Score in PCA

Obtaining the PCA Scores. In PCA, the matrix of measured instrumental responses of I calibration samples R (/ x J) (often column-centred or autoscaled) is decomposed into the product of two smaller matrices ... 

Like the scores in PCA, the Fourier coefficients can be used as a new compressed representation of the data, and thus can be directly used with regression techniques to provide quantitative prediction models.30... 

Scatter plots in PCA have special properties because the scores are plotted on the base P, and the columns of P are orthonormal vectors. Hence, the scores in PCA are plotted on an orthonormal base. This means that Euclidean distances in the space of the original variables, apart from the projection step, are kept intact going to the scores in PCA. Stated otherwise, distances between two points in a score plot can be understood in terms of Euclidian distances in the space of the original variables. This is not the case for score plots in PARAFAC and Tucker models, because they are usually not expressed on an orthonormal base. This issue was studied by Kiers [2000], together with problems of differences in horizontal and vertical scales. The basic conclusion is that careful consideration should be given to the interpretation of scatter plots. This is illustrated in Example 8.3. 

Partial least squares (PLS) analysis allows the simultaneous investigation of the relationships between a multitude of activity data (F matrix) and a set of chemical descriptors (X matrix) through latent variables (Wold et aL, 1984 Geladi and Kowalski, 1986 Hellberg, 1986 Geladi and Tosato, 1990). The latent variables correspond to the component scores in PCA and the respective coefficients to the PCA loading vectors. The PLS model can also be applied when the number of (collinear) descriptors exceeds the number of compounds in the data set. The main difference between PCA and PLS concerns the criteria for extracting the principal components and the latent variables, respectively PCA is based on the maximum variance criterion, whereas PLS uses covariance with another set of variables (X matrix). 

Sample leverages are calculated from the factor scores in PCA/PCR and PLS models. It is a relatively simple calculation ... 

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. 

The decomposition in eqn (3.30) is general for PCR, PLS and other regression methods. These methods differ in the criterion (and the algorithm) used for calculating P and, hence, they characterise the samples by different scores T. In PCR, T and P are found from the PCA of the data matrix R. Both the NIPALS algorithm [3] and the singular-value decomposition (SVD) (much used, see Appendix) of R can be used to obtain the T and P used in PCA/PCR. In PLS, other algorithms are used to obtain T and P (see Chapter 4). 

The criterions have considered the distribution of the samples in PCA score plot, and 10 samples among outliers and those that represent most part of the analysed samples were selected. 

As can be seen from Figures 6.5 and 6.6, there are several similarities between PLS and PCA. For example, both methods make a linear model of the data table X by means of a score vector, t (one score for each object), and a loading vector, p, which measures the importance of the variables. However, in PCA, neither t nor p is influenced (computationally) by anything but the variation in the measurements. Hence, if it is attempted to relate the measurements X to some external event (for example, drug treatment) via the PC t-scores, it must be realised that, unless this external event is a sufficiently large... 

The loading and scores for PCA can be generated by singular value decomposition (SVD). Instead of expressing the matrix containing the mixture spectra, A, as a product of two matrices as in Equation (4.4), SVD expresses it as a product of three matrices... 

In the preceding description of the Mahalanobis distance, the number of coordinates in the distance metric is equal to the number of spectral frequencies. As discussed earlier in the section on principal component analysis, the intensities at many frequencies are dependent, and by using the full spectrum, we fit the noise in addition to the real information. In recent years, Mahalanobis distance has been defined with PCA or PLS scores instead of the spectral frequencies because these techniques eliminate or at least reduce most of the overfitting problem. The overall application of the Mahalanobis distance metric is the same except that the rt intensity values are replaced by the scores from PCA or PLS. An example of a Mahalanobis distance calculation on a set of Raman spectra for 25 carbohydrates is shown in Fig. 5-11. The 25 spectra were first subjected to PCA, and it was found that the first three principal components could account for most of the variance in the spectra. It was first assumed that all 25 spectra belonged to the same class because they were all carbohydrates. However, as shown in the three-dimensional plot in Fig. 5-11, the spectra can be clearly divided into three separate classes, with two of the spectra almost equal distance from each of the three classes. Most of the components in the upper left class in the two-dimensional plot were sugars however, some sugars were found in the other two classes. For unknowns, scores have to be calculated from the principal components and processed in the same way as the spectral intensities. 

A very useful method of discriminating between samples from different classes is to plot PCA or PLS scores in two or three dimensions. This is very similar to the Mahalanobis distance discussed earlier in Fig. 5-11, except that it is limited to two or three dimensions, and the Mahalanobis distance can be constructed for n dimensions. Score plots do provide a good visual understanding of the underlying differences between data from samples belonging to different classes. 

Figure 5-13 Score plots 1 = PET 2 = HDPE 3 = PVC 4 = LDPE 5 = PP 6 = PS. (a) All samples used in PCA. (b) Samples 2 and 4 only in PCA. (Reprinted with permission from Ref. 4.)...

Three-dimensional analogies to principal components are required. There are no direct analogies to scores and loadings as in PCA, so the components in each of the diree dimensions are often called weights . There are a number of methods available to tackle this problem. 

Principle components regression (PCR) is one of the supervised methods commonly employed to analyze NMR data. This method is typically used for developing a quantitative model. In simple terms, PCR can be thought of as PCA followed by a regression step. In PCR, the scores matrix (T) obtained in PCA (Section 3.1) is related to an external variable in a least squares sense. Recall that the data matrix can be reconstructed or estimated using a limited number of factors (/ffact), such that only the fc = Mfaet PCA loadings (l fc) are required to describe the data matrix. Eq. (15) can be reconstructed as... 

PRC Nelson, PA Taylor, and JF MacGregor. Missing data methods in PCA and PLS score calculations with incomplete observations. Chemometrics Intell. Lah. Sys., 35 45-65, 1996. 

Determine the geometric center of mean for active compounds per target in PCA score space. 

Fig. 4. Schematic representation of the decomposition of X into scores (t), loadings (p) and a residual as performed in PCA.

Two-dimensional plots (score and loading plots in PCA and for three-way analysis)... 

In PCA, because the data are often centered, scores are usually nicely centered around zero, meaning that the origin of the coordinate system is very often in the scatter plot. For... 

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