scholarly journals Tail Heterogeneity for Dynamic Covariance-Matrix-Valued Random Variables: the F-Riesz Distribution

2021 ◽  
Author(s):  
Francisco Blasques ◽  
Andre Lucas ◽  
Anne Opschoor ◽  
Luca Rossini
Keyword(s):  
Covariance Matrix ◽  
Random Variables ◽  
Riesz Distribution

ANOVA models with random effects: an approach via symmetry

1986 ◽  
Vol 23 (A) ◽  
pp. 355-368 ◽  
Author(s):  
T. P. Speed
Keyword(s):  
Covariance Matrix ◽  
Random Effects ◽  
Spectral Decomposition ◽  
Linear Models ◽  
Random Variables ◽  
Components Of Variance

The standard ANOVA models with random effects for multi-indexed arrays of random variables with an arbitrary nesting structure on the indices are considered from the viewpoint of symmetry. It is found that the covariance matrix of such an array has sufficient symmetry to permit viewing the usual components of variance as a generalised spectrum and the linear models of random effects as a generalised spectral decomposition.

scholarly journals Bayesian statistics and Monte Carlo methods

2018 ◽  
Vol 8 (1) ◽  
pp. 18-29 ◽  
Author(s):  
K. R. Koch
Keyword(s):  
Monte Carlo ◽  
Bayesian Statistics ◽  
Covariance Matrix ◽  
Random Vector ◽  
Monte Carlo Methods ◽  
Confidence Region ◽  
Random Variables ◽  
Point Estimation ◽  
Unknown Parameters ◽  
Multivariate Distributions

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.

scholarly journals Capturing a Change in the Covariance Structure of a Multivariate Process

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 156
Author(s):  
Andriette Bekker ◽  
Johannes T. Ferreira ◽  
Schalk W. Human ◽  
Karien Adamski
Keyword(s):  
Covariance Matrix ◽  
Random Variables ◽  
Covariance Structure ◽  
Multivariate Normal ◽  
New Approach ◽  
Downward Shift ◽  
Consecutive Time ◽  
Time Periods ◽  
Unknown Covariance ◽  
Mean Vector

This research is inspired from monitoring the process covariance structure of q attributes where samples are independent, having been collected from a multivariate normal distribution with known mean vector and unknown covariance matrix. The focus is on two matrix random variables, constructed from different Wishart ratios, that describe the process for the two consecutive time periods before and immediately after the change in the covariance structure took place. The product moments of these constructed random variables are highlighted and set the scene for a proposed measure to enable the practitioner to calculate the run-length probability to detect a shift immediately after a change in the covariance matrix occurs. Our results open a new approach and provides insight for detecting the change in the parameter structure as soon as possible once the underlying process, described by a multivariate normal process, encounters a permanent/sustained upward or downward shift.

ANOVA models with random effects: an approach via symmetry

1986 ◽  
Vol 23 (A) ◽  
pp. 355-368 ◽  
Author(s):  
T. P. Speed
Keyword(s):  
Covariance Matrix ◽  
Random Effects ◽  
Spectral Decomposition ◽  
Linear Models ◽  
Random Variables ◽  
Components Of Variance

The standard ANOVA models with random effects for multi-indexed arrays of random variables with an arbitrary nesting structure on the indices are considered from the viewpoint of symmetry. It is found that the covariance matrix of such an array has sufficient symmetry to permit viewing the usual components of variance as a generalised spectrum and the linear models of random effects as a generalised spectral decomposition.

scholarly journals Principal Component Analysis In Radar Polarimetry

2005 ◽  
Vol 3 ◽  
pp. 399-400 ◽  
Author(s):  
A. Danklmayer ◽  
M. Chandra ◽  
E. Lüneburg
Keyword(s):  
Covariance Matrix ◽  
Random Variables ◽  
Joint Probability ◽  
Principal Component ◽  
Component Analysis ◽  
Distribution Functions ◽  
Linear Transformations ◽  
Multivariate Gaussian Distribution ◽  
Data Set ◽  
Radar Polarimetry

Abstract. Second order moments of multivariate (often Gaussian) joint probability density functions can be described by the covariance or normalised correlation matrices or by the Kennaugh matrix (Kronecker matrix). In Radar Polarimetry the application of the covariance matrix is known as target decomposition theory, which is a special application of the extremely versatile Principle Component Analysis (PCA). The basic idea of PCA is to convert a data set, consisting of correlated random variables into a new set of uncorrelated variables and order the new variables according to the value of their variances. It is important to stress that uncorrelatedness does not necessarily mean independent which is used in the much stronger concept of Independent Component Analysis (ICA). Both concepts agree for multivariate Gaussian distribution functions, representing the most random and least structured distribution. In this contribution, we propose a new approach in applying the concept of PCA to Radar Polarimetry. Therefore, new uncorrelated random variables will be introduced by means of linear transformations with well determined loading coefficients. This in turn, will allow the decomposition of the original random backscattering target variables into three point targets with new random uncorrelated variables whose variances agree with the eigenvalues of the covariance matrix. This allows a new interpretation of existing decomposition theorems.

scholarly journals On determining the undrained bearing capacity coefficients of variation for foundations embedded on spatially variable soil

2020 ◽  
Vol 42 (2) ◽  
pp. 125-136
Author(s):  
Marcin Chwała
Keyword(s):  
Shear Strength ◽  
Random Field ◽  
Bearing Capacity ◽  
Covariance Matrix ◽  
Three Dimensional ◽  
Random Variables ◽  
Undrained Shear Strength ◽  
Spatial Averaging ◽  
The Matrix ◽  
Undrained Shear

AbstractThis paper presents an efficient method and its usage for the three-dimensional random bearing capacity evaluation for square and rectangular footings. One of the objectives of the study is to deliver graphs that can be used to easily estimate the approximated values of coefficients of variations of undrained bearing capacity. The numerical calculations were based on the proposed method that connects three-dimensional failure mechanism, simulated annealing optimization scheme and spatial averaging. The random field is used for describing the spatial variability of undrained shear strength. The proposed approach is in accordance with a constant covariance matrix concept, that results in a highly efficient tool for estimating the probabilistic characteristics of bearing capacity. As a result, numerous three-dimensional simulations were performed to create the graphs. The considered covariance matrix is a result of Vanmarcke’s spatial averaging discretization of a random field in the dissipation regions to the single random variables. The matrix describes mutual correlation between each dissipation region (or between those random variables). However, in the presented approach, the matrix was obtained for the expected value of undrained shear strength and keep constant during Monte Carlo simulations. The graphs were established in dimensionless coordinates that vary in the observable in practice ranges of parameters (i.e., values of fluctuation scales, foundation sizes and shapes). Examples of usage were given in the study to illustrate the application possibility of the graphs. Moreover, the comparison with the approach that uses individually determined covariance matrix is shown.

Array Variate Random Variables with Multiway Kro- necker Delta Covariance Matrix Structure

2011 ◽  
Vol 2 (1) ◽  
Author(s):  
Deniz Akdemir ◽  
Arjun K. Gupta
Keyword(s):  
Principal Component Analysis ◽  
Covariance Matrix ◽  
Random Variables ◽  
Principal Component ◽  
Random Variable ◽  
High Dimensional ◽  
Underlying Structure ◽  
Data Sets ◽  
Matrix Structure ◽  
Background Material

Standard statistical methods applied to matrix random variables often fail to describethe underlying structure in multiway data sets. After a review of the essential background material,this paper introduces the notion of array variate random variable. A normal array variate randomvariable is dened and a method for estimating the parameters of array variate normal distributionis given. We introduce a technique called slicing for estimating the covariance matrix of highdimensional data. Finally, principal component analysis and classication techniques are developedfor array variate observations and high dimensional data.

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