Scores, principal components

Fig. 3. Principal component score plots based on the antibiotic resistance profiles. The diamond ( ), the open square (o) and the triangle (A) indicate BG, DDF and DEF, respectively. The value in the parenthesis indicates the percentage of the variability explained by the principal component.

Table 5. Linear regression between principal component scores antibiotic resistance MPNs and soil physico-chemical characteristics based on the...

Fig. 2. Plots of 2nd principal component scores (x-axis) verses 3rd principal component scores (y-axis) showing the regions of space where unknown spessartine data points would be classified as originating from one of the four geographic locations Brazil, Virginia, Australia, and China, clockwise form upper right, respectfully.

More definitive information may also be gleaned from PCA. In Fig. 6.9, three principal component scores are graphed to show how the amount of roasting in coffee beans may be ascertained. 

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... 

An examination of the sample distributions observed in principal components projections using isomer concentration data expressed as fractional composition, as well as the clustering of samples by similar values of their second principal component score term, revealed consistent differences existed in sample profiles. The next step in this data evaluation is to statistically analyze correlations of the PLS components from analyses with the external variables such as percent sand, clay and silt, and total organic matter in samples. These correlations may play an important role in identifying factors resulting in changes in PCB composition and enable one to more clearly understand the forces determining the distribution and fate of PCB in a complex ecosystem. 

Figure 3. 3-D Plot of Principal Components Scores (Theta-1,-2,-3) Representing Normalized Isomer Composition Data for Aroclors 1242, 1248, 1254, 1260, and their mixtures. The points for each Aroclor represent individual sample analyses. The plot in the upper right quadrant is the view parallel to the Z-axis.

Figure 4.70. Principal component score plot of ail samples in class B (after remov-i si=i-=--ing the mislabeled samples). The samples in the calibration set are Xs and the vali- dation samples are Os. 

Figure 7.2 Principal component scores plot for a set of dopamine mimetics. Compounds with teratogenic activity are indicated by filled circles. (From Ridings, J.E., Manallack, D.T., Saunders, M.R., Baldwin, J.A., and Livingstone, D.J., Toxicology, 76, 209-217, 1992. With permission.)...

Regression on principal components (PCR) is another from of regression modeling that may be used for continuous response data. Here, the independent variables (the x set) are computed from the descriptor variables using PC A as shown in Equation 7.1. These are the principal component scores and they have several advantages ... 

With PCA, it is possible to build an empirical mathematical model of the data as described by Equation 4.4 where Tk is the n x k matrix of principal component scores and k is the m x A matrix of eigenvectors. 

As the first pure component begins to elute, the principal component scores increase in Figure 4.15 along the axis labeled pure component 1. As the second component begins to elute, the points shift way from the component 1 axis and toward the component 2 axis. As the concentration of the second component begins to decrease, the principal component scores decrease along the axis labeled pure component 2. Points that lie between the two pure-component... 

FIGURE 4.15 Scatter plot of the principal component scores from the analysis of the HPLC-UV/visible data set shown in Figure 4.1. The principal component axes are orthogonal, whereas the pure-component axes are not. Distances from the origin along the pure-component axes are proportional to concentration. Pure spectra lie on the pure-component axes. Mixture spectra he between the two pure-component axes. Dashed lines show the coordinates (e.g., concentrations) of one point on the pure-component axes. 

To conduct a search for the E-optimal design directly to the NIR data, we need a methodology that is robust with respect to correlation between the variables (wavelengths). As previously noted, by using the principal component scores, it is possible to use the E-optimal approach to reduce the number of samples and minimize the number of time-consuming GC measurements while also improving the quality of the calibration model. 

The resultant principal components scores and loadings plots are given in Figure 4.16. Several conclusions are possible. 

Figure 4. An S-space defined by the two first principal component scores (principal properties t, and t2) of the 78 ketones. The rings indicate the selection of 9 compounds according to a D-optimal design and a quadratic model in t, and t2. The design has selected three aliphatic ketones (prefix a. filled black circles), and six aromatic ketones (prefix ar, filled grey squares). No alicyclic ketones (open triangles) were selected by the design.

Preference mapping can be accomplished with projection techniques such as multidimensional scaling and cluster analysis, but the following discussion focuses on principal components analysis (PCA) [69] because of the interpretability of the results. A PCA represents a multivariate data table, e.g., N rows ( molecules ) and K columns ( properties ), as a projection onto a low-dimensional table so that the original information is condensed into usually 2-5 dimensions. The principal components scores are calculated by forming linear combinations of the original variables (i.e., properties ). These are the coordinates of the objects ( molecules ) in the new low-dimensional model plane (or hyperplane) and reveal groups of similar... 

The coordinates of each solvent point are (i) the factor (or principal component) scores F, and (ii) the factor (or principal component) loadings L. They give the information necessary to reconstitute the original physical properties D of any solvent according to Eq. (3-15). 

Thus, the elements of the eigenvectors become the required coefficients for the original variables, and are referred to as loadings. The individual elements of the new variables (PCI and PCI) are derived from X and XI and are termed the jcorej. " The principal components scores for the chromium and nickel data are given in Table ... 

Analysis of diversity in response to polar and nonpolar vapors of all screened polymers was performed using PCA analysis followed by cluster analysis. The scores plots of the first three PCs (Fig. 5.12) illustrate the diversity of performance of all sensing polymers. The larger the distance between polymer data points, the larger the difference in the response pattern between the respective CdSe/polymer nanocomposites. To quantify this diversity, cluster analysis was further performed where the distances based on principal component scores were adjusted to unit variance.45 This distance measure, known as Mahalanobis distance, accounts for the different amount of variation in different directions. An example of such difference is shown in Fig. 5.12a, b, where the distance between polymer 4 and other polymers is much larger on the plot of PC 1 vs. PC 2 when compared with the plot of PC 2 vs. PC 3. 

Fig. 5.11 Multivariate analysis (principal components scores plots of the first three principal components) of the response of the two-size CdSe nanocrystals sensor film (a) PC 1 vs. PC 2 and (B) PC 1 vs. PC 3. Regions numbered 1 and 2 are data points of dynamic response from replicate (n = 3) film exposures to methanol and toluene, respectively. Unlabeled data points result from times when the film was exposed to a blank (dry air). Reprinted with permission from Leach and Potyrailo26 copyright 2006 Materials Research Society...

The principal component score is a linear combination of the descriptors. 

Appendix 15A Tables of descriptors and principal component scores... 

This appendix contains descriptor data and the corresponding principal component scores of the classes of compounds described in the text. The score values are measures of the principal properties. 

---