Loadings plot

Figure 9-8. A loading plot. Summary of the relationships among the variables P1 (descriptors).

A second piece of important information obtained by PCA is the loadings, which are denoted by PI, P2, etc. They indicate which variables influence a model and how the variables are correlated In algebraic terms the loadings indicate how the variables are combined to build the scores. Figure 9-8 shows a loading plot each point is a feature of the data set, and features that are close in the plot are correlated. 

To investigate the variance structure in the raw physical/chemical data material a PCA was performed on the autoscaled Y-data. Figure 3 shows a loading plot of the Y-data as a function of the two first PC s describing together 57 % of the total variance. 

Fig. 31.1. (a) Score plot in which the distances between representations of rows (wind directions) are reproduced. The factor scaling coefficient a equals 1. Data are listed in Table 31.1. (b) Loading plot in which the distances between representations of columns (trace elements) are preserved. The factor scaling coefficient P equals 1. Data are defined in Table 31.1. 

These loadings have been used for the construction of the so-called loading plot in Fig. 31.1b which shows the positions of the three trace elements in 2-dimensional factor-space. The elements Na and Cl are clearly related, while Si takes a position of its own in this plot. 

Fig. 31.2. Geometrical example of the duality of data space and the concept of a common factor space, (a) Representation of n rows (circles) of a data table X in a space Sf spanned by p columns. The pattern P" is shown in the form of an equiprobabi lity ellipse. The latent vectors V define the orientations of the principal axes of inertia of the row-pattern, (b) Representation of p columns (squares) of a data table X in a space y spanned by n rows. The pattern / is shown in the form of an equiprobability ellipse. The latent vectors U define the orientations of the principal axes of inertia of the column-pattern, (c) Result of rotation of the original column-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the score matrix S and the geometric representation is called a score plot, (d) Result of rotation of the original row-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the loading table L and the geometric representation is referred to as a loading plot, (e) Superposition of the score and loading plot into a biplot.

Since U and V express one and the same set of latent vectors, one can superimpose the score plot and the loading plot into a single display as shown in Fig. 31,2e. Such a display was called a biplot (Section 17.4), as it represents two entities (rows and columns of X) into a single plot [10]. The biplot plays an important role in the graphic display of the results of PCA. A fundamental property of PCA is that it obviates the need for two dual data spaces and that instead of these it produces a single space of latent variables. 

Fig. 31.3. (a,b) Reproduction of distances D and angular distances 0 in a score plot (a = 1) or loading plot (p = 1) in the common factor-space (c,d) Unipolar axis through the representation of a row or column and through the origin 0 of space. Reproduction of the data X is obtained by perpendicular projection of the column- or row-pattern upon the unipolar axis (a + P = 1). (e,0 Bipolar axis through the representation of two rows or two columns. Reproduction of differences (contrasts) in the data X is obtained by perpendicular projection of the column- or row-pattern upon the bipolar axis (a + P = 1). 

Fig. 32.6. (a) Generalized score plot derived by correspondence factor analysis (CFA) from Table 32.4. The figure shows the distance of Triazolam from the origin, and the distance between Triazolam and Lorazepam. (b) Generalized loading plot derived by CFA from Table 32.4. The figure shows the distance of epilepsy from the origin, and the distance between epilepsy and anxiety. 

Fig. 32.7. CFA biplot resulting from the superposition of the score and loading plots of Figs. 32.6a and b. The coordinates of the products and the disorders are contained in Table 32.9.

By decomposing the HPLC data matrix of spectra shown in Fig. 34.2 according to eq. (34.4), a matrix V is obtained containing the two significant columns of V. Evidently the loading plots shown in Fig. 34.4 do not represent the two pure spectra, though each mixture spectrum can be represented as a linear combination of these two PCs. Therefore, these two PCs are called abstract spectra. Equations... 

Fig. 37.2. Principal components loading plot of 7 physicochemical substituent parameters, as obtained from the correlations in Table 37.5 [39,40]. The horizontal and vertical axes account for 46 and 31%, respectively, of the correlations. Most of the residual correlation is along the perpendicular to the plane of the diagram. The line segments define clusters of parameters that have been computed by means of cluster analysis.

Fig. 10.1. Loadings plot for the variables in the partial least squares (PLS) analysis. Abscissa first component ordinate second component.

Figure 8.7 PCA loading plot of GC/MS data for the 53 samples of waterproofing material...

PCA results are summarized in Fig. 5, which shows the loading plots characterizing the main contamination patterns in every analyzed data set and their explained... 

Fig. 5.5 Variation of the Tafel slope and the anode potential with Ru02 loading (plotted from the data in Ref. [18]). 

The duration of testing is not specified, but ISO 11403-1 [36] proposes that the loads should be 20%, 40%, 60% and 80% of the maximum load for the respective temperature and that strains should be tabulated after 1,10,100,1,000 and 10,000 h (10,000 h equals 13.7 months). This data will enable creep strain curves and an isochronous diagram to be prepared (load plotted against strain for each duration) with sufficient accuracy for design. 

A biplot combines a score plot and a loading plot. It contains points for the objects and points (or arrows) for the variables, and can be an instructive display of PC A results however, the number of objects and the number of variables should not be too high. An appropriate scaling of the scores and the loadings is necessary, and mean-centering of the variables is useful. The biplot gives information about clustering of objects and of variables (therefore bi ), and about relationships between objects and variables. 

The described projection method with scores and loadings holds for all linear methods, such as PCA, LDA, and PLS. These methods are capable to compress many variables to a few ones and allow an insight into the data structure by two-dimensional scatter plots. Additional score plots (and corresponding loading plots) provide views from different, often orthogonal, directions. 

Projection of the variable space on to a plane (defined by two loading vectors) is a powerful approach to visualize the distribution of the objects in the variable space, which means detection of clusters and eventually outliers. Another aim of projection can be an optimal separation of given classes of objects. The score plot shows the result of a projection to a plane it is a scatter plot with a point for each object. The corresponding loading plot (with a point for each variable) indicates the relevance of the variables for certain object clusters. 

Factor analysis with the extraction of two factors and varimax rotation can be carried out in R as described below. The factor scores are estimated with a regression method. The resulting score and loading plots can be used as in PCA. 

FIGURE 3.21 Three-way methods Unfolding Tucker3 and PARAFAC. Resulting loading plots and score plot are shown for using the first two components. 

FIGURE 3.27 Loading plot of PCI and PC2 computed from the original scaled data, corresponding to the score plot in Figure 3.26, left. The variable numbers 1-14 are as given in Table 3.5. PAHs in the left upper part are characteristic for Vienna, in the right part for Linz. 

In the principal components plots presented in this paper, the number plotted corresponds to the sample identification number given in the appendix. If more than one sample has the same locus in the score (Theta s) or loading plots (Beta s), the letter M is plotted. The values for the sample coordinates in the principal components plots can be listed by the SIMCA-3B program. 

---