Finite Normal Mixture Model Research Articles

The time series modelling on exchange rate and inflation rate: finite normal mixture model

The time series modelling on exchange rate and inflation rate: finite normal mixture model

The time series modelling on exchange rate and inflation rate: finite normal mixture model

The time series modelling on exchange rate and inflation rate: finite normal mixture model

Finite Normal Mixture Models for the Ensemble Learning of Recurrent Neural Networks with Applications to Currency Pairs

Finite Normal Mixture Models for the Ensemble Learning of Recurrent Neural Networks with Applications to Currency Pairs

A Likelihood Ratio Test of a Homoscedastic Multivariate Normal Mixture Against a Heteroscedastic Multivariate Normal Mixture

The multivariate finite normal mixture model is one of the most commonly used tools to analyze a heterogeneous data. When using the multivariate finite normal mixture model, one is usually interested in knowing whether a homoscedastic mixture model can be used to simplify the model. The likelihood ratio test (LRT) is the most popular statistic tool to choose between two nested models. Under the null model of a homoscedastic multivariate normal mixture, the asymptotic χ2 distribution is commonly used to approximate the null distribution of the LRT statistic. However, it is demonstrated using numerical studies that the χ2 distribution approximation is not satisfactory and fails to control the nominal type I error unless the sample size is larger than 2000, the mixture components are well-separated, and the singular solutions are avoided. A parametric bootstrap method is further proposed to approximate the distribution of the LRT statistic and its effectiveness is evaluated through extensive numerical studies.

Genotype calling of triploid offspring from diploid parents

BackgroundPolyploidy is widespread in animals and especially in plants. Different kinds of ploidies exist, for example, hexaploidy in wheat, octaploidy in strawberries, and diploidy, triploidy, tetraploidy, and pseudo-tetraploidy (partly tetraploid) in fish. Triploid offspring from diploid parents occur frequently in the wild in Atlantic salmon (Salmo salar) and, as with triploidy in general, the triploid individuals are sterile. Induced triploidy in Atlantic salmon is common practice to produce sterile fish. In Norwegian aquaculture, production of sterile triploid fish is an attempt by government and industry to limit genetic introgression between wild and farmed fish. However, triploid fish may have traits and properties that differ from those of diploids. Investigating the genetics behind traits in triploids has proved challenging because genotype calling of genetic markers in triploids is not supported by standard software. Our aim was to develop a method that can be used for genotype calling of genetic markers in triploid individuals.ResultsAllele signals were produced for 381 triploid Atlantic salmon offspring using a 56 K Thermo Fisher GeneTitan genotyping platform. Genotypes were successfully called by applying finite normal mixture models to the (transformed) allele signals. Subsets of markers were filtered by quality control statistics for use with downstream analyses. The quality of the called genotypes was sufficient to allow for assignment of diploid parents to the triploid offspring and to discriminate between maternal and paternal parents from autosomal inheritance patterns. In addition, as the maternal inheritance in triploid offspring is identical to gynogenetic inheritance, the maternal recombination pattern for each chromosome could be mapped by using a similar approach as that used in gene-centromere mapping.ConclusionsWe show that calling of dense marker genotypes for triploid individuals is feasible. The resulting genotypes can be used in parentage assignment of triploid offspring to diploid parents, to discriminate between maternal and paternal parents using autosomal inheritance patterns, and to map the maternal recombination pattern using an approach similar to gene-centromere mapping. Genotyping of triploid individuals is important both for selective breeding programs and unravelling the underlying genetics of phenotypes recorded in triploids. In principle, the developed method can be used for genotype calling of other polyploid organisms.

A simple root selection method for univariate finite normal mixture models

–It is well known that there exist multiple roots of the likelihood equations for finite normal mixture models. Selecting a consistent root for finite normal mixture models has long been a challenging problem. Simply using the root with the largest likelihood will not work because of the spurious roots. In addition, the likelihood of normal mixture models with unequal variance is unbounded and thus its maximum likelihood estimate (MLE) is not well defined. In this paper, we propose a simple root selection method for univariate normal mixture models by incorporating the idea of goodness of fit test. Our new method inherits both the consistency properties of distance estimators and the efficiency of the MLE. The new method is simple to use and its computation can be easily done using existing R packages for mixture models. In addition, the proposed root selection method is very general and can be also applied to other univariate mixture models. We demonstrate the effectiveness of the proposed method and compare it with some other existing methods through simulation studies and a real data application.

Consistency of the MLE under Mixture Models

The large-sample properties of likelihood-based statistical inference under mixture models have received much attention from statisticians. Although the consistency of the nonparametric MLE is regarded as a standard conclusion, many researchers ignore the precise conditions required on the mixture model. An incorrect claim of consistency can lead to false conclusions even if the mixture model under investigation seems well behaved. Under a finite normal mixture model, for instance, the consistency of the plain MLE is often erroneously assumed in spite of recent research breakthroughs. This paper streamlines the consistency results for the nonparametric MLE in general, and in particular for the penalized MLE under finite normal mixture models.

Testing the Number of Components in Normal Mixture Regression Models

Testing the number of components in finite normal mixture models is a long-standing challenge because of its nonregularity. This article studies likelihood-based testing of the number of components in normal mixture regression models with heteroscedastic components. We construct a likelihood-based test of the null hypothesis of m0 components against the alternative hypothesis of m0 + 1 components for any m0. The null asymptotic distribution of the proposed modified EM test statistic is the maximum of m0 random variables that can be easily simulated. The simulations show that the proposed test has very good finite sample size and power properties. Supplementary materials for this article are available online.

Semiparametric Bayesian Density Estimation With Disparate Data Sources: A Meta-Analysis of Global Childhood Undernutrition

Undernutrition, resulting in restricted growth, and quantified here using height-for-age z-scores, is an important contributor to childhood morbidity and mortality. Since all levels of mild, moderate, and severe undernutrition are of clinical and public health importance, it is of interest to estimate the shape of the z-scores’ distributions. We present a finite normal mixture model that uses data on 4.3 million children to make annual country-specific estimates of these distributions for under-5-year-old children in the world’s 141 low- and middle-income countries between 1985 and 2011. We incorporate both individual-level data when available, as well as aggregated summary statistics from studies whose individual-level data could not be obtained. We place a hierarchical Bayesian probit stick-breaking model on the mixture weights. The model allows for nonlinear changes in time, and it borrows strength in time, in covariates, and within and across regional country clusters to make estimates where data are uncertain, sparse, or missing. This work addresses three important problems that often arise in the fields of public health surveillance and global health monitoring. First, data are always incomplete. Second, different data sources commonly use different reporting metrics. Last, distributions, and especially their tails, are often of substantive interest.

Determination of natural backgrounds and thresholds of nitrate in South Korean groundwater using model-based statistical approaches

Increased nitrate loading of groundwater has emerged as a major environmental problem in many countries, including South Korea. This study aims to evaluate the nitrate levels of South Korean groundwater on a regional (national) scale and specifically to demonstrate the procedure to better estimate the natural background level (NBL) and threshold of nitrate as the basis of groundwater management. For this work, nitrate data of groundwater (n=8510) in two major hydrogeologic units (alluvium and bedrock) were collected from the National Groundwater Monitoring Network (NGMN) of South Korea. Four supplementary datasets (n=1074) were also used to test the rationality of estimated thresholds by comparing them with NGMN datasets. Compared with the data reported in many countries, the nitrate concentrations in NGMN groundwater in 2009 are high, with median values of 12.2 and 8.7mg/L, respectively, for alluvial groundwater and bedrock groundwater. The nitrate levels of South Korean groundwater seem to have been historically steady at these high levels between 1997 and 2009, suggesting widespread diffusive contamination since the 1980s. The NBLs and anthropogenic polluted levels (APLs) of nitrate on a regional (national) scale are statistically established by the model-based approach using a finite normal (Gaussian) two-component mixture model, because (1) the sample size (frequency) of the natural background group is much smaller than that of the polluted group, as a result of widespread nitrate contamination, and (2) nitrate concentrations are more or less affected by natural attenuation processes. Accordingly, thresholds of nitrate (as the concentration level indicating groundwater pollution) are selected as the lower limits (i.e., 10th percentile) of the polluted group, which are 3.0 and 5.5mg/L NO3−, respectively, for bedrock groundwater and alluvial groundwater. This study provides a practical guideline for national groundwater management, based on a heuristic procedure to statistically determine the NBLs and thresholds in the case of groundwater systems with pervasive contamination. Compared with the other classical methods to estimate NBLs, the model-based approach using a finite normal-mixture model can be more effective to reasonably separate the polluted samples from a regional (or national) dataset.

Capabilities ofRPackagemixAKfor Clustering Based on Multivariate Continuous and Discrete Longitudinal Data

R package mixAK originally implemented routines primarily for Bayesian estimation of finite normal mixture models for possibly interval-censored data. The functionality of the package was considerably enhanced by implementing methods for Bayesian estimation of mixtures of multivariate generalized linear mixed models proposed in Komárek and Komárková (2013). Among other things, this allows for a cluster analysis (classification) based on multivariate continuous and discrete longitudinal data that arise whenever multiple outcomes of a different nature are recorded in a longitudinal study. This package also allows for a data-driven selection of a number of clusters as methods for selecting a number of mixture components were implemented. A model and clustering methodology for multivariate continuous and discrete longitudinal data is overviewed. Further, a step-by-step cluster analysis based jointly on three longitudinal variables of different types (continuous, count, dichotomous) is given, which provides a user manual for using the package for similar problems.

Southeastern United States summer rainfall framework and its implication for seasonal prediction

A new rainfall framework is constructed to describe the complex probability distribution of southeastern United States (SE US) summer (June–July–August) rainfall, which cannot be well represented by traditional kernel fitting methods. The new framework is based on the configuration of a three-cluster finite normal mixture model and is realized by Bayesian inference and a Markov Chain Monte Carlo (MCMC) algorithm. The three rainfall clusters reflect the probability distribution of light, moderate, and heavy rainfall in summer, and are linked to different climate factors. The variation of light rainfall intensity is likely associated with the combined effects of La Niña and the tri-pole sea surface temperature anomaly (SSTA) over the North Atlantic. Heavy rainfall concurs with a ‘horseshoe-like’ SSTA over the North Atlantic. In contrast, moderate rainfall is less correlated with the SSTA and likely caused by atmospheric internal dynamics. Rainfall characteristics and their linkages with SSTAs help improve seasonal predictions of regional climate. Such a new framework has an important implication in understanding the response of regional hydrology to climate variability and climate change; and our study suggest that it can be extended to other regions and seasons with similar climate.

Inference on the Order of a Normal Mixture

Finite normal mixture models are used in a wide range of applications. Hypothesis testing on the order of the normal mixture is an important yet unsolved problem. Existing procedures often lack a rigorous theoretical foundation. Many are also hard to implement numerically. In this article, we develop a new method to fill the void in this important area. An effective expectation-maximization (EM) test is invented for testing the null hypothesis of arbitrary order m 0 under a finite normal mixture model. For any positive integer m 0 ⩾ 2, the limiting distribution of the proposed test statistic is . We also use a novel computer experiment to provide empirical formulas for the tuning parameter selection. The finite sample performance of the test is examined through simulation studies. Real-data examples are provided. The procedure has been implemented in R code. The p-values for testing the null order of m 0 = 2 or m 0 = 3 can be calculated with a single command. This article has supplementary materials available online.

Root selection in normal mixture models

Finite mixtures of normal distributions are attractive in identifying the underlying group structure in the data. However, it is a challenging task to do statistical inference in normal mixture models using the method of maximum likelihood, due to the unbounded likelihood and the existence of multiple roots to the likelihood equation including a so-called spurious root. In this article we propose a new likelihood-based method for selecting a statistically reasonable root when there exist multiple roots of the likelihood equation for a finite normal mixture model. We first prove that our proposed methodology can choose a root to the mixture likelihood equation with consistency. We then show, by simulation studies and real examples, that the proposed methods can greatly reduce the risk of choosing problematic roots that have the same features as spurious roots.

Analysis of rounded data in mixture normal model

Rounding errors have a considerable impact on statistical inferences, especially when the data size is large and the finite normal mixture model is very important in many applied statistical problems, such as bioinformatics. In this article, we investigate the statistical impacts of rounding errors to the finite normal mixture model with a known number of components, and develop a new estimation method to obtain consistent and asymptotically normal estimates for the unknown parameters based on rounded data drawn from this kind of models.

Fund Rating Model Based on Finite Normal Mixture Distribution

This paper proposes a new method of fund rating based on the cross-sectional distribution of fund performance measured by alpha. This distribution-based fund rating model is more flexible and provides more interesting results than current commercial fund rating method used by Morningstar. Unlike Morningstar's rating, this method does not use preset percentiles to rate funds. It is the distribution of alpha that dictates the number of performance groups in a given fund category and time period. The framework is based on the crucial assumption that the expected fund performance may be different, and the difference of the expected fund performance arises from the segmented market information and/or the differentiated ability of mangers to acquire and analyze information. The multimodal shape and formal normality tests prompt us to model the distribution of alpha by finite normal mixture model. We introduce the parametric bootstrap procedure to determine the number of performance groups in the model. We then employ expectation and maximization (EM) algorithm to estimate the model. Based on the estimated posterior probabilities of the fund, we assign the rating to funds. Our empirical results show that the number of performance groups is not fixed and varies across time and fund categories. We observe a clear tendency of the merging of information sets, which implies that the fund market has become gradually more efficient over time as information was well transmitted and analyzed.

Flexible mixture modeling via the multivariate t distribution with the Box-Cox transformation: an alternative to the skew-t distribution

Cluster analysis is the automated search for groups of homogeneous observations in a data set. A popular modeling approach for clustering is based on finite normal mixture models, which assume that each cluster is modeled as a multivariate normal distribution. However, the normality assumption that each component is symmetric is often unrealistic. Furthermore, normal mixture models are not robust against outliers; they often require extra components for modeling outliers and/or give a poor representation of the data. To address these issues, we propose a new class of distributions, multivariate t distributions with the Box-Cox transformation, for mixture modeling. This class of distributions generalizes the normal distribution with the more heavy-tailed t distribution, and introduces skewness via the Box-Cox transformation. As a result, this provides a unified framework to simultaneously handle outlier identification and data transformation, two interrelated issues. We describe an Expectation-Maximization algorithm for parameter estimation along with transformation selection. We demonstrate the proposed methodology with three real data sets and simulation studies. Compared with a wealth of approaches including the skew-t mixture model, the proposed t mixture model with the Box-Cox transformation performs favorably in terms of accuracy in the assignment of observations, robustness against model misspecification, and selection of the number of components.

Hypothesis test for normal mixture models: The EM approach

Normal mixture distributions are arguably the most important mixture models, and also the most technically challenging. The likelihood function of the normal mixture model is unbounded based on a set of random samples, unless an artificial bound is placed on its component variance parameter. Moreover, the model is not strongly identifiable so it is hard to differentiate between over dispersion caused by the presence of a mixture and that caused by a large variance, and it has infinite Fisher information with respect to mixing proportions. There has been extensive research on finite normal mixture models, but much of it addresses merely consistency of the point estimation or useful practical procedures, and many results require undesirable restrictions on the parameter space. We show that an EM-test for homogeneity is effective at overcoming many challenges in the context of finite normal mixtures. We find that the limiting distribution of the EM-test is a simple function of the 0.5χ02+0.5χ12 and χ12 distributions when the mixing variances are equal but unknown and the χ22 when variances are unequal and unknown. Simulations show that the limiting distributions approximate the finite sample distribution satisfactorily. Two genetic examples are used to illustrate the application of the EM-test.

Interval Mapping for Quantitative Trait Loci Detection in Finite Mixture Models with General Kernel Functions

The advent of complete genetic linkage maps of DNA markers has made the systematic study of mapping quantitative trait loci (QTL) in experimental organisms feasible. In recent years, extensive methodological research on QTL mapping has been performed, Chen and Chen (2005) recently considered the statistical inference problems in the interval mapping method under finite normal mixture models. This article examines the asymptotic null distribution of the likelihood ratio test in finite mixture models with general kernel functions. The limiting null distribution is found to be free of the choice of a kernel function. Technical details are given to the kernel functions with a single parameter under backross populations. Extension to the finite mixture models with presence of a structure parameter is discussed and proposed. The EM algorithm for approximating the MLE's is also discussed.

Assessing Heterogeneity in Discrete Choice Models Using a Dirichlet Process Prior

The finite normal mixture model has emerged as a dominant methodology for assessing heterogeneity in choice models. Although it extends the classic mixture models by allowing within component variablility, it requires that a relatively large number of models be separately estimated and fairly difficult test procedures to determine the “correct” number of mixing components. We present a very general formulation, based on Dirichlet Process Piror, which yields the number and composition of mixing components a posteriori, obviating the need for post hoc test procedures and is capable of approximating any target heterogeneity distribution. Adapting Stephens’ (2000) algorithm allows the determination of ‘substantively’ different clusters, as well as a way to sidestep problems arising from label-switching and overlapping mixtures. These methods are illustrated both on simulated data and A.C. Nielsen scanner panel data for liquid detergents. We find that the large number of mixing components required to adequately represent the heterogeneity distribution can be reduced in practice to a far smaller number of segments of managerial relevance.