Multivariate correlation

Gonzalez-Miret, M. L., Terrab, A., Hernanz, D., Femandez-Recamales, M. A., and Heredia, F. J. (2005). Multivariate correlation between color and mineral composition of honeys and by their botanical origin. /. Agric. Food Chem. 53, 2574-2580. 

During the last two or three decades, chemists became used to the application of computers to control their instruments, develop analytical methods, analyse data and, consequently, to apply different statistical methods to explore multivariate correlations between one or more output(s) (e.g. concentration of an analyte) and a set of input variables (e.g. atomic intensities, absorbances). 

Solvatochromic Approach Solvatochromic relationships are multivariate correlations between a property, usually solubility or partitioning property (see Sections 11.4 and 13.3), and solvatochromic parameters, parameters that account for the solutes interaction with the solvent. In the case of vapor pressure, the solvatochromic parameters only have to account for intermolecular interaction such as selfassociation between the solute (i.e., pure compound) molecules themselves. The following model has been reported for liquid and solid compounds, including hydrocarbons, halogenated hydrocarbons, alkanols, dialkyl ethers, and compounds such as dimethyl formamide, dimethylacetamide, pyridine, and dimethyl sulfoxide... 

For this purpose the well-known univariate correlation analysis was changed to the more general multivariate case [GEISS and EINAX, 1991 1996]. Multivariate correlation analysis enables inclusion of all interactions within the variables and the exclusion of the share of the variance resulting from the variable noise. 

One advantage of multivariate correlation is the possibility of simultaneous handling of all variables in the time or local series. This enables all interactions within the variables in the series and between the series which are dependent upon lag to be taken into consideration. 

When testing the significance of the multivariate correlation value it is necessary to take three influences into consideration ... 

In the multivariate case, the significant cross-correlation or autocorrelation coefficients for each variable add up to the significant multivariate correlation value. 

In order to declare a multivariate probability P = 0.977 for testing the multivariate correlation coefficient, the following univariate probability is required ... 

The following critical multivariate correlation coefficient results ... 

In comparison with Fig. 7-18 the MACF for the fifth fraction (dmedium = 0.45 pm) (Fig. 7-19) is less scattered and the duration of correlation is definitely longer - significant multivariate correlation with the impact of the emissions can be found, for up to 12 weeks. 

Fig. 8-5. Multivariate correlation coefficients between the sampling points on the river Saale for the heavy metals cadmium, chromium, copper, iron, nickel, and zinc...

The computation of the multivariate autocorrelation function (MACF) is useful if the simultaneous consideration of all measured variables and their interactions is of interest. The mathematical fundamentals of multivariate correlation analysis are described in detail in Section 6.6.3. The computed multivariate autocorrelation function Rxx according to Eqs. 6-30-6-37 is demonstrated in Fig. 9-6. The periodically encountered... 

The multivariate autocorrelation function of the measured values compared with the highest randomly possible correlation value shows significant correlation up to Lag 7. So, the range of multivariate correlation is more extended than that of univariate correlation (see Section 9.1.3.3.1). This fact must be understood because the computation of the MACF includes the whole data matrix with all interactions between the measured parameters. For characterization of the multivariate heavy metal load of the test area only 14 samples in the screen are necessary. 

Tauler, R., Barceld, D., and Thurman, E.M., Multivariate correlations between concentration of selected herbicides and derivatives in outflows from selected U.S. midwestem reservoirs, Environ. Sci. Technol., 34, 3307-3314, 2000. 

Tables 2 and 3 report proximate and ultimate analyses of the wastes and derived char sampleS) respectively. Heating values (HV) of the wastes and chars were estimated by an own multivariate correlation in terms of samples elemental compositions, previously reported (5). Estimated heating values on a diy-ash free basis are listed in Table 3.

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 - embedded correlation A emb in the Free-Wilson matrix, i.e. the intrinsic multivariate correlation of the data set, is ... 

The multivariate correlation index calculated by diagonalization of the correlation matrix obtained from the -> molecular matrix M is also among the -> WHIM shape descriptors. 

A rule based on the -> multivariate K correlation index, which compares the multivariate correlation index Kx of the X-block of the predictor variables with the multivariate correlation index Kxy obtained by the augmented X-block matrix by adding the column of the response variable [Todeschini et al, 1998]. Only regression models having multivariate correlation Kxy greater than multivariate correlation Kx can fulfill the QUIK rule, a necessary condition for the model validity, i.e. 

A method consisting in the iterative procedure of the elimination of one variable at a time, based on the -> multivariate Kcorrelation index [Todeschini etal.,1998]. All the variables are removed one at a time and the K multivariate correlation Kp/j (/ = 1, p) of the set of p - 1 variables is calculated. The j th variable associated with the minimum Kp/j correlation is removed from the set of p variables (i.e. the variable which is maximally correlated with all the others), and the procedure is repeated on the remaining variables. The elimination procedure ends when the minimum Kp/j is greater than the correlation Kp of the whole set of the remaining variables or when a standardized correlation value, called the K inflation factor (KIF), is less than 0.6 -0.5. 

Some basic concepts and definitions of statistics, chemometrics, algebra, graph theory, similarity/diversity, which are fundamental tools in the development and application of molecular descriptors, are also presented in the Handbook in some detail. More attention has been paid to information content, multivariate correlation, model complexity, variable selection, and parameters for model quality estimation, as these are the characteristic components of modern QSAR/QSPR modelling. 

When there are many response variables to consider, simultaneous evaluation of response surface models from each response becomes cumbersome. In such cases, a considerable simplification can often be achieved by multivariate analysis of the response matrix. For such purposes, principal components analysis and/or multivariate correlation by PLS are useful. These methods are discussed in Chapters 15 and 17. 

Table in shows, for the 21 structures in the analysis, the first canonical correlation coefficients relating the shape descriptors with the five measures of biological activity. In the case of a single descriptor (Kier s index, for example) the canonical correlation coefficient is the same as the simple correlation coefficient, so univariate and multivariate correlations can be compared directly. 

Supervised variable elimination might also be regarded as variable selection. Whether we consider this to be the third major section of how to treat multivariate datasets is a matter of semantics, however. It is possible to eliminate variables in a supervised manner rather than to select them. One obvious way is to eliminate variables that have a zero or very low correlation with the response variable or variables. In the case of classified response data, this selection means those descriptors that have the same distribution (mean and standard deviation) for the two or more classes. The danger in this selection process is the possibility that a variable might have a low correlation with the response but contribute to a multivariate correlation. Although this is possible, in practice, it is unlikely. 

During the last two or three decades atomic spectroscopists have become used to the application of computers to control their instruments, develop analytical methods, analyse data and, consequently, to apply different statistical methods to explore multivariate correlations between one or more output(s) e.g. concentration of an analyte) and a set of input variables e.g. atomic intensities, absorbances). On the other hand, the huge efforts made by atomic spectroscopists to resolve interferences and optimise the instrumental measuring devices to increase accuracy and precision have led to a point where many of the difficulties that have to be solved nowadays cannot be described by simple univariate linear regression methods (Chapter 1 gives an extensive review of some typical problems shown by several atomic techniques). Sometimes such problems cannot even be addressed by multivariate regression methods based on linear relationships, as is the case for the regression methods described in the previous two chapters. 

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