Principal components plot description

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 and partial least squares analysis are chemometric tools for extracting and rationalizing the information from any multivariate description of a biological system. Complexity reduction and data simplification are two of the most important features of such tools. PCA and PLS condense the overall information into two smaller matrices, namely the score plot (which shows the pattern of compounds) and the loading plot (which shows the pattern of descriptors). Because the chemical interpretation of score and loading plots is simple and straightforward, PCA and PLS are usually preferred to other nonlinear methods, especially when the noise is relatively high. ... 

For the determination of the key process parameters and their respective values, fluorescence spectra from five 96-microreactor arrays were collected and processed. The normalized spectra are presented in Figure 5.7. The spectral features of the polymeric materials in the microreactors contain a wealth of information about the chemical properties of the materials that were extracted using PCA. According to the PCA results, the first two principal components (PCs) accounted for more than 95% of the spectral variation among all spectra. Thus the first two PCs were used for an adequate description of the fluorescence spectra. Results of the principal components analysis of the spectra from all 96-microreactor arrays as a function of catalyst concentration C are presented in Figure 5.8. The plot demonstrates the existence of the major general trend in the spectral descriptors where the variation in scores of both PCs strongly depends on concentration of component C for all screened process parameters. 

Data generated with the EOS are elaborated by Exploratory Data Analysis (EDA) software, a written-in-house software package based on MATLAB [22]. The EDA software includes the usual (univariate or multivariate) descriptive statistics functions among which Principal Component Analysis (PCA) [23], with the additional utilities for easy data manipulation (e.g. data sub sampling, data set fusion) and plots customization. 

PCA is by far the most important method in multivariate data analysis and has two main applications (a) visualization of multivariate data by scatter plots as described above (b) data reduction and transformation, especially if features are highly correlating or noise has to be removed. For this purpose instead of the original p variables X a subset of uncorrelated principal component scores U can be used. The number of principal components considered is often determined by applying a threshold for the score variance. For instance, only principal components with a variance greater than 1% of the total variance may be selected, while the others are considered as noise. The number of principal components with a non-negligible variance is a measure for the intrinsic dimensionality of the data. As an example consider a data set with three features. If all object points are situated exactly on a plane, then the intrinsic dimensionality is two. The third principal component in this example has a variance of zero. Therefore two variables (the scores of PCI and PC2) are sufficient for a complete description of the data structure. 

Figure 8 Two-dimensional scatter plots of the factor loading (A) and principal component score (B) using the total Charm values of 10 aroma descriptions (above 50% OSV). E, Ethiopia T, Tanzania I, Indonesia. Numbers (26, 23, and 18) refer to roast degrees. OSV, odor spectrum value.

In place of Charm values of 10 aroma descriptions mentioned, the PCA was conducted by using the mean Charm values of 35 potent odorants, which showed an OSV of 50 or above listed in Table 4. The two-dimensional scatter plots of the factor loading and principal component score are shown in Fig. 9. The PCI and PC2 explained 38.4% and 21.7% of the total GCO information, respectively. The factor loading of odorants (Fig. 9A) was plotted similarly to that of the 10 aroma descriptions into... 

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