Mixture Of Statistics Research Articles

Analyzing 2D gel images using a two-component empirical Bayes model.

BackgroundTwo-dimensional polyacrylomide gel electrophoresis (2D gel, 2D PAGE, 2-DE) is a powerful tool for analyzing the proteome of a organism. Differential analysis of 2D gel images aims at finding proteins that change under different conditions, which leads to large-scale hypothesis testing as in microarray data analysis. Two-component empirical Bayes (EB) models have been widely discussed for large-scale hypothesis testing and applied in the context of genomic data. They have not been implemented for the differential analysis of 2D gel data. In the literature, the mixture and null densities of the test statistics are estimated separately. The estimation of the mixture density does not take into account assumptions about the null density. Thus, there is no guarantee that the estimated null component will be no greater than the mixture density as it should be.ResultsWe present an implementation of a two-component EB model for the analysis of 2D gel images. In contrast to the published estimation method, we propose to estimate the mixture and null densities simultaneously using a constrained estimation approach, which relies on an iteratively re-weighted least-squares algorithm. The assumption about the null density is naturally taken into account in the estimation of the mixture density. This strategy is illustrated using a set of 2D gel images from a factorial experiment. The proposed approach is validated using a set of simulated gels.ConclusionsThe two-component EB model is a very useful for large-scale hypothesis testing. In proteomic analysis, the theoretical null density is often not appropriate. We demonstrate how to implement a two-component EB model for analyzing a set of 2D gel images. We show that it is necessary to estimate the mixture density and empirical null component simultaneously. The proposed constrained estimation method always yields valid estimates and more stable results. The proposed estimation approach proposed can be applied to other contexts where large-scale hypothesis testing occurs.

Exact nonparametric meta-analysis for multiple independent doubly Type-II censored samples

Nonparametric inferential methods are discussed for the situation of multiple independent doubly Type-II censored samples. The basic distribution theory for the pooled sample is given assuming that the underlying distribution is continuous and it is demonstrated how the weights in the mixture representations of the pooled order statistics as a mixture of the usual order statistics can be obtained. This is used to construct nonparametric prediction intervals, tolerance intervals for a future sample, and confidence intervals for a population quantile. A small simulation study compares the exact coverage probabilities for population quantiles, to those obtained where the mixture weights are generated by simulation.

Extending the multivariate generalised [formula omitted] and generalised [formula omitted] distributions

The G G H family of multivariate distributions is obtained by scale mixing on the Exponential Power distribution using the Extended Generalised Inverse Gaussian distribution. The resulting G G H family encompasses the multivariate generalised hyperbolic ( G H ), which itself contains the multivariate t and multivariate Variance-Gamma ( V G ) distributions as special cases. It also contains the generalised multivariate t distribution [O. Arslan, Family of multivariate generalised t distribution, Journal of Multivariate Analysis 89 (2004) 329–337] and a new generalisation of the V G as special cases. Our approach unifies into a single G H -type family the hitherto separately treated t -type [O. Arslan, A new class of multivariate distribution: Scale mixture of Kotz-type distributions, Statistics and Probability Letters 75 (2005) 18–28; O. Arslan, Variance–mean mixture of Kotz-type distributions, Communications in Statistics-Theory and Methods 38 (2009) 272–284] and V G -type cases. The G G H distribution is dual to the distribution obtained by analogous mixing on the scale parameter of a spherically symmetric stable distribution. Duality between the multivariate t and multivariate V G [S.W. Harrar, E. Seneta, A.K. Gupta, Duality between matrix variate t and matrix variate V.G. distributions, Journal of Multivariate Analysis 97 (2006) 1467–1475] does however extend in one sense to their generalisations.

Problems and Techniques

This issue's Problems and Techniques section contains three papers that treat areas that are simultaneously diverse and connected. The first paper considers a favorite topic in linear algebra---Krylov subspace methods, which made the widely cited "Top 10 Algorithms" (of the twentieth century) list published by Computing in Science and Engineering in January 2000. Thousands of papers have been written about these iterative techniques for solving large linear systems Ax = b. The issue considered by V. Simoncini and D. Szyld is the "superlinear convergence" of Krylov subspace methods, where the term means linear convergence that accelerates as the iterations proceed. The authors propose an analytic model of superlinear convergence derived from proximity of an invariant subspace to a crucial subspace in the algorithm. The paper contains a clear summary of previous work on analytic models of superlinear convergence; it then analyzes the convergence of specific methods, including GMRES, the conjugate gradient method, block Krylov subspace methods, and inexact Krylov subspace methods, accompanied in each case by illuminating numerical results. A particularly interesting result for inexact methods involves the effects of using invariant subspaces of the implicitly perturbed matrix rather than the original matrix. The second and third papers involve problems that might be seen as variations on a theme---the collision sequences generated by beads on a frictionless loop, and the stationary density for the traffic flow from random walks on a circle where passing is forbidden. (A visual similarity is evident in the first figures in both papers.) The second paper, by B. Cooley and P. K. Newton, studies beads on a ring. The authors note that the number of collisions is bounded for a finite number of particles colliding elastically on a line, and that several results are known for elastic particles moving in space. But on a ring there are usually an infinite number of collisions, and thus the emphasis shifts to the long-term properties of the dynamics, which strongly depend on assumptions about the nature of the collisions. The paper examines basic properties of the general problem in terms of matrix products, explaining in addition the relationship between a collision sequence and a billiard trajectory in a right-angle table with nonstandard reflection rules. (Figure 9 summarizes computational results that express the limiting results of this analysis.) The authors present a clear and self-contained exposition of a deceptively simple problem with extensions to questions about the connection between randomness and chaotic deterministic systems. Our third paper, by J. D. Skufca, starts by noting the similarities among the slowdown of cars in dense traffic, the enforced single-file motion of protein-controlled diffusion through a membrane, and the seemingly inevitable delays in serving oneself at a buffet dinner. The author then turns to the underlying general problem---analyzing the single-file flow of particles on a closed loop when the particles are unable to pass one another. Beginning with a Markov chain description transforming the problem to that of a generalized random walk on a bounded hyperplane, the author constructs a digraph representation that allows transition probabilities to be obtained via a shortest path determination. The ultimate goal of this analysis is to determine the efficiency of the system, measured by the expected fraction of the number of time steps when an individual worker is blocked (unable to move) because of the presence of another worker. The paper's results are derived based on a mixture of probability, statistics, graph theory, and geometry. We hope that SIAM Review readers will learn something new from the insights of these papers into several fields of applied mathematics.

Scaling of Structural Strength

Scaling of Structural Strength

Why the South Lost the Civil War (review)

BOOK REVIEWS Why the South Lost the Civil War. By Richard E. Beringer, Herman Hattaway, Archer Jones, and William N. Still, Jr. (Athens, Georgia: The University of Georgia Press. 1986, Pp. xi, 582. $29.95.) Although they reject P. G. T. Beauregard's explanation for Confederate defeat, the authors of Why the South Lost the Civil War agree with the general's contention that more than a "mere material contrast" was involved . Their book, a thorough and sophisticated analysis of the reasons why the struggle ended as it did, constitutes a major contribution to Civil War scholarship. The volume is also an important historiographical survey. In developing the thesis that defective nationalism caused the Confederate failure, Beringer and the others criticize much of the pertinent historical literature , paying special attention to the works of Frank L. Owsley, David Donald, T. Harry Williams, Grady McWhiney, and Perry D. Jamieson. Yet Why the South Lost the Civil War is more than a revisionist exercise. The authors use an imaginative combination of comparative history, social psychology, statistics, and cultural history to argue their own case as well as to challenge the premises of other scholars. An intriguing example of this interpretative skill is the way Beringer and his colleagues dispatch the Jominian myth espoused by T. Harry Williams, David Donald, and others. For far too long historians have accepted the false proposition that Jomini and Clausewitz, his Prussian rival, had derived antithetical principles from their studies of Napoleonic warfare. Scholars have applied this inaccurate generalization to the Civil War, claiming that unwavering devotion to Jomini's teachings ruined the Confederates, whereas their Union counterparts, though starting from the sameJominian base, finally won by unwittingly adopting Clausewitz's precepts. Beringer et al. expose this fallacy by demonstrating that the two military philosophers' views were more harmonious than commonly supposed. They also show how both sides fought according to principles that were as much Jominian as Clausewitzian. Indeed , Grant, whom historians usually portray as an inadvertentherald of Clausewitz, emerges in this study as an exemplar of Jomini. Why the South Lost the Civil War also attacks Frank L. Owsley's states'-rights thesis and its modern derivative, excessive democracy. The BOOK REVIEWS89 authors maintain that personal bias, together with the bombast of President Jefferson Davis's opponents, led Owsley to overestimate the strength of southern particularism. Actually, the state governments, despite occasional clashes with and protests against the administration in Richmond, rendered crucial support to the cause not only by providing troops and materiel to the Confederate field armies but by shouldering the onerous tasks of local defense and civilian welfare. Even Governors Zebulon Vance and Joseph Brown, Owsley's pet villains, cooperated with Davis to a much greater extent than for which they receive credit. In this regard the authors make a significant point: the states'-rights issue served as a needed safety valve in a political system which lacked an opposition party. In Attack and Die: Civil War Military Tactics and the Southern Heritage , Grady McWhiney and Perry D. Jamieson call on comparative history and statistics to assert that ethnic characteristics drove the Confederates to "murder themselves" in repeated frontal assaults. Beringer and his colleagues employ the same weapons to dispute that theory. According to them, McWhiney and Jamieson mistakenly assumed the style of warfare practiced by the Celtic ancestors of the Confederates was unique. Moreover, when the sample of battles is expanded and the number of casualties more precisely classified, it becomes apparent that southerners were no more addicted to the offensive than their enemies. Likewise, the rebels suffered heavier relative losses, not because of a proclivity for headlong assaults, but because Union forces had greater combat power. In responding to the question of why the South lost, the authors resort to a mixture of social psychology, religious history, statistics, and common sense. As they see it, manpower shortages, the blockade, the loss of the Mississippi, and industrial inferiority were at most secondary factors . The basic cause was a nationalism too feeble to withstand the strains of protracted war. The southern people were insufficiently committed to the concept of a Confederate nation to carry on the fight past the spring of 1865...