Topological Features of Multivariate Distributions: Dependency on the Covariance Matrix

Abstract The Bayesian approach allows an intuitive way to derive the methods of statistics. Probability is defined as a measure of the plausibility of statements or propositions. Three rules are sufficient to obtain the laws of probability. If the statements refer to the numerical values of variables, the so-called random variables, univariate and multivariate distributions follow. They lead to the point estimation by which unknown quantities, i.e. unknown parameters, are computed from measurements. The unknown parameters are random variables, they are fixed quantities in traditional statistics which is not founded on Bayes’ theorem. Bayesian statistics therefore recommends itself for Monte Carlo methods, which generate random variates from given distributions. Monte Carlo methods, of course, can also be applied in traditional statistics. The unknown parameters, are introduced as functions of the measurements, and the Monte Carlo methods give the covariance matrix and the expectation of these functions. A confidence region is derived where the unknown parameters are situated with a given probability. Following a method of traditional statistics, hypotheses are tested by determining whether a value for an unknown parameter lies inside or outside the confidence region. The error propagation of a random vector by the Monte Carlo methods is presented as an application. If the random vector results from a nonlinearly transformed vector, its covariance matrix and its expectation follow from the Monte Carlo estimate. This saves a considerable amount of derivatives to be computed, and errors of the linearization are avoided. The Monte Carlo method is therefore efficient. If the functions of the measurements are given by a sum of two or more random vectors with different multivariate distributions, the resulting distribution is generally not known. TheMonte Carlo methods are then needed to obtain the covariance matrix and the expectation of the sum.

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The Ordering of Shannon Entropies for the Multivariate Distributions and Distributions of Eigenvalues

Entropy ◽

10.3390/e21020201 ◽

2019 ◽

Vol 21 (2) ◽

pp. 201

Author(s):

Ming-Tien Tsai ◽

Feng-Ju Hsu ◽

Chia-Hsuan Tsai

Keyword(s):

General Population ◽

Covariance Matrix ◽

Shannon Entropy ◽

Distribution Functions ◽

Multivariate Distributions ◽

Convex Ordering ◽

Totally Positive ◽

Entropy Measures ◽

Set Up ◽

Entropy Inequalities

In this paper, we prove the Shannon entropy inequalities for the multivariate distributions via the notion of convex ordering of two multivariate distributions. We further characterize the multivariate totally positive of order 2 ( M T P 2 ) property of the distribution functions of eigenvalues of both central Wishart and central MANOVA models, and of both noncentral Wishart and noncentral MANOVA models under the general population covariance matrix set-up. These results can be directly applied to both the comparisons of two Shannon entropy measures and the power monotonicity problem for the MANOVA problem.

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Imaging of platinum-shadowed macromolecular complexes in dark field

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100110891 ◽

1984 ◽

Vol 42 ◽

pp. 146-147

Author(s):

D.W. Andrews ◽

F.P. Ottensmeyer

Keyword(s):

Thin Film ◽

Three Dimensional ◽

Average Diameter ◽

Dark Field ◽

Bright Field ◽

Field Mode ◽

Macromolecular Complexes ◽

Average Mass ◽

Topological Features ◽

Projection Images

Shadowing with heavy metals has been used for many years to enhance the topological features of biological macromolecular complexes. The three dimensional features present in directionaly shadowed specimens often simplifies interpretation of projection images provided by other techniques. One difficulty with the method is the relatively large amount of metal used to achieve sufficient contrast in bright field images. Thick shadow films are undesirable because they decrease resolution due to an increased tendency for microcrystalline aggregates to form, because decoration artefacts become more severe and increased cap thickness makes estimation of dimensions more uncertain.The large increase in contrast provided by the dark field mode of imaging allows the use of shadow replicas with a much lower average mass thickness. To form the images in Fig. 1, latex spheres of 0.087 μ average diameter were unidirectionally shadowed with platinum carbon (Pt-C) and a thin film of carbon was indirectly evaporated on the specimen as a support.

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