scholarly journals Normal Distribution on the Rotation Group So(3)

1997 ◽  
Vol 29 (3-4) ◽  
pp. 201-233 ◽  
Author(s):  
Dmitry I. Nikolayev ◽  
Tatjana I. Savyolov
Keyword(s):  
Normal Distribution ◽  
Orientation Distribution ◽  
Rotation Group ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Normal Distributions ◽  
The Central Limit Theorem ◽  
Special Cases ◽  
Wrapped Normal ◽  
Circular Normal Distribution

We study the normal distribution on the rotation group SO(3). If we take as the normal distribution on the rotation group the distribution defined by the central limit theorem in Parthasarathy (1964) rather than the distribution with density analogous to the normal distribution in Eucledian space, then its density will be different from the usual (1/2πσ) exp⁡(−(x−m)2/2σ2) one. Nevertheless, many properties of this distribution will be analogous to the normal distribution in the Eucledian space. It is possible to obtain explicit expressions for density of normal distribution only for special cases. One of these cases is the circular normal distribution.The connection of the circular normal distribution SO(3) group with the fundamental solution of the corresponding diffusion equation is shown. It is proved that convolution of two circular normal distributions is again a distribution of the same type. Some projections of the normal distribution are obtained. These projections coincide with a wrapped normal distribution on the unit circle and with the Perrin distribution on the two-dimensional sphere. In the general case, the normal distribution on SO(3) can be found numerically. Some algorithms for numerical computations are given. These investigations were motivated by the orientation distribution function reproduction problem described in the Appendix.

scholarly journals Optimal Calculation of ODF With the Canonical Normal Distribution on the Rotation Group

1999 ◽  
Vol 33 (1-4) ◽  
pp. 337-341
Author(s):  
T. I. Savyolova ◽  
E. A. Davidzhan ◽  
T. M. Ivanova
Keyword(s):  
Experimental Data ◽  
Distribution Function ◽  
Physical Properties ◽  
Normal Distribution ◽  
Orientation Distribution Function ◽  
Orientation Distribution ◽  
Rotation Group ◽  
Polycrystalline Materials ◽  
Pole Figures ◽  
Optimal Calculation

Macroscopic physical properties of most polycrystalline materials are controlled by orientation distribution of their grains. The orientation distribution function (ODF) of a polycrystal is seldom if ever determined directly from an experiment. Usually experimental data are represented by a set of pole figures (PFs), these latter are some integral projections of the ODF. The main problem of quantitative texture analysis is to recover ODF from its corresponding PFs. With any set of PFs the solution of this problem is non-unique. That is why some assumptions about ODF structure are necessary. We consider ODF as superposition of the canonical normal distribution (CND) on the rotation group SO(3).

The computational approaches to calculate normal distributions on the rotation group

2007 ◽  
Vol 40 (3) ◽  
pp. 449-455 ◽  
Author(s):  
Maxim Borovkov ◽  
Tatjana Savyolova
Keyword(s):  
Normal Distribution ◽  
Electron Backscatter Diffraction ◽  
Classical Method ◽  
Short Review ◽  
Rotation Group ◽  
Statistical Simulation ◽  
Computational Approaches ◽  
Normal Distributions ◽  
Method Of Calculation ◽  
Advantages And Disadvantages

There are several kinds of probability distribution widely used in quantitative texture analysis. One of them is the normal (Gaussian) distribution. The main application of the normal distribution is the orientation distribution function and pole-figure approximation on the rotation group SO(3) and sphere S2accordingly. The calculation of the normal distribution is a complicated computational task. There are currently several methods for calculating the normal distribution. Each of these methods has its advantages and disadvantages. The classical method of calculation by Fourier series summation is effective enough only in the case of continuous texture approximation. In the case of sharp texture approximation, the analytical approach is more suitable and effective. These two calculation methods result in a continuous function. The other method allows a discrete orientation set to be obtained, corresponding to a random sample of normal distribution similar to experimental electron backscatter diffraction data. This algorithm represents a statistical simulation by the particularized Monte Carlo method. A short review of these computational approaches to the calculation of normal distributions on the rotation group is presented.

scholarly journals A Logistic Normal Mixture Model for Compositional Data Allowing Essential Zeros

2016 ◽  
Vol 45 (4) ◽  
pp. 3-23 ◽  
Author(s):  
John Bear ◽  
Dean Billheimer
Keyword(s):  
Normal Distribution ◽  
Compositional Data ◽  
Normal Mixture ◽  
Dispersion Parameters ◽  
Normal Distributions ◽  
A Value ◽  
Common Location ◽  
Logistic Normal Distribution ◽  
Special Cases ◽  
Support Methods

The usual candidate distributions for modeling compositions, the Dirichlet and the logistic normal distribution, do not include zero components in their support. Methods have been developed and refined for dealing with zeros that are rounded, or due to a value being below a detection level. Methods have also been developed for zeros in compositions arising from count data. However, essential zeros, cases where a component is truly absent, in continuous compositions are still a problem.The most promising approach is based on extending the logistic normal distribution to model essential zeros using a mixture of additive logistic normal distributions of different dimension, related by common parameters. We continue this approach, and by imposing an additional constraint, develop a likelihood, and show ways of estimating parameters for location and dispersion. The proposed likelihood, conditional on parameters for the probability of zeros, is a mixture of additive logistic normal distributions of different dimensions whose location and dispersion parameters are projections of a common location or dispersion parameter. For some simple special cases, we contrast the relative efficiency of different location estimators.

scholarly journals Approximation of the Pole Figures and the Orientation of Distribution of Grains in Polycrystalline Samples by Means of Canonical Normal Distributions

1993 ◽  
Vol 22 (1) ◽  
pp. 17-27 ◽  
Author(s):  
T. I. Savyolova
Keyword(s):  
Distribution Function ◽  
Orientation Distribution Function ◽  
Hexagonal Lattice ◽  
Rolling Texture ◽  
Polycrystalline Sample ◽  
Orientation Distribution ◽  
Rotation Group ◽  
Normal Distributions ◽  
Lattice Symmetry ◽  
Pole Figures

The orientation distribution function (ODF) as determined from experimental pole figures (PF) in a polycrystalline sample by classical spherical harmonics analysis can have ghost effects and regions of negative values. The regions of negative values and the ghosts are a consequence of the loss of information on the “odd” part of ODF.In the present paper the canonical normal distributions (CND) on the rotation group SO(3) and on the sphere S2 in R3 used in texture analysis are discussed.The examples of CND on SO(3), S2 and their PF calculated for hexagonal lattice symmetry and for a rolling texture of beryllium are demonstrated.

scholarly journals Application of Normal Distributions on SO(3) and Sn for Orientation Distribution Function Approximation

1993 ◽  
Vol 21 (2-3) ◽  
pp. 161-176 ◽  
Author(s):  
T. I. Bucharova ◽  
T. I. Savyolova
Keyword(s):  
Distribution Function ◽  
Orientation Distribution Function ◽  
Function Approximation ◽  
Rolling Texture ◽  
Polycrystalline Sample ◽  
Orientation Distribution ◽  
Rotation Group ◽  
Normal Distributions ◽  
Pole Figures ◽  
Ill Posed

The orientation distribution function (ODF) in a polycrystalline sample is of special interest in texture analysis. Its determination from pole figures leads to an ill-posed problem, the solution of which is non-unique.In the present paper the properties of normal distributions on the rotation group SO(3) proposed by Parthasarathy (1964), Savyolova (1984) are discussed. A method for ODF determination based on the superposition of the normal distributions is proposed. The parameters of normal distributions are determined from the experimental pole figures. The application of this method is demonstrated for a rolling texture of beryllium.

scholarly journals Central and Local Limit Theorems for Numbers of the Tribonacci Triangle

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 880
Author(s):  
Igoris Belovas
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Rate Of Convergence ◽  
Limit Theorems ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Constructive Proof ◽  
The Central Limit Theorem ◽  
Local Limit Theorems

In this research, we continue studying limit theorems for combinatorial numbers satisfying a class of triangular arrays. Using the general results of Hwang and Bender, we obtain a constructive proof of the central limit theorem, specifying the rate of convergence to the limiting (normal) distribution, as well as a new proof of the local limit theorem for the numbers of the tribonacci triangle.

scholarly journals Normal approximation for mixtures of normal distributions and the evolution of phenotypic traits

2021 ◽  
Vol 53 (1) ◽  
pp. 162-188
Author(s):  
Krzysztof Bartoszek ◽  
Torkel Erhardsson
Keyword(s):  
Normal Distribution ◽  
Phenotypic Trait ◽  
Phenotypic Traits ◽  
Normal Distributions ◽  
Average Value ◽  
Conditional Moments ◽  
Mixture Of Normal Distributions ◽  
Explicit Bounds ◽  
Mixtures Of Normal Distributions ◽  
Parameter Values

AbstractExplicit bounds are given for the Kolmogorov and Wasserstein distances between a mixture of normal distributions, by which we mean that the conditional distribution given some $\sigma$ -algebra is normal, and a normal distribution with properly chosen parameter values. The bounds depend only on the first two moments of the first two conditional moments given the $\sigma$ -algebra. The proof is based on Stein’s method. As an application, we consider the Yule–Ornstein–Uhlenbeck model, used in the field of phylogenetic comparative methods. We obtain bounds for both distances between the distribution of the average value of a phenotypic trait over n related species, and a normal distribution. The bounds imply and extend earlier limit theorems by Bartoszek and Sagitov.

scholarly journals Approximation of the Orientation Distribution of Grains in Polycrystalline Samples by Means of Gaussians

1992 ◽  
Vol 19 (1-2) ◽  
pp. 9-27 ◽  
Author(s):  
D. I. Nikolayev ◽  
T. I. Savyolova ◽  
K. Feldmann
Keyword(s):  
Rolling Texture ◽  
Expansion Method ◽  
Operating Mode ◽  
Orientation Distribution ◽  
New Method ◽  
Gaussian Distributions ◽  
Positivity Condition ◽  
The Central Limit Theorem ◽  
Pole Figures ◽  
Series Expansion Method

The orientation distribution function (ODF) obtained by classical spherical harmonics analysis may be falsified by ghost influences as well as series truncation effects. The ghosts are a consequence of the inversion symmetry of experimental pole figures which leads to the loss of information on the “odd” part of ODF.In the present paper a new method for ODF reproduction is proposed. It is based on the superposition of Gaussian distributions satisfying the central limit theorem in the SO(3)-space as well as the ODF positivity condition. The kind of ODF determination offered here is restricted to the fit of Gaussian parameters and weights with respect to the experimental pole figures. The operating mode of the new method is demonstrated for a rolling texture of copper. The results are compared with the corresponding ones obtained by the series expansion method.

NON-NORMAL STATISTICAL TOLERANCE ANALYSIS USING ANALYTICAL CONVOLUTION METHOD

2008 ◽  
Vol 07 (01) ◽  
pp. 127-130 ◽  
Author(s):  
S. G. LIU ◽  
P. WANG ◽  
Z. G. LI
Keyword(s):  
Normal Distribution ◽  
Closed Loop ◽  
Tolerance Analysis ◽  
Accurate Method ◽  
Probability Density Functions ◽  
Triangular Distribution ◽  
Normal Distributions ◽  
Statistical Tolerance ◽  
Statistical Tolerance Analysis ◽  
Convolution Method

In statistical tolerance analysis, it is usually assumed that the statistical tolerance is normally distributed. But in practice, there are many non-normal distributions, such as uniform distribution, triangular distribution, etc. The simple way to analyze non-normal distributions is to approximately represent it with normal distribution, but the accuracy is low. Monte-Carlo simulation can analyze non-normal distributions with higher accuracy, but is time consuming. Convolution method is an accurate method to analyze statistical tolerance, but there are few reported works about it because of the difficulty. In this paper, analytical convolution is used to analyze non-normal distribution, and the probability density functions of closed loop component are obtained. Comparing with other methods, convolution method is accurate and faster.

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