Rank analyses

Setting the boundaries of windows of the different components can only be done if we are able to know how the number and nature of the components change in the data set. Obtaining this information is the main goal of local-rank analysis methods, which are used to locate and describe the evolution of each component in a system. This is accomplished by combining the information obtained from multiple rank analyses performed locally on limited zones (row or column windows) of the data set. 

Figure 19.3 shows typical heats of combustion, and Fig. 19.4 shows typical proximate analyses (reported on an ash-free basis) of various coal ranks. Analyses listed by coal... 

Methods for evolutionary rank analysis are explained and discussed in this section. The different approaches of evolutionary rank analysis have in common that the two-way data stmcture is analyzed piece-wise to locally reveal the presence of the analytes. A reference to the review of evolutionary methods by Toft et al. is included in the additional recommended reading list at the end of this chapter. 

Local rank analysis of suspected selective regions in a LC-DAD data set (see Fig. 34.34)... 

J. Toft, Tutorial Evolutionary rank analysis applied to multidetectional chromatographic structures. Chemom. Intell. Lab. Syst., 29 (1995) 189-212. 

Apparently similar flowstream universal buffers have been developed by Alibrandi and others [128,129] for assessing kinetic parameters, such as the pH-dependent hydrolysis of acetylsalicylic acid. The pH-time curves are not as linear as in the SGA system. Other reports of continuous flow pH gradient spectrophotometric data have been described, with application to rank-deficient resolution of solution species, where the number of components detected by rank analysis is lower than the real number of components of the system [130]. The linear pH-time gradient was established in the flowstream containing 25 mM H3PO4 by the continuous addition of 100 mM Na3P04. 

The factor extraction according to Eq. (8.20) in the course of which the number of common factors are estimated by rank analysis and coefficients of the factors (factor loadings) are calculated. 

An important simplifying consequence of the use of inverted concentration ratios is that the reaction is independent of O2 concentration, which means that unintended 02 contamination should not distort the data. Because of the complexity of the reaction, the relatively new technique of Matrix Rank Analysis was used to sort out the speciation. This analysis led to the identification of two sulfur-containing intermediates [Fe2(0H)S03]3+ and [Fe(S03]+. Other reactant species known to be present under these conditions include S02, HS03, Fe3+, Fe(OH)2+, and... 

The number of linearly independent columns (or rows) in a matrix is called the rank of that matrix. The rank can be seen as the dimension of the space that is spanned by the columns (rows). In the example of Figure 4-15, there are three vectors but they only span a 2-dimensional plane and thus the rank is only 2. The rank of a matrix is a veiy important property and we will study rank analysis and its interpretation in chemical terms in great detail in Chapter 5, Model-Free Analyses. 

So far in this chapter, all our elaborations were completely abstract there has been no attempt at an interpretation or understanding of the results of Factor Analysis in chemical terms. Abstract Factor Analysis is the core of most applications of Factor Analysis within chemistry, but, nevertheless, much more insight can be gained than the results of the rank analysis we have seen so far. How can we relate the factors U and V to something chemically meaningful Very sensibly these factors are called abstract factors, in contrast to real factors such as the matrices C and A containing the concentration profiles and pure component spectra. Is there a useful relationship between U, V, C and A ... 

D. Wienke, W. van den Broek and L. Buydens, Identification of plastics among nonplastics in mixed waste hy remote sensing near-infrared imaging spectroscopy. 2. Multivariate rank analysis for rapid classification. Anal. Ghent., 67, 3760-3766 (1995). 

Wienke, D. van den Broek, W. Buydens, L., Identification of Plastics among Nonplastics in Mixed Waste by Remote Sensing Near-Infrared Imaging Spectroscopy. 2. Multivariate Rank Analysis for Rapid Classification Anal. Chem. 1995, 67, 3760-3766. 

Some of the local-rank analysis methods, such as evolving-factor analysis (EFA) [27-29], are more process oriented and rely on the sequential evolution of the components as a function of time or any other variable in the data set, while others, such as fixed-size moving-window-evolving-factor analysis (FSMW-EFA) [30, 31], can be applied to processes and mixtures. EFA and FSMW-EFA are the two pioneering local-rank analysis methods and can still be considered the most representative and widely used. 

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. 

The ranking analysis discussed in the remainder of this section used benzene exposure at a nearby residence as a proxy for the risk associated with population exposure to refinery releases. In Table X, the share each option represents of the total benzene exposure reduction achieved by implementing all options is given in the column labeled Benzene exposure reduction. The barge loading option accounts for 55% of the benzene exposure reduction attributable to all options. In cost-effectiveness terms, the cost for a 1% benzene exposure reduction ranges from 9000 for secondary seals to 1.48 million for upgrading the wastewater treatment plant. 

In a properly constructed ranking analysis, each assumption has to be documented. A sensitivity analysis can be performed, investigating the impacts of ranking decisions upon the final outcome. Uncertainties can also be quantified and data gathered to make the ranking more based upon data-derived rules. 

Although these pills were supposed to be formed by two constituents, a FSIW-EFA analysis detected the presence of a third compound (impurity) in some cases [65]. According to the theoretical composition of the piU and to the local rank analysis, information on the presence/absence of constituents in the different images could be introduced in the multi-image resolution process. From the pure spectra resolved and the distribution maps in Figure 2.17, the positive influence of the pill with the largest amount of impurity was noticeable when this compound was modeled in pills where it was present in very few pixels only. 

Finally, we considered the use of three dimensional analytical techniques such as the GC-MS-MS and video fluorometric monitoring of BPLC effluents. As with two dimensional techniques, we can use Rank Analysis, Rank Annihilation and Factor Analysis. However, in this case the data can be shown to be unambiguously decomposible, even when there is severe overlap among conq>onents. 

Evaluating of the Distribution of Components in the Peak by Local Rank Analysis. Complementary information about the evolution of the components inside the CE peak system can be obtained from local rank analysis. In this case, instead of estimating the rank of the whole D matrix, a succession of smaller submatrices derived from D is analyzed to get the evolution of the mathematical factors throughout the system. The most widely used evolutionary methods are as follows ... 

In a related study, Latorre et al. applied exploratory rank analysis to ascertain the number of components of complex nonresolved electrophoretic peaks of some amino acid derivatives (32). The performance of EFA, WFA, and MCR-ALS for following the evolution of overlapping species in the system was compared. It was found that MCR-ALS provided the best results in the case of strongly overlapping contributions. The simultaneous treatment of the sample mixture with data from standards of interest permitted the analytes to be successfully quantihed. 

A pseudo-rank analysis on the three matricized matrices X(IxJK), X(J /iK) and X(K/U) was performed and the results are shown in Table 10.8. This table shows that the same number of components always explains the highest percentage of the variation in X(J/iK) (/-mode, variables), the second highest percentage in XiKyu (A--mode, times) and the lowest percentage in X(IyJK (/-mode, samples). 

Under these conditions the rank of the matrix in eq. (5.2) equals according to definition. That means 5 equals the number of linear independent concentrations and the number of linear independent partial steps of reaction according to eq. (5.1), respectively. The numerical rank analysis is explained by an example [15]. 

Example 5.1 Example of a matrix rank analysis The following mechanism is assumed ... 

The formalism of a numerical rank analysis can be substituted by graphical methods. Even though this graphical approach is formally equivalent to the numerical rank analysis, diagrams frequently are more descriptive and result in more information. 

In the same way the A/, values can be taken for a graphical rank analysis. Instead of the K-diagrams related M-diagrams are obtained. 

In the previous section a graphical rank analysis has been demonstrated to... 

Rank analysis allows the determination of the number of linear independent steps of reaction. Graphical approaches are preferable since deviations of a correct determination of this number become better visible than fuzzy results which make the decision delicate. Absorbance diagrams are constructed at different orders. 

For FA in its genuine sense, additional criteria are used for rank analysis, that is, for determination of the number of significant factors. Most frequently, an empirical indicator function, IND, introduced by Malinowski [2] is used. It is computed from the real error, RE, or the residual standard deviation, RSD, as follows ... 

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