Exponential Mixture Research Articles

Multiple rogue wave, double-periodic soliton and breather wave solutions for a generalized breaking soliton system in (3 + 1)-dimensions

We focused on solitonic phenomena in wave propagation which was extracted from a generalized breaking soliton system in (3 + 1)-dimensions. The model describes the interaction phenomena between Riemann wave and long wave via two space variable in nonlinear media. Abundant double-periodic soliton, breather wave and the multiple rogue wave solutions to a generalized breaking soliton system by the Hirota bilinear form and a mixture of exponentials and trigonometric functions are presented. Periodic-soliton, breather wave and periodic are studied with the usage of symbolic computation. In addition, the symbolic computation and the applied methods for governing model are investigated. Through three-dimensional graph, density graph, and two-dimensional design using Maple, the physical features of double-periodic soliton and breather wave solutions are explained all right. The findings demonstrate the investigated model’s broad variety of explicit solutions. All outcomes in this work are necessary to understand the physical meaning and behavior of the explored results and shed light on the significance of the investigation of several nonlinear wave phenomena in sciences and engineering.

Continuous Erlang Mixtures and their Relation to Exponential Mixtures and Poisson Mixtures

Continuous Erlang Mixtures and their Relation to Exponential Mixtures and Poisson Mixtures

Exponential stochastic volatility model with Laplace returns and its variants

This paper explores the exponential stochastic volatility model generated by first-order exponential Markov sequences to model financial time series. The stationary exponential Markov sequences used to generate the volatility model are the linear exponential autoregressive model, minification model, new exponential autoregressive time series model, and an exponential mixture of autoregressive and minification processes. The statistical properties of the models are studied and the estimation of model parameters is carried out using the generalized method of moments. An approximate Kalman filtering method and a non-linear filtering approach are employed to diagnose the models. To verify the correctness of the estimations, simulation studies are conducted, and two real datasets are used to demonstrate how the models are applied. Furthermore, a comparison of the suggested models is performed using the measures root mean square error, mean absolute error, and Value at Risk via a backtesting procedure.

A dynamic count process

The current paper aims to complement the recent development of the observation-driven models of dynamic counts with a parametric-driven one for a general case, particularly discrete two parameters exponential family distributions. The current paper proposes a finite semiparametric exponential mixture of SETAR processes of the conditional mean of counts to capture the nonlinearity and complexity. Because of the intrinsic latency of the conditional mean, the general additive state-space representation of dynamic counts is firstly proposed then stationarity and geometric ergodicity are established under a mild set of conditions. We also propose to estimate the unknown parameters by using quasi maximum likelihood estimation and establishes the asymptotic properties of the quasi maximum likelihood estimators (QMLEs), particularly T-consistency and normality under the relatively mild set of conditions. Furthermore, the finite sample properties of the QMLEs are investigated via simulation exercises and an illustration of the proposed process is presented by applying the proposed method to the intraday transaction counts per minute of AstraZeneca stock.

Shape-margin knowledge augmented network for thyroid nodule segmentation and diagnosis

Background and ObjectiveThyroid nodule segmentation is a crucial step in the diagnostic procedure of physicians and computer-aided diagnosis systems. However, prevailing studies often treat segmentation and diagnosis as independent tasks, overlooking the intrinsic relationship between these processes. The sequencial steps of these independent tasks in computer-aided diagnosis systems may lead to the accumulation of errors. Therefore, it is worth combining them as a whole by exploring the relationship between thyroid nodule segmentation and diagnosis. According to the diagnostic procedure of thyroid imaging reporting and data system (TI-RADS), the assessment of shape and margin characteristics is the prerequisite for radiologists to discriminate benign and malignant thyroid nodules. Inspired by TI-RADS, this study aims to integrate these tasks into a cohesive process, leveraging the insights from TI-RADS, thereby enhancing the accuracy and interpretability of thyroid nodule analysis. MethodsSpecifically, this paper proposes a shape-margin knowledge augmented network (SkaNet) for simultaneous thyroid nodule segmentation and diagnosis. Due to the visual feature similarities between segmentation and diagnosis, SkaNet shares visual features in the feature extraction stage and then utilizes a dual-branch architecture to perform thyroid nodule segmentation and diagnosis tasks respectively. In the shared feature extraction, the combination of convolutional feature maps and self-attention maps allows to exploitation of both local information and global patterns in thyroid nodule images. To enhance effective discriminative features, an exponential mixture module is introduced, combining convolutional feature maps and self-attention maps through exponential weighting. Then, SkaNet is jointly optimized by a knowledge augmented multi-task loss function with a constraint penalty term. The constraint penalty term embeds shape and margin characteristics through numerical computations, establishing a vital relationship between thyroid nodule diagnosis results and segmentation masks. ResultsWe evaluate the proposed approach on a public thyroid ultrasound dataset (DDTI) and a locally collected thyroid ultrasound dataset. The experimental results reveal the value of our contributions and demonstrate that our approach can yield significant improvements compared with state-of-the-art counterparts. ConclusionsSkaNet highlights the potential of combining thyroid nodule segmentation and diagnosis with knowledge augmented learning into a unified framework, which captures the key shape and margin characteristics for discriminating benign and malignant thyroid nodules. Our findings suggest promising insights for advancing computer-aided diagnosis joint with segmentation.

A generalized Budan-Fourier approach to generalized Gaussian and exponential mixtures

<p>In the literature, finite mixture models were described as linear combinations of probability distribution functions having the form $ f(x) = \Lambda \sum\limits_{i = 1}^n w_i f_i(x) $, $ x \in \mathbb{R} $, where $ w_i $ were positive weights, $ \Lambda $ was a suitable normalising constant, and $ f_i(x) $ were given probability density functions. The fact that $ f(x) $ is a probability density function followed naturally in this setting. Our question was: <italic>if we removed the sign condition on the coefficients $ w_i $, how could we ensure that the resulting function was a probability density function?</italic></p><p>The solution that we proposed employed an algorithm which allowed us to determine all zero-crossings of the function $ f(x) $. Consequently, we determined, for any specified set of weights, whether the resulting function possesses <italic>no</italic> such zero-crossings, thus confirming its status as a probability density function.</p><p>In this paper, we constructed such an algorithm which was based on the definition of a suitable sequence of functions and that we called a <italic>generalized Budan-Fourier sequence</italic>; furthermore, we offered theoretical insights into the functioning of the algorithm and illustrated its efficacy through various examples and applications. Special emphasis was placed on generalized Gaussian mixture densities.</p>

Continuous Erlang Mixtures and their Relation to Exponential Mixtures and Poisson Mixtures

Continuous Erlang Mixtures and their Relation to Exponential Mixtures and Poisson Mixtures

Double-Periodic Soliton Solutions of the (2+1)-Dimensional Ito Equation

In this work, a (2 + 1)-dimensional Ito equation is investigated, which represents the generalization of the bilinear KdV equation. Abundant double-periodic soliton solutions to the (2 + 1)-dimensional Ito equation are presented by the Hirota bilinear form and a mixture of exponentials and trigonometric functions. The dynamic properties are described through some 3D graphics and contour graphics.

Estimation of the Xrama Distribution Parameter Under Complete and Progressive Type-II Censored Schemes

In this paper, a two-component mixture of Exponential and Rama distribution named Xrama distribution is introduced and discussed. The distributional properties such as moment and the related measures including other characteristics namely quantile function, mean residual life function, Bonferroni curve, Lorenz curve, order statistics, stochastic ordering, and entropy were derived. Simulation study for both complete and progressive type-2 censored schemes is used to illustrate and analyze the behavior of the model. The proposed model’s performance is assessed through a comparison with various other lifetime distributions using negative log-likelihood, Akaike Information Criterion, Corrected Akaike Information Criterion, Bayesian Information Criterion, and Hannan–Quinn information criterion on an actual dataset. The results reveal that, when modeling real lifetime data, the suggested model exhibits greater flexibility than the competing models examined in this study.

Fitting with matrix exponential mixtures generated by discrete probabilistic scaling

Matrix exponential (ME) distributions generalize phase-type distributions; however, their use in queueing theory is hampered by the difficulty of checking their feasibility. We propose a novel ME fitting algorithm that produces a valid distribution by construction. The ME distribution used during the fitting is a product of independent random variables that are easy to control in isolation. Consequently, the calculation of the CDF and the Mellin transform factorizes, making it possible to use these measures for the fitting without significant restriction on the distribution order. Trace-driven queueing simulations indicate that the resulting distributions yield highly accurate results.

Binomial-discrete Erlang-truncated exponential mixture and its application in cancer disease

Among diseases, cancer exhibits the fastest global spread, presenting a substantial challenge for patients, their families, and the communities they belong to. This paper is devoted to modeling such a disease as a special case. A newly proposed distribution called the binomial-discrete Erlang-truncated exponential (BDETE) is introduced. The BDETE is a mixture of binomial distribution with the number of trials (parameter n) taken after a discrete Erlang-truncated exponential distribution. A comprehensive mathematical treatment of the proposed distribution and expressions of its density, cumulative distribution function, survival function, failure rate function, Quantile function, moment generating function, Shannon entropy, order statistics, and stress-strength reliability, are provided. The distribution's parameters are estimated using the maximum likelihood method. Two real-world lifetime count data sets from the cancer disease, both of which are right-skewed and over-dispersed, are fitted using the proposed BDETE distribution to evaluate its efficacy and viability. We expect the findings to become standard works in probability theory and its related fields.

Recursive Selection of Hyperexponential Distributions in Approximation of Distributions with "Heavy Tails"

It is known that many quantities that determine the network characteristics of the functioning of an infocommunication network have probability distributions with "heavy tails", which can have a significant impact on network performance. Models with heavy-tailed distributions tend to be difficult to analyze. The analysis can be simplified by using an algorithm to approximate a heavy-tailed distri-bution by a hyperexponential distribution (a finite mixture of exponentials). The paper presents a algorithm for calculating the parameters of the hyperexponential distribution components, which is based on a recursive selection of parameters. This algorithm allows you to analyze various models of queues, including G/G/1. It is shown that the approach under consideration is applicable to the approxi-mation of monotonically decreasing distributions, including those with a "heavy tail". Examples of approximation of Pareto and Weibull distributions are given.

Retracted: Bayesian Analysis of Cancer Data Using a 4-Component Exponential Mixture Model.

[This retracts the article DOI: 10.1155/2021/6289337.].

Bayesian quantile regression for longitudinal count data

ABSTRACT This work introduces Bayesian quantile regression modelling framework for the analysis of longitudinal count data. In this model, the response variable is not continuous and hence an artificial smoothing of counts is incorporated. The Bayesian implementation utilizes the normal–exponential mixture representation of the asymmetric Laplace distribution for the response variable. An efficient Gibbs sampling algorithm is derived for fitting the model to the data. The model is illustrated through simulation studies and implemented in an application drawn from neurology. Model comparison demonstrates the practical utility of the proposed model.

A Fully Bayesian Inference with Gibbs Sampling for Finite and Infinite Discrete Exponential Mixture Models

ABSTRACT In this paper, we propose clustering algorithms based on finite mixture and infinite mixture models of exponential approximation to the Multinomial Generalized Dirichlet (EMGD), Multinomial Beta-Liouville (EMBL) and Multinomial Shifted-Scaled Dirichlet (EMSSD) with Bayesian inference. The finite mixtures have already shown superior performance in real data sets clustering using the Expectation–Maximization approach. The proposed approaches in this paper are based on a Monte Carlo simulation technique namely Gibbs sampling algorithm including an additional Metropolis–Hastings step, and we utilize exponential family conjugate prior information to construct their posterior relying on Bayesian theory. Furthermore, we also present the infinite models based on Dirichlet processes, which results in clustering algorithms that do not require the specification of the number of mixture components to be given in advance and selects it in a principled manner. The performance of our Bayesian approaches was evaluated in some challenging real-world applications concerning text sentiment analysis, fake news detection, and human face gender recognition.

Exponential Mixture Density Based Approximation to Posterior Cramér-Rao Lower Bound for Distributed Target Tracking

The posterior Cramr-Rao lower bound (PCRLB) and a number of its extensions including the unconditional PCRLB (U-PCRLB) and the conditional PCRLB (C-PCRLB) have been widely studied in multi-sensor target tracking (MSTT). Previous MSTT works assume that the measurements are conditionally independent, which is often not the case in practice. When the correlations of the measurements are unknown, the standard Bayes update and PCRLB computations cannot be performed. Inspired by geometric average (GA) fusion that implements data fusion with unknown data correlations by employing exponential mixture density (EMD) to compute the global posterior, EMD-based approximations to U-PCRLB and C-PCRLB are derived in this paper for two classic distributed fusion architectures, namely the hierarchical and consensus architectures. The PCRLB can be decomposed into two parts, one coming from the prior information and the other from the information contained in the data. The data information part of PCRLB is approximated via EMD-based posterior, leading to the approximation of the PCRLB. We present a sequential Monte Carlo solution to recursively compute the proposed approximation to the PCRLB for nonlinear non-Gaussian estimation problems. Numerical simulations are provided to show that the proposed approximation to the bound on the estimation mean square error (MSE) is tighter compared to the existing bounds based on likelihood fusion obtained under the measurement independence assumption.

Bayesian Analysis of Cancer Data Using a 4-Component Exponential Mixture Model.

Cancer is among the major public health problems as well as a burden for Pakistan. About 148,000 new patients are diagnosed with cancer each year, and almost 100,000 patients die due to this fatal disease. Lung, breast, liver, cervical, blood/bone marrow, and oral cancers are the most common cancers in Pakistan. Perhaps smoking, physical inactivity, infections, exposure to toxins, and unhealthy diet are the main factors responsible for the spread of cancer. We preferred a novel four-component mixture model under Bayesian estimation to estimate the average number of incidences and death of both genders in different age groups. For this purpose, we considered 28 different kinds of cancers diagnosed in recent years. Data of registered patients all over Pakistan in the year 2012 were taken from GLOBOCAN. All the patients were divided into 4 age groups and also split based on genders to be applied to the proposed mixture model. Bayesian analysis is performed on the data using a four-component exponential mixture model. Estimators for mixture model parameters are derived under Bayesian procedures using three different priors and two loss functions. Simulation study and graphical representation for the estimates are also presented. It is noted from analysis of real data that the Bayes estimates under LINEX loss assuming Jeffreys' prior is more efficient for the no. of incidences in male and female. As far as no. of deaths are concerned again, LINEX loss assuming Jeffreys' prior gives better results for the male population, but for the female population, the best loss function is SELF assuming Jeffreys' prior.

Incorporating spatial structure into inclusion probabilities for Bayesian variable selection in generalized linear models with the spike-and-slab elastic net

Spike-and-slab priors model predictors as arising from a mixture of distributions: those that should (slab) or should not (spike) remain in the model. The spike-and-slab lasso (SSL) is a mixture of double exponentials, extending the single lasso penalty by imposing different penalties on parameters based on their inclusion probabilities. The SSL was extended to Generalized Linear Models (GLM) for application in genetics/genomics, and can handle many highly correlated predictors of a scalar outcome, but does not incorporate these relationships into variable selection. When images/spatial data are used to model a scalar outcome, relevant parameters tend to cluster spatially, and model performance may benefit from incorporating spatial structure into variable selection. We propose to incorporate spatial information by assigning intrinsic autoregressive priors to the logit prior probabilities of inclusion, which results in more similar shrinkage penalties among spatially adjacent parameters. Using MCMC to fit Bayesian models can be computationally prohibitive for large-scale data, but we fit the model by adapting a computationally efficient coordinate-descent-based EM algorithm. A simulation study and an application to Alzheimer’s Disease imaging data show that incorporating spatial information can improve model fitness.

Influence Function and Bootstrap Methods of Estimating the Standard Errors of the Estimators of Mixture Exponential Distribution Parameter

This work estimated the standard error of the maximum likelihood estimator (MLE) and the robust estimators of the exponential mixture parameter (θ) using the influence function and the bootstrap approaches. Mixture exponential random samples of sizes 10, 15, 20, 25, 50, and 100 were generated using 3 mixture exponential models at 2%, 5% and 10% contamination levels. The selected estimators namely: mean, median, alpha-trimmed mean, Huber M-estimate and their standard errors (Tn ) were estimated using the two approaches at the indicated sample sizes and contamination levels. The results were compared using the coefficient of variation, confidence interval and the asymptotic relative efficiency of Tn in order to find out which approach yields the more reliable, precise and efficient estimate of Tn. The results of the analysis show that the two approaches do not equally perform at all conditions. From the results, the bootstrap method was found to be more reliable and efficient method of estimating the standard error of the arithmetic mean at all sample sizes and contamination levels. In estimating the standard error of the median, the influence function method was found to be more effective especially when the sample size is small and yet contamination is high. The influence function based approach yielded more reliable, precise and efficient estimates of the standard errors of the alpha-trimmed mean and the Huber M-estimate for all sample sizes and levels of contamination although the reliability of the bootstrap method improved better as sample size increased to 50 and above. All simulations and analysis were carried out in R programming language.

Sequential Test for a Mixture of Finite Exponential Distribution

Testing the number of components in a finite mixture is considered one of the challenging problems. In this paper, exponential finite mixtures are used to determine the number of components in a finite mixture. A sequential testing procedure is adopted based on the likelihood ratio test (LRT) statistic. The distribution of the test statistic under the null hypothesis is obtained using a resampling technique based on B bootstrap samples. The quantiles of the distribution of the test statistic are evaluated from the B bootstrap samples. The performance of the test is examined through the empirical power and application on two real datasets. The proposed procedure is not only used for testing the number of components but also for estimating the optimal number of components in a finite exponential mixture distribution. The innovation of this paper is the sequential test, which tests the more general hypothesis of a finite exponential mixture of k components versus a mixture of k + 1 components. The special case of testing an exponential mixture of one component versus two components is the one commonly used in the literature.