Multilinear Regression Analysis

Another problem is to determine the optimal number of descriptors for the objects (patterns), such as for the structure of the molecule. A widespread observation is that one has to keep the number of descriptors as low as 20 % of the number of the objects in the dataset. However, this is correct only in case of ordinary Multilinear Regression Analysis. Some more advanced methods, such as Projection of Latent Structures (or. Partial Least Squares, PLS), use so-called latent variables to achieve both modeling and predictions. 

Furthermore, QSPR models for the prediction of free-energy based properties that are based on multilinear regression analysis are often referred to as LFER models, especially, in the wide field of quantitative structure-activity relationships (QSAR). 

The models are applicable to large data sets with a rapid calculation speed, a wide range of compounds can be processed. Neural networks provided better models than multilinear regression analysis. 

More than just a few parameters have to be considered when modelling chemical reactivity in a broader perspective than for the well-defined but restricted reaction sets of the preceding section. Here, however, not enough statistically well-balanced, quantitative, experimental data are available to allow multilinear regression analysis (MLRA). An additional complicating factor derives from comparison of various reactions, where data of quite different types are encountered. For example, how can product distributions for electrophilic aromatic substitutions be compared with acidity constants of aliphatic carboxylic acids And on the side of the parameters how can the influence on chemical reactivity of both bond dissociation energies and bond polarities be simultaneously handled when only limited data are available ... 

Multilinear Regression Analysis. As an entry to the problem we have selected simple gas phase reactions involving proton or hydride ion transfer which are influenced by only a few effects and for which reactivity data of high accuracy are available. In these situations where a larger set of numerial data are available multilinear regression analysis (MLRA) was applied. Thus, the simplest mathematical form, a linear equation is chosen to describe the relationship between reactivity data and physicochemical factor. The number of parameters (factors) simultaneously applied was always kept to a minimum, and a particular parameter was only included in a MLRA study if a definite indication of its relevance existed. 

Algebraic expressions for terms M and C were derived using Dewar s PMO method (for C in a version similar to the co-technique [57] in order to calculate carbocation stabilization energies). The size factor S is simply a cubic function of the number of carbon atoms [97], The three independent variables of the model were assumed to be linearly related to the experimental Iball indices (vide supra). By multilinear regression analysis (sample size = 26) an equation was derived for calculating Iball indices from the three theoretical parameters. The correlation coefficient for the linear relation between calculated and experimental Iball indices is r = 0.961. 

It is clear that for an unsymmetrical data matrix that contains more variables (the field descriptors at each point of the grid for each probe used for calculation) than observables (the biological activity values), classical correlation analysis as multilinear regression analysis would fail. All 3D QSAR methods benefit from the development of PLS analysis, a statistical technique that aims to find the multidimensional direction in the X space that explains the maximum multidimensional variance direction in the F space. PLS is related to principal component analysis (PCA)." ° However, instead of finding the hyperplanes of maximum variance, it finds a linear model describing some predicted variables in terms of other observable variables and therefore can be used directly for prediction. Complexity reduction and data... 

Multilinear regression analysis (MLRA) Principal component analysis (PCA) Partial least squares regression (PLS) Principal component regression (PCR). 

Since the composition of the sample is variable, and therefore so is its mass absorption coefficient, it is sometimes necessary to carry out interelement corrections. The values of the correction factors can either be calculated from fundamental theory using commercially available software, or by multilinear regression analysis using the calibration samples. The former is preferred since the latter can force a mathematical fit to the data which bears no relation to X-ray theory. 

Through the multilinear regression analysis, we developed a fomiula for minimum film thickness, for the range of data tabulated in Tables 2 and 3, as... 

Multilinear Regression Analysis for the Derivation of CLND Response Factors... 

Multilinear regression analysis was carried out with the Excel data analysis addin. 

The empirical correlations do not differentiate between configurational isomers or polymorphs, but they are useful in order to estimate the overall expected values of PE. As can be seen from Table 8-2, the values derived from the multilinear regression analysis (MLRA) are at average 28 % ( ) lower than the PEs calculated by the force field, which indicates a likely overestimation of polar forces by the chosen charge model. In contrast, the values for the least polar molecule listed in Table 8-2, P.B.16, are almost identical. 

Charlton, M. H., Docherty, R., and Hutchings, M. G., Quantitative structure-sublimation enthalpy relationships studied by neural networks, theoretical crystal packing calculations and multilinear regression analysis, J. Chem. Soc. Perkin Trans. 2, 2203, 1995. 

Chapter 2 (Statistical Space for Multivariate Correlations) Aims to prepare the conceptual-computational ground for correlating chemical structure with biological activity by the celebrated quantitative stractuie-activity relationships (QSARs). Additionally, the fundamental statistical advanced frameworks are detailed to best understand the classical multilinear regression analysis generalized by an algebraic (in quantum Hilbert space) reformulation in terms of data vectors and orthogonal conditions (explained in see Chapter 3). 

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