Factor loadings plot

Figure 2 Factor loadings plot of the first three factors computed from a set of 58 calculated chemical descriptors.

It can be seen from the figure that a group of size descriptors, shown in bold on the left, is clustered just as was shown in the factor loadings plot in Figure 2. 

Analysis ofLoading Factor for Each Principle Component Facilitates Identification of Responsible Molecules Which Differentiate Control and Diseased Samples As a next step, an analysis of the factor loading plot would identify peaks that were differentially expressed between regions. Since a component score defined for each spectrum is a sum of the value of the factor loading value, multiplied by peak intensity, when numbers (= m) of mass peaks were used in the analysis, the component score will be ... 

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.

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. 3. Biplots of log-ratio factor loading for Aegean Sea sediments. Upper plot, complete dataset lower plot, offshore samples. Percentages indicate variance accounted for by factor.

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. 

Two factors characterized most of the waters sampled in the monitoring program. The factor loadings for Factor one indicate that the following chemical species participate in correlated behavior that permits the separations and distinctions described above alkalinity, bicarbonate, B, Cl, conductance, F, Li, Mo, and Na. To simplify discussions in the plots shown earlier this group of species was called the salinity factor. Specific conductance in natural waters usually correlates well with hardness and not as well with bicarbonate, but the current study finds specific conductance closely related to bicarbonate and unrelated to hardness (Ca, Mg, sulfate). This indicates that the ions responsible for increased conductance are probably not calcium or magnesium, rather they are more likely sodium, fluoride, and chloride. 

Loadings Plot (Model and Variable Diagnostic) Loadings 1-4 are displayed in Figure 4.67. The first three have a definite nonrandom behavior compared with the fourth. There is a limited amount of structure in loading four, indicating a model of size 3 or 4 factors. 

Loadings Plot (Model and Sample Diag io tiL) The iouding.s can he used to help determine the optimal number of factors to consider for the model. For spectroscopic and chromatographic data, the point at which the loading displays random behavior can indicate the maximum number to consider. Numerical evaluation of the randomness of the loadings has been proposed as a method for determination of the rank of a data matrix for spectroscopic data... 

With the molecular descriptors as the X-block, and the senso scores for sweet as the Y-block, PLS was used to calculate a predictive model using the Unscrambler program version 3.1 (CAMO A/S, Jarleveien 4, N-7041 Trondheim, Norway). When the full set of 17 phenols was us, optimal prediction of sweet odour was shown with 1 factor. Loadings of variables and scores of compounds on the first two factors are shown in Fig es 1 and 2 respectively. Figure 3 shows predicted sweet odour score plotted against that provid by the sensory panel. Vanillin, with a sensory score of 3.3, was an obvious outlier in this set, and so the model was recalculated without it. Again 1 factor was r uired for optimal prediction, shown in Figure 4. 

Fig. 15.5. Factor analysis results for the C-H stretching region (2800-3050 cm 1 region) in human skin and in cultured skin model (Epiderm ). Data from human skin (8 x 12 pixels) and cultured skin (7 x 12 pixels) have been concatenated. Pixels marked with x s were excluded from the analysis, a Factor loadings for the methylene stretching region. The dashed vertical line marks 2876 cm-1 and emphasizes the shift in frequency between factors 1 and 2. b Score plots for factor 1 are depicted for human skin in the left set of 8 X 12 pixels and for cultured skin in the right set of 7 x 12 pixels, c Score plots for factor 2 are depicted for human skin in the left set of 8 x 12 pixels and for cultured skin in the right set of 7 x 12 pixels...

The interpretation of the matrix of the factor loadings (Tab. 9-4) is verified unambiguously by the plot of the scores of the two factors with the highest eigenvalues (Fig. 9-11). 

Figure 3 Plots of (a) factor loadings and (b) factor scores of factors FI and F2 for the surface siliceous ooze samples from Wahine survey area near Hawaii (Calvert et al, 1978). (c) and (d) are similar plots for surface pelagic sediment samples from the equatorial Pacific (Calvert and Price, 1977).

Figure 12 Plots of factor loadings (a), factor scores (c and d), and Fe versus Co (b) for ferromanganese nodules from the equatorial Pacific. See the details in the text (source Usui and Mochizuki, 1984).

It is convenient to make the component matrices orthogonal, i.e. A A = B B = C C = I. This allows for an easy interpretation of the elements of the core-array and of the loadings by the loading plots. The sum of the squared elements of the core-array associated with the combination of certain factors then represents the amount of variation explained by that combination of factors in the different modes. If X (/ x / x K) is the three-way array modeled by a (P,Q,R) Tucker3 model2 and if X represents the fitted part of X, then the following holds [Kroonenberg 1984],... 

Fig. 3.75. Plot of the factor loadings of the variables on three PCs (Zhang et at, 2009) (With permission from Elsevier s Copyright Clearance Center)...

---