Finite Mixture Distribution Research Articles

Joint genome-wide prediction in several populations accounting for randomness of genotypes: A hierarchical Bayes approach. II: Multivariate spike and slab priors for marker effects and derivation of approximate Bayes and fractional Bayes factors for the complete family of models

This study corresponds to the second part of a companion paper devoted to the development of Bayesian multiple regression models accounting for randomness of genotypes in across population genome-wide prediction. This family of models considers heterogeneous and correlated marker effects and allelic frequencies across populations, and has the ability of considering records from non-genotyped individuals and individuals with missing genotypes in any subset of loci without the need for previous imputation, taking into account uncertainty about imputed genotypes. This paper extends this family of models by considering multivariate spike and slab conditional priors for marker allele substitution effects and contains derivations of approximate Bayes factors and fractional Bayes factors to compare models from part I and those developed here with their null versions. These null versions correspond to simpler models ignoring heterogeneity of populations, but still accounting for randomness of genotypes. For each marker loci, the spike component of priors corresponded to point mass at 0 in RS, where S is the number of populations, and the slab component was a S-variate Gaussian distribution, independent conditional priors were assumed. For the Gaussian components, covariance matrices were assumed to be either the same for all markers or different for each marker. For null models, the priors were simply univariate versions of these finite mixture distributions. Approximate algebraic expressions for Bayes factors and fractional Bayes factors were found using the Laplace approximation. Using the simulated datasets described in part I, these models were implemented and compared with models derived in part I using measures of predictive performance based on squared Pearson correlations, Deviance Information Criterion, Bayes factors, and fractional Bayes factors. The extensions presented here enlarge our family of genome-wide prediction models making it more flexible in the sense that it now offers more modeling options.

A Study on The Mixture of Exponentiated-Weibull Distribution

Mixtures of measures or distributions occur frequently in the theory and applications of probability and statistics. In the simplest case it may, for example, be reasonable to assume that one is dealing with the mixture in given proportions of a finite number of normal populations with different means or variances. The mixture parameter may also be denumerable infinite, as in the theory of sums of a random number of random variables, or continuous, as in the compound Poisson distribution. The use of finite mixture distributions, to control for unobserved heterogeneity, has become increasingly popular among those estimating dynamic discrete choice models. One of the barriers to using mixture models is that parameters that could previously be estimated in stages must now be estimated jointly: using mixture distributions destroys any additive reparability of the log likelihood function. In this thesis, the maximum likelihood estimators have been obtained for the parameters of the mixture of exponentiated Weibull distribution when sample is available from censoring scheme. The maximum likelihood estimators of the parameters and the asymptotic variance covariance matrix have been also obtained. A numerical illustration for these new results is given.

Maximum a-posteriori estimation of autoregressive processes based on finite mixtures of scale-mixtures of skew-normal distributions

ABSTRACTThis article investigates maximum a-posteriori (MAP) estimation of autoregressive model parameters when the innovations (errors) follow a finite mixture of distributions that, in turn, are scale-mixtures of skew-normal distributions (SMSN), an attractive and extremely flexible family of probabilistic distributions. The proposed model allows to fit different types of data which can be associated with different noise levels, and provides a robust modelling with great flexibility to accommodate skewness, heavy tails, multimodality and stationarity simultaneously. Also, the existence of convenient hierarchical representations of the SMSN random variables allows us to develop an EM-type algorithm to perform the MAP estimates. A comprehensive simulation study is then conducted to illustrate the superior performance of the proposed method. The new methodology is also applied to annual barley yields data.

Statistical analysis and probabilistic modeling of WIM monitoring data of an instrumented arch bridge

Traffic load and volume is one of the most important physical quantities for bridge safety evaluation and maintenance strategies formulation. This paper aims to conduct the statistical analysis of traffic volume information and the multimodal modeling of gross vehicle weight (GVW) based on the monitoring data obtained from the weigh-in-motion (WIM) system instrumented on the arch Jiubao Bridge located in Hangzhou, China. A genetic algorithm (GA)-based mixture parameter estimation approach is developed for derivation of the unknown mixture parameters in mixed distribution models. The statistical analysis of one-year WIM data is firstly performed according to the vehicle type, single axle weight, and GVW. The probability density function (PDF) and cumulative distribution function (CDF) of the GVW data of selected vehicle types are then formulated by use of three kinds of finite mixed distributions (normal, lognormal and Weibull). The mixture parameters are determined by use of the proposed GA-based method. The results indicate that the stochastic properties of the GVW data acquired from the field-instrumented WIM sensors are effectively characterized by the method of finite mixture distributions in conjunction with the proposed GA-based mixture parameter identification algorithm. Moreover, it is revealed that the Weibull mixture distribution is relatively superior in modeling of the WIM data on the basis of the calculated Akaike

Reliability Analysis for Monte Carlo Simulation Using the Expectation- Maximization Algorithm for a Weibull Mixture Distribution Model

This paper presents a simulation study of a finite Weibull mixture distribution (WMD) for modelling life data related to system components with different failure modes. The main aim of this study is to compare two analytical methods for estimating the parameters of WMD models, the maximum likelihood estimation (MLE) method using the expectation-maximization (EM) algorithm, [A1] and the non-linear median rank regression (NLMRR) method with the Levenberg-Marquardt algorithm. To perform this comparison, the Monte Carlo simulation technique is implemented to generate several replicates for complete failure data and censored data based on samples of different sizes that follow a two-component WMD. This study showed that MLE using the EM algorithm yields more accurate parameter estimates than the NLMRR method for small or moderate complete failure data samples. This method also converges faster than the NLMRR method for large samples that include censored data.

A Bayesian mixture model for short-term average link travel time estimation using large-scale limited information trip-based data

Accurate estimation and prediction of urban link travel times are important for urban traffic operations and management. This paper develops a Bayesian mixture model to estimate short-term average urban link travel times using large-scale trip-based data with partial information. Unlike typical GPS trajectory data, trip-based data from taxies or other sources provide limited trip level information, which only contains the trip origin and destination locations, trip travel times and distances, etc. The focus of this study is to develop a robust probabilistic short-term average link travel time estimation model and demonstrate the feasibility of estimating network conditions using large-scale trip level information. In the model, the path taken by each trip is considered as latent and modeled using a multinomial logit distribution. The observed trip data given the possible path set and the mean and variance of the average link travel times can thus be characterized using a finite mixture distribution. A transition model is also introduced to serve as an informative prior that captures the temporal and spatial dependencies of link travel times. A solution approach based on the expectation–maximization (EM) algorithm is proposed to solve the problem. The model is tested on estimating the mean and variance of the average link travel times for 30 min time intervals using a large-scale taxi trip dataset from New York City. More robust estimation results are obtained owing to the adoption of the Bayesian framework.

A self-adaptive multi-view framework for multi-source information service in cloud ITS

In traditional intelligent transportation system (ITS), the information source is often collected from a single view. However, as ITS becoming increasingly complicated, there is a need to describe one object from several different information views/domains. Information from multiple sources can provide extra worthwhile information for ITS users especially in cloud environment. Moreover, existing local ITSs usually provide information by fixed algorithms, which is not adequate to the dynamic transportation scenarios that produce big traffic data with time series. In this paper, we propose a complete self-adaptive multi-view framework for multi-source information service in cloud ITS, which mainly consists of a Newton multi-parameter optimization, a multi-layer feed-forward neural network and a finite multi-view mixture distribution. A simulation on real-world application, with six different types of information views, demonstrates the underlying effectiveness of the proposed framework.

Likelihood Ratio Test for Multi-Sample Mixture Model and Its Application to Genetic Imprinting

Genomic imprinting is a known aspect of the etiology of many diseases. The imprinting phenomenon depicts differential expression levels of the allele depending on its parental origin. When the parental origin is unknown, the expression level has a finite normal mixture distribution. In such applications, a random sample of expression levels consists of three subsamples according to the number of minor alleles an individual possesses, of which one is the mixture and the other two are homogeneous. This understanding leads to a likelihood ratio test (LRT) for the presence of imprinting. Because of the nonregularity of the finite mixture model, the classical asymptotic conclusions on likelihood-based inference are not applicable. We show that the maximum likelihood estimator of the mixing distribution remains consistent. More interestingly, thanks to the homogeneous subsamples, the LRT statistic has an elegant and rather distinct 0.5χ21 + 0.5χ22 null limiting distribution. Simulation studies confirm that the limiting distribution provides precise approximations of the finite sample distributions under various parameter settings. The LRT is applied to expression data. Our analyses provide evidence for imprinting for a number of isoform expressions.

A novel approach to compare pinniped populations across a broad geographic range

We utilized aerial images and employed photogrammetric methodologies to collect standardized lengths of Steller sea lions (Eumetopias jubatus) terrestrially hauled out. We conducted comparisons among all site types and separately for rookery and haulout site-types between the two distinct population segments (DPSs; eastern and western) and two broad regions within the western DPS experiencing contrasting population abundance trends. An observed adult female index was created from measurements of reproductive females — in the presence of a pup or juvenile — and was applied as a model constraint for “adult females”. We fitted a finite mixture distribution model to the length-frequency data to estimate the proportion population for three delineated age–sex classes (juveniles, adult females, and adult males) and mean length for juveniles and adult males. Estimated proportions reflected what we expected; however, the broad region within the western DPS exhibiting substantial population declines had greater proportion of all age–sex classes on rookery sites than increasing broad region. Adult sea lions were significantly shorter in the eastern DPS than the western area, providing further evidence of morphological differences between the DPSs. We also introduce a less resource-demanding method for estimating population demographics, and potentially vital rates, for pinnipeds across a vast geographic range.

Rossi, Peter E.: Bayesian non- and semi-parametric methods and applications

This book demonstrates the flexibility and accuracy of the Bayesian approach towards nonand semi-parametric statistical modelling. As the Bayesian approach offers flexible yet parsimoniously parameterized statistical model specifications, it is well suited for empirical analysis. Although the book is not always perfect stringent with regard to notation, it provides a concise and very readable access for readers who would like to become familiar with the application of Bayesian techniques in the context of nonand semi-parametric statistical modelling. The presented material documents the author’s profound mastership in providing access to the most advanced topics of statistical analysis without stepping over operational complexity or losing rigour. It highlights all involved important topics that arise in the context of nonand semiparametric Bayesian settings and it is well suited to motivate empirical researchers to adapt these techniques. Using finite mixture distributions as one core element, Rossi elaborates thoroughly on advantages as well as on caveats of the Bayesian approach and its implementation in such a way that the book serves also as a source of reference related to the abundant illustrated applications, carefully provided from microeconomics, behavioral. The book contains six chapters. Chapter 1 introduces the concept of finite normal mixture distributions. Next to a short review of Maximum Likelihood estimation, facilitated via the Expectation-Maximization algorithm, the Bayesian estimation approach, facilitated byMarkovChainMonteCarlo (MCMC) techniques, is discussed. Thereby, attention is paid to the important issue of label switching occurring in the absence of identifying constraints on the parameter space. Relabeling algorithms, as discussed by Stephens (2000) and Celeux et al. (2000), are shortly discussed as devices to base inference on the output of the unconstrained sampler. Also the different

Identifying connected components in Gaussian finite mixture models for clustering

Model-based clustering associates each component of a finite mixture distribution to a group or cluster. Therefore, an underlying implicit assumption is that a one-to-one correspondence exists between mixture components and clusters. In applications with multivariate continuous data, finite mixtures of Gaussian distributions are typically used. Information criteria, such as BIC, are often employed to select the number of mixture components. However, a single Gaussian density may not be sufficient, and two or more mixture components could be needed to reasonably approximate the distribution within a homogeneous group of observations. A clustering method, based on the identification of high density regions of the underlying density function, is introduced. Starting with an estimated Gaussian finite mixture model, the corresponding density estimate is used to identify the cluster cores, i.e. those data points which form the core of the clusters. Then, the remaining observations are allocated to those cluster cores for which the probability of cluster membership is the highest. The method is illustrated using both simulated and real data examples, which show how the proposed approach improves the identification of non-Gaussian clusters compared to a fully parametric approach. Furthermore, it enables the identification of clusters which cannot be obtained by merging mixture components, and it can be straightforwardly extended to cases of higher dimensionality.

Bayesian Prediction of Future Generalized Order Statistics from a Class of Finite Mixture Distributions

This article is concerned with the problem of prediction for the future generalized order statistics from a mixture of two general components based on doubly type II censored sample. We consider the one sample prediction and two sample prediction techniques. Bayesian prediction intervals for the median of future sample of generalized order statistics having odd and even sizes are obtained. Our results are specialized to ordinary order statistics and ordinary upper record values. A mixture of two Gompertz components model is given as an application. Numerical computations are given to illustrate the procedures.

Hierarchical Failure Time Regression Using Mixtures for Classification of the Immune Response of Atlantic Salmon

This work presents a Bayesian hierarchical model with the dual objective to analyze stratified survival data and to automatically classify each stratum into a finite number of groups. This is achieved by specifying parametric as well as piecewise stratum-specific baseline hazards and a finite mixture distribution for the stratum-specific shape parameters. A proportional hazards or accelerated failure time regression component allows to identify the influence of covariates on the survival distribution. We illustrate the model using a dataset of Atlantic salmon, stratified by families, that have been challenged with infectious pancreatic necrosis virus (IPNV). The main objectives are to model the survival time in terms of certain covariates as well as to classify the salmon families into either an IPNV susceptible or resistant group with the ultimate goal of improving resistance to IPNV through a selective breeding program. We compare the fit of different models that include stratum-specific baselines and covariate effects. The classifications show a certain degree of robustness with respect to model choice.

Regime-dependent Characteristics of KOSPI Return

Stylized facts on asset return are fat-tail, asymmetry, volatility clustering and structure changes. This paper simultaneously captures these characteristics by introducing a multi-regime models: Finite mixture distribution and regime switching GARCH model. Analyzing the daily KOSPI return from <TEX>$4^{th}$</TEX> January 2000 to <TEX>$30^{th}$</TEX> June 2014, we find that a two-component mixture of t distribution is a good candidate to describe the shape of the KOSPI return from unconditional and conditional perspectives. Empirical results suggest that the equality assumption on the shape parameter of t distribution yields better discrimination of heterogeneity component in return data. We report the strong regime-dependent characteristics in volatility dynamics with high persistence and asymmetry by employing a regime switching GJR-GARCH model with t innovation model. Compared to two sub-samples, Pre-Crisis (January 2003 ~ December 2007) and Post-Crisis (January 2010 ~ June 2014), we find that the degree of persistence in the Pre-Crisis is higher than in the Post-Crisis along with a strong asymmetry in the low-volatility (high-volatility) regime during the Pre-Crisis (Post-Crisis).

A new approach for Weibull modeling for reliability life data analysis

This paper presents a proposed approach for modeling the life data for system components that have failure modes by different Weibull models. This approach is applied for censored, grouped and ungrouped samples. To support the main idea, numerical applications with exact failure times and censored data are implemented. The parameters are obtained by different computational statistical methods such as graphic method based on Weibull probability plot (WPP), maximum likelihood estimates (MLE), Bayes estimators, non-linear Benard’s median rank regression. This paper also presents a parametric estimation method depends on expectation–maximization (EM) algorithm for estimation the parameters of finite Weibull mixture distributions. GOF is used to determine the best distribution for modeling life data. The performance of the proposed approach to model lifetime data is assessed. It’s an efficient approach for moderate and large samples especially with a heavily censored data and few exact failure times.

Applications of stochastic models and geostatistical analyses to study sources and spatial patterns of soil heavy metals in a metalliferous industrial district of China

An extensive soil survey was conducted to study pollution sources and delineate contamination of heavy metals in one of the metalliferous industrial bases, in the karst areas of southwest China. A total of 597 topsoil samples were collected and the concentrations of five heavy metals, namely Cd, As (metalloid), Pb, Hg and Cr were analyzed. Stochastic models including a conditional inference tree (CIT) and a finite mixture distribution model (FMDM) were applied to identify the sources and partition the contribution from natural and anthropogenic sources for heavy metal in topsoils of the study area. Regression trees for Cd, As, Pb and Hg were proved to depend mostly on indicators of anthropogenic activities such as industrial type and distance from urban area, while the regression tree for Cr was found to be mainly influenced by the geogenic characteristics. The FMDM analysis showed that the geometric means of modeled background values for Cd, As, Pb, Hg and Cr were close to their background values previously reported in the study area, while the contamination of Cd and Hg were widespread in the study area, imposing potentially detrimental effects on organisms through the food chain. Finally, the probabilities of single and multiple heavy metals exceeding the threshold values derived from the FMDM were estimated using indicator kriging (IK) and multivariate indicator kriging (MVIK). The high probabilities exceeding the thresholds of heavy metals were associated with metalliferous production and atmospheric deposition of heavy metals transported from the urban and industrial areas. Geostatistics coupled with stochastic models provide an effective way to delineate multiple heavy metal pollution to facilitate improved environmental management.

Construction of Single Sampling Plans by Attributes for Heterogeneous Lots Using a Mixture of Two Poisson Distributions

Abstract In many manufacturing processes, production takes place in more than one stream, with the objective of increasing the production in order to meet the demands of the consumers for the products. The lots or batches of finished products that can be constructed by grouping the products manufactured from various streams are generally heterogeneous (non-homogeneous) in nature. In such a situation, the lot quality might vary as a result of the machines used, the skill of working personnel, and the raw materials, and existing sampling plans are not suitable or viable for carrying out inspection activity in sentencing such lots. When the inspection lots are non-homogeneous, a finite mixture distribution is an appropriate probability model for determining the quality variation of the product. In this paper, the design of single’ sampling plans by attributes under a mixture of two Poisson distributions is considered under the assumption that the products are manufactured in two independent streams. Tables providing plan parameters protecting the interests of the producer and the consumer for specified strengths are presented. The risk efficiency of Poisson bi-mixture plans relative to Poisson sampling plans is studied. The selection of plan parameters is illustrated.

On the Exponentiated Weibull Distribution for Modeling Wind Speed in South Western Nigeria

Summary Statistics Min 1st Quarter Median Mean 3rd Quarter Mae 1.54 2.94 4.06 4.578 5.88 9.81 Note: Kurtosis = 2.502187, Skewness = 0.6333066 Conclusion The performance of Exponentiated Weibull and Weibull distribution functions to model wind energy was systematically compared. It was observed that the log SHITTU & ADEPOJU 441 likelihood values and the Akaike information criterion (AIC) for the Exponentiated Weibull was always smaller for the Weibull distribution for each month except the month of November. This indicates that the proposed Exponentiated Weibull distribution outperformed the existing Weibull distribution for wind speed data in terms of minimum AIC and likelihood function over the months of the years under review. Thus, the exponentiated Weibull can be used as an alternative distribution that adequately describes wind speed, and may provide better representation of the potentials of wind energy. References Akpinar, S. & Akpinar, E. K. (2009). Estimation of wind energy potential using finite mixture distribution models. Energy Conversion and Management, 50(4): 877–84. Bivona, S., Burlon, R., & Leone, C.. (2003). Hourly wind speed analysis in Sicily. Renewable Energy, 28(9): 1371-1385. Burton, T., Sharpe, D., Jenkins, N., & Bossanyi, E. (2001). Wind energy handbook. Wiley. Carta, J. A., Ramirez, P., & Velazquez, S. (2009). A review of wind speed probability distributions used in wind energy analysis Case studies in the Canary Islands. Renewable and Sustainable Energy Reviews, 13(5): 933–55. Carta, J. A., Ramirez, P., & Velazquez, S. (2008) Influence of the level of fit of a density probability function to wind-speed data on the WECS means power output estimation. Energy Conversion and Management, 49(10):2647–55. Carta, J. A. & Ramirez, P. (2007). Analysis of two-component mixture Weibull statistics for estimation of wind speed distributions. Renewable Energy, 32(3): 518–31. Carta, J. A. & Ramirez, P. (2007). Use of finite mixture distribution models in the analysis of wind energy in the Canarian Archipelago. Energy Conversion and Management, 48(1): 281–91. Celik, A. N. (2004). A statistical analysis of wind power density based on the Weibull and Rayleigh models at the southern region of Turkey. Renewable Energy, 29(4): 593–604. Chang, T-J & Tu, Y-L. (2007). Evaluation of monthly capacity factor of WECS using Chronological and probabilistic wind speed data: a case study of Taiwan. Renewable Energy, 32(12): 1999–2010.

An Updated EM Algorithm for Classification in Protein Interaction Networks

We provide an updated Expectation-Maximization (UEM) algorithm to estimate parameters of finite Gaussian mixture distributions in Bayesian networks. The UEM algorithm is composed by three steps: an expectation (E) and a maximization (M) steps, akin to the EM algorithm, where a created function for log-likelihood is valuated using the current estimate for the parameter after that the parameters estimated are maximized. Given a graph specifying the relationship between nodes and based on Lauritzen formula, the third step (U) updates the M-estimates. We demonstrate that UEM algorithm is useful to system biology data, by considering protein interaction networks where arcs represent probabilistic relationships (interaction) between nodes (proteins). The UEM algorithm provides the classification of the data to each Gaussian distribution cluster. We apply our algorithm on a simulation study of the EGFR protein interaction network.

EM algorithms for estimating the Bernstein copula

A method that uses order statistics to construct multivariate distributions with fixed marginals and which utilizes a representation of the Bernstein copula in terms of a finite mixture distribution is proposed. Expectation–maximization (EM) algorithms to estimate the Bernstein copula are proposed, and a local convergence property is proved. Moreover, asymptotic properties of the proposed semiparametric estimators are provided. Illustrative examples are presented using three real data sets and a 3-dimensional simulated data set. These studies show that the Bernstein copula is able to represent various distributions flexibly and that the proposed EM algorithms work well for such data.