PCA scores

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 selection of relevant effects for the MLR in PCR can be quite a complex task. A straightforward approach is to take those PCA scores which have a variance above a certain threshold. By varying the number of PCA components used, the... 

Fig. 17.5. PCA score plot for the example on metabolic stability. Filled points represent compounds with metabolic stability <20% open points refer to compounds with metabolic stability >90%. (See text for explanation.)...

Figure 9.2 PCA score plot of amino acidic profiles obtained in the GC/MS analysis of samples from the collection of paint reference materials of Opificio delle Pietre Dure (+), containing egg, casein and animal glue as binders, and of samples from the OL17bis series (x ) from the Leonetto Tintori Collection [ 10]...

Figure 3 shows the PCA score plot of the same data of figure 2 after the application of equation 4. The application of linear normalization to an array of linear sensors should produce, on the PCA score plot, one point for each compound, independent of its concentration, and achieve the highest possible recognition. Deviations from ideal behaviour, as shown in figure 3, are due to the presence of measurement errors, and to the non-linear relationship between sensor response and concentration. 

Note i, Object number t and are the PCA scores of PCI and PC2, respectively x, mean v, variance v%, variance in percent of total variance. 

The PCA scores have a very powerful mathematical property. They are orthogonal to each other, and since the scores are usually centered, any two score vectors are uncorrelated, resulting in a zero correlation coefficient. No other rotation of the coordinate system except PCA has this property. 

The A-matrix can be reconstructed from the PCA scores, T. Usually, only a few PCs are used (the maximum number is the minimum of n and m), corresponding to the main structure of the data. This results in an approximated A-matrix with reduced noise (Figure 3.3). If all possible PCs would be used, the error (residual) matrix E would be zero. 

FIGURE 3.3 Approximate reconstruction, Aappr, of the A-matrix from PCA scores T and the loading matrix P using a components E is the error (residual) matrix, see Equation 3.7. 

If the PCA scores are used in subsequent methods as uncorrelated new variables, the optimum number of PCs can be estimated by several techniques. The strategies applied use different criteria and usually give different solutions. Basics are the variances of the PCA scores, for instance, plotted versus the PC number (Figure 3.5, left). According to the definition, the PCI must have the largest variance, and the variances decrease with increasing PC number. For many data sets, the plot shows a steep descent after a few components because most of the variance is covered by the first components. In the example, one may conclude that the data structure is mainly influenced by two driving factors, represented by PCI and PC2. The other... 

FIGURE 3.5 Scree plot for an artificial data set with eight variables, v, variance of PCA scores (percent of total variance) v climul. cumulative variance of PCA scores. 

The cumulative variance, vCumul of the PCA scores shows how much of the total variance is preserved by a set of PCA components (Figure 3.5, right). As a rule of thumb, the number of considered PCA components should explain at least 80%, eventually 90% of the total variance. 

Pearson s correlation coefficient of different robust PCA scores is usually not zero. 

For mean-centered X the matrix To has size nxm and contains the PCA scores normalized to a length of 1. S is a diagonal matrix of size mxm containing the so-called singular values in its diagonal which are equal to the standard deviations of the scores. PT is the transposed PCA loading matrix with size mxm. The PCA scores, T. as defined above are calculated by... 

FIGURE 3.26 Plot of the first and second PCA scores for original scaled data (left) and the ILR transformed data (right). The different symbols correspond to the samples of Vienna and Linz, respectively, and the symbol size is proportional to the temperature. For both data sets PCA is able to separate the samples from the two cities. Also clusters of different temperatures are visible. 

For comparison also a dendrogram (Figure 3.28) and a nonlinear mapping (NLM) (Figure 3.29) have been performed on the PAH data. Results from these methods show a clear separation of the samples from Linz and Vienna, but not much more details. The clusters in the NLM plots are very similar to the clusters in the PCA score plots. Thus, preserving the distances using two dimensions—the goal of... 

PCA transforms a data matrix X(n x m)—containing data for n objects with m variables—into a matrix of lower dimension T(n x a). In the matrix T each object is characterized by a relative small number, a, of PCA scores (PCs, latent variables). Score ti of the /th object xt is a linear combination of the vector components (variables) of vector x, and the vector components (loadings) of a PCA loading vector/ in other formulation the score is the result of a scalar product xj p. The score vector tk of PCA component k contains the scores for all n objects T is the score matrix for n objects and a components P is the corresponding loading matrix (see Figure 3.2). 

PCA scores and loadings have unique properties as follows ... 

PCA score 1 (PCI, first principal component) is the linear latent variable with the maximum possible variance. The direction of PC2 is orthogonal to the direction of PCI and again has maximum possible variance of the scores. Subsequent PCs follow this mle. 

Depending on the clustering method, cluster results can be displayed in the form of a dendrogram or in a PCA score plot (Section 3.8.2). Such graphic allows abetter visual impression about the relations between the variables and about the clustering structure. It is thus a suitable tool for selecting the variables. 

PCR is an alternative method to the much more used regression method PLS (Section 4.7). PCR is a strictly defined method and the model often gives a very similar performance as a PLS model. Usually PCR needs more components than PLS because no information ofy is used for the computation of the PCA scores this is not necessarily a disadvantage because more variance of X is considered and the model may gain stability. 

Often simple strategies for the selection of a good set of PCA scores (for PCR) are applied (a) selection of the first PCA scores which cover a certain percentage of the total variance of X (for instance, 99%) (b) selection of the PCA scores with maximum correlation to y. Application of PCR within R is easy for a given number of components. For an example and a comparison of PCR and PLS, see Section 4.9.1. 

FIGURE 6.5 PCA score plot (a) of n — 20 standard amino acid structures characterized by m — 8 binary descriptors (27.1% and 20.5% of the total variance preserved in PCI and PC2). In the lower plots (b) presence/absence of selected four substructures is indicated. 

Four pairs of structures with identical descriptors merge at a distance of zero. From the chemist s point of view clustering appears more satisfying than the linear projection method PCA (with only 47.6% of the total variance preserved by the first two PCA scores). A number of different clustering algorithms have been applied to the 20 standard amino acids by Willet (1987). 

FIGURE 6.25 Plots of the first two PCA scores obtained from the original data (Figure 6.19, left). The symbol sizes are proportional to the concentrations of 1,8-cineole (left) and fenchone (right). Since the values are quite different for the groups, these variables are useful for cluster interpretation. 

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