Loadings matrix

What UV-scaling does is to concentrate the relevant information into the same range for all the variables (or, at least, for those subjected to this method). Then, the loading matrix yielded by PCA will show the importance of the initial variables. 

Here W is diagonal matrix of singular values, is the transpose of the second re-sultant matrix, being actually the same as the loading matrix in PCA, and X is the matrix, which is applied for further modeling. 

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. ... 

Enantiomer separation factors (a values) for valine and phenylalanine as well as their esters of 5-10 for phenylalanine and 4-10 for valine have been shown at the 0.1-1 g ChiraLig scale. These a values vary as a function of solvent and other loading matrix factors (pH, salts, etc.). However, all of these cases show a values high enough to obtain reasonable enantiometric purity in less than or equal to three stages. The system with a value of = 6 for the valine methyl ester enantiomers has the ability to load the valine onto the resin in H,0 containing LiClO and also to... 

In the same way, we consider the special case where P = 1, which yields for the loading matrix L the expression ... 

The columns of the loading matrix L contain the principal components of X in column-space S . 

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 ... 

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. 

Figure 22 Insulin release rate (normalized to mg/hr per 160 mg device) from insulin-loaded matrix of polyethyloxazolin-poly(methacrylic acid) complex with the application of step-function electric current in 0.9% saline solution (mean from three measurements). ( ) Current on (5mA) (O) current off. (From Ref. 47.)...

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)... 

V V = 0.3576 -0.9339 0.9339 0.3576 Display the V matrix or the right singular values (RSV) matrix (Note this is also known as the P matrix or Loadings matrix)... 

According to the magnitude of the retained variance and the contribution that the original variables make to each component, the environmental meaning of the identified components can be deduced, and the approximate level of error contained in the experimental data can also be determined. In this context, the displaying of scores (matrix X) and loadings (matrix YT) obtained from PCA decomposition of the original data matrix D are extremely useful. 

In this equation, whereas the same loading matrix (YT matrix) is common for the different individual data matrices Dt, k = 1, 2, 3, 4, four different score matrices Xjt, k = 1, 2, 3, 4 are considered to explain the variation in Daug. Since these four D. matrices have equal sizes (same number of rows or samples and of columns or variables) they can also be arranged in a three-way data cube, with the four data matrices in the different slabs of this cube. However, in the frame of the MCR-ALS method and of the general bilinear model in (10), it is preferable to consider them to be arranged in the column-wise augmented data matrix Daug. 

The loading matrix FA1/2 given by equation (4.4.26) can be found in Table 4.14. Lead isotopes have strong correlation coefficients on the first component. They are decoupled from Sr and Nd isotopes which strongly correlate and anticorrelate, respectively, with the second component. On a global scale, Pb isotopic variations in oceanic islands seem to be decoupled from Sr and Nd isotopic variations. 

T is often called the score matrix and L the loadings matrix. The relationship between decompositions (5.1) and (5.3) is... 

If several linear latent variables are calculated, the corresponding loading vectors are collected in a loading matrix B, and the scores form a score matrix V (Figure 2.16). 

All loading vectors are collected as columns in the loading matrix, P, and all score vectors in the score matrix, T (Figure 3.2). 

FIGURE 3.3 Approximate reconstruction, Aappr, of the A-matrix from PCA scores T and the loading matrix P using a components E is the error (residual) matrix, see Equation 3.7. 

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... 

The interpretation of the resulting factors is based on the loading matrix Pfa which are the coefficients for the linear combinations similar to PCA... 

---