Principal components analysis results

Spectral features and their corresponding molecular descriptors are then applied to mathematical techniques of multivariate data analysis, such as principal component analysis (PCA) for exploratory data analysis or multivariate classification for the development of spectral classifiers [84-87]. Principal component analysis results in a scatter plot that exhibits spectra-structure relationships by clustering similarities in spectral and/or structural features [88, 89]. 

Measurement of the experimental uncertainties in our rain sampling procedures and the application of these uncertainties in a screening procedure to eliminate questionable samples from the data set increases the confidence in the interpretation of spatial variations in rainwater composition and of the principal component analysis. Results presented here for a storm collected during smelter operation suggest that it was the major source of the downwind excess SO. elevation above background and the pH depress ion oelow the background value of 5. 

One of the main attractions of normal mode analysis is that the results are easily visualized. One can sort the modes in tenns of their contributions to the total MSF and concentrate on only those with the largest contributions. Each individual mode can be visualized as a collective motion that is certainly easier to interpret than the welter of information generated by a molecular dynamics trajectory. Figure 4 shows the first two normal modes of human lysozyme analyzed for their dynamic domains and hinge axes, showing how clean the results can sometimes be. However, recent analytical tools for molecular dynamics trajectories, such as the principal component analysis or essential dynamics method [25,62-64], promise also to provide equally clean, and perhaps more realistic, visualizations. That said, molecular dynamics is also limited in that many of the functional motions in biological molecules occur in time scales well beyond what is currently possible to simulate. 

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). 

Malinowski, E.R., Theory of die Distribution of Error Eigenvalues Resulting from Principal Component Analysis with Applications to Spectroscopic Data",... 

The data from sensory evaluation and texture profile analysis of the jellies made with amidated pectin and sunflower pectin were subjected to Principal component analysis (PC) using the statistical software based on Jacobi method (Univac, 1973). The results of PC analysis are shown in figure 7. The plane of two principal components (F1,F2) explain 89,75 % of the variance contained in the original data. The attributes related with textural evaluation are highly correlated with the first principal component (Had.=0.95, Spr.=0.97, Che.=0.98, Gum.=0.95, Coe=0.98, HS=0.82 and SP=-0.93). As it could be expected, spreadability increases along the negative side of the axis unlike other textural parameters. 

An alternative and illuminating explanation of reduced rank regression is through a principal component analysis of Y, the set of fitted F-variables resulting from an unrestricted multivariate multiple regression. This interpretation reveals the two least-squares approximations involved projection (regression) of Y onto X, followed by a further projection (PCA) onto a lower dimensional subspace. 

Sets of spectroscopic data (IR, MS, NMR, UV-Vis) or other data are often subjected to one of the multivariate methods discussed in this book. One of the issues in this type of calculations is the reduction of the number variables by selecting a set of variables to be included in the data analysis. The opinion is gaining support that a selection of variables prior to the data analysis improves the results. For instance, variables which are little or not correlated to the property to be modeled are disregarded. Another approach is to compress all variables in a few features, e.g. by a principal components analysis (see Section 31.1). This is called... 

Figure 1. Results of a principal component analysis (PCA) performed on soil properties under greenhouse conditions...

Probability that the analyte A is present in the test sample Conditional probability probability of an event B on the condition that another event A occurs Probability that the analyte A is present in the test sample if a test result T is positive Score matrix (of principal component analysis)... 

An important application field of factor and principal component analysis is environmental analysis. Einax and Danzer [1989] used FA to characterize the emission sources of airborne particulates which have been sampled in urban screening networks in two cities and one single place. The result of factor analysis basing on the contents of 16 elements (Al, B, Ba, Cr, Cu, Fe, Mg, Mn, Mo, Ni, Pb, Si, Sn, Ti, V, Zn) determined by Optical Atomic Emission Spectrography can be seen in Fig. 8.17. In Table 8.3 the common factors, their essential loadings, and the sources derived from them are given. 

Musumarra et al. [43] identified miconazole and other drugs by principal components analysis of standardized thin-layer chromatographic data in four eluent systems. The eluents, ethylacetate methanol 30% ammonium hydroxide (85 10 15), cyclohexane-toluene-diethylamine (65 25 10), ethylacetate chloroform (50 50), and acetone with the plates dipped in potassium hydroxide solution, provided a two-component model that accounts for 73% of the total variance. The scores plot allowed the restriction of the range of inquiry to a few candidates. This result is of great practical significance in analytical toxicology, especially when account is taken of the cost, the time, the analytical instrumentation and the simplicity of the calculations required by the method. 

We now have the data necessary to calculate the singular value decomposition (SVD) for matrix A. The operation performed in SVD is sometimes referred to as eigenanal-ysis, principal components analysis, or factor analysis. If we perform SVD on the A matrix, the result is three matrices, termed the left singular values (LSV) matrix or the V matrix the singular values matrix (SVM) or the S matrix and the right singular values matrix (RSV) or the V matrix. 

In our original column on this topic [1] we had only done a principal component analysis to compare with the MLR results. One of the comments made, and it was made by all the responders, was to ask why we did not also do a PLS analysis of the synthetic linearity data. There were a number of reasons, and we offered to send the data to any or all of the responders who would care to do the PLS analysis and report the results. Of the original responders, Paul Chabot took us up on our offer. In addition, at the 1998 International Diffuse Reflectance Conference (The Chambersburg meeting), Susan Foulk also offered to do the PLS analysis of this data. 

The results show that DE-MS alone provides evidence of the presence of the most abundant components in samples. On account of the relatively greater difficulty in the interpretation of DE-MS mass spectra, the use of multivariate analysis by principal component analysis (PCA) of DE-MS mass spectral data was used to rapidly differentiate triterpene resinous materials and to compare reference samples with archaeological ones. This method classifies the spectra and indicates the level of similarity of the samples. The output is a two- or three-dimensional scatter plot in which the geometric distances among the various points, representing the samples, reflect the differences in the distribution of ion peaks in the mass spectra, which in turn point to differences in chemical composition of... 

Different categories of Zonyl polymers are studied by ToF-SIMS both in the positive and negative ion mode. Studies have shown that, for each polymer, a specific fingerprint is obtained and the peaks corresponding to the specific chemical moieties of each polymer are detected (Figure 15.4). To represent this good selectivity, Principal Component Analysis is performed on the obtained spectra. The result clearly discriminates the different polymers. ToF-SIMS is then suited to the characterization of these materials. 

Figure 3.13 Principal component analysis of repetitive GC/MS profiles of M. truncatula root (R), stem (S) and leaves (L). The first and second principal component of each GC/MS analysis were calculated and plotted. The relative distance between points is a measure of similarity or difference. The clustering shows good reproducibility within the independent tissues but clear differentiation of tissues. The results also show that roots and stems are more similar to each other than to leaves.

A sample may be characterized by the determination of a number of different analytes. For example, a hydrocarbon mixture can be analysed by use of a series of UV absorption peaks. Alternatively, in a sediment sample a range of trace metals may be determined. Collectively, these data represent patterns characteristic of the samples, and similar samples will have similar patterns. Results may be compared by vectorial presentation of the variables, when the variables for similar samples will form clusters. Hence the term cluster analysis. Where only two variables are studied, clusters are readily recognized in a two-dimensional graphical presentation. For more complex systems with more variables, i.e. //, the clusters will be in -dimensional space. Principal component analysis (PCA) explores the interdependence of pairs of variables in order to reduce the number to certain principal components. A practical example could be drawn from the sediment analysis mentioned above. Trace metals are often attached to sediment particles by sorption on to the hydrous oxides of Al, Fe and Mn that are present. The Al content could be a principal component to which the other metal contents are related. Factor analysis is a more sophisticated form of principal component analysis. 

The authors wanted to select indicators that specifically tap melancholic depression. To evaluate this construct, a principal components analysis of the joint pool of K-SADS and BDI items was performed. Two independent statistical tests suggested a two-component solution, but the resulting components appeared to reflect method factors, rather than substantive factors. Specifically, all of the BDI items loaded on the first component (except for three items that did not load on either component) and nearly all of the K-SADS items loaded on the second component. In fact, the first component correlated. 98 with the BDI and the second component correlated. 93 with the K-SADS. Ambrosini et al., however, concluded that the first component reflected depression severity and the second component reflected melancholic depression. This interpretation was somewhat at odds with the data. Specifically, the second component included some K-SADS items that did not tap symptoms of melancholia (e.g., irritability and anger) and did not include some BDI items that measure symptoms of melancholia (e.g., loss of appetite). 

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