Principal score plot

The coordinate of an object when projected onto an axis given by a principal component is called its score. Scores arc usually denoted by Tl, T2,. ... Figure 9-7 is a sketch of a score plot the points are the objects in the coordinate system... 

As described above, PCA can be used for similarity detection The score plot of two principal components can be used to indicate which objects are similar. 

Initially, the first two principal components were calculated. This yielded the principal components which are given in Figure 9-9 (left) and plotted in Figure 9-9 (right). The score plot shows which mineral water samples have similar mineral concentrations and which are quite different. For e3oimple, the mineral waters 6 and 7 are similar whUe 4 and 7 are rather dissimilar. 

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.

The score matrix T gives the location of the spectra in the space defined by the two principal components. Figure 34.5 shows a scores plot thus obtained with a clear structure (curve). The cause of this structure is explained in Section 34.2.1. 

As explained before, the scores of the spectra can be plotted in the space defined by the two principal components of the data matrix. The appearance of the scores plot depends on the way the rows (spectra) and the columns have been normalized. If the spectra are not normalized, all spectra are situated in a plane (see Fig. 34.5). From the origin two straight lines depart, which are connected by a curved line. We have already explained that the straight line segments correspond with the pure spectra which are located in the wings of the elution bands (selective retention time... 

Fig. 34.34. The three first principal components obtained by a local PCA (a) zero component region, (b) up-slope selective region, (c) down-slope selective region (d) three-component region. The spectra included in the local PCA are indicated in the score plot and in the chromatogram.

Two significant principal components were extracted. The score plot of the two principal components is shown in Fig. 17.5, where the compounds are color-coded according to their metabolic stability (filled points represent unstable compounds,... 

Musumarra et al. [43] identified miconazole and other drugs by principal components analysis of standardized thin-layer chromatographic data in four eluent systems. The eluents, ethylacetate methanol 30% ammonium hydroxide (85 10 15), cyclohexane-toluene-diethylamine (65 25 10), ethylacetate chloroform (50 50), and acetone with the plates dipped in potassium hydroxide solution, provided a two-component model that accounts for 73% of the total variance. The scores plot allowed the restriction of the range of inquiry to a few candidates. This result is of great practical significance in analytical toxicology, especially when account is taken of the cost, the time, the analytical instrumentation and the simplicity of the calculations required by the method. 

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.

A cluster analysis of the amino acid structures by PCA of the A -matrix is shown in Figure 6.5a note that PCA optimally represents the Euclidean distances. The score plot for the first two principal components (preserving 27.1% and 20.5% of the total variance) shows some clustering of similar structures. Four structure pairs have identical variables 1 (Ala) and 8 (Gly), 5 (Cys) and 13 (Met), 10 (He) and 11 (Leu), and 16 (Ser) and 17 (Thr). Objects with identical variables of course have identical scores, but for a better visibility the pairs have been artificially... 

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. 

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... 

Pattern recognition studies on complex data from capillary gas chromatographic analyses were conducted with a series of microcomputer programs based on principal components (SIMCA-3B). Principal components sample score plots provide a means to assess sample similarity. The behavior of analytes in samples can be evaluated from variable loading plots derived from principal components calculations. A complex data set was derived from isomer specific polychlorinated biphenyl (PCBS) analyses of samples from laboratory and field studies. 

The principal components model of the Aroclor seunples (Table i) preserves greater than 95% of the sample variance of the entire data set. From the 3-D seunple score plot (Figure 3) one can make these observations PCB mixtures of two Aroclors form a straight line three Aroclor mixtures form a plane and that possible mixtures of the four Aroclors are bounded by the intersection of the four planes. Samples not bounded by or inside the volume formed by the intersection of the four planes may... 

The fitness function of the pattern recognition GA scores the principal component plots and thereby identifies a set of features that optimize the separation of the classes in a plot of the two or three largest principal components of the data. To facilitate the tracking and scoring of the principal component plots, class and sample weights, which are an integral part of the fitness function, are computed ... 

Scores Pla (Sample Diagnostic) The scores are the coordinates of the samples in the new coordinate system where the axes are defined by the principal components. These new axes are used to view the relevant variation in the data see m a smaller number of dimensions. The plot reveals how the samples arc rela d to each other given the measurements that have been made. Samples that are close to each other on a given score plot are similar with respect to the original measurements provided the plot displays a sufficient amount of the total variation. This mathematical proximity translates to chemical similaritySmeaningful measurements have been made. 

Scores Plot (Sample Diagnostic) Figure 4.39 shows the plane spanned by principal components i and 2, representing 97% of the variation in the data. The axis labels display the amount of variation that each PC describes. There is a tigjit cluster of points from which two lines of points extend. PCI discriminates the tight cluster from the lines while PC2 primarily distinguishes the... 

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. 

The sensor responses were collected and elaborated by PCA performed on scaled data to achieve a partial visualisation of the set in a reduced dimension. The two first principal components represented 98% of the total variance and their score plot (Fig. 31.1) allowed a separation of the samples according to the storage conditions. Samples were distributed along PCI and PC2 according to the storage time and storage temperature, respectively. 

Fig. 31.1. PCA score plot of Crescenza cheese samples in the plane defined by the first two principal components ( ) samples stored at 8°C and (A) samples stored at 15°C.

The four first principal components represent 69% of the total variance. On examining the score plot (Fig. 31.4) in the area defined by the first two principal components (51.7% of the total variance), a clear... 

In Fig. 19.1, the electronic nose fingerprint of ripened (Asiago d Allevo) cheese samples of the winter period (26 samples) and of summer period (24 samples) is shown. On examining the score plot in the space defined by the first two principal components (98.0% of the total variability), the major part of the samples was found in an area of the plot close to the intersection of the axes, denoting that the samples have an equal gaseous... 

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.)...

Table 3.4 shows the loadings of the first, second and third principal component the others have very small variances. In the principal component plots in Figure 3.9 the samples are represented by different symbols related to the average temperature during the sampling time. The scores computed for the first principal component clearly distinguish the cold and warm season. 

An important application of PCA is classification and pattern recognition. This particular application of PCA is described in detail in Chapter 9. The fundamental idea behind this approach is that data vectors representing objects in a high-dimensional space can be efficiently projected into a low-dimensional space by PCA and viewed graphically as scatter plots of PC scores. Objects that are similar to each other will tend to cluster in the score plots, whereas objects that are dissimilar will tend to be far apart. By efficient, we mean the PCA model must capture a large fraction of the variance in the data set, say 70% or more, in the first few principal components. 

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