Matrix score

The profits from using this approach are dear. Any neural network applied as a mapping device between independent variables and responses requires more computational time and resources than PCR or PLS. Therefore, an increase in the dimensionality of the input (characteristic) vector results in a significant increase in computation time. As our observations have shown, the same is not the case with PLS. Therefore, SVD as a data transformation technique enables one to apply as many molecular descriptors as are at one s disposal, but finally to use latent variables as an input vector of much lower dimensionality for training neural networks. Again, SVD concentrates most of the relevant information (very often about 95 %) in a few initial columns of die scores matrix. 

First, one can check whether a randomly compiled test set is within the modeling space, before employing it for PCA/PLS applications. Suppose one has calculated the scores matrix T and the loading matrix P with the help of a training set. Let z be the characteristic vector (that is, the set of independent variables) of an object in a test set. Then, we first must calculate the scores vector of the object (Eq. (14)). 

In matrix notation PCA approximates the data matrix X, which has n objects and m variables, by two smaller matrices the scores matrix T (n objects and d variables) and the loadings matrix P (d objects and m variables), where X = TPT... 

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 second line results from the first, because 0 and A are independent a priori, whereas the score of the alignment A depends on A, and the likelihood of the sequences, given the alignment, depends only on the scoring matrix 0. 

See Kepner and Tregoe (1981) or CCPS (1995a) for additional discussion, particularly on how potential negative consequences may impact the scoring matrix. 

In the special case where a = 1 we can express the score matrix S in the form ... 

A very special case arises when a + P equals 1. If we form the product of the score matrix S with the transpose of the loading matrix L, then we obtain the original measurement table X ... 

Fig. 31.2. Geometrical example of the duality of data space and the concept of a common factor space, (a) Representation of n rows (circles) of a data table X in a space Sf spanned by p columns. The pattern P" is shown in the form of an equiprobabi lity ellipse. The latent vectors V define the orientations of the principal axes of inertia of the row-pattern, (b) Representation of p columns (squares) of a data table X in a space y spanned by n rows. The pattern / is shown in the form of an equiprobability ellipse. The latent vectors U define the orientations of the principal axes of inertia of the column-pattern, (c) Result of rotation of the original column-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the score matrix S and the geometric representation is called a score plot, (d) Result of rotation of the original row-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the loading table L and the geometric representation is referred to as a loading plot, (e) Superposition of the score and loading plot into a biplot.

Factor scores for each of the n objects with respect to the r computed factors are then defined in the usual way by means of the nxr orthogonal factor score matrix S ... 

Using D as input we apply principal coordinates analysis (PCoA) which we discussed in the previous section. This produces the nxn factor score matrix S. The next step is to define a variable point along they th coordinate axis, by means of the coefficient kj and to compute its distance d kj) from all n row-points ... 

From the latent vectors and singular values one can compute the nxr generalized score matrix S and the pxr generalized loading matrix L. These matrices contain the coordinates of the rows and columns in the space spanned by the latent vectors ... 

Figure 32.8 shows the biplot constructed from the first two columns of the scores matrix S and from the loadings matrix L (Table 32.11). This biplot corresponds with the exponents a = 1 and p = 1 in the definition of scores and loadings (eq. (39.41)). It is meant to reconstruct distances between rows and between columns. The rows and columns are represented by circles and squares respectively. Circles are connected in the order of the consecutive time intervals. The horizontal and vertical axes of this biplot are in the direction of the first and second latent vectors which account respectively for 86 and 13% of the interaction between rows and columns. Only 1% of the interaction is in the direction perpendicular to the plane of the plot. The origin of the frame of coordinates is indicated... 

In Chapter 31 we stated that any data matrix can be decomposed into a product of two other matrices, the score and loading matrix. In some instances another decomposition is possible, e.g. into a product of a concentration matrix and a spectrum matrix. These two matrices have a physical meaning. In this chapter we explain how a loading or a score matrix can be transformed into matrices to which a physical meaning can be attributed. We introduce the subject with an example from environmental chemistry and one from liquid chromatography. 

The score matrix T gives the location of the spectra in the space defined by the two principal components. Figure 34.5 shows a scores plot thus obtained with a clear structure (curve). The cause of this structure is explained in Section 34.2.1. 

Having derived a solution for two-component systems, we could try and extend this solution to three-component systems. A PCA of a data set of spectra of three-component mixtures yields three significant eigenvectors and a score matrix with three scores for each spectrum. Therefore, the spectra are located in a three-dimensional space defined by the eigenvectors. For the same reason, explained for the two-component system, by normalization, the ternary spectra are found on a surface with one dimension less than the number of compounds, in this case, a plane. 

PCR is based on a PCA input data transformation that by definition is independent of the Y-data set. The approach to defining the X-Y relationship is therefore accomplished in two steps. The first is to perform PCA on the. Y-data, yielding a set of scores for each measurement vector. That is, if xk is the fcth vector of d measurements at a time k, then zk is the corresponding kth vector of scores. The score matrix Z is then regressed onto the Y data, generating the predictive model... 

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)... 

The complete principal component decomposition of the data matrix X into a score matrix P and a loading matrix P is given by... 

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... 

The data matrix A x the Loadings matrix V = Scores matrix T (22-2)... 

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. 

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