Score vector analysis

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 ... 

Partial least square (PLS) regression model describes the dependences between two variables blocks, e.g. sensor responses and time variables. Let the X matrix represent the sensor responses and the Y matrix represent time, the X and Y matrices could be approximated to few orthogonal score vectors, respectively. These components are then rotated in order to get as good a prediction of y variables as possible [25], Linear discriminant analysis (LDA) is among the most used classification techniques. The method maximises the variance between... 

The wa in equation (6) are the PLS loading weights. They are explained in the theory in references 53 - 62. Equation (7) shows how X is decomposed bilinearly (as in principal component analysis) with its own residual Epls A. T is the matrix with the score vectors as columns, P is the matrix having the PLS loadings as columns. Also the vectors of P and wa can be used to construct scatter plots. These can reveal the data structure of the variable space and relations between variables or groups of variables. Since PLS mainly looks for sources of variance, it is a very good dirty data technique. Random noise will not be decomposed into scores and loadings, and will be stored in the residual matrices (E and F), which contain only non-explained variance . 

Outliers due to real differences in data will also be detected by plotting the first two score vectors against each other. It depends on the specific problem whether or not they should be included in the final analysis. Nevertheless, they can be detected at an early stage of the invesitgation. 

In principal components analysis, the score vectors Zj are eigenvectors to XX , whereas in PLS the first score vector of the X block is an eigenvector to XX YY . 

A plot of the score vectors against each other, e.g. against tj, shows the positions of the projected object points in the plane spanned by the PLS vectors in ths X space, and a plot of U2 against Uj shows the same for the Y space. These plots are therefore similar to the score plots from principal components analysis. 

Fig. 4.5. MPCA analysis. Temporal patterns of Butyric acid and Isoamylacetate. Score vectors for direct stimulus (A,B), during inhibition with lectin (C,D), and after removal of the inhibitor (E,F).

The same types of graphic analysis for any choice of the matrix V are used. Here an example of the graphic analysis based on an industrial on-line data is presented. The figures illustrate results of static linear PLS model, in which NIR data fix>m an oil refinery is used to model density of the product The vectors in the algorithm are displayed graphically to illustrate the structure and variation in data. The basic plots are 1. ta versus it, The vectors (t,) decompose the data matrix X. Therefore the plots of t, versus tb show us the sample (time) variation in data. Fig 1 reveals that the process (samples) is changing with the time. Arrows in Fig 1 visualise the drift on 1.-4. PLS components. The dynamic behaviour can be clearly seen even on the first two score vectors. Therefore, it cannot be expected that the same model will be valid at the... 

Ta versus ri, The transformation vectors (Fj) are generated from Pa=Sra. They also satisfy da ta=Xra. Thus, these vectors tell us how the variables contribute to the analysis, and, how the covariance structure in S has been used. We can also multiply element-wise X and r, to see which variable contribute most to the score vectors. 

In PCR, a principal component analysis (PCA) is first made of the X matrix (properly transformed and scaled), giving as the result the score matrix T and the loading matrix P. Then in a second step a few of the first score vectors tg are used as predictor variables in a multiple linear regression with Y as the response matrix. In the case that the few first components of PCA indeed contain most of the information of X related to Y, PCR indeed works as well as PLS. This is often the case in spectroscopic data, and here PCR is an often used alternative. In more complicated applications, however, such as QSAR and process modeling, the first few principal components of X rarely contain a sufficient part of the relevant information, and PLS works much better than PCR. ... 

FIGURE 4.27 Canonical correlation analysis (CCA), x-scores are uncorrelated v-scores are uncorrelated pairs of x- and y-sores (for instance t and Ui) have maximum correlation loading vectors are in general not orthogonal. Score plots are connected projections of x- and y-space. 

It is, furthermore, possible to interpret the latent vectors t or u. The latent vectors have got scores for each object, as in factor analysis. These scores can be used to display the objects. Another possibility is to compute the correlation between original features and the latent vectors to assess the kind of interacting features for both data sets. 

In PLS, the response matrix X is decomposed in a fashion similar to principal component analysis, generating a matrix of scores, T, and loadings or factors, P. (These vectors can also be referred to as basis vectors.) A similar analysis is performed for Y, producing a matrix of scores, U, and loadings, Q. 

Perform PCA to give loadings and scores matrices T and P for the x data, then obtain a vector r for the c data using standard regression techniques. Note that these arrays will differ according to which sample is removed from the analysis. 

To compare aroma profiles of the 12 glycoside hydrolysate samples, a principal component (PC) analysis of the mean data was performed the attributes (plotted as vectors) and wine factor scores are plotted for the first two rotated components in Figure 2. The first component contrasted differences in intensity of the samples for the apple attribute compared to that of the tobacco and dried fig attributes. The second component contrasted chocolate and honey with the floral attribute. 

Figure 2. Principal component biplot of rotated components 1 and 2 for mean descriptive analysis ratings (n=14 judges x 2 reps). Vectors for the aroma attributes, and the scores for the fifteen samples are shown. Open symbols indicate juice samples, while closed symbols indicate skin extracts. For sample codes, see Table II.

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