Experimental models basic statistical measures

Quantifying predictive performance is basic to statistics and has therefore naturally been a part of QSAR work. Accepted statistical measures of performance are routinely reported with QSAR models (see Chapter 8), and models for which the measures are not provided or for which they show poor performance are not trusted. Some of the mathematical methods that are used, both to develop QSAR models and to assess their performance, relate to predictions of potency. Others assess performance in terms of the correctness of simple binary (yes/no) prediction of whether a chemical is active or inactive, where chemicals are classed as active experimentally if the relevant measure exceeds a predetermined threshold. 

In the following we will thus present some basic statistical methods useful for determining turbulence quantities from experimental data, and show how these measurements of turbulence can be put into the statistical model framework. Usually, this involves separating the turbulent from the non-turhulent parts of the flow, followed by averaging to provide the statistical descriptor. We will survey some of the basic methods of statistics, including the mean, variance, standard deviation, covariance, and correlation (e.g., [66], chap 1 [154], chap 2 [156]). 

In physical chemistry the most important application of the probability arguments developed above is in the area of statistical mechanics, and in particular, in statistical thermodynamics. This subject supplies the basic connection between a microscopic model of a system and its macroscopic description. The latter point of view is of course based on the results of experimental measurements (necessarily carried out in each experiment on a very large number of particle ) which provide the basis of classical thermodynamics. With the aid of a simple example, an effort now be made to establish a connection between the microscopic and macroscopic points of view. 

Also shown in Figs. 40 and 41 are experimental data measured by Neyer et al. [244]. The agreement is excellent and the basic theoretical predictions are confirmed. In contrast to the quantum mechanical calculations, all statistical models predict — for J = 0 — constant distributions, which abruptly vanish at a specific maximum value of j and thus fail to describe the details of the quantum mechanical and experimental distributions. The same is also true for the classical distributions. 

Alternatively, tests can be used to obtain the basic stiffness properties of the material form and their corresponding range measured by some statistical property such as the standard deviation. In two-dimensional cases where there are no significant loads in the out-of-plane direction, the basic orthotropic stiffness properties in Eqn (6.1) can be measured experimentally. Then, the classical laminated plate theory described in previous sections for determination of stiffness can be used effectively to model these sttuctures. Alternatively, the four basic stiffiiesses for 3-D woven composites can be... 

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