Nonlinear correlator

Instead of the definition in Eq. (7-82), the selectivity is often written as log k,). Another way to consider a selectivity-reactivity relationship is to compare the relative effects of a series of substituents on a pair of reactions. This is what is done when Hammett plots are made for a pair of reactions and their p values are compared. The slope of an LEER is a function of the sensitivity of the process being correlated to structural or solvent changes. Thus, in a family of closely related LFERs, the one with the steepest slope is the most selective, and the one with the smallest slope is the least selective.Moreover, the intercept (or some arbitrarily selected abscissa value, usually log fco for fhe reference substituent) should be a measure of reactivity in each reaction series. Thus, a correlation should exist between the slopes (selectivity) and intercepts (reactivity) of a family of related LFERs. It has been suggested that the slopes and intercepts should be linearly related, but the conditions required for linearity are seldom met, and it is instead common to find only a rough correlation, indicative of normal selectivity-reactivity behavior. The Br nsted slopes, p, for the halogenation of a series of carbonyl compounds catalyzed by carboxylate ions show a smooth but nonlinear correlation with log... 

The coupling constants increase with increasing sum of the electronegativities (E EN) of the substituents. A least square fit results in the nonlinear correlation (Pauling electronegativities)... 

In 1986, Puchot et al.104 studied the nonlinear correlation between the enantiomeric excess of a chiral auxiliary and the optical yield in an asymmetric synthesis, either stoichiometric or catalytic. Negative NLEs [(—)-NLEs] were observed in the asymmetric oxidation of sulfide and in [.S ]-proline-mediated asymmetric Robinson annulation reactions, while a positive NLE [(+)-NLEs]... 

The energy profiles for an enantioselective and a diasteroselective synthesis are compared in Figure 14.2. An interesting feature of the asymmetric catalytic synthesis is the nonlinear correlation between the optical purity of the chiral catalyst or auxiliary and that of the reaction product, reported... 

There are more sophisticated methods to evaluate AvapH (and thus, Avap2/°) than by using equation 2.34. They all assume nonlinear correlations between Inp and 1/T. For instance, the vapor pressures in figure 2.4 can be fitted with the Antoine equation, which has three adjustable parameters (A, B, and C equation 2.38), or even to expressions involving a larger number of parameters, such as Riedel s equation (equation 2.39) [41]. 

As a first step, SGS turbulence-radiation interactions associated with the nonlinear correlations in Eqs. (10.8) and (10.9) are neglected, thus ... 

Another critical advance responsible for the success of MP2-R12 is the introduction of nonlinear correlation functions, in particular, the Slater-type correlation function, 1— exp(—yr 2), of Ten-no [29]. It is asymptotically linear in ru near the coalescence point and, hence, satisfies the cusp condition in the leading order. Unlike the linear correlation function, the Slater-type correlation function... 

Multilinear regressions Multipredictor nonlinear correlations Quantitative property-property relationships and quantitative structure-property relationships... 

The quantities A or A/mim2 are available from isopiestic and binary data. Then the constants Ap>q of Equation 12 are calculated by means of the nonlinear correlation and if transferred into Bp>q... 

The constants of Equation 22 were calculated by means of nonlinear correlation. The results are summarized in Tables III and IV. 

Model of Bodor, Gabanyi, and Wong The model Bodor et al. [43] is based on a nonlinear correlation of log K0w with molecular descriptors such as molecular surface, volume, weight, and charge densities on N and O atoms in the molecule. The equation is... 

PCM modeling aims to find an empirical relation (a PCM equation or model) that describes the interaction activities of the biopolymer-molecule pairs as accurate as possible. To this end, various linear and nonlinear correlation methods can be used. Nonlinear methods have hitherto been used to only a limited extent. The method of prime choice has been partial least-squares projection to latent structures (PLS), which has been found to work very satisfactorily in PCM. PCA is also an important data-preprocessing tool in PCM modeling. Modeling includes statistical model-validation techniques such as cross validation, external prediction, and variable-selection and signal-correction methods to obtain statistically valid models. (For general overviews of modeling methods see [10]). 

The terms transformation, convolution, and correlation are used over and over again in NMR spectroscopy and imaging in different contexts and sometimes with different meanings. The transformation best known in NMR is the Fourier transformation in one and in more dimensions [Bral]. It is used to generate one- and multi-dimensional spectra from experimental data as well as ID, 2D, and 3D images. Furthermore, different types of multi-dimensional spectra are explicitly called correlation spectra [Eml]. It is shown below how these are related to nonlinear correlation functions of excitation and response. 

Similar to convolutions, correlations can be computed via a detoiu into the frequency domain. However, for the nonlinear correlation functions c ( > a ) this requires lifting the causal time order, just as for calculation of the nonlinear convolutions,... 

Level A correlation This correlation represents a point-to-point relationship between in vitro dissolution and in vivo dissolution (input/absorption rate). Level A IVIVC is also viewed as a predictive model for the relationship between the entire in vitro release time course and entire in vivo response time course. In general, correlations are linear at this level. Although a concern of acceptable nonlinear correlation has been addressed, no formal guidance on the nonlinear IVIVC has been established. Level A correlation is the most informative and very useful from a regulatory perspective. 

A set of observed data points is assumed to be available as samples from an unknown probability density function. Density estimation is the construction of an estimate of the density function from the observed data. In parametric approaches, one assumes that the data belong to one of a known family of distributions and the required function parameters are estimated. This approach becomes inadequate when one wants to approximate a multi-model function, or for cases where the process variables exhibit nonlinear correlations [127]. Moreover, for most processes, the underlying distribution of the data is not known and most likely does not follow a particular class of density function. Therefore, one has to estimate the density function using a nonparametric (unstructured) approach. 

The approach presented here is based on Kramer s work [150] where his method uncovers both linear and nonlinear correlations independent of the... 

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