Principal components plot example

Figure 6.2 An example of the spectral shift arising from unique production lines is apparent in this principal component plot where significant displacement is observed between pilot and production batches of product containing identical drug concentrations. Each point represents the spectrum of a single tablet.

Figure 38 shows the variance explained by the two principal component (PC) model as a percentage of each of the two indices batch number and time. The lower set of bars in Fig. 38a are the explained variances for the first PC, while the upper set of bars reflects the additional contribution of the second PC. The lower line in Fig. 38b is the explained variance over time for the first PC and the upper line is the combination of PC 1 and 2. Figure 38a indicates, for example, that batch numbers 13 and 30 have very small explained variances, while batch numbers 12 and 33 have variances that are captured very well by the reference model after two PCs. It is impossible to conclude from this plot alone, however, that batches 13 and 30 are poorly represented by the reference model. 

Classification To illustrate the use of SIMCA in classification problems, we applied the method to the data for 23 samples of Aroclors and their mixtures (samples 1-23 in Appendix I). In this example, the Aroclor content of the three samples of transformer oil was unknown. Samples 1-4, 5-8, 9-12 and 13-16, were Aroclors 1242, 1248, 1254, and 1260, respectively. Samples 17-20 were 1 1 1 1 mixtures of the Aroclors. Application of SIMCA to these data generated a principal components score plot (Figure 12) that shows the transformer oil is similar, but not... 

Principal components analysis can be best understood using a simple m o-variable example. With only two variables it is possible to plot the row space without the need to reduce the number of variables. Although this docs not fully present the utilit> of PCA. it is a good demonstration of how it functions. A two-dimensional plot of the row space of an example data set is shown in Figure 4.23. The data matrix consists of two columns, representing the two measurements, and 40 rows, representing the samples. Each row of the matrix is represented as a point (O) on the graph. 

In this example, the first principal component describes 98% of the variation. Successive PCs can be estimated that describe a portion of the remaining variation in this example, the second PC contains 2%. This illustnites another property of PCs, that is, successive PCs describe decreasing amounts of variation. Knowing the percent variation described is veiy important when interpreting the plots. For example, if close to 100% of the variation is described using two PCs, a two-dimensional plot can effectively be used to study the variation in the data set. However, a two-dimensional plot will not be adequate... 

The peptide example illustrates these points. The PC r-scores of the first three principal components of the physicochemical properties of amino acids are plotted in Figure 6.14. This space is well spanned by arg, asp, gly, ile and trp (Figure 6.24), which may therefore form a basis for construction of a fractional factorial design of a QSAR as previously suggested [6]. 

In the preceding description of the Mahalanobis distance, the number of coordinates in the distance metric is equal to the number of spectral frequencies. As discussed earlier in the section on principal component analysis, the intensities at many frequencies are dependent, and by using the full spectrum, we fit the noise in addition to the real information. In recent years, Mahalanobis distance has been defined with PCA or PLS scores instead of the spectral frequencies because these techniques eliminate or at least reduce most of the overfitting problem. The overall application of the Mahalanobis distance metric is the same except that the rt intensity values are replaced by the scores from PCA or PLS. An example of a Mahalanobis distance calculation on a set of Raman spectra for 25 carbohydrates is shown in Fig. 5-11. The 25 spectra were first subjected to PCA, and it was found that the first three principal components could account for most of the variance in the spectra. It was first assumed that all 25 spectra belonged to the same class because they were all carbohydrates. However, as shown in the three-dimensional plot in Fig. 5-11, the spectra can be clearly divided into three separate classes, with two of the spectra almost equal distance from each of the three classes. Most of the components in the upper left class in the two-dimensional plot were sugars however, some sugars were found in the other two classes. For unknowns, scores have to be calculated from the principal components and processed in the same way as the spectral intensities. 

Principal component analysis of the aldehydes and the ketones, respectively, afforded two significant components which accounted for 78% (aldehydes) and 88% (ketones) of the total variance. A score plot of the ketones is shown in the example of the Fischer indole synthesis given in Sect. 5.3.2. For a score plot of the aldehydes, see [61]. 

However, what if we had more than one variable to consider In other words, we have multivariate data. For example, what if we want to identify trends in the properties of a range of organic molecules The variables we might want to consider could be melting point, boiling point, M, solubility in a solvent and vapour pressure. We can, of course, tabulate the data, as before, but this does not allow us to consider any trends in the data. To do this we need to be able to plot the data. However, once we exceed three variables (which we need to be able to plot in three dimensions) it becomes impossible to produce a straightforward plot. It is in this context that chemometrics offers a solution, reducing the dimensionality to a smaller number of dimensions and hence the ability to display multivariate data. The most important technique in this context is called principal component analysis (PCA). 

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