Updating direct methods

2019 ◽  
Vol 75 (1) ◽  
pp. 142-157 ◽  
Author(s):  
Carmelo Giacovazzo
Keyword(s):  
Probability Distribution ◽  
Prior Information ◽  
Structural Model ◽  
Joint Probability ◽  
Direct Methods ◽  
Distribution Functions ◽  
Joint Probability Distribution ◽  
Conditional Distributions ◽  
Interatomic Distances ◽  
Different Types

The standard method of joint probability distribution functions, so crucial for the development of direct methods, has been revisited and updated. It consists of three steps: identification of the reflections which may contribute to the estimation of a given structure invariant or seminvariant, calculation of the corresponding joint probability distribution, and derivation of the conditional distribution of the invariant or seminvariant phase given the values of some diffracted amplitudes. In this article the conditional distributions are derived directly without passing through the second step. A good feature of direct methods is that they may work in the absence of any prior information: that is also their weakness. Different types of prior information have been taken into consideration: interatomic distances, interatomic vectors, Patterson peaks, structural model. The method of directly deriving the conditional distributions has been applied to those cases. Some new formulas have been obtained estimating two-, three- and four-phase invariants. Special attention has been dedicated to the practical aspects of the new formulas, in order to simplify their possible use in direct phasing procedures.

Joint probability distribution functions when a model is available: Fourier syntheses

2013 ◽  
Author(s):  
Carmelo Giacovazzo
Keyword(s):  
Probability Distribution ◽  
Current Model ◽  
Joint Probability ◽  
Distribution Functions ◽  
Joint Probability Distribution ◽  
Conditional Distributions ◽  
Target Structure ◽  
Probability Distribution Functions ◽  
Structure Factors ◽  
Joint Probability Distributions

The title of this chapter may seem a little strange; it relates Fourier syntheses, an algebraic method for calculating electron densities, to the joint probability distribution functions of structure factors, which are devoted to the probabilistic estimate of s.i.s and s.s.s. We will see that the two topics are strictly related, and that optimization of the Fourier syntheses requires previous knowledge and the use of joint probability distributions. The distributions used in Chapters 4 to 6 are able to estimate s.i. or s.s. by exploiting the information contained in the experimental diffraction moduli of the target structure (the structure one wants to phase). An important tool for such distributions are the theories of neighbourhoods and of representations, which allow us to arrange, for each invariant or seminvariant Φ, the set of amplitudes in a sequence of shells, each contained within the subsequent shell, with the property that any s.i. or s.s. may be estimated via the magnitudes constituting any shell. The resulting conditional distributions were of the type, . . . P(Φ| {R}), (7.1) . . . where {R} represents the chosen phasing shell for the observed magnitudes. The more information contained within the set of observed moduli {R}, the better will be the Φ estimate. By definition, conditional distributions (7.1) cannot change during the phasing process because prior information (i.e. the observed moduli) does not change; equation (7.1) maintains the same identical algebraic form. However, during any phasing process, various model structures progressively become available, with different degrees of correlation with the target structure. Such models are a source of supplementary information (e.g. the current model phases) which, in principle, can be exploited during the phasing procedure. If this observation is accepted, the method of joint probability distribution, as described so far, should be suitably modified. In a symbolic way, we should look for deriving conditional distributions . . . P (Φ| {R}, {Rp}) , (7.2) . . . rather than (7.1), where {Rp} represents a suitable subset of the amplitudes of the model structure factors. Such an approach modifies the traditional phasing strategy described in the preceding chapters; indeed, the set {Rp} will change during the phasing process in conjunction with the model changes, which will continuously modify the probabilities (7.2).

scholarly journals Multiple-wavelength anomalous dispersion techniques applied to powder data: a probabilistic method for finding the substructureviajoint probability distribution functions

2008 ◽  
Vol 42 (1) ◽  
pp. 30-35 ◽  
Author(s):  
Angela Altomare ◽  
Benny Danilo Belviso ◽  
Maria Cristina Burla ◽  
Gaetano Campi ◽  
Corrado Cuocci ◽  
...  
Keyword(s):  
Probability Distribution ◽  
Probability Distribution Function ◽  
Probabilistic Method ◽  
Joint Probability ◽  
Direct Methods ◽  
Distribution Functions ◽  
Joint Probability Distribution ◽  
Probability Distribution Functions ◽  
Multiple Wavelength ◽  
Joint Probability Distribution Function

A new joint probability distribution function method is described to find the anomalous scatterer substructure from powder data. The method requires two wavelengths; the conclusive formulas provide estimates of the substructure structure factor moduli, from which the anomalous scatterer positions can be found by Patterson or direct methods. The theory has been preliminarily applied to two compounds, the first having Pt and the second having Fe as anomalous scatterer. Both substructures were correctly identified.

scholarly journals The Degree Analysis of an Inhomogeneous Growing Network with Two Types of Vertices

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Huilin Huang
Keyword(s):  
Probability Distribution ◽  
Law Of Large Numbers ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Degree Sequences ◽  
Strong Law ◽  
Large Numbers ◽  
Different Types ◽  
Azuma's Inequality ◽  
Growing Network

We consider an inhomogeneous growing network with two types of vertices. The degree sequences of two different types of vertices are investigated, respectively. We not only prove that the asymptotical degree distribution of typesfor this process is power law with exponent2+1+δqs+β1-qs/αqs, but also give the strong law of large numbers for degree sequences of two different types of vertices by using a different method instead of Azuma’s inequality. Then we determine asymptotically the joint probability distribution of degree for pairs of adjacent vertices with the same type and with different types, respectively.

scholarly journals Optimal compromise between incompatible conditional probability distributions, with application to Objective Bayesian Kriging

2019 ◽  
Vol 23 ◽  
pp. 271-309
Author(s):  
Joseph Muré
Keyword(s):  
Probability Distribution ◽  
Gibbs Sampler ◽  
Prediction Interval ◽  
Probability Distributions ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Conditional Distributions ◽  
Stationary Probability Distribution ◽  
Gibbs Sampling Algorithm ◽  
Correlation Parameters

Models are often defined through conditional rather than joint distributions, but it can be difficult to check whether the conditional distributions are compatible, i.e. whether there exists a joint probability distribution which generates them. When they are compatible, a Gibbs sampler can be used to sample from this joint distribution. When they are not, the Gibbs sampling algorithm may still be applied, resulting in a “pseudo-Gibbs sampler”. We show its stationary probability distribution to be the optimal compromise between the conditional distributions, in the sense that it minimizes a mean squared misfit between them and its own conditional distributions. This allows us to perform Objective Bayesian analysis of correlation parameters in Kriging models by using univariate conditional Jeffreys-rule posterior distributions instead of the widely used multivariate Jeffreys-rule posterior. This strategy makes the full-Bayesian procedure tractable. Numerical examples show it has near-optimal frequentist performance in terms of prediction interval coverage.

The approach of the joint probability distribution functions: the SIR-MIR, SAD-MAD and SIRAS-MIRAS, cases

2002 ◽  
Vol 217 (12) ◽  
Author(s):  
C. Giacovazzo ◽  
M. Ladisa ◽  
D. Siliqi
Keyword(s):  
Probability Distribution ◽  
Joint Probability ◽  
Distribution Functions ◽  
Joint Probability Distribution ◽  
Probability Distribution Functions

AbstractThe method of the joint probability distribution functions has been recently applied to SIR-MIR, SAD-MAD and SIRAS-MIRAS cases. The capacity of the method to treat various forms of errors (i.e., errors in measurements, possible lack of isomorphism, errors in a substructure model when a model is

Reliability analysis of a telecommunication tower

1999 ◽  
Vol 26 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Kamal El-Fashny ◽  
Luc E Chouinard ◽  
Ghyslaine McClure
Keyword(s):  
Wind Speed ◽  
Probability Distribution ◽  
Reliability Analysis ◽  
Structural Reliability ◽  
Joint Probability ◽  
Distribution Functions ◽  
Sensitivity Analyses ◽  
Ice Thickness ◽  
Joint Probability Distribution ◽  
Type I

This study presents a structural reliability analysis of a microwave tower subject to wind and freezing-rain hazards. The tower (name code CEBJ, owned by Hydro-Québec) is a 66 m tall, three-legged, steel lattice structure located in the James Bay area. The reliability analysis is performed conditionally with respect to wind speed and ice thickness accretion, and the results are integrated over the domain of wind and ice values using their joint probability distribution. This approach makes it possible to perform sensitivity analyses with respect to various assumptions on the joint probability distribution function of the climatological variable, without having to repeat the detailed coupled reliability - structural analysis of the tower. The probability distribution functions assumed for the wind speed and the ice thickness accretion on the tower members are both extreme-value type I (Gumbel) distributions. Adopting a weakest link model, the failure of the tower is assumed to occur when any of the members fails either in tension, compression, or global buckling. Without loss of generality, the proposed procedure can be applied with more refined probability distribution functions.Key words: reliability, telecommunication towers, wind, ice.

The probability distribution function of structure factors with non-integral indices. III. The joint probability distribution in the P1¯ case

1999 ◽  
Vol 55 (2) ◽  
pp. 322-331 ◽  
Author(s):  
Carmelo Giacovazzo ◽  
Dritan Siliqi ◽  
Angela Altomare ◽  
Giovanni Luca Cascarano ◽  
Rosanna Rizzi ◽  
...  
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Probability Distribution Function ◽  
Function Method ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Conditional Distributions ◽  
Structure Factors ◽  
Joint Probability Distribution Function ◽  
Atomic Parameters

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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