Principal components analysis chemical factors

We now consider a type of analysis in which the data (which may consist of solvent properties or of solvent effects on rates, equilibria, and spectra) again are expressed as a linear combination of products as in Eq. (8-81), but now the statistical treatment yields estimates of both a, and jc,. This method is called principal component analysis or factor analysis. A key difference between multiple linear regression analysis and principal component analysis (in the chemical setting) is that regression analysis adopts chemical models a priori, whereas in factor analysis the chemical significance of the factors emerges (if desired) as a result of the analysis. We will not explore the statistical procedure, but will cite some results. We have already encountered examples in Section 8.2 on the classification of solvents and in the present section in the form of the Swain et al. treatment leading to Eq. (8-74). 

More detailed statistical analyses (chemical element balance, principal component analysis and factor analysis) demonstrate that soil contributes >50% to street dust, iron materials, concrete/cement and tire wear contribute 5-7% each, with smaller contributions from salt spray, de-icing salt and motor vehicle emissions (5,93-100). A list is given in Table VII of the main sources of the elements which contribute to street dust. 

Principal components analysis (PCA), factor analysis (FA), and derivative methods are tools used to analyze multivariate data. The data are in the form of an m X n matrix of data, X, where m is the number of observations and n is the number of variables per observation. The matrix is analyzed to find the number of underlying factors that influences the chemical system. With further analysis, a quantitative model can be generated based on the factors that have been discovered. A goal in the development of these factors and of the model is that they should have maximal chemical or physical significance. 

In some cases a principal components analysis of a spectroscopic- chromatographic data-set detects only one significant PC. This indicates that only one chemical species is present and that the chromatographic peak is pure. However, by the presence of noise and artifacts, such as a drifting baseline or a nonlinear response, conclusions on peak purity may be wrong. Because the peak purity assessment is the first step in the detection and identification of an impurity by factor analysis, we give some attention to this subject in this chapter. 

The application of principal components regression (PCR) to multivariate calibration introduces a new element, viz. data compression through the construction of a small set of new orthogonal components or factors. Henceforth, we will mainly use the term factor rather than component in order to avoid confusion with the chemical components of a mixture. The factors play an intermediary role as regressors in the calibration process. In PCR the factors are obtained as the principal components (PCs) from a principal component analysis (PC A) of the predictor data, i.e. the calibration spectra S (nxp). In Chapters 17 and 31 we saw that any data matrix can be decomposed ( factored ) into a product of (object) score vectors T(nxr) and (variable) loadings P(pxr). The number of columns in T and P is equal to the rank r of the matrix S, usually the smaller of n or p. It is customary and advisable to do this factoring on the data after columncentering. This allows one to write the mean-centered spectra Sq as ... 

How is dimension reduction of chemical spaces achieved There are a number of different concepts and mathematical procedures to reduce the dimensionality of descriptor spaces with respect to a molecular dataset under investigation. These techniques include, for example, linear mapping, multidimensional scaling, factor analysis, or principal component analysis (PCA), as reviewed in ref. 8. Essentially, these techniques either try to identify those descriptors among the initially chosen ones that are most important to capture the chemical information encoded in a molecular dataset or, alternatively, attempt to construct new variables from original descriptor contributions. A representative example will be discussed below in more detail. 

Multivariate models have been successful in identifying source contributions in urban areas. They are not independent of Information on source composition since the chemical component associations they reveal must be verified by source emissions data. Linear regressions can produce the typical ratio of chemical components in a source but only under fairly restrictive conditions. Factor and principal components analysis require source composition vectors, though it is possible to refine these source composition estimates from the results of the analysis (6.17). 

Figure 2. Plot of the first two factors (accounting for 69 percent of the variance) from a principal components analysis of the residential soil sample data (n = 38) from El Coyote using 11 chemical elements (Al, Ba, Ca, Fe, K, Mg Mn, Na, P, Sr, Ti). Ovens tend to have higher concentrations ofBa, Fe, and Na (which are all highly correlated, r>.7) compared to hearths, while hearths tend to have higher concentrations of P, K, Al, Mg and Ti (which are all highly correlated, r>0.7) compared to ovens.

Figure 5. Scatter plot showing principal component analysis for all 225 samples after best-relative-fit factors were applied (symbols indicate chemical subgroups o =outliers/loners A = Subgroup 4A B = Subgroup 4B).

The approach taken to observe the impact of the copper smelter on mesoscale variations rainwater composition was to determine the spatial, temporal, and experimental components of the variability of a number of appropriate chemical species in the rainwater. This paper presents results for 1985, during smelter operation, and includes (1) estimates of the experimental variability in chemical composition, (2) an approach for a two step chemical and statistical screening of the data set, (3) the spatial variation in rainwater composition for a storm collected on February 14-15, and (4) a principal component analysis of the rainwater concentrations to help identify source factors influencing our samples. 

In order to determine the chemical characteristics of Japanese rain, the major chemical components were determined at eleven stations throughout Japan for two years. The principal component analysis showed that nitrate and calcium can be used to characterize the local factors. The deposition of sulfate is discussed in relation to its origin. Some typical differences were observed between the stations on the Pacific side and the Japan Sea side of Honshu Is. 

Comparison and ranking of sites according to chemical composition or toxicity is done by multivariate nonparametric or parametric statistical methods however, only descriptive methods, such as multidimensional scaling (MDS), principal component analysis (PCA), and factor analysis (FA), show similarities and distances between different sites. Toxicity can be evaluated by testing the environmental sample (as an undefined complex mixture) against a reference sample and analyzing by inference statistics, for example, t-test or analysis of variance (ANOVA). 

Recently, Riviere and Brooks (2007) published a method to improve the prediction of dermal absorption of compounds dosed in complex chemical mixtures. The method predicts dermal absorption or penetration of topically applied compounds by developing quantitative structure-property relationship (QSPR) models based on linear free energy relations (LFERs). The QSPR equations are used to describe individual compound penetration based on the molecular descriptors for the compound, and these are modified by a mixture factor (MF), which accounts for the physical-chemical properties of the vehicle and mixture components. Principal components analysis is used to calculate the MF based on percentage composition of the vehicle and mixture components and physical-chemical properties. 

Statisticians do not always distinguish between factor analysis and principal components analysis, but for chemists factors often have a physical significance, whereas PCs are simply abstract entities. However, it is possible to relate PCs to chemical information, such as elution profiles and spectra in HPLC-DAD by... 

Chemometrics stands in this context for analysis of multivariate chemical data by means of statistical methods such as principal component analysis (PCA) or factor analysis (FA) cf. Section 3.5. 

Steps (a) to (c) are as for principal components analysis. However, as the final aim is usually to interpret the results of the analysis in terms of chemical or spectroscopic properties, the method adopted at each step should be selected with care and forethought. A simple example will serve to illustrate the principles of factor analysis and the application of some of the options available at each stage. 

Fig. 6-15 Principal component analysis of multidimensional, chemical-genetic data, (a) Eigenvalues and associated variance, and eigenvectors and associated factor scores computed from the data in Fig. 6-14(a). The matrix of eigenvectors...

In this section we shall consider the rather general case where for a series of chemical compounds measurements are made in a number of parallel biological tests and where a set of descriptor variables is believed to be related to the biological potencies observed. In order to imderstand the data in their entirety and to deal adequately with the mathematical properties of such data, methods of multivariate statistics are required. A variety of such methods is available as, for example, multivariate regression, canonical correlation, principal component analysis, principal component regression, partial least squares analysis, and factor analysis, which have all been applied to biological or chemical problems (for reviews, see [1-11]). Which method to choose depends on the ultimate objective of an analysis and the property of the data. We have found principal component and factor analysis particularly useful. For this reason and also since many multivariate methods make use of components for factors we will start with these methods in some detail, while the discussion of other approaches will be less extensive. 

Multiple intercorrelations between descriptors of chemical structures are illustrated best using multivariate statistics (section 3.2.2). A principal component analysis of the data set of 18 descriptors (Table 1.6, Figure 1.11) revealed that > 80% of the information content of these descriptors is expressed by four factors that explain 54.7%, 15.8%, 8.1% and 5.6% of the total variance, respectively. 

One aim of chemometrics is to obtain these predictions after first treating the chromatogram as a multivariate data matrix, and then performing principal component analysis (PCA). Each compound in the mixture is a chemical factor with its associated spectra and elution profile, which can be related to principal components, or abstract factors, by a mathematical transformation. 

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