Statistical partitioning methods

Despite the conceptual elegance of partitioning in low-dimensional descriptor spaces, dimensional reduction is not essential for effective partitioning, as has been shown, for example, by application of statistical partitioning methods (4). 

In addition to cell-based partitioning, statistical partitioning methods are widely used for compound classification. One of the most popular approaches is recursive partitioning (Rusinko et al. 1999), a decision tree method, as illustrated in Figure 1.8. Recursive partitioning divides data sets along decision trees formed by sequences of molecular descriptors. At each node of the tree, a descriptor-based decision is made and the molecular data set is subdivided. For example, a chosen descriptor could simply detect the presence or absence of a structural fragment in a molecule. Alternatively, the... 

To reiterate a point that we made earlier, these problems of accurately calculating the free energy and entropy do not arise for isolated molecules that have a small number of well-characterised minima which can all be enumerated. The partition function for such systems can be obtained by standard statistical mechanical methods involving a summation over the mini mum energy states, taking care to include contributions from internal vibrational motion. 

The computation of quantum many-body effects requires additional effort compared to classical cases. This holds in particular if strong collective phenomena such as phase transitions are considered. The path integral approach to critical phenomena allows the computation of collective phenomena at constant temperature — a condition which is preferred experimentally. Due to the link of path integrals to the partition function in statistical physics, methods from the latter — such as Monte Carlo simulation techniques — can be used for efficient computation of quantum effects. 

Partitioning methods occasionally struggle to provide the accuracy associated with more powerful, albeit less informative techniques such as machine learning and statistical approaches. For this reason, there is a continuing need for the application of more accurate and informative classification techniques to QSAR analysis. The goal of a classifier is to produce a model that can separate new, untested compounds into classes with a training set of already classified compounds. 

The thermodynamic functions of the gaseous lanthanide trihalides have been calculated using standard statistical thermodynamic methods which relate the functions Cp, S, and H to the molecular partition function Q (Lewis et al., 1961) ... 

An example is the distribution equilibrium of a single atom between two phases, where the distribution coefficient Kd is defined by the probabilities of finding the single atom in one phase or in the other. If a one-step partition method is applied, Ka must be measured repeatedly many times, in order to obtain a statistically relevant result. Much more favourable is the use of a multi-stage method, particularly a chromatographic method, in which the partition of the single atom is taking place many times successively. 

Several methods have been described to evaluate dandruff, such as brushing off the hair and/or the scalp with various devices and weighing the scruf [162,173], However, the most popular approach involves partitioning the scalp into several areas, rating each area for dandruff severity, and analyzing the combined data statistically [174], The scalp-partitioning method using appropriate statistical procedures provides a powerful tool to evaluate dandruff severity and the efficacy of antidandruff products. 

Lipkovich 1, Dmitrienko A, Denne J, Enas G. Subgroup identification based on differential effect search A recursive partitioning method for establishing response to treatment in patient subpopulations. Statistics in Medicine, 30 2601-2621, 2011. 

The molecular properties, such as geometry, vibrational frequencies, and rotational constants, are needed to compute thermodynamic properties such as enthalpy, entropy, and Gibbs free energy through calculation of the partition functions of the substances using statistical mechanics methods. 

TST calculations apply in the high-pressure hmit pressure-dependent reactions can be analyzed via more complex kinetic analyses such as Pace—Ramsperger—Kassel—Marcus (PJkKM) theory. The RRKM theory is used to model the energy transfer in the transition state (or activated complex) of a molecule, using statistical mechanical methods that describe the density of energy states, which depend upon the vibrational and rotational partition functions for the molecule. 

Several methods have been developed for the quantitative description of such systems. The partition function of the polymer is computed with the help of statistical thermodynamics which finally permits the computation of the degree of conversion 0. In the simplest case, it corresponds to the linear Ising model according to which only the nearest segments interact cooperatively149. The second possibility is to start from already known equilibrium relations and thus to compute the relevant degree of conversion 0. 

Again, therefore, all thermodynamic properties of a system in quantum statistics can be derived from a knowledge of the partition function, and since this is the trace of an operator, we can choose any convenient representation in which to compute it. The most fruitful application of this method is probably to the theory of imperfect gases, and is well covered in the standard reference works.23... 

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