PCA loading

Figure 33-1 presents the plot of the PLS loadings. Paul and Susan each computed both loadings. Note that the first loading is indistinguishable to the eye from the first PCA loading (see our original column on this topic [1]). 

Figure 8.7 PCA loading plot of GC/MS data for the 53 samples of waterproofing material...

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... 

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). 

All PCA loading vectors are orthogonal to each other PCA is a rotation of the original orthogonal coordinate system resulting in a smaller number of axes. 

As the above example illustrates, PCA can be an effective exploratory tool. However, it can also be used as a predictive tool in a PAT context. A good example of this nsage is the case where one wishes to determine whether newly collected analyzer responses are normal or abnormal with respect to previously collected responses. An efficient way to perform snch analyses wonld be to construct a PCA model using the previously collected responses, and apply this model to any analyzer response (Xp) generated by a subse-qnently-collected sample. Such PCA model application involves hrst a mnltiplication of the response vector with the PCA loadings (P) to generate a set of PCA scores for the newly collected response ... 

Typical parameters are the means and standard deviations of each class in the specified space, and the number of samples used to define each class. However, some methods might require other parameters, such as the number of nearest neighbors (KNN), the set of PCA loadings associated with each class (SIMCA), or specific... 

Earlier it was mentioned, and demonstrated using the Fisher Iris example (Section 12.2.5), that the PCA scores (T) can be used to assess relationships between samples in a data set. Similarly, the PCA loadings (P) can be used to assess relationships between variables in a data set. For PCA, the first score vector and the first loading vector make up the first principal component (PC), which represents the most dominant source of variability in the original x data. Subsequent pairs of scores and loadings ([score vector 2, loading vector 2], [score vector 3, loading vector 3]...) correspond to the next most dominant sources of variability. 

It shonld be noted that Equations 12.21 and 12.22 refer to the and Q statistics for samples in the x data, not for variables. However, the and Q valnes for the variables in the x data are calculated in a very similar manner. The T values for x variables are calculated using the PCA loadings, rather than the scores ... 

The advantage of supervised classification is that it avoids a wavelength selection step or the interpretation of PCA loading. It also rapidly extracts several chemical components. These methods provide accurate results provided the sample spectra are similar to the reference spectra. In our case the powder agglomeration was heterogeneous and the layers had a high content of excipient, making it possible to apply supervised classification. 

For such outliers, detection and assessment can actually be accomplished using some of the modeling tools themselves.1,3 In this work, the use of PCA and PLS for outlier detection is discussed. Since the PCA method only operates on the X-data, it can be used to detect X-sample and X-variable outliers. The three entities in the PCA model that are most commonly used to detect such outliers are the estimated PCA scores (T), the estimated PCA loadings (P), and the estimated PCA residuals (E), which are calculated from the estimated PCA scores and loadings ... 

Similarly, the leverages for each of the M X-variables can be calculated using the PCA loadings ... 

Figure 13.2 Analysis of raspberries by NIR imaging (third PC image) showing a grading in the maturity (Berries i = low maturity ii = medium maturity iii = ripe), (a) The figure shows the third PC image and (b) is a plot of the raw spectrum and the third PCA loading. From Walloon Agricultural Research Centre, Belgium.

Calculate the first principal component of ZZT and compute the normalized weight vector w from the PCA loading vector p (the first eigenvector of ZZT with eigenvalue A). 

FIGURE 11.17 PCA loadings for raw mean-centered augmented data matrix from top to bottom, first to fifth (PCI to PC5) principal components. Compound names and abbreviations are as follows alachlor (ALA), atrazine (ATR), bentazone (BEN), biphenyl (BIF), 3-chlorophenol (3-CP), 4-chlorophenol (4-CP), (2,4-dichlorophenosy)acetic acid (2,4-D), dichloroprop (DCP), dimethoate (DIM), linuron (LIN), h-chloro-z-methyphenoxyacetic acid (MCPA), mecoprop (MEC), 4-chloro-3-methylphenol (MEP), metholachlor (MET), pen-tachlorophenol (PCP), simazine (SIM), (2,4,5-trichlorophenoxy)acetic acid (2,4,5-T), tribu-tylphosphate (TBP), 2,4,6-trichlorophenol (TCP). 

The ethanol concentration in the medium of a Saccharomyces cerevisiae cultivation can be monitored from its content in the gas phase by directly recording the current from a chemical MOS sensor [28]. The accuracy of such a measurement was significantly improved by using an electronic nose with five sensors in the array and recognizing the response pattern with ANN [29, 30]. The sensors were a combination of MOS and MOSFET sensors selected from a PCA loading plot. Data sets from three cultivations were used to train the ANN. When the trained net was applied on new cultivations the ethanol was predicted with a mean square error (RMSE) of 4.6% compared to the off-line determined ethanol (Fig. 6). With only one sensor the RMSE was 18%. 

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... 

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