Multinomial Distribution Research Articles

Automated Extraction of Mutual Independence Patterns Using Bayesian Comparison of Partition Models.

Mutual independence is a key concept in statistics that characterizes the structural relationships between variables. Existing methods to investigate mutual independence rely on the definition of two competing models, one being nested into the other and used to generate a null distribution for a statistic of interest, usually under the asymptotic assumption of large sample size. As such, these methods have a very restricted scope of application. In this article, we propose to change the investigation of mutual independence from a hypothesis-driven task that can only be applied in very specific cases to a blind and automated search within patterns of mutual independence. To this end, we treat the issue as one of model comparison that we solve in a Bayesian framework. We show the relationship between such an approach and existing methods in the case of multivariate normal distributions as well as cross-classified multinomial distributions. We propose a general Markov chain Monte Carlo (MCMC) algorithm to numerically approximate the posterior distribution on the space of all patterns of mutual independence. The relevance of the method is demonstrated on synthetic data as well as two real datasets, showing the unique insight provided by this approach.

Endmember Extraction From Hyperspectral Imagery Based on Probabilistic Tensor Moments

This letter presents a novel hyperspectral endmember extraction approach that integrates a tensor-based decomposition scheme with a probabilistic framework in order to take advantage of both technologies when uncovering the signatures of pure spectral constituents in the scene. On the one hand, statistical unmixing models are generally able to provide accurate endmember estimates by means of rather complex optimization algorithms. On the other hand, tensor decomposition techniques are very effective factorization tools which are often constrained by the lack of physical interpretation within the remote sensing field. In this context, this letter develops a new hybrid endmember extraction approach based on the decomposition of the probabilistic tensor moments of the hyperspectral data. Initially, the input image reflectance values are modeled as a collection of multinomial distributions provided by a family of Dirichlet generalized functions. Then, the unmixing process is effectively conducted by the tensor decomposition of the third-order probabilistic tensor moments of the multivariate data. Our experiments, conducted over four hyperspectral data sets, reveal that the proposed approach is able to provide efficient and competitive results when compared to different state-of-the-art endmember extraction methods.

Subspace-Based Representation and Learning for Phonotactic Spoken Language Recognition

Phonotactic constraints can be employed to distinguish languages by representing a speech utterance as a multinomial distribution or phone events. In the present study, we propose a new learning mechanism based on subspace-based representation, which can extract concealed phonotactic structures from utterances, for language verification and dialect/accent identification. The framework mainly involves two successive parts. The first part involves subspace construction. Specifically, it decodes each utterance into a sequence of vectors filled with phone-posteriors and transforms the vector sequence into a linear orthogonal subspace based on low-rank matrix factorization or dynamic linear modeling. The second part involves subspace learning based on kernel machines, such as support vector machines and the newly developed subspace-based neural networks (SNNs). The input layer of SNNs is specifically designed for the sample represented by subspaces. The topology ensures that the same output can be derived from identical subspaces by modifying the conventional feed-forward pass to fit the mathematical definition of subspace similarity. Evaluated on the “General LR” test of NIST LRE 2007, the proposed method achieved up to 52%, 46%, 56%, and 27% relative reductions in equal error rates over the sequence-based PPR-LM, PPR-VSM, and PPR-IVEC methods and the lattice-based PPR-LM method, respectively. Furthermore, on the dialect/accent identification task of NIST LRE 2009, the SNN-based system performed better than the aforementioned four baseline methods.

Multinomial, Poisson and Gaussian statistics in count data analysis

It is generally known that counting statistics is not correctly described by a Gaussian approximation. Nevertheless, in neutron scattering, it is common practice to apply this approximation to the counting statistics; also at low counting numbers. We show that the application of this approximation leads to skewed results not only for low-count features, such as background level estimation, but also for its estimation at double-digit count numbers. In effect, this approximation is shown to be imprecise on all levels of count. Instead, a Multinomial approach is introduced as well as a more standard Poisson method, which we compare with the Gaussian case. These two methods originate from a proper analysis of a multi-detector setup and a standard triple axis instrument. We devise a simple mathematical procedure to produce unbiased fits using the Multinomial distribution and demonstrate this method on synthetic and actual inelastic scattering data. We find that the Multinomial method provide almost unbiased results, and in some cases outperforms the Poisson statistics. Although significantly biased, the Gaussian approach is in general more robust in cases where the fitted model is not a true representation of reality. For this reason, a proper data analysis toolbox for low-count neutron scattering should therefore contain more than one model for counting statistics.

The probabilities of large deviations for a certain class of statistics associated with multinomial distribution

Let η = (η1, …, ηN) be a multinomial random vector with parameters n = η1 + ⋯ + ηN and pm > 0, m = 1, …, N, p1 + ⋯ + pN = 1. We assume that N →∞ and maxpm → 0 as n →∞. The probabilities of large deviations for statistics of the form h1(η1) + ⋯ + hN(ηN) are studied, where hm(x) is a real-valued function of a non-negative integer-valued argument. The new large deviation results for the power-divergence statistics and its most popular special variants, as well as for several count statistics are derived as consequences of the general theorems.

A Metropolis within Gibbs algorithm for knowledge discovery in language assessments

The reduced reparameterised unified model (RRUM) has been frequently used in language assessment studies. The objective of this research is to advance an MCMC algorithm for the Bayesian RRUM. The algorithm starts with estimating correlated attributes. Using a saturated model and a binary decimal conversion, the algorithm transforms possible attribute patterns to a multinomial distribution. Along with the likelihood of an attribute pattern, a Dirichlet distribution is used as the prior to sample from the posterior. The Dirichlet distribution is constructed using Gamma distributions. Correlated attributes of examinees are generated using the inverse transform sampling. Model parameters are estimated using the Metropolis within Gibbs sampler sequentially. Two simulation studies are conducted to evaluate the performance of the algorithm. The simulation studies show that as sample size increases, the measure of accuracy increases. The algorithm developed in this research is implemented in R.

A Metropolis within Gibbs algorithm for knowledge discovery in language assessments

The reduced reparameterised unified model (RRUM) has been frequently used in language assessment studies. The objective of this research is to advance an MCMC algorithm for the Bayesian RRUM. The algorithm starts with estimating correlated attributes. Using a saturated model and a binary decimal conversion, the algorithm transforms possible attribute patterns to a multinomial distribution. Along with the likelihood of an attribute pattern, a Dirichlet distribution is used as the prior to sample from the posterior. The Dirichlet distribution is constructed using Gamma distributions. Correlated attributes of examinees are generated using the inverse transform sampling. Model parameters are estimated using the Metropolis within Gibbs sampler sequentially. Two simulation studies are conducted to evaluate the performance of the algorithm. The simulation studies show that as sample size increases, the measure of accuracy increases. The algorithm developed in this research is implemented in R.

SUPERPOSITIONED STATIONARY COUNT TIME SERIES

AbstractThis paper probabilistically explores a class of stationary count time series models built by superpositioning (or otherwise combining) independent copies of a binary stationary sequence of zeroes and ones. Superpositioning methods have proven useful in devising stationary count time series having prespecified marginal distributions. Here, basic properties of this model class are established and the idea is further developed. Specifically, stationary series with binomial, Poisson, negative binomial, discrete uniform, and multinomial marginal distributions are constructed; other marginal distributions are possible. Our primary goal is to derive the autocovariance function of the resulting series.

An eight-layer convolutional neural network with stochastic pooling, batch normalization and dropout for fingerspelling recognition of Chinese sign language

Fingerspelling recognition of Chinese sign language rendered an opportunity to smooth the communication barriers of hearing-impaired people and health people, which occupies an important position in sign language recognition. This study proposed an eight-layer convolutional neural network, combined with three advanced techniques: batch normalization, dropout, and stochastic pooling. The output of the stochastic pooling was obtained via sampling from a multinomial distribution formed from the activations of each pooling region. In addition, we used data augmentation method to enhance the training set. In total 10 runs were implemented with the hold-out randomly set for each run. Our method achieved the highest accuracy of 90.91% and overall accuracy of 89.32 ± 1.07%, which was superior to three state-of-the-art approaches compared.

PSIV-B-30 Late-Breaking: Effects of wheat straw bedding usage on hide tag scores during winter and spring in finishing feedlot cattle fed in eastern South Dakota

Abstract Hide tag scores (TS; 1 to 5 point scale; 1=no tag, clean hide and 5=lumps of manure attached to the hide continuously on the underbelly and side of the animal from brisket to rear quarter) were collected on March 22 or March 25, 2019 (winter), and on May 10, 2019 (spring) from pens of 5 to 8 head of steers and heifers housed in 7.6 × 7.6 m concrete pens at the Ruminant Nutrition Center in Brookings, SD. Treatments included: 1) 2.5 kg (as is basis) of bedding/animal/d (n = 8 pens; HI); 2) 1.8 kg (as is basis) of bedding/animal/d (n = 15 pens; LO); or 3) no bedding (n = 15 pens; NO). Pen TS distribution was analyzed using a multinomial distribution approach appropriate for a CRD. Winter: There was greater (P = 0.01) TS 2 in HI compared to LO and NO (42.5 vs. 4.2 and 0.0 ± 7.27%). Cattle in HI had greater (P = 0.01) TS 3 compared to LO and NO (55.0 vs. 25.8 and 5.0 ± 10.20%). Cattle in LO and NO had greater (P = 0.01) TS 4 compared to HI (59.2 and 45.6 vs. 2.5 ± 10.48%). Cattle in NO had greater (P = 0.01) TS 5 compared to LO and HI (49.4 vs 10.8 and 0.0 ± 8.80%). Spring: HI had the greatest (P = 0.01; 5.0 vs. 0.0 and 0.0 ± 1.56%) TS 1. Cattle in HI had the greatest (P = 0.01) TS 2 compared to NO and LO (55.0 vs. 23.7 and 21.4 ± 7.58%). Cattle in HI had lower (P = 0.01) TS 3 compared to NO and LO (37.5 vs. 69.5 and 65.6 ± 8.40%). No differences (P ≥ 0.13) for TS’s 4 and 5 were detected during the spring. These data indicate that greater amounts of bedding results in less severe TS and TS became less severe as the season changed from winter to spring.

Hypothesis testing of Geographically weighted bivariate logistic regression

In this study, the hypothesis testing of geographically weighted bivariate logistic regression (GWBLR) procedure is proposed. The GWBLR model is a bivariate logistic regression (BLR) model which all of the regression parameters depend on the geographical location in the study area. The geographical location is expressed as a point coordinate in two-dimensional geographic space (longitude and latitude). The response variables of BLR model are constructed from a (2 × 2) contingency table and it follows the multinomial distribution. The purpose of this study is to test the GWBLR model parameters. There are three hypothesis tests. The first is a parameters similarity test using the Vuong test method. The test is to obtain a significant difference between GWBLR and BLR. The second is a simultaneous test using the likelihood ratio test method. The simultaneous test is to obtain the simultaneous significance of the regression parameters. The last is a partial test using Wald test method. The result showed that the Vuong statistic and Wald statistic have an asymptotic standard normal distribution, whereas the likelihood ratio statistic has an asymptotic chi-squared distribution.

SCOPIT: sample size calculations for single-cell sequencing experiments

BackgroundIn single cell DNA and RNA sequencing experiments, the number of cells to sequence must be decided before running an experiment, and afterwards, it is necessary to decide whether sufficient cells were sampled. These questions can be addressed by calculating the probability of sampling at least a defined number of cells from each subpopulation (cell type or cancer clone).ResultsWe developed an interactive web application called SCOPIT (Single-Cell One-sided Probability Interactive Tool), which calculates the required probabilities using a multinomial distribution (www.navinlab.com/SCOPIT). In addition, we created an R package called pmultinom for scripting these calculations.ConclusionsOur tool for fast multinomial calculations provide a simple and intuitive procedure for prospectively planning single-cell experiments or retrospectively evaluating if sufficient numbers of cells have been sequenced. The web application can be accessed at navinlab.com/SCOPIT.

MDiNE: a model to estimate differential co-occurrence networks in microbiome studies.

MotivationThe human microbiota is the collection of microorganisms colonizing the human body, and plays an integral part in human health. A growing trend in microbiome analysis is to construct a network to estimate the co-occurrence patterns among taxa through precision matrices. Existing methods do not facilitate investigation into how these networks change with respect to covariates.ResultsWe propose a new model called Microbiome Differential Network Estimation (MDiNE) to estimate network changes with respect to a binary covariate. The counts of individual taxa in the samples are modeled through a multinomial distribution whose probabilities depend on a latent Gaussian random variable. A sparse precision matrix over all the latent terms determines the co-occurrence network among taxa. The model fit is obtained and evaluated using Hamiltonian Monte Carlo methods. The performance of our model is evaluated through an extensive simulation study and is shown to outperform existing methods in terms of estimation of network parameters. We also demonstrate an application of the model to estimate changes in the intestinal microbial network topology with respect to Crohn’s disease.Availability and implementation MDiNE is implemented in a freely available R package: https://github.com/kevinmcgregor/mdine.Supplementary information Supplementary data are available at Bioinformatics online.

The Prevalence of Lumbar Paraspinal Muscle Fatty Degeneration in Patients with Modic Type I and I/II End Plate Changes.

Study DesignRetrospective case control.PurposeThe authors of this study assessed whether the prevalence of paraspinal fatty degeneration correlates with the presence of Modic type I and I/II change in patients with low back pain (LBP).Overview of LiteratureModic changes are bone marrow and end plate changes visible on magnetic resonance imaging.MethodsA consecutive series of 141 patients who attended the neurosurgery outpatient clinic between April 2017 and September 2017 for nonspecific LBP were evaluated. Sixty-one patients with single-level Modic type I or I/II change constituted the patient group. Eighty age-, gender-, and body mass index (BMI)-matched patients without any Modic changes were recruited as the control group. A retrospective review was performed in 61 patients with Modic changes and 80 controls without Modic changes. The percentage of fatty muscle degeneration was graded by two reviewers using T2-weighted axial images at the L4–L5 level. The system was graded as follows: grade 0, normal; grade I, minimal focal or linear fat deposition; grade II, up to 25%; grade III, 25%–50%; and grade IV, more than 50%.ResultsSixty-one patients with nonspecific LBP and Modic type I or I/II change and 80 patients without Modic changes were evaluated. There was no difference between these groups in terms of age, gender, and BMI distribution. The mean muscle cross-sectional area in the patient and control groups were 1,507.37±410.63 and 1,681.64±379.69. Regarding fatty degeneration, a chi-square test of homogeneity was run, and the two multinomial probability distributions were not equal within the population analyzed.ConclusionsThe novel finding of this investigation is that patients with Modic type I and I/II changes have greater amounts of fatty degeneration in their lumbar paraspinal musculature.

On an asymptotic relative efficiency concept based on expected volumes of confidence regions

ABSTRACTThe paper deals with an asymptotic relative efficiency concept for confidence regions of multidimensional parameters that is based on the expected volumes of the confidence regions. Under standard conditions the asymptotic relative efficiencies of confidence regions are seen to be certain powers of the ratio of the limits of the expected volumes. These limits are explicitly derived for confidence regions associated with certain plugin estimators, likelihood ratio tests and Wald tests. Under regularity conditions, the asymptotic relative efficiency of each of these procedures with respect to each one of its competitors is equal to 1. The results are applied to multivariate normal distributions and multinomial distributions in a fairly general setting.

Construction of Fuzzy Control Chart with Multinomial Quality using Process Capability

Control charts are the effective and quietest form of statistical process control methods. Many of the times, data are obtained in quantitative form; however there are many quality characteristics that cannot be expressed in numerical measure, such as characteristics for appearance, smoothness and colour, etc. Fuzzy sets theory is an impressive mathematical methodology to evaluate the vagueness related uncertainty that can linguistically express data in these situations. In this paper, we construct a fuzzy control limits under fixed and varying sample size with various quality levels for observing a manufacturing process based on the multinomial distribution using degrees of membership and process capability.

Thematic Accuracy Quality Control by Means of a Set of Multinomials

The error matrix has been adopted as both the “de facto” and the “de jure” standard way to report on the thematic accuracy assessment of any remotely sensed data product. This perspective assumes that the error matrix can be considered as a set of values following a unique multinomial distribution. However, the assumption of the underlying statistical model falls down when true reference data are available for quality control. To overcome this problem, a new method for thematic accuracy quality control is proposed, which uses a multinomial approach for each category and is called QCCS (quality control column set). The main advantage is that it allows us to state a set of quality specifications for each class and to test if they are fulfilled. These requirements can be related to the percentage of correctness in the classification for a particular class but also to the percentage of possible misclassifications or confusions between classes. In order to test whether such specifications are achieved or not, an exact multinomial test is proposed for each category. Furthermore, if a global hypothesis test is desired, the Bonferroni correction is proposed. All these new approaches allow a more flexible way of understanding and testing thematic accuracy quality control compared with the classical methods based on the confusion matrix. For a better understanding, a practical example of an application is included for classification with four categories.

Computing the exact distributions of some functions of the ordered multinomial counts: maximum, minimum, range and sums of order statistics

Starting from seminal neglected work by Rappeport (Rappeport 1968 Algorithms and computational procedures for the application of order statistics to queuing problems. PhD thesis, New York University), we revisit and expand on the exact algorithms to compute the distribution of the maximum, the minimum, the range and the sum of the J largest order statistics of a multinomial random vector under the hypothesis of equiprobability. Our exact results can be useful in all those situations in which the multinomial distribution plays an important role, from goodness-of-fit tests to the study of Poisson processes, with applications spanning from biostatistics to finance. We describe the algorithms, motivate their use in statistical testing and illustrate two applications. We also provide the codes and ready-to-use tables of critical values.

A New Method for Positional Accuracy Control for Non-Normal Errors Applied to Airborne Laser Scanner Data

A new statistical method for the quality control of the positional accuracy, useful in a wide range of data sets, is proposed and its use is illustrated through its application to airborne laser scanner (ALS) data. The quality control method is based on the use of a multinomial distribution that categorizes cases of errors according to metric tolerances. The use of the multinomial distribution is a very novel and powerful approach to the problem of evaluating positional accuracy, since it allows for eliminating the need for a parametric model for positional errors. Three different study cases based on ALS data (infrastructure, urban, and natural cases) that contain non-normal errors were used. Three positional accuracy controls with different tolerances were developed. In two of the control cases, the tolerances were defined by a Gaussian model, and in the third control case, the tolerances were defined from the quantiles of the observed error distribution. The analysis of the test results based on the type I and type II errors show that the method is able to control the positional accuracy of freely distributed data.

Mixture variational autoencoders

Variational autoencoders (VAEs) combine a generative model and a recognition model, and jointly train them to maximize a variational lower bound. VAEs play an important role in unsupervised learning and representation learning. But the isotropic generative model in VAEs cannot sufficiently utilize the latent representative space. In this paper, we present mixture variational autoencoders (MVAEs) which suppose observed data is generated from mixture models. We use a continuous variable with prior of Normal distribution as the latent representation, and introduce a discrete variable with prior of Multinomial distribution as latent indicator for mixture models. Two latent variables are both approximated by recognition models computed from neural networks. Furthermore, we combine the reparameterization trick and stick-breaking parameterization to realize stochastic gradient variational Bayes (SGVB) estimator on our variational objective. The MVAEs improve the generative performance by enlarging the latent representative space. Finally, we demonstrate the performance of MVAEs compared with the state-of-the-art models on MNIST, OMNIGLOT, and Fashion-MNIST.