Correlation matrix array

There are three stages in factor analysis (1) generation of a correlation matrix for all the variables consisting of an array of correlation coefficients of the variables (2) extraction of factors from the correlation matrix based on the correlation coefficients of the variables and (3) rotation of factors to maximize the relationship between the variables and some of the factors. FA requires a set of data points in matrix form, and the data must be bilinear — that is, the rows and columns must be independent of each other. 

Some of the first ideas on multi-way analysis were published by Raymond Cattell [1944,1952], Thurstone s principle of parsimony states that a simple structure should be found to describe a data matrix or its correlation matrix with the help of factors [Thurstone 1935], For the simultaneous analysis of several matrices together, Cattell proposed to use the principle of parallel proportional profiles [Cattell 1944], The principle of parallel proportional profiles states that a set of common factors should be found that can be fitted with different dimension weights to many data matrices at the same time. This is the same as finding a common set of factors for a stack of matrices, a three-way array. To quote Cattell ... 

The principal relationships between descriptors of chemical structures in a large array of compounds can be demonstrated. The correlation matrix (Table 1.5) obtained for a variety of parameters for several hundred compounds in more than 20 chemical classes (Nendza and Russom, 1991) reveals that hardly any descriptor is truly independent and that several highly significant correlations result. 

The aim of all the foregoing methods of factor analysis is to decompose a data-set into physically meaningful factors, for instance pure spectra from a HPLC-DAD data-set. After those factors have been obtained, quantitation should be possible by calculating the contribution of each factor in the rows of the data matrix. By ITTFA (see Section 34.2.6) for example, one estimates the elution profiles of each individual compound. However, for quantitation the peak areas have to be correlated to the concentration by a calibration step. This is particularly important when using a diode array detector because the response factors (absorptivity) may considerably vary with the compound considered. Some methods of factor analysis require the presence of a pure variable for each factor. In that case quantitation becomes straightforward and does not need a multivariate approach because full selectivity is available. 

With the rapid increase in understanding of the mechanisms of cell injury and repair, a number of new substances have been identified that may prove to be useful markers of acute injury or disease activity. These include various cytokines and growth factors, several lipid mediators, a complex array of extracellular matrix components and cell adhesion molecules, plus a variety of miscellaneous compounds. At the present time, the clinical utility of their measurement in biologic samples is unknown, although in selected instances, clinical correlates have emerged. Unfortunately, not all of these markers are present in urine or blood samples. For some, detection involves histologic or histochemical techniques applied to renal tissue samples. Nonetheless, the substances discussed below are intimately involved in the control and modification of cell function, the response to stress and/or the processes of repair. It is anticipated that with proper amplification, one or more may be useful as a marker of susceptibility, exposure or effect. 

Odor and taste perception permits recognition and discrimination between a large number of different molecules. The detection mechanism is based on the processing of signals from several neurons in an array processing system, and does not require the presence of specific receptor proteins. The same mechanism has been proposed for the action of eye irritant molecules and studies have been made to correlate the absorption behavior of eye irritant compounds in the lipid matrix not only by hydrophobicity, but through a more complicated procedure related to phase transition phenomena in the lipid matrix. 

A 2-D NMR spectrum is obtained after carrying out two Fourier transformations on a matrix of data (as opposed to one Fourier transform on an array of data for a 1-D NMR spectrum). A 2-D NMR spectrum will generate cross peaks that correlate information on one axis with information on the other usually, both axes are chemical shift axes, but this is not always the case. 

For the purpose of adding a simple model to the following discussion. Figs. 5.1 and 5.2 present an overview of the covariance transformation types 1—4 on a selected set of matrices. The arrays used are of very small dimensions as compared to experimentally recorded NMR data. Still they reflect the essential features of 2D NMR spectra such as a H-H COSY or a H—TOCSY with a short mixing time, cf Fig. 5.1A, and an H-C HSQC, cf. Fig. 5.IB. The values of the matrix elements were arbitrarily chosen and indicate a signal or a correlation at that position. The relative signs should be interpreted as signal phases, cf. Fig. 5.IE as an example for a multiphcity-edited HSQC. 

Covariance NMR has established itself as a valuable tool in the ranks ofNMR methodologies. As direct and indirect covariance of 2-dimensional NMR data sets, it can replace the second Fourier transformation, whereas as unsymmetrical and GIC it follows Fourier transformation. Based on matrix algebra and statistical mathematics, covariance transformations were extended to doubly indirect and 3- and 4-dimensional covariance NMR. Since the parent or component data arrays can originate from any type of 2D NMR experiment, 2D NMR spectra were co-processed to yield C-N and C-P correlations of non-isotopicaUy labelled small molecules. Further heterospectroscopic covariance was used to concatenate NMR and MS data allowing to allocate the information of both to a compound. 

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