Probability models

Lancet D, Sadovsky E and Seidemann E 1993 Probability model for molecular recognition in biological receptor repertoires significance to the olfactory system Proc. Natl Acad. Sci. USA 90 3715-19... 

Setting up the probability model for the data and parameters of the system under smdy. This entails defining prior distributions for all relevant parameters and a likelihood function for the data given the parameters. 

A general methodology has been developed for the treatment of NMR data of polymer mixtures. The methodology is based on reaction probability models and computer optimization methods, resulting in a family of computer programs called MIXCO. The use of MIXCO programs enabled three components to be resolved from NMR tacticity data of fractionated polybutylene. 

An excellent way to treat such data is to use reaction probability models.(1,2) In the NMR analysis of tacticity, it is frequently possible to distinguish whether the configuration is chain-end controlled or catalytic-site controlled during polymerization. Various statistical models have been proposed. The chain-end controlled models include Bemoullian (B), and first- and second-order Markovian (Ml and M2) statistics.(1) The simplest catalytic-site controlled model is the enantiomorphic site (E) model.(3) The relationship between the chain-end and catalytic-site controlled models and possible hybrid models have been delineated in a recent article.(4)... 

Multi-State Models. In studies of copolymerization kinetics and polymer microstructure, the use of reaction probability models can provide a convenient framework whereby the experimental data can be organized and interpreted, and can also give insight on reaction mechanisms. (1.,2) The models, however, only apply to polymers containing one polymer component. For polymers with mixtures of different components, the one-state simple models cannot be used directly. Generally multi-state models(11) are needed, viz. 

Statistical methods can also be utilized to form probability models or to estimate the likelihood of particular descriptors forming the known classes. Chemical Computing Group Inc. has recently developed a new technology. 

Figure 9.3. Concept of using a state estimator to Figure 9.4. A probable model for a state estimator,...

The next task is to seek a model for the observer. We stay with a single-input single-output system, but the concept can be extended to multiple outputs. The estimate should embody the dynamics of the plant (process). Thus one probable model, as shown in Fig. 9.4, is to assume that the state estimator has the same structure as the plant model, as in Eqs. (9-13) and (9-14), or Fig. 9.1. The estimator also has the identical plant matrices A and B. However, one major difference is the addition of the estimation error, y - y, in the computation of the estimated state x. 

Statistical studies of MALDI MS applied to bacterial samples show that some biomarker peaks are highly reproducible and appear very consistently, while others appear much less reliably.1719 In Jarman et al.20 and Wahl et al.21 a probability model for MALDI signatures is proposed that takes into account the variability in appearance of biomarker peaks. This method constructs MALDI reference signatures from the set of peak locations for reproducible biomarker peaks, along with a measure of the reproducibility of each peak. 

Studies of the influence of total pressure on the initial reaction rate for pure reactants present in stoichiometric proportions provide a means of discriminating between various classes of Hoqgen-Watson models. Isolation of a class of probable models by means of plots of initial reaction rate versus total pressure, feed composition, and temperature constitutes the first step n developing a Hougen-Watson rate model. Hougen (14) has considered the influence of total pressure for unimolecular and bimolecular surface reactions the analysis that follows is adopted from his monograph. 

Since an assumed distribution of structures per unit cell allows one to calculate the relative intensities of the five 29Si lines, one can choose the combination of structures which will best fit the NMR data. We have chosen to use the Box complex alogrithm to determine the distribution of structures that minimize the variance a2 between the experimental and computed relative 29Si intensities. Tables III, IV and V list results for sieves of various Si/Al ratios. The 29Si NMR data in Table III has appeared previously in the literature (2) and the data in Tables IV and V are new (Table I). At the lowest Si/Al ratio of 1.145, the best fit distribution is nearly the same as the maximum probability model with a low a2 = 28. At higher Si/Al ratios >1.9 the best fit result continues to give a low a2 between 0 and 18. 

Miller, R. S., S. H. Frankel, C. K. Madnia, andP. Givi (1993). Johnson-Edgeworth translation for probability modeling of binary scalar mixing in turbulent flows. Combustion Science and Technology 91, 21-52. 

The connectivity is not known for the seven-helix bundle of purple membrane protein (Henderson and Unwin, 1975), but on the basis of its resemblance to other antiparallel a proteins the most likely topologies would be either up-and-down or Greek key (see below). An analysis based on the sequence and the relative electron-densities of the helices (Engelman et ah, 1980) considers a left-handed up-and-down topology as the most probable model. 

Henstenburg, R.B. and Phoenix, S.L. (1989). Interfaeial shear strength studies using the single-filament-compositc test, part II, A probability model and Monte Carlo simulation. Polym. Composites 10, 389-408. 

The reaction probability model produces a complete calculated C nmr spectrum as a best fit to the observed experimental spectrum. The deduced probabilities provide the following derived quantities (1) "rira", a measure of monomer sequence randomness, (2) the distribution of methylene sequence lengths,... 

Hierarchical model A model consisting of multiple parameters that can be regarded as related or connected in some way by the structure of the problem, implying that a joint probability model for these parameters should reflect the dependence among them. 

Figure 15.7 Near approach (or avoided crossing) of two electronic states as a function of nuclear coordinate Q. The inset expands the region of the avoided crossing to facilitate the definition of quantities appearing in the Landau-Zener surface-hopping-probability model...

A whole series of known anomalies in. the static and mnetie properties of the proton in aqueous solution may he explained by the assumption that it is present as the hydrated hydroninm ion. Earlier considerations [1 j have already resulted in the following most probable model for the hydration shell of the proton (Pig. I). H3(F as centre is strongly... 

Classroom exercise to derive Beer s law R. W. Ricci, M. A. Ditzler, and L. P. Nestor, Discovering the Beer-Lambert Law, J. Chem. Ed. 1994, 71, 983. An alternate derivation W. D. Bare, A More Pedagogically Sound Treatment of Beer s Law A Derivation Based on a Corpuscular-Probability Model, J. Chem. Ed. 2000, 77, 929. 

A binomial probability model is to be based on the following index function model ... 

Suppose that a linear probability model is to be fit to a set of observations on a dependent variable, y, which takes values zero and one, and a single regressor, x, which varies continuously across observations. Obtain the exact expressions for the least squares slope in the regression in terns of the mean(s) and variance of x and interpret the result. 

Figure 10. Pairing of two disclination lines of opposite signs (lamellar details are not featured) (top) a less probable model for the core of a dislocation (middle) and focal line appearing on the dislocation in order to release locally deformation energy (bottom)...

The statistical classification uses probability models to classify objects. 

Choose a probability model and a random variable associated with it. This choice may be based on previous experience or intuition. 

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