Multivariate Uniform Distribution Research Articles

Bayesian group bridge composite quantile regression

Bayesian regularized composite quantile regression (CQR) method with group bridge penalty is adopted to conduct covariate selection and estimation in CQR. MCMC algorithm was improved for posterior inference employing a scale mixture of normal of the asymmetric Laplace distribution (ALD). The suggested algorithm uses priors for the coefficients of regression, which are scale mixtures of multivariate uniform distributions with a particular Gamma distribution as a mixing distribution. Simulation results and analyses of real data show that the suggested MCMC sampler has excellent mixing feature and outperforms the current approaches in terms of prediction accuracy and model selection.

Stochastic Order for a Multivariate Uniform Distributions Family

In this article we give sufficient conditions for stochastic order of multivariate uniform distributions on closed convex sets.

Monte Carlo methods

Monte Carlo methods deal with generating random variates from probability density functions in order to estimate unknown parameters or general functions of unknown parameters and to compute their expected values, variances and covariances. One generally works with the multivariate normal distribution due to the central limit theorem. However, if random variables with the normal distribution and random variables with a different distribution are combined, the normal distribution is not valid anymore. The Monte Carlo method is then needed to get the expected values, variances and covariances for the random variables with distributions different from the normal distribution. The error propagation by Monte Carlo methods is discussed and methods for generating random variates from the multivariate normal distribution and from the multivariate uniform distribution. The Monte Carlo integration is presented leading to the sampling–importance-resampling algorithm. Markov chain Monte Carlo methods provide by the Metropolis algorithm and the Gibbs sampler additional ways of generating random variates. A special topic is the Gibbs sampler for computing and propagating large covariance matrices. This task arises, for instance, when the geopotential is determined from satellite observations. The example of the minimal detectable outlier shows, how the Monte Carlo method is used to determine the power of a hypothesis test.

Multivariate Uniformity Test Based on Incomplete Samples

The purpose of this paper is to propose an approach for testing multivariate uniformity. The proposed test procedure works for both complete sample case and incomplete sample case. What multivariate uniformity test does is to check whether the population distribution, from which a random sample is drawn is a multivariate uniform distribution. Particularly, it is desired to check whether the population distribution is a uniform distribution on a [Formula: see text]-dimensional unit cube. There are some existing research papers in the literature. The proposed method works when only some arbitrarily selected order statistics are available for the test.

Item usage in a multidimensional computerized adaptive test (MCAT) measuring health-related quality of life

PurposeExamining item usage is an important step in evaluating the performance of a computerized adaptive test (CAT). We study item usage for a newly developed multidimensional CAT which draws items from three PROMIS domains, as well as a disease-specific one.MethodsThe multidimensional item bank used in the current study contained 194 items from four domains: the PROMIS domains fatigue, physical function, and ability to participate in social roles and activities, and a disease-specific domain (the COPD-SIB). The item bank was calibrated using the multidimensional graded response model and data of 795 patients with chronic obstructive pulmonary disease. To evaluate the item usage rates of all individual items in our item bank, CAT simulations were performed on responses generated based on a multivariate uniform distribution. The outcome variables included active bank size and item overuse (usage rate larger than the expected item usage rate).ResultsFor average θ-values, the overall active bank size was 9–10%; this number quickly increased as θ-values became more extreme. For values of −2 and +2, the overall active bank size equaled 39–40%. There was 78% overlap between overused items and active bank size for average θ-values. For more extreme θ-values, the overused items made up a much smaller part of the active bank size: here the overlap was only 35%.ConclusionsOur results strengthen the claim that relatively short item banks may suffice when using polytomous items (and no content constraints/exposure control mechanisms), especially when using MCAT.

Systematic Effects in Laser Scanning and Visualization by Confidence Regions

AbstractA new method for dealing with systematic effects in laser scanning and visualizing them by confidence regions is derived. The standard deviations of the systematic effects are obtained by repeatedly measuring three-dimensional coordinates by the laser scanner. In addition, autocovariance and cross-covariance functions are computed by the repeated measurements and give the correlations of the systematic effects. The normal distribution for the measurements and the multivariate uniform distribution for the systematic effects are applied to generate random variates for the measurements and random variates for the measurements plus systematic effects. Monte Carlo estimates of the expectations and the covariance matrix of the measurements with systematic effects are computed. The densities for the confidence ellipsoid for the measurements and the confidence region for the measurements with systematic effects are obtained by relative frequencies. They only depend on the size of the rectangular volume elements for which the densities are determined. The problem of sorting the densities is solved by sorting distances together with the densities. This allows a visualization of the confidence ellipsoid for the measurements and the confidence region for the measurements with systematic effects.

A Bayesian Network-Based Method to Alleviate the Ill-Posed Inverse Problem: A Case Study on Leaf Area Index and Canopy Water Content Retrieval

Retrieval of vegetation parameters from remotely sensed data using a radiative transfer model is generally hampered by the ill-posed inverse problem, which dramatically decreases the precision level of retrieved parameters. The purpose of this study was to use a Bayesian network-based method to allow the alleviation of the ill-posed inverse problem. This was achieved by introducing the correlations between the model free parameters into their prior joint probability distribution (PJPD), allowing the reduction of the probabilities of unrealistic combinations. Three sampling strategies intended to design three types of PJPDs that considered different correlations (represented by a correlation matrix) were presented. They were multivariate uniform distribution composed by independent free parameters, multivariate uniform distribution based on a simple correlation matrix, and multivariate Gaussian distribution based on a complicated correlation matrix, respectively. A case study of the presented method to retrieve leaf area index (LAI) and canopy water content (CWC) using the PROSAIL_5B (PROSPECT-5 + 4SAIL) model from Landsat 8 products was implemented. Results indicate that the presented method greatly improves the precision level of target parameters, with the coefficient of determination $R^2$ of 0.69, 0.77, and 0.82 and root-mean-square error (RMSE) of 0.55, 0.51, and 0.44 $\mbox{m}^{2}\cdot\mbox{m}^{-2}$ for LAI and $R^{2}=0.68, 0.78, \mbox{and} 0.84$ and $\mbox{RMSE}=230, 198, \mbox{and} 166 \mbox{g}\cdot\mbox{m}^{-2}$ for CWC, respectively. Hence, the ill-posed inverse problem can be alleviated by the presented method, which can be widely applied for vegetation parameters retrieval.

A comparison between the Gumbel-Hougaard and distorted Frank copulas for drought frequency analysis

Drought is a global phenomenon and is a common characteristic of extreme climate. It is widely considered as the world’s costliest natural disaster since their effects are especially devastating to the agricultural and social economy. Effective water resource infrastructure and management is required for the mitigation of drought, and this requires drought risk assessment. Essential characteristics of drought are severity, average intensity and duration, which are highly correlated. Copulas are multivariate uniform distribution functions with uniform marginals, which provide a flexible approach to model their relationship. For an extreme drought event to be modelled, the upper-tail dependence must be investigated. The construction of an alternative copula form is introduced through a transformation called a distortion, to allow for more versatile modelling of tail dependence. Data from a rainfall district in New South Wales is considered and the Gumbel-Hougaard copula and distorted Frank copulas are fitted to the drought characteristics. Goodness-of-fit and upper tail correlations of these copulas are analysed and discussed. Finally, conditional annual drought recurrence intervals are calculated and compared for these fitted copulas.

Motivational typologies of drinkers: do enhancement and coping drinkers form two distinct groups?

This study used a person-centered approach to test whether drinking motive typologies could be identified. Longitudinal study of college students within the intensive multivariate prospective alcohol college-transitions (IMPACTS) data set. University campus in the United States. University students (baseline n reporting alcohol motives = 2158; baseline age = 18.60 years old). The drinking motives questionnaire-revised (DMQ-R). Using Steinley & Brusco's cluster analysis approach [based on the theoretical ratio expected between the within sum of squares and the total sum of squares when the data are divided into two clusters when no cluster structure is present; the cut-off for the ratio is 0.25 for uniform (multivariate uniform) distributions and 0.36 for normal (multivariate normal) distributions], we examined whether there was evidence for distinct clusters of individuals that differed on their overall level of motives to drink. We tested the fit of a one-group (cluster) solution compared to multi-cluster solutions. Both cross-sectionally and prospectively, the data could not be partitioned into two or more clusters [regardless of whether the cut-off assuming a multivariate uniform distribution (i.e. 0.25) or the more liberal multivariate normal distribution (i.e., 0.36) was used]. These findings showed that enhancement and coping drinkers do not form two distinct groups but, rather, these motives exist on a continuum such that individuals who are high in one internal motive tend to be high in the other motive. Coping and enhancement drinkers do not form two distinct groups. Variable-centered approaches to drinking motives may be a better alternative to classifying all drinkers as either enhancement or coping drinkers for both clinical and research endeavors.

Model-Based Learning Using a Mixture of Mixtures of Gaussian and Uniform Distributions

We introduce a mixture model whereby each mixture component is itself a mixture of a multivariate Gaussian distribution and a multivariate uniform distribution. Although this model could be used for model-based clustering (model-based unsupervised learning) or model-based classification (model-based semi-supervised learning), we focus on the more general model-based classification framework. In this setting, we fit our mixture models to data where some of the observations have known group memberships and the goal is to predict the memberships of observations with unknown labels. We also present a density estimation example. A generalized expectation-maximization algorithm is used to estimate the parameters and thereby give classifications in this mixture of mixtures model. To simplify the model and the associated parameter estimation, we suggest holding some parameters fixed-this leads to the introduction of more parsimonious models. A simulation study is performed to illustrate how the model allows for bursts of probability and locally higher tails. Two further simulation studies illustrate how the model performs on data simulated from multivariate Gaussian distributions and on data from multivariate t-distributions. This novel approach is also applied to real data and the performance of our approach under the various restrictions is discussed.

Trivariate copulas for characterisation of droughts

Australian agriculture is at serious risk from drought, and water resource infrastructure and management can mitigate the effects. The consequences of droughts depend on their intensity, duration and severity. These variables are correlated and the dependence structure is here described by copulas. Copulas are multivariate uniform distributions which allow for the dependence structure to be modelled independently of the marginal distributions. Trivariate Gaussian and Gumbel copulas are fitted to the data from a rainfall district in NSW. We assess the goodness of fit of the data to the different forms using several criteria. The data are best described by a Gumbel copula and three parameter Weibull marginal distributions. References P. Embrechets, F. Lindskog and A. J. McNeil. Modelling dependence with copulas and applications to risk management. In: Rachev S. T., editor. Handbook of heavy tailed distributions in finance. Amsterdam, North--Holland: Elsevier, 2003. A-C Favre, S. E. Adlouni, L. Perreault, N. Thiemonge and B. Bobee. Multivariate hydrological frequency analysis using copulas. Water Resources Research , 2004, 40 , W01101. doi:10.1029/2003WR002456 C. Genest and J. MacKay. The joy of copulas: bivariate distributions with uniform marginals. The American Statistician , 1986, 40 (4), 280--283. C. Genest and A-C Favre. Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering , 2006, 12 (4), 347--368. doi:10.1061/(ASCE)1084-0699(2007)12:4(347) S. Grimaldi and F. Serinaldi. Asymmetric copula in multivariate flood frequency analysis. Advances in Water Resources , 2006, 29 , 1155--1167. doi:10.1016/j.advwatres.2005.09.005 H. Joe. Multivariate models and dependence concepts . New York: Chapman and Hall, 1997. T. B. McKee, N. J. Doesken and J. Kliest. The relationship of drought frequency and duration to time scales. In Proceeding of the 8th Conference on Applied Climatology , 17--22 January 1993, Anaheim, CA, American Meteorological Society: Boston, MA, USA, 179--184. R Development Core Team. R: A Language and Environment for Statistical Computing . R Foundation for Statistical Computing. 2007. http://www.R-project.org J. T. Shiau. Fitting Drought Duration and Severity with Two-Dimensional Copulas. Water Resources Management , 2006, 20 , 795--815. doi:10.1007/s11269-005-9008-9 A. Sklar. Fonctions de repartitions a $n$ dimensions et leurs marges. Publications de l'Institut de Statistique de l'Universite de Paris , 1959, 8 , 229--231. The Australian. El Nino bad news for drought-stricken east . The Australian, 14th September. Accessed 19th September. http://www.theaustralian.news.com.au/printpage/0,5942,2041027 L. Zhang and V. P. Singh. Bivariate Flood Frequency Analysis using Copula Method. Journal of Hydrologic Engineering , 2006, 11 (2), 150--164. doi:10.1061/(ASCE)1084-0699(2006)11:2(150)

A survey on continuous elliptical vector distributions

In this paper it is taken up a revision and characterization of the class of abso- lutely continuous elliptical distributions upon a parameterization based on the density function. Their properties (probabilistic characteristics, aÆne trans- formations, marginal and conditional distributions and regression) are shown in a practical and easy to interpret way. Two examples are fully undertaken: the multivariate double exponential distribution and the multivariate uniform distribution. 2000 Mathematics Su

A Stochastic Model for Predicting the Velocity Distribution of Warhead Fragments

It has been the practice of those who study the velocity of fragments from a warhead to average the velocity of the fragment over its path rather than to examine in greater detail the underlying stochastic process. In this paper we present a more careful analysis of the pertinent statistical process by taking into account the correlations that exist along the trajectory of the fragment. One assumes that the fragments generally tumble uniformly and that drag coefficient depends only on presented area. This seems to be a good approximation in the supersonic regime (from Mach 3 to Mach 6, for example). A brief discussion of the case when drag, lift, and gravity effects are all assumed to be present is given. Also, we present a new family of “multivariate uniform distributions” geared to the physics of the problem.