Scores 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. 

Different product sorts (20-22,24-26) are marked in the score plot of spectral NIR measurements (Fig. 4). Sorts 23 and 31 are in separate classes outside the range of this plot. A gradient is seen in the plot, indicating the chemical differences among the sorts the %DE are increasing in the 20->25/26 direction while the opposite holds for the Ca-based gel strength measm-ements. 

Figure 4. Score plots from a PCA of NIR R spectral data (2nd derivative). Left PC 1 versus PC 3 shows a clear segregation of sorts 23 and 31. Right PC 1 versus PC 2 (without sorts 23 and 31) shows a more distinct sort gradient/classification than the one produced by the chemical data.

In Fig. 31.1a these scores are used as the coordinates of the four wind directions in 2-dimensional factor-space. From this so-called score plot one observes a large degree of association between the wind directions of 90, 180 and 270 degrees, while the one at 0 degrees stemds out from the others. 

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. 

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). 

These factor scores can be used for the construction of a score plot in which each object is represented as a point in the plane of the first two dominant factors. 

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. 

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. 

Fig. 34.5. Score plot (PC score vs PC2 score) of the mixture spectra given in Fig. 34.2.

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. 34.13. Score plot (ti vs 12) of the spectra given in Fig. 34.2 after normalisation. The points A and B are the purest spectra in the data set. The points A and B are the spectra at the boundaries of the non-negativity constraint.

Fig. 34.33. Simulated three-component system (a) spectra (b) elution profiles and (c) scores plot obtained by a global PC A with HELP.

The method also provides what is called a data-scope, which zooms in on a particular part of the data set. The functioning of the data-scope is illustrated with a simulated three-component system given in Figs. 34.33a and b. The scores plot (Fig. 34.33c) obtained by a global PCA in wavelength space shows the usual line structures. In this case the data-scope technique is applied to evaluate the purity of the up-slope and down-slope elution zones of the peak. Therefore, data-scope performs a local PCA on the up-slope and down-slope regions of the data. 

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.

Figure 36.5 shows the scores plots for PC2 v. PCI (A) and PC4 v. PC3 (B). Such plots are useful in indicating a possible clustering of samples in subsets or the presence of influential observations. Again, a spectrum with a spike may show up as an outlier for that sample in one of the scores plots. If outliers are indicated, one should try and identify the cause of the outlying behaviour. Only when a satisfactory explanation is found can the outlier be safely omitted. In practice, one will... 

Fig. 36.5. Score plot of the samples in PC space (a) scatterplot of PC2-scores vs. PCl-scores, (b) scatterplot of PC4-scores vs. PC3-scores.

Fig. 38.18. Score plot (PC2 v. PCI) based on non-centered PCA of Tl-curves from 9 panellists for a bitter (caffeine) solution.

Fig. 37. Score plot of batch polymer reactor data (numbers indicate batch numbers).

This interpretation of Fig. 38b leads to the conclusion that better insight into the variations can be obtained by separating the data into two time histories—the first covers time k = 1 to 53, and the second covers time k = 54 to 113 (all time units are in minutes). Incorporating this extended perspective, Fig. 39 provides score plots for two PC models where the score plots are now divided into these two main processing phases. Observe the... 

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