Pure variable

Pure variables are fully selective for one of the factors. This means that only one pure factor contributes to the values of that variable. When the pure variables or selective wavelengths for each factor are known then the pure spectra can be calculated in a straightforward memner from the mixture spectra by solving ... 

The pure variable technique can be applied in the column space (wavelength) as well as in the row space (time). When applied in the column space, a pure column is one of the column factors. In LC-DAD this is the elution profile of the compound which contains that selective wavelength in its spectrum. When applied in the row space, a pure row is a pure spectrum measured in a zone where only one compound elutes. 

In Section 34.2 we explained that factor analysis consists of a rotation of the principal components of the data matrix under certain constraints. When the objects in the data matrix are ordered, i.e. the compounds are present in certain row-windows, then the rotation matrix can be calculated in a straightforward way. For non-ordered spectra with three or less components, solution bands for the pure factors are obtained by curve resolution, which starts with looking for the purest spectra (i.e. rows) in the data matrix. In this section we discuss the VARDIA [27,28] technique which yields clusters of pure variables (columns), for a certain pure factor. 

The variance diagram obtained for the example discussed before is quite simple. Clusters of pure variables are found at 30 degrees (var = 0.5853) and at 300 degrees (var = 0.4868) (see Fig. 34.36). The distance from the centre of the diagram to each point is proportional to the variance value. Neighbouring points are connected by solid lines. All values were scaled in such a way that the highest variance is full scale. As can be seen from Fig. 34.36, two clusters of pure variables are found. The... 

Determination of pure variables from mixture spectra... 

Once the pure variables have been identified, the data set can be resolved into the pure spectra by solving eq. (34.11). For the mixture spectra in Table 34.4, this gives ... 

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. 

Exploration of a data set before resolution is a golden rule fully applicable to image analysis. In this context, there are two important domains of information in the data set the spectral domain and the spatial domain. Using a method for the selection of pure variables like SIMPLISMA [53], we can select the pixels with the most dissimilar spectra. As in the resolution of other types of data sets, these spectra are good initial estimates to start the constrained optimization of matrices C and ST. The spatial dimension of an image is what makes these types of measurement different from other chemical data sets, since it provides local information about the sample through pixel-to-pixel spectral variations. This local character can be exploited with chemometric tools based on local-rank analysis, like FSMW-EFA [30, 31], explained in Section 11.3. 

Another related aspect involves using diese graphs to select pure variables. This is often useful in spectroscopy, where certain masses or frequencies are most diagnostic of different components in a mixture, and finding these helps in the later stages of the analysis. 

If pure variables such as spectral frequencies or mlz values can be determined, even if there are embedded peaks, it is also possible to use these to obtain first estimates of... 

More difficult situations occur when only some components exhibit selectivity. A common example is a completely embedded peak in HPLC-DAD. In the case of LC-MS or LC-NMR, this problem is often solved by finding pure variables, but because UV/vis spectra are often completely overlapping it is not always possible to treat data in this manner. 

ACD/AutoChrom uses the mutual automated peak matching [33] or UV-MAP approach based on extraction of pure variables from diode array data. The UV-MAP algorithm applies abstract factor analysis (AFA) followed by iterative key set factor analysis to the augmented data matrix in order to extract retention times for each of the selected experiments. 

An exploratory analysis performed by FSIW-EFA provides an estimate of the number of components in each pixel. For resolution purposes, only those pixels in the partial local rank map will be potentially constrained, because these are the pixels for which a robust estimation of the number of missing components can be obtained. However, the FSIW-EFA information is not sufficient to identify which components are absent from the constrained pixels. For identification purposes, the local rank information should be combined with reference spectral information, the ideal reference being the pure spectra of the constituents, although in most images not all of these are known. For the image components with no pure spectrum available, the reference taken is an approximation of this pure spectrum. These approximate pure spectra can be obtained by pure variable selection methods, or they may be the result of a simpler MCR-ALS analysis where only non-negativity constraints have been applied. 

Those cases for which the pile of the two alternating non-commensurate layers is periodically sheared, kinked or modified by antiphase boundaries, represent combination of the accretional (separately for each layer) and variable-fit (for the interlayer match) principles. The geometrical constraints in these structures are much more severe than in the pure variable-fit structures and result in a very reduced number of homologues (usually homologous pairs). 

A separate set of approaches relates to determine selective or pure variables. These are variables that most characterize each individual component in a mixture. For example, in the GC-MS of mixtures there is Hkely to be a specific miz value most characteristic of each component. In HPLC-DAD, we look for selective elution times for each compound. 

Construction of non-random initial estimates of matrix C or S. Local rank analysis methods, such as evolving factor analysis (EFA) [1,2], or methods based on the selection of pure variables, such as simple-to-use-interactive self-modelling mixture analysis (SIMPLISMA) [18,19], can be used for this purpose. 

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