Principal component analysis scores

Figure 6. Principal component analysis scores for the suljury attributes of the...

Fig. 1. Principal component analysis scores scatter plot of the FTIR data set in the spectral window of 3000-600 cm-i wavenumber of landrace maize flours of whole and degermed grain cultivated in the southern Brazil.

Fig. 6. Principal component analysis scores scatter plot of the UV-visible data set in the spectral window of 200 qm to 700 qm (450 data points) of propolis samples produced in the southern Brazil (Santa Catarina State). CRi, CR2, and FW refer to propolis samples originated from coastal (BG and FLN Counties) and far-west (C County) regions, respectively, of Santa Catarina State. The sample grouping of propolis with similar UV-visivel scanning profiles regarding their (poly)phenolic composition is detached in the PCl-i-/PC2-i-quadrant. PCI and PC2 resolved 96% of the total variability of the spectral data set.

Figure 10.13 Principal components analysis scores plots (a) using all three example variables (first two principal components discriminations of the varieties, particularly sample Le2, would be very difficult) (b) using the best two variables, unweighted w selected [equation (10.31) in text] (discrimination of the varieties is now possible using only the first principal component) (c) discarding the best variable, unweighted w selected (linear discrimination of the varieties would not appear to be possible from this chart). Details of the variables are given in Table 10.3.

The diagnosis of colorectal cancer has been the focus of several studies [232-234]. Researchers used the first and second overtone C-H stretching to discriminate between cancer and normal tissue. The use of linear discriminant analysis, artificial neural networks, and clustering analysis was compared and very similar results obtained. While the former results were performed on resected samples, Shao et al. implemented an endoscope-based detection method to identify in vivo hyperplastic and adenomatous polyps [235]. Using a simple linear discriminant analysis based on principal component analysis scores, very good diagnostic sensitivities and specificities were obtained. 

FIGU RE 11.9 Principal component analysis score plots from IM-MS data collected in positive and negative mode for rat lymph samples collected in hourly intervals before and after feeding. (Reprinted from Kaplan, K Dwivedi, P. Davidson, S. Yang, Q. Tso, P. Siems, W. Hill, H. H. Jr. Anal. Chem. 2009, 81, 7944-7953. Copyright 2009, American Chemical... 

Figure 17 Cluster analysis of 44 isomers with molecular formula C6H3CI3 by principal component analysis (score plot of the first and second principal component containing 41.2% and 18.8% of the total variance, respectively). The chemical structures have been characterized by 20 binary molecular descriptors. The common structural properties within each cluster are characterized by the maximum common substructure (MCS). [Reproduced from Ref. 133 with kind permission of Gesellschaft Deutscher Chemiker]...

Figure 6,6 First three principal component analysis scores (correlation) for the analysis of truffles taken from six regions in Italy (Langhe, Marche, Umbria, Lazio, Toscana and Molise). The results for truffles taken from Marche, Umbria and Toscana are particularly well separated. Reproduced with permission from [ 10]. Copyright 2007 John Wiley Sons, Ltd.

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 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 ... 

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)... 

In general, the evaluation of interlaboratory studies can be carried out in various ways (Danzer et al. [1991]). Apart from z-scores, multivariate data analysis (nonlinear mapping, principal component analysis) and information theory (see Sect. 9.2) have been applied. 

Musumarra et al. [43] identified miconazole and other drugs by principal components analysis of standardized thin-layer chromatographic data in four eluent systems. The eluents, ethylacetate methanol 30% ammonium hydroxide (85 10 15), cyclohexane-toluene-diethylamine (65 25 10), ethylacetate chloroform (50 50), and acetone with the plates dipped in potassium hydroxide solution, provided a two-component model that accounts for 73% of the total variance. The scores plot allowed the restriction of the range of inquiry to a few candidates. This result is of great practical significance in analytical toxicology, especially when account is taken of the cost, the time, the analytical instrumentation and the simplicity of the calculations required by the method. 

We now have enough information to find our Scores matrix and Loadings matrix. First of all the Loadings matrix is simply the right singular values matrix or the V matrix this matrix is referred to as the P matrix in principal components analysis terminology. The Scores matrix is calculated as... 

Note the Scores matrix is referred to as the T matrix in principal components analysis terminology. Let us look at what we have completed so far by showing the SVD calculations in MATLAB as illustrated in Table 22-1. 

Principal component analysis (PCA) of the soil physico-chemical or the antibiotic resistance data set was performed with the SPSS software. Before PCA, the row MPN values were log-ratio transformed (ter Braak and Smilauer 1998) each MPN was logio -transformed, then, divided by sum of the 16 log-transformed values. Simple linear regression analysis between scores on PCs based on the antibiotic resistance profiles and the soil physico-chemical characteristics was also performed using the SPSS software. To find the PCs that significantly explain variation of SFI or SEF value, multiple regression analysis between SFI or SEF values and PC scores was also performed using the SPSS software. The stepwise method at the default criteria (p=0.05 for inclusion and 0.10 for removal) was chosen. 

For principal component analysis (PCA), the criterion is maximum variance of the scores, providing an optimal representation of the Euclidean distances between the objects. 

Principal Component Analysis (PCA) is the most popular technique of multivariate analysis used in environmental chemistry and toxicology [313-316]. Both PCA and factor analysis (FA) aim to reduce the dimensionality of a set of data but the approaches to do so are different for the two techniques. Each provides a different insight into the data structure, with PCA concentrating on explaining the diagonal elements of the covariance matrix, while FA the off-diagonal elements [313, 316-319]. Theoretically, PCA corresponds to a mathematical decomposition of the descriptor matrix,X, into means (xk), scores (fia), loadings (pak), and residuals (eik), which can be expressed as... 

As in many such problems, some form of pretreatment of the data is warranted. In all applications discussed here, the analytical data either have been untreated or have been normalized to relative concentration of each peak in the sample. Quality Assurance. Principal components analysis can be used to detect large sample differences that may be due to instrument error, noise, etc. This is illustrated by using samples 17-20 in Appendix I (Figure 6). These samples are replicate assays of a 1 1 1 1 mixture of the standard Aroclors. Fitting these data for the four samples to a 2-component model and plotting the two first principal components (Theta 1 and Theta 2 [scores] in... 

Then using these 91 peaks only, the original data set was reexamined by principal components analysis. Eigenvalues greater than one were plotted to determine how many factors should be retained. After variraax rotation, the factor scores were plotted and interpreted. 

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