distribution offers the advantage of being able to handle this kind of data sets, while still maintaining similar characteristics as the traditional negative binomial. The closed form and the factorial moment of the NB-GE distribution are derived. The NB-GE distribution appropriate when we find the one related to count data which have a large number of zeros and overdispersion in Poisson distribution. In addition, we present the basic properties of the new distribution such as mean, variance, skewness and kurtosis. Including, the negative binomial-exponential distributions are presented as special cases of this NB-GE distribution. Parameters estimation is also implemented using maximum likelihood method and the usefulness of the NB-GE distribution is illustrated by real data set. When using a goodness of fit test, the results show that the NB-GE distribution can provide a better fit than the NB distribution and Poisson distribution for count data that contain a large number of zeros.

Over the past two decades, seismic hazard modeling has advanced along two complementary frontiers: empirical probabilistic frameworks, which systematically capture uncertainty through statistical inference, and physics-based simulation platforms, which directly compute ground motions from the governing equations of wave propagation. This project seeks to unify these two worlds by developing an end-to-end integration between OpenQuake and CyberShake, thereby creating a new generation of seismic hazard models that are globally extensible, probabilistically complete, and physically consistent. CyberShake has been under active development for more than a decade, demonstrating its robustness and scientific maturity through extensive implementations in California. It performs a physics-based probabilistic seismic hazard analysis (PSHA), replacing traditional empirical Ground Motion Prediction Equations (GMPEs) with full 3D numerical simulations of seismic wave propagation. Built upon the UCERF2/3 Earthquake Rupture Forecasts, CyberShake computes hazard curves directly from synthetic seismograms generated via Strain Green’s Tensors and thousands of stochastic rupture variations. This approach enables non-ergodic, site-specific hazard estimation and has set a global benchmark for high-fidelity hazard computation. However, its application has remained geographically limited: both the ERF and 3D velocity models were designed specifically for California, requiring extensive datasets that are rarely available elsewhere. Conversely, OpenQuake, developed by the Global Earthquake Model (GEM) Foundation, provides a fully open-source, Python-based framework for probabilistic seismic hazard and risk analysis. It serves as the computational backbone of large-scale hazard models such as the European Seismic Hazard Model 2020 (ESHM20), which integrates decades of regional expertise into a unified and statistical representation. OpenQuake provides a complete probabilistic framework to build Earthquake Rupture Forecasts (ERFs) that combine declustered catalogs, background seismicity, and multi-branch logic trees, ensuring a balanced and uncertainty-aware representation of regional tectonics. Furthermore, its ecosystem extends seamlessly to vulnerability and exposure modules, enabling the translation of hazard into actionable risk assessments and resilience planning.This project will establish a direct pipeline from OpenQuake’s event-based results to the generation of an ERF compatible with CyberShake’s simulation framework, ensuring moment–rate consistency. By doing so, it will enable CyberShake simulations to be performed for regions beyond California, extending its use to Europe based on the knowledge contained in the ESHM20. The first pilot region is Istanbul, Turkey, a densely populated metropolis located near the western termination of the North Anatolian Fault. Our initial results show that the workflow is already functioning at the prototype level: we have developed a unified 3D velocity model for the Istanbul region by combining available tomographic models with local datasets; generated preliminary event-based rupture catalogs from ESHM20 using OpenQuake; and demonstrated early convergence behavior in hazard curves, indicating that the rupture sampling strategy is statistically robust. These initial results demonstrate the feasibility of the integration approach and indicate that the essential elements needed for a CyberShake-ready ERF are already in place.

Holtekamp bridge which is located in Jayapura region between Hamadi and Holtekamp Village in the Northern part of Papua Island. The region constitutes an active earthquake zone with the recurrence frequency and magnitude of the earthquake are relatively high. The region is located on an active fault zone. The seismotectonic setting in the region is directly implied to the calculations of the seismic design for engineering and design purposes on building and non-building structures including Holtekamp bridge in Jayapura city. To recognize a characteristic of the earthquake at the site of interest, the historical earthquakes (background) data surrounding the study area. Local microtremor measurements data are analyzed by means of a probabilistic seismic hazard analysis (PSHA). Deterministic seismic hazard analysis (DSHA) using some ground-motion models in attenuation relationship equations. In resulting of seismic hazard parameter as represented by peak ground acceleration (PGA), values in earthquake scenario at operating basis earthquake (OBE), maximum design earthquake (MDE), and maximum credible earthquake (MCE) conditions. These values are used as a reference to evaluate the compliance of the current technical aspects with the new required design facing the updated seismicity parameters.

This paper is concerned with filtering of hidden Markov processes (HMP) which possess (or approximately possess) the property of lumpability. This property is a generalization of the property of lumpability of a Markov chain which has been previously addressed by others. In essence, the property of lumpability means that there is a partition of the (atomic) states of the Markov chain into aggregated sets which act in a similar manner as far as the state dynamics and observation statistics are concerned. We prove necessary and sufficient conditions on the HMP for exact lumpability to hold. For a particular class of hidden Markov models (HMM), namely finite output alphabet models, conditions for lumpability of all HMP representable by a specified HMM are given. The corresponding optimal filter algorithms for the aggregated states are then derived. The paper also describes an approach to efficient suboptimal filtering for HMP which are approximately lumpable. By this we mean that the HMM generating the process may be approximated by a lumpable HMM. This approach involves directly finding a lumped HMM which approximates the original HMM well, in a matrix norm sense. An alternative approach for model reduction based on approximating a given HMM by an exactly lumpable HMM is also derived. This method is based on the alternating convex projections algorithm. Some simulation examples are presented which illustrate the performance of the suboptimal filtering algorithms.

The hidden Markov model (HMM) and the high order Markov model have higher prediction accuracy than the first order Markov model, and then widely used in pattern recognition such as speech, handwriting and gesture recognition. In the high-order Markov chain, the number of parameters grows exponentially with respect to the order, and hampers the parameter estimation. To solve these problems, Raftery introduced the mixture transition distribution (MTD) model in 1985 as a parsimonious model for high-order Markov chains. However, the parameter estimation of MTD model is still not easy when using the EM algorithm. In this paper we propose a new Markov model with high-order hidden process and MTD. We show that, by assuming that the latent process follows a second-order Markov chain, the class of high-order Markov models can be generalized in an advisable way. This combination generalizes not only the MTD model, but also the HMM. To reduce some unnecessary errors in parameter estamation, we use the scaling procedure. Moreover, an application using an impulsive noise sequence shows that the generalization can lead to better results than its nested models.

The Poisson-lognormal distribution was suggested as a model for commonness of species by Preston. The analysis leading him to suggest this distribution was as follows: Suppose that the number of each species caught in a trap is assumed to be a single sample from a Poisson distribution with mean λ. R. S. Uhler and P. G. Bradley proposed two models to describe a phenomenon in petroleum exploration. These two models used the sum of lognormal random variables as a compounder of Poisson and negative binomial distributions. P. Holgate noted that a clear distinction between the P-LN and the lognormal has not always been maintained. R. M. Cassie indicated that for most plankton populations, samples with μ = 100 would not depart significantly from a lognormal distribution.