Joint Probability Distribution Function for the Electric Microfield and its Ion-Octupole Inhomogeneity Tensor

The joint probability distribution function method has been developed in P1¯ for reflections with rational indices. The positional atomic parameters are considered to be the primitive random variables, uniformly distributed in the interval (0, 1), while the reflection indices are kept fixed. Owing to the rationality of the indices, distributions like P(F p 1 , F p 2 ) are found to be useful for phasing purposes, where p 1 and p 2 are any pair of vectorial indices. A variety of conditional distributions like P(|F p 1 | | |F p 2 |), P(|F p 1 | |F p 2 ), P(\varphi_{{\bf p}_1}|\,|F_{{\bf p}_1}|, F_{{\bf p}_2}) are derived, which are able to estimate the modulus and phase of F p 1 given the modulus and/or phase of F p 2 . The method has been generalized to handle the joint probability distribution of any set of structure factors, i.e. the distributions P(F 1, F 2,…, F n+1), P(|F 1| |F 2,…, F n+1) and P(\varphi1| |F|1, F 2,…, F_{n+1}) have been obtained. Some practical tests prove the efficiency of the method.

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EVALUATION OF ELECTROMYOGRAPHIC NOISE STATISTICAL CHARACTERISTICS IN MULTICHANNEL ECG RECORDINGS

Journal of the Russian Universities Radioelectronics ◽

10.32603/1993-8985-2018-21-6-118-125 ◽

2019 ◽

pp. 118-125

Author(s):

Evgene B. Grigoriev ◽

Alexander S. Krasichkov ◽

Evgeny M. Nifontov

Keyword(s):

Distribution Function ◽

Probability Distribution ◽

Probability Distribution Function ◽

Joint Probability ◽

Bivariate Normal Distribution ◽

Statistical Characteristics ◽

Joint Probability Distribution ◽

Electrocardiogram Signal ◽

Ecg Leads ◽

Joint Probability Distribution Function

Electromyographic noise is one of the most common noises in electrocardiogram. In case of several electrocardiogram leads, electromyographic noise affects each lead to different extent. It can be taken into account when developing algorithms for multilead electrocardiogram record processing. However, in the existing literature, there is no information about the relationship of electromyographic noise in various ECG leads and their joint probability distribution. The purpose of this paper is to study statistical characteristics of electromyographic noise in ECG signal, from which the electromyographic noise is extracted. The paper proposes a method for extracting electromyographic noise from electrocardiogram signal, based on a polynomial approximation of electrocardiogram signal fragments in sliding window with overlapping fragment subsequent weight averaging. Using this method, fragments of electromyographic noise are extracted from multichannel electrocardiogram records. Based on the obtained data, a joint probability distribution function of electromyographic noise in two adjacent leads is selected, and the correlation relationships between the electromyographic noise in various ECG leads are investigated. The results show that the joint probability distribution function of electromyographic noise in two adjacent leads in the first approximation can be described using bivariate normal distribution. In addition, between the samples of electromyographic noise from two adjacent leads quite strong correlation relationships can be observed.

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On the distribution of an arbitrary subset of the eigenvalues for some finite dimensional random matrices

Random Matrices Theory and Application ◽

10.1142/s2010326320400043 ◽

2019 ◽

Vol 09 (01) ◽

pp. 2040004

Author(s):

Marco Chiani ◽

Alberto Zanella

Keyword(s):

Distribution Function ◽

Probability Distribution ◽

Random Matrices ◽

Probability Distribution Function ◽

Joint Distribution ◽

Joint Probability ◽

Joint Probability Distribution ◽

Arbitrary Subset ◽

Finite Dimensional ◽

Joint Probability Distribution Function

We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant operator. Specifically, we derive compact expressions for the joint probability distribution function of the eigenvalues and the expectation of functions of the eigenvalues, including joint moments, for the case of both ordered and unordered eigenvalues.

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The joint probability distribution function of structure factors with rational indices. VI. Cases with reduced dimensionality

Acta Crystallographica Section A Foundations of Crystallography ◽

10.1107/s010876739900690x ◽

1999 ◽

Vol 55 (6) ◽

pp. 984-990

Author(s):

Carmelo Giacovazzo ◽

Dritan Siliqi ◽

Cristina Fernández-Castaño ◽

Giovanni Luca Cascarano ◽

Benedetta Carrozzini

Keyword(s):

Distribution Function ◽

Probability Distribution ◽

Probability Distribution Function ◽

Mixed Type ◽

Three Dimensional ◽

Joint Probability ◽

Joint Probability Distribution ◽

Reduced Dimensionality ◽

Structure Factors ◽

Joint Probability Distribution Function

The probabilistic formulas relating standard and mixed type reflections (these last show integral and half-integral indices) are derived. It is shown that probabilistic estimates can be obtained by using particular sections of the three-dimensional reciprocal space. The concept of structure invariant is extended to define the wider class of structure quasi-invariant. Their statistical behaviour is briefly discussed with the help of some practical tests.

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The method of joint probability distribution functions, neighbourhoods, and representations

Phasing in Crystallography ◽

10.1093/oso/9780199686995.003.0009 ◽

2013 ◽

Author(s):

Carmelo Giacovazzo

Keyword(s):

Distribution Function ◽

Probability Distribution ◽

Probability Distribution Function ◽

Joint Probability ◽

Distribution Functions ◽

Joint Probability Distribution ◽

Phase Problem ◽

Additional Information ◽

Structure Factors ◽

Joint Probability Distribution Function

Wilson statistics, described in Chapter 2, aims at calculating the distribution of the structure factor P(F) ≡ P(|F|, φ) when nothing is known about the structure; the positivity and atomicity of the electron density (both promoted by the positive nature of the atomic scattering factors fj) are the only necessary assumptions. Wilson results may be synthesized as follows: . . . the modulus R = |E| is distributed according to equations (2.7) or (2.8), while no prevision is possible about φ, which is distributed with constant probability 1/(2π). . . . In other words, knowledge of the R moduli does not provide information about a phase; this agrees well with Section 3.3, according to which experimental data only allow an estimate of s.i. (and also s.s. if the algebraic form of the symmetry operators has been fixed). Let us now consider P(Fh1 , Fh2 ) ≡ P(|Fh1 |, |Fh2 |, φh1 , φh2 ), the joint probability distribution function of two structure factors. If the two structure factors are uncorrelated (i.e. no relation is expected between their moduli and between their phases), P will coincide with the product of two Wilson distributions (2.7) or (2.8), say, . . . P(Fh1 , Fh2 ) ≡ P(|Fh1 |, φh1 ) · P(|Fh2 |, φh2 ) = 1/4π2 P(|Fh1 |)P(|Fh2 |), . . . which is useless (because the two Wilson distributions are useless) for solving the phase problem; indeed, the relation does not provide any phase information. The question is now: if two structure factors are correlated, may their joint probability distribution function be used for solving the phase problem? Let us first use a simple example to show how much additional information (i.e. that is not present in the two elementary distributions) may be stored in a joint probability distribution function; then we will answer the question. Let us suppose that the human population of a village has been submitted to statistical analysis to define how weight and height are distributed.

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