The joint probability distribution of any set of phases given any set of diffraction magnitudes. IV. The active use of psi-zero triplets

1995 ◽  
Vol 51 (6) ◽  
pp. 820-825 ◽  
Author(s):  
G. Cascarano ◽  
C. Giacovazzo
Keyword(s):  
Probability Distribution ◽  
Joint Probability ◽  
Joint Probability Distribution

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.

Improved Locality Preserving Projection for Hyperspectral Image Classification in Probabilistic Framework

2021 ◽  
Author(s):  
Reza Seifi Majdar ◽  
Hassan Ghassemian
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Probability Distribution Function ◽  
Spatial Information ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Spectral Features ◽  
Probabilistic Framework ◽  
Locality Preserving ◽  
Training Samples

Unlabeled samples and transformation matrix are two main parts of unsupervised and semi-supervised feature extraction (FE) algorithms. In this manuscript, a semi-supervised FE method, locality preserving projection in the probabilistic framework (LPPPF), to find a sufficient number of reliable and unmixed unlabeled samples from all classes and constructing an optimal projection matrix is proposed. The LPPPF has two main steps. In the first step, a number of reliable unlabeled samples are selected based on the training samples, spectral features, and spatial information in the probabilistic framework. In this way, the spectral and spatial probability distribution function is calculated for each unlabeled sample. Therefore, the spectral features and spatial information are integrated together with a joint probability distribution function. Finally, a sufficient number of unlabeled samples with the highest joint probability distribution are selected. In the second step, the selected unlabeled samples are applied to construct the transformation matrix based on the spectral and spatial information of the unlabeled samples. The adjacency graph is improved by using new weights based on spectral and spatial information. This method is evaluated on three data sets: Indian Pines, Pavia University, and Kennedy Space Center (KSC) and compared with some recent and well-known supervised, semi-supervised, and unsupervised FE methods. Various experiments demonstrate the efficiency of the LPPPF in comparison with the other FE methods. LPPPF has also considerable performance with limited training samples.

Fuzzy Multi-Objective Programming With Joint Probability Distribution

2019 ◽  
pp. 263-295
Keyword(s):  
Probability Distribution ◽  
Goal Programming ◽  
Joint Probability ◽  
Chance Constraints ◽  
Fuzzy Goal Programming ◽  
Probabilistic Constraints ◽  
Joint Probability Distribution ◽  
Multi Objective ◽  
Fuzzy Goal ◽  
The Difference

In this chapter, a fuzzy goal programming (FGP) model is employed for solving multi-objective linear programming (MOLP) problem under fuzzy stochastic uncertain environment in which the probabilistic constraints involves fuzzy random variables (FRVs) following joint probability distribution. In the preceding chapters, the authors explain about linear, fractional, quadratic programming models with multiple conflicting objectives under fuzzy stochastic environment. But the chance constraints in these chapters are considered independently. However, in practical situations, the decision makers (DMs) face various uncertainties where the chance constraints occur jointly. By considering the above fact, the authors presented a solution methodology for fuzzy stochastic MOLP (FSMOLP) with joint probabilistic constraint following some continuous probability distributions. Like the other chapters, chance constrained programming (CCP) methodology is adopted for handling probabilistic constraints. But the difference is that in the earlier chapters chance constraints are considered independently, whereas in this chapter all the chance constraints are taken jointly. Then the transformed problem involving possibilistic uncertainty is converted into a comparable deterministic problem by using the method of defuzzification of the fuzzy numbers (FNs). Objectives are now solved independently under the set of modified system constraints to obtain the best solution of each objective. Then the membership function for each objective is constructed, and finally, a fuzzy goal programming (FGP) model is developed for the achievement of the highest membership goals to the extent possible by minimizing group regrets in the decision-making context.

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).

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