Principal scores

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... 

PCR is a combination of PCA and MLR, which are described in Sections 9.4.4 and 9.4.3 respectively. First, a principal component analysis is carried out which yields a loading matrix P and a scores matrix T as described in Section 9.4.4. For the ensuing MLR only PCA scores are used for modeling Y The PCA scores are inherently imcorrelated, so they can be employed directly for MLR. A more detailed description of PCR is given in Ref. [5. ... 

The procedure is as follows first, the principal components for X and Yare calculated separately (cf. Section 9.4.4). The scores of the matrix X are then used for a regression model to predict the scores of Y, which can then be used to predict Y. 

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.

Using D as input we apply principal coordinates analysis (PCoA) which we discussed in the previous section. This produces the nxn factor score matrix S. The next step is to define a variable point along they th coordinate axis, by means of the coefficient kj and to compute its distance d kj) from all n row-points ... 

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.

Fig. 35.3. Scatter plot of 16 olive oils scored by two sensory panels (Dutch panel lower case British panel upper case). The combined data are shown after Procrustes matching and projection onto the principal plane of the average configuration.

The application of principal components regression (PCR) to multivariate calibration introduces a new element, viz. data compression through the construction of a small set of new orthogonal components or factors. Henceforth, we will mainly use the term factor rather than component in order to avoid confusion with the chemical components of a mixture. The factors play an intermediary role as regressors in the calibration process. In PCR the factors are obtained as the principal components (PCs) from a principal component analysis (PC A) of the predictor data, i.e. the calibration spectra S (nxp). In Chapters 17 and 31 we saw that any data matrix can be decomposed ( factored ) into a product of (object) score vectors T(nxr) and (variable) loadings P(pxr). The number of columns in T and P is equal to the rank r of the matrix S, usually the smaller of n or p. It is customary and advisable to do this factoring on the data after columncentering. This allows one to write the mean-centered spectra Sq as ... 

The suffix in T (nxA) and P (< xA) indicates that only the first A columns of T and P are used, A being much smaller than n and q. In principal component regression we use the PC scores as regressors for the concentrations. Thus, we apply inverse calibration of the property of interest on the selected set of factor scores ... 

Fig. 35. Scores of Principal Component 1 vs scores of Principal Component 2 PCA model.

FIGURE 23.5 Effect of feeding captive male ring-necked pheasant (Ph. colchicus) young a high- or low-protein feed for the first three weeks of life on the expression of wattle coloration (mean+SE) at 20 (open circles) and 40 (filled circles) weeks of age. Coloration was determined using a principal components analysis (PCA) of tristimulus scores (hue, saturation, and brightness) obtained with a Colortron II reflectance spectrophotometer. 

Probability that the analyte A is present in the test sample Conditional probability probability of an event B on the condition that another event A occurs Probability that the analyte A is present in the test sample if a test result T is positive Score matrix (of principal component analysis)... 

The complete principal component decomposition of the data matrix X into a score matrix P and a loading matrix P is given by... 

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