Probability distribution maps

An important statistical characterization of long-term motion is provided by the invariant measure (also called the invariant distribution, or simply the probability distribution) of a mapping M x f ), Pm - satisfies two conditions ... 

NCu was calculated for 55 boundaries from four quantitative X-ray maps, and the probability distribution is shown in Figure 5.26. The mean coverage of this distribution is 2.2 atoms/nm2, and these data illustrate the feasibility of measuring segregation by X-ray mapping in the AEM in the case of fine-grained, thin film materials. 

According to the latter model, the crystal is described as formed of anumber of equal scatterers, all randomly, identically and independently distributed. This simplified picture and the interpretation of the electron density as a probability distribution to generate a statistical ensemble of structures lead to the selection of the map having maximum relative entropy with respect to some prior-prejudice distribution m(x) [27, 28],... 

Clearly for the procedure outlined in Fig. 10.1 to work, we need to take care of several critical steps next to reasonable initial phase estimates required to formulate the initial restraints, we need a statistically valid procedure for the combination of the phases obtained by back transformation of the real space restrained map and the initial phase probability distribution. This recombination step is discussed below. 

Fig. 11.9. The valence map of Mg2Si04 transformed to simulate the probability distribution function of at (a) room temperature, (b) high temperature. The...

This quantity averages the Shannon entropies conditional on the Gamma and lognormal models, with weights given by their posterior probabilities. In Appendix B, we show that the average entropy is a concave function on the space of probability distributions which is monotone under contractive maps (Sebastiani and... 

In agreement with the Heisenberg uncertainty principle, the model cannot specify the detailed electron motions. Instead, the square of the wave function represents the probability distribution of the electron in that orbital. This approach allows us to picture orbitals in terms of probability distributions, or electron density maps. 

We see then that for any given choice of the reagent charges m and n we obtain a parameter characterizing the distribution of the electrostatic potential. With a number of experiments of this type we should be in a position to map out the probability distribution of X (or of Y) in considerable detail (25). 

Intuitively, a probabilistic schema mapping describes a probability distribution of a set of possible schema mappings between a source schema and a target schema. For completeness, we repeat its definition as follows (also see Definition 3 in Chap. 3). 

In the general case, among the many p-mappings that are consistent with a set of weighted correspondences C, we choose the one with the maximum entropy, that is, the p-mappings whose probability distribution obtains the maximum value of Y i= Pi log Pi In the above example, pM obtains the maximum entropy. 

We define a conditional p-mapping as a set cpM = (pMi,C ),. .., (pMn,Cn), where pM, ..., pM are p-mappings, and Ci,..., C are pairwise disjoint conditions. Intuitively, for each i e [l,n], pMi describes the probability distribution of possible mappings when condition Cj holds. Conditional mappings make more sense for by-tuple semantics. The following theorem shows that the complexity results carry over to such mappings. 

Journel(7-9) provides a means for generating the desired mappings in situations such as the example site which are free from any undue assumptions regarding the distributional form of concentrations within blocks. This approach simply transforms the data into the sample cumulative probability distribution. This nonparametric geostatistical approach is sometimes called "indicator kriging."... 

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