Estimating the Components of a Mixture of Extremal Distributions under Strong Dependence

In this paper, we provide a method based on quantiles to estimate the parameters of a finite mixture of Fréchet distributions, for a large sample of strongly dependent data. This is a situation that appears when dealing with environmental data and there was a real need of such method. We validate our approach by means of estimation and goodness-of-fit testing over simulated data, showing an accurate performance.

A Bayesian Mixture Model Approach to Disparity Testing

The topic of this article is one-sided hypothesis testing for disparity, i.e., the mean of one group is larger than that of another when there is uncertainty as to which group a datum is drawn. For each datum, the uncertainty is captured with a given discrete probability distribution over the groups. Such situations arise, for example, in the use of Bayesian imputation methods to assess race and ethnicity disparities with certain insurance, health, and financial data. A widely used method to implement this assessment is the Bayesian Improved Surname Geocoding (BISG) method which assigns a discrete probability over six race/ethnicity groups to an individual given the individual’s surname and address location. Using a Bayesian framework and Markov Chain Monte Carlo sampling from the joint posterior distribution of the group means, the probability of a disparity hypothesis is estimated. Four methods are developed and compared with an illustrative data set. Three of these methods are implemented in an R-code and one method in WinBUGS. These methods are programed for any number of groups between two and six inclusive. All the codes are provided in the appendices.

Distribution performance of gas–liquid mixture in the shell side of spiral-wound heat exchangers

The non-uniformity of gas–liquid mixture is a critical issue which leads to the heat transfer deterioration of spiralwound heat exchangers(SWHEs).Two-phase mass flow rate and the content of gas are important parameters as well as structural parameters which have prominent influences on flow distribution uniformity of SWHE shell side.In order to investigate the influences of these parameters,an experimental test system was built using water and air as mediums and a novel distributor named"tubes distributor"was designed.The effects of mass flow rate and the content of gas on two-phase distribution performance were analyzed,where the mass flow rate ranged from 28.4 to 171.9 kg·h-1 and the content of gas changed from 0.2 to 0.8,respectively.The results showed that the mixture mass flow rate considerably influenced the liquid distribution than that of gas phase and the larger mass flow rate exhibited the better distribution uniformity of two-phase flow.It was also found that the tubes distributor had the better two-phase uniformity when the content of gas was around 0.4.Tube diameter played an important role in the distribution of gas phase and slit width was more significant for the uniformity of liquid phase.

Effect of Particle Size Distribution on Radiative Heat Transfer in High-Temperature Homogeneous Gas-Particle Mixtures

The weighted-sum-of-gray-gas(WSGG)model and Mie theory are applied to study the influents of particle size on the radiative transfer in high temperature homogeneous gas-particle mixtures,such as the flame in aero-engine combustor.The radiative transfer equation is solved by the finite volume method.The particle size is assumed to obey uniform distribution and logarithmic normal(L-N)distribution,respectively.Results reveal that when particle size obeys uniform distribution,increasing particle size with total particle volume fraction fvunchanged will result in the decreasing of the absolute value of radiative heat transfer properties,and the effect of ignoring particle scattering will also be weakened.Opposite conclusions can be obtained when total particle number concentration N0 is unchanged.Moreover,if particle size obeys L-N distribution,increasing the narrowness indexσor decreasing the characteristic diameter Dˉwith the total particle volume fraction fvunchanged will increase the absolute value of radiative heat transfer properties.With total particle number concentration N0 unchanged,opposite conclusions for radiative heat source and incident radiation terms can be obtained except for radiative heat flux term.As a whole,the effects of particle size on the radiative heat transfer in the high-temperature homogeneous gas-particle mixtures are complicated,and the particle scattering cannot be ignoring just according to the particle size.

Prediction Based on Generalized Order Statistics from a Mixture of Rayleigh Distributions Using MCMC Algorithm

This article considers the problem in obtaining the maximum likelihood prediction (point and interval) and Bayesian prediction (point and interval) for a future observation from mixture of two Rayleigh (MTR) distributions based on generalized order statistics (GOS). We consider one-sample and two-sample prediction schemes using the Markov chain Monte Carlo (MCMC) algorithm. The conjugate prior is used to carry out the Bayesian analysis. The results are specialized to upper record values. Numerical example is presented in the methods proposed in this paper.

Evaluating Traffic Congestion Using the Traffic Occupancy and Speed Distribution Relationship: An Application of Bayesian Dirichlet Process Mixtures of Generalized Linear Model

Accurate classification and prediction of future traffic conditions are essential for developing effective strategies for congestion mitigation on the highway systems. Speed distribution is one of the traffic stream parameters, which has been used to quantify the traffic conditions. Previous studies have shown that multi-modal probability distribution of speeds gives excellent results when simultaneously evaluating congested and free-flow traffic conditions. However, most of these previous analytical studies do not incorporate the influencing factors in characterizing these conditions. This study evaluates the impact of traffic occupancy on the multi-state speed distribution using the Bayesian Dirichlet Process Mixtures of Generalized Linear Models (DPM-GLM). Further, the study estimates the speed cut-point values of traffic states, which separate them into homogeneous groups using Bayesian change-point detection (BCD) technique. The study used 2015 archived one-year traffic data collected on Florida’s Interstate 295 freeway corridor. Information criteria results revealed three traffic states, which were identified as free-flow, transitional flow condition (congestion onset/offset), and the congested condition. The findings of the DPM-GLM indicated that in all estimated states, the traffic speed decreases when traffic occupancy increases. Comparison of the influence of traffic occupancy between traffic states showed that traffic occupancy has more impact on the free-flow and the congested state than on the transitional flow condition. With respect to estimating the threshold speed value, the results of the BCD model revealed promising findings in characterizing levels of traffic congestion.

Evaluation of the Effectiveness of a 4-Months Continuous Injection of a Gas Mixture (CO₂and Inert Gases) on <i>Legionella</i>Contamination of a Hot Water Distribution System

Within the biofilm and scales Legionella is less far susceptible to the effects of the most frequently used biocides. The objective of this study was to evaluate the effect of a 4-months continuous injection of a gas mixture (CO2 and inert gas) in the hot water distribution system of a large hotel colonized by L. pneumophila sg3 on limiting biofilm formation and scales and in turn Legionella growth. Before the continuous injection of the gas mixture, out of the 15 sampling points examined every month 60% were colonized by Legionella (mean concentrations of 102 cfu/L in the boilers and the return loop, and 104 cfu/L in taps and showers). One week after the injection of the gas mixture and daily fluxing of the distal outlets, the level of colonization decreased (3 cfu/L). When it was decided to flux all the distal outlets only 1 day per week the mean concentration of Legionella increased again (>104 cfu/L) in all the sampling points. Thus, cleaning of the boilers was performed and distal outlets were again fluxed daily. One week after the level of contamination decreased again (2 cfu/L). Nonetheless, the colonization was not eliminated and when fluxing of the distal outlets was not performed every day the mean concentrations of Legionella raised up to >104 cfu/L. Results indicate that the gas mixture was able to reduce the level of colonization by Legionella only because associated to the fluxing of the distal outlets.

Adaptive Sparse Group Variable Selection for a Robust Mixture Regression Model Based on Laplace Distribution

The traditional estimation of Gaussian mixture model is sensitive to heavy-tailed errors;thus we propose a robust mixture regression model by assuming that the error terms follow a Laplace distribution in this article. And for the variable selection problem in our new robust mixture regression model, we introduce the adaptive sparse group Lasso penalty to achieve sparsity at both the group-level and within-group-level. As numerical experiments show, compared with other alternative methods, our method has better performances in variable selection and parameter estimation. Finally, we apply our proposed method to analyze NBA salary data during the period from 2018 to 2019.

Inference and Properties of Mixture Two Extreme Lower Bound Distributions

In this paper, we discuss the mixture model of two extreme lower bound distributions. First, some properties we obtain of the model with hazard function are discussed. In addition, the estimates of the unknown parameters via the EM algorithm are obtained. The performance of the findings in the paper is showed by demonstrating some numerical illustrations through Monte Carlo simulation.

Inference on Constant-Partially Accelerated Life Tests for Mixture of Pareto Distributions under Progressive Type-II Censoring

The main purpose of this paper is to obtain the inference of parameters of heterogeneous population represented by finite mixture of two Pareto (MTP) distributions of the second kind. The constant-partially accelerated life tests are applied based on progressively type-II censored samples. The maximum likelihood estimates (MLEs) for the considered parameters are obtained by solving the likelihood equations of the model parameters numerically. The Bayes estimators are obtained by using Markov chain Monte Carlo algorithm under the balanced squared error loss function. Based on Monte Carlo simulation, Bayes estimators are compared with their corresponding maximum likelihood estimators. The two-sample prediction technique is considered to derive Bayesian prediction bounds for future order statistics based on progressively type-II censored informative samples obtained from constant-partially accelerated life testing models. The informative and future samples are assumed to be obtained from the same population. The coverage probabilities and the average interval lengths of the confidence intervals are computed via a Monte Carlo simulation to investigate the procedure of the prediction intervals. Analysis of a simulated data set has also been presented for illustrative purposes. Finally, comparisons are made between Bayesian and maximum likelihood estimators via a Monte Carlo simulation study.

A Special Weight for Inverse Gaussian Mixing Distribution in Normal Variance Mean Mixture with Application

Normal Variance-Mean Mixture (NVMM) provides a general framework for deriving models with desirable properties for modelling financial market variables such as exchange rates, equity prices, and interest rates measured over short time intervals, i.e. daily or weekly. Such data sets are characterized by non-normality and are usually skewed, fat-tailed and exhibit excess kurtosis. The Generalised Hyperbolic distribution (GHD) introduced by Barndorff-Nielsen (1977) which act as Normal variance-mean mixtures with Generalised Inverse Gaussian (GIG) mixing distribution nest a number of special and limiting case distributions. The Normal Inverse Gaussian (NIG) distribution is obtained when the Inverse Gaussian is the mixing distribution, i.e., the index parameter of the GIG is . The NIG is very popular because of its analytical tractability. In the mixing mechanism, the mixing distribution characterizes the prior information of the random variable of the conditional distribution. Therefore, considering finite mixture models is one way of extending the work. The GIG is a three parameter distribution denoted by and nest several special and limiting cases. When , we have which is called an Inverse Gaussian (IG) distribution. When , , , we have , and distributions respectively. These distributions are related to and are called weighted inverse Gaussian distributions. In this work, we consider a finite mixture of and and show that the mixture is also a weighted Inverse Gaussian distribution and use it to construct a NVMM. Due to the complexity of the likelihood, direct maximization is difficult. An EM type algorithm is provided for the Maximum Likelihood estimation of the parameters of the proposed model. We adopt an iterative scheme which is not based on explicit solution to the normal equations. This subtle approach reduces the computational difficulty of solving the complicated quantities involved directly to designing an iterative scheme based on a representation of the normal equation. The algorithm is easily programmable and we obtained a monotonic convergence for the data sets used.

A Finite Mixture of Generalised Inverse Gaussian with Indexes -1/2 and -3/2 as Mixing Distribution for Normal Variance Mean Mixture with Application

Mixture models have become more popular in modelling compared to standard distributions. The mixing distributions play a role in capturing the variability of the random variable in the conditional distribution. Studies have lately focused on finite mixture models as mixing distributions in the mixing mechanism. In the present work, we consider a Normal Variance Mean mixture model. The mixing distribution is a finite mixture of two special cases of Generalised Inverse Gaussian distribution with indexes -1/2 and -3/2. The parameters of the mixed model are obtained via the Expectation-Maximization (EM) algorithm. The iterative scheme is based on a presentation of the normal equations. An application to some financial data has been done.

Variable Selection for Robust Mixture Regression Model with Skew Scale Mixtures of Normal Distributions

In this paper, we propose a robust mixture regression model based on the skew scale mixtures of normal distributions (RMR-SSMN) which can accommodate asymmetric, heavy-tailed and contaminated data better. For the variable selection problem, the penalized likelihood approach with a new combined penalty function which balances the SCAD and l2 penalty is proposed. The adjusted EM algorithm is presented to get parameter estimates of RMR-SSMN models at a faster convergence rate. As simulations show, our mixture models are more robust than general FMR models and the new combined penalty function outperforms SCAD for variable selection. Finally, the proposed methodology and algorithm are applied to a real data set and achieve reasonable results.

Gaussian-Student's t mixture distribution PHD robust filtering algorithm based on variational Bayesian inference

Aiming at the problem of filtering precision degradation caused by the random outliers of process noise and measurement noise in multi-target tracking(MTT) system,a new Gaussian-Student’s t mixture distribution probability hypothesis density(PHD) robust filtering algorithm based on variational Bayesian inference(GST-vbPHD) is proposed.Firstly,since it can accurately describe the heavy-tailed characteristics of noise with outliers,Gaussian-Student’s t mixture distribution is employed to model process noise and measurement noise respectively.Then Bernoulli random variable is introduced to correct the likelihood distribution of the mixture probability,leading hierarchical Gaussian distribution constructed by the Gaussian-Student’s t mixture distribution suitable to model non-stationary noise.Finally,the approximate solutions including target weights,measurement noise covariance and state estimation error covariance are obtained according to variational Bayesian inference approach.The simulation results show that,in the heavy-tailed noise environment,the proposed algorithm leads to strong improvements over the traditional PHD filter and the Student’s t distribution PHD filter.

Gaussian mixture model clustering with completed likelihood minimum message length criterion

An improved Gaussian mixture model （GMM）- based clustering method is proposed for the difficult case where the true distribution of data is against the assumed GMM. First, an improved model selection criterion, the completed likelihood minimum message length criterion, is derived. It can measure both the goodness-of-fit of the candidate GMM to the data and the goodness-of-partition of the data. Secondly, by utilizing the proposed criterion as the clustering objective function, an improved expectation- maximization （EM） algorithm is developed, which can avoid poor local optimal solutions compared to the standard EM algorithm for estimating the model parameters. The experimental results demonstrate that the proposed method can rectify the over-fitting tendency of representative GMM-based clustering approaches and can robustly provide more accurate clustering results.

Establishment Method of a Mixture Model and Its Practical Application for Transmission Gears in an Engineering Vehicle

Highly versatile machines, such as wheel loaders, forklifts, and mining haulers, are subject to many kinds of working conditions, as well as indefinite factors that lead to the complexity of the load. The load probability distribution function （PDF） of transmission gears has many distributions centers; thus, its PDF cannot be well represented by just a single-peak function. For the purpose of representing the distribution characteristics of the complicated phenomenon accurately, this paper proposes a novel method to establish a mixture model. Based on linear regression models and correlation coefficients, the proposed method can be used to automatically select the best-fitting function in the mixture model. Coefficient of determination, the mean square error, and the maximum deviation are chosen and then used as judging criteria to describe the fitting precision between the theoretical distribution and the corresponding histogram of the available load data. The applicability of this modeling method is illustrated by the field testing data of a wheel loader. Meanwhile, the load spectra based on the mixture model are compiled. The comparison results show that the mixture model is more suitable for the description of the load-distribution characteristics. The proposed research improves the flexibility and intelligence of modeling, reduces the statistical error and enhances the fitting accuracy, and the load spectra complied by this method can better reflect the actual load characteristic of the gear component.

A WYNER-ZIV VIDEO CODING METHOD UTILIZING MIXTURE CORRELATION NOISE MODEL

In Wyner-Ziv (WZ) Distributed Video Coding (DVC), correlation noise model is often used to describe the error distribution between WZ frame and the side information. The accuracy of the model can influence the performance of the video coder directly. A mixture correlation noise model in Discrete Cosine Transform (DCT) domain for WZ video coding is established in this paper. Different correlation noise estimation method is used for direct current and alternating current coefficients. Parameter estimation method based on expectation maximization algorithm is used to estimate the Laplace distribution center of direct current frequency band and Mixture Laplace-Uniform Distribution Model (MLUDM) is established for alternating current coefficients. Experimental results suggest that the proposed mixture correlation noise model can describe the heavy tail and sudden change of the noise accurately at high rate and make significant improvement on the coding efficiency compared with the noise model presented by DIStributed COding for Video sERvices (DISCOVER).

Location Based Generalized Akash Distribution: Properties and Applications

In recent years, several statistical finite mixture models have been proposed to model the lifetime data with heterogeneity. The Lindley distribution has been highlighted by many authors for these types of lifetime data analysis. This paper introduces a new Lindley family distribution called location-based generalized Akash distribution (NGAD) with monotonic increasing and bathtub?failure rates. The density function of NGAD is flexible to cover the left-skewed, right-skewed and symmetrical shapes with different tail-weights. Its fundamental structural properties and its ability to provide a suitable statistical model for various types of data sets are studied. The maximum likelihood (ML) method is used to estimate its unknown parameters and the performance of ML estimates are examined by a simulation study. Finally, several real-data sets with different characteristics are used to illustrate its flexibility. It is observed that NGAD provides a better fit than some other existing modified Lindley distributions.

Microscopic Pore Structure of Asphalt Mixture Incorporating with MFL

In order to analyze the microstructure of salt anti-freezing asphalt concrete, i e, MFL（Mafilon） modified asphalt concrete, MIP（mercury intrusion porosity） method was used to obtain the data including porosity and pore size distribution in micro scale. Results show that the porosity grows up with the increase of immersion duration and the salt content. During the immersion, the amount of large pores（60-200 μm） grow up gradually and porosity also grows up correspondingly. Even with different immersion duration, most pores＇ size distribute is beyond 7000 nm.

Estimation under a Finite Mixture of Exponentiated Exponential Components Model and Balanced Square Error Loss

By exponentiating each of the components of a finite mixture of two exponential components model by a positive parameter, several shapes of hazard rate functions are obtained. Maximum likelihood and Bayes methods, based on square error loss function and objective prior, are used to obtain estimators based on balanced square error loss function for the parameters, survival and hazard rate functions of a mixture of two exponentiated exponential components model. Approximate interval estimators of the parameters of the model are obtained.