Parameter correlation matrix

In this approach a particular rate function is assumed and nonlinear least-squares parameter optimization techniques are used to calculate rate coefficients. Many techniques are available, and a computer program developed by Parker and Van Genuchten (1984) is excellent for this purpose. It is basically the maximum neighborhood method of Marquardt (1963). Various statistics are used to evaluate goodness-of-fit of the rate functions to the data including r-square, root mean square, 95% confidence intervals for computed parameters, and the parameter correlation matrix. The rate function(s) I hat give the best fit to the data are then assumed to be the most nearly correct. 

More objective procedures in series design are clustering methods in multidimensional parameter space substituents from different clusters are selected for synthesis (chapter 3) [50, 154, 403]. As this approach cannot automatically avoid collinearity or multicollinearity, several different standard sets of aromatic substituents have been proposed e.g. [652, 653]). A distance mapping technique may be used to select further substituents on the basis of a nj ximum distance to the substituents which already are included [652]. A modification [654] of this approach uses the determinant of the parameter correlation matrix as the criterion for substituent selection. 

The covariances between the parameters are the off-diagonal elements of the covariance matrix. The covariance indicates how closely two parameters are correlated. A large value for the covariance between two parameter estimates indicates a very close correlation. Practically, this means that these two parameters may not be possible to be estimated separately. This is shown better through the correlation matrix. The correlation matrix, R, is obtained by transforming the co-variance matrix as follows... 

The lion s share of the computer-time for the least-squares process has to be provided for forming the Z-matrix. The elements of this matrix are evaluated partly numerically and partly analytically in the calculations of Lifson and Warshel (17). In certain cases, strong parameter correlations may occur. Therefore caution is demanded when inverting the matrix C. Also from investigations other than consistent force-field calculations it is known that such correlations frequently occur among the parameters for the nonbonded interactions (34,35). Another example of force field parameter correlations was encountered by Ermer and Lifson (19) in the course of the calculation of olefin properties. When... 

The covariance between the three model parameters i and the five observations y can be calculated through equation (5.4.31). Their 3x5 correlation matrix / (i, j) can be shown to be... 

We will follow the guidance of Albert Einstein to make everything as simple as possible, but not simpler. The reader will find practical formulae to compute results like the correlation matrix, but will also be reminded that there exist other possibilities of parameter estimation, like the robust or nonparametric estimation of a correlation. 

This approach can, in many instances, be extended even to cases where the basis is comprised of physicochemical, topological, or other such parameters. The similarity matrix is replaced in these cases by the correlation matrix computed with respect to the basis set of parameters (vide supra). 

The first part of the output contains the principal component analysis of the correlation matrix discussed later in Section 3.5. In addition to the residuals, goodness-of-fit, parameter estimates and bounds, the Durbin-Wattson D statistics is also printed by the module. 

Table IV. Correlation Matrix for Areas of FTIR Absorption Bands and Parameters of Coalification...

The effects of the culturing variables, interaction coefficients (95%), correlation matrix for estimated parameters, respective confidence inter-... 

Calculate the correlation matrix between each of the five fundamental parameters. How does this relate to clustering in the loadings plot ... 

To obtain a statistically sound QSAR, it is important that certain caveats be kept in mind. One needs to be cognizant about col-linearity between variables and chance correlations. Use of a correlation matrix ensures that variables of significance and/or interest are orthogonal to each other. With the rapid proliferation of parameters, caution must be exercised in amassing too many variables for a QSAR analysis. Topliss has elegantly demonstrated that there is a high risk of ending up with a chance correlation when too many variables are tested (62). 

It is fairly evident that because of the complex interactions of deposi-tionally influenced and metamorphically influenced properties, the fundamental chemical-structural properties will need to be related to each other in a complex statistical fashion. A multivariate correlation matrix such as that pioneered by Waddell (8) appears to be an absolute requirement. However, characterization parameters far more sophisticated than those employed by Waddell are required. One can hope that, as correlations between parameters become evident, certain key properties will be discovered that will allow coal scientists and technologists to identify and classify vitrinites uniquely. Measurement of reflectance or other optical properties, if carried out properly, possibly on somewhat modified samples, might prove valuable in this respect. It then would not be necessary for every laboratory to have supersophisticated analytical equipment at its disposal in order to classify a coal properly. By properly identifying and classifying the vitrinite in a coal, one then could estimate accurately the many other vitrinite properties available in the multivariate correlation matrix. 

The corresponding equation with it instead of molar refractivity resulted in a less statistically significant correlation. Table XIII shows the correlation matrix for para parameters for the para substituted compounds. 

Table VIII. Correlation Matrix for Para Parameters for Para Substituted Compounds...

---