New methods for deriving joint probability distributions of structure factors. II. Strengthening the triplet relationship inP1

1989 ◽  
Vol 45 (7) ◽  
pp. 463-468
Author(s):  
J. Brosius
Keyword(s):  
Probability Distributions ◽  
Joint Probability ◽  
New Methods ◽  
Structure Factors ◽  
Triplet Relationship ◽  
Joint Probability Distributions

The joint probability distribution function of structure factors with rational indices. IV. The P1 case

1999 ◽  
Vol 55 (3) ◽  
pp. 512-524
Author(s):  
Carmelo Giacovazzo ◽  
Dritan Siliqi ◽  
Cristina Fernández-Castaño
Keyword(s):  
Probability Distribution ◽  
Probability Distributions ◽  
Probabilistic Approach ◽  
Joint Probability ◽  
Distribution Functions ◽  
Accurate Estimation ◽  
Joint Probability Distribution ◽  
Structure Factors ◽  
Joint Probability Distributions ◽  
The Hilbert Transform

The method of the joint probability distribution functions of structure factors has been extended to reflections with rational indices. The most general case, space group P1, has been considered. The positional parameters are the primitive random variables of our probabilistic approach, while the reflection indices are kept fixed. Quite general joint probability distributions have been considered from which conditional distributions have been derived: these proved applicable to the accurate estimation of the real and imaginary parts of a structure factor, given prior information on other structure factors. The method is also discussed in relation to the Hilbert-transform techniques.

An alternative method for the calculation of joint probability distributions. Application to the expectation of the triplet invariant

2015 ◽  
Vol 71 (1) ◽  
pp. 76-81
Author(s):  
J. Brosius
Keyword(s):  
Probability Distributions ◽  
Random Variables ◽  
Joint Probability ◽  
New Method ◽  
Expectation Value ◽  
Patterson Function ◽  
First Approximation ◽  
Alternative Method ◽  
Structure Factors ◽  
Joint Probability Distributions

This paper presents a completely new method for the calculation of expectations (and thus joint probability distributions) of structure factors or phase invariants. As an example, a first approximation of the expectation of the triplet invariant (up to a constant) is given and acomplexnumber is obtained. Instead of considering the atomic vector positions or reciprocal vectors as the fundamental random variables, the method samples over all functions (distributions) with a given number of atoms and given Patterson function. The aim of this paper was to explore the feasibility of the method, so the easiest problem was chosen: the calculation of the expectation value of the triplet invariant inP1. Calculation of the jointprobabilitydistribution of the triplet is not performed here but will be done in the future.

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