Cross-product matrix

From the 4x2 matrix X of our previous illustration we already derived V and from the eigenvalue decomposition of the 2x2 cross-product matrix X X ... 

In a similar way we can derive the eigenvalue decomposition of the corresponding 4x4 cross-product matrix XX ... 

A special form of cross-product matrix is the variance-covariance matrix (or covariance matrix for short) Cp, which is based on the column-centered matrix Yp derived from an original matrix X ... 

In the case where a equals 1 we can reconstruct the diagonalized cross-product matrix A ... 

A key operation in the power algorithm is the calculation of the deflated cross-product matrix which is independent of the contribution by the first eigenvector. This is achieved by means of the instmction ... 

The second critical fact that comes from equation 70-20 can be seen when you look at the Chemometric cross-product matrices used for calibrations (least-squares regression, for example, as we discussed in [1]). What is this cross-product matrix that is often so blithely written in matrix notation as ATA as we saw in our previous chapter Let us write one out (for a two-variable case like the one we are considering) and see ... 

And note a fine point we have deliberately ignored until now that in equation 70-20 the (Statistical) cross-product term was multiplied by two. This translates into the two appearances of that term in the (chemometrics) cross-product matrix. 

This is where we see the convergence of Statistics and Chemometrics. The cross-product matrix, which appears so often in Chemometric calculations and is so casually used in Chemometrics, thus has a very close and fundamental connection to what is one of the most basic operations of Statistics, much though some Chemometricians try to deny any connection. That relationship is that the sums of squares and cross-products in the (as per the Chemometric development of equation 70-10) cross-product matrix equals the sum of squares of the original data (as per the Statistics of equation 70-20). These relationships are not approximations, and not within statistical variation , but, as we have shown, are mathematically (algebraically) exact quantities. 

B60 IF EI=1 THEN LPRINT " REQUIRED THRESHOLD NOT ATTAINED" LPRINT iLPRINT 4B62 IF ES=1 THEN LPRINT " SINGULAR CROSS PRODUCT MATRIX" iLPRINT iLPRINT 4864 FOR 1=1 TO NT... 

PRINCIPAL COMPONENT ANALYSIS OF NORMED CROSS PRODUCT MATRIX... 

Overparameterization and frequently its sources are revealed by an eigenvalue-eigenvector analysis. In the module 1445 the matrix JT( )WJ(jB) is investigated. We call it normalized cross product matrix, because the partial derivatives are computed with respect to the normalized parameters... 

In this way, once the results for any given LS state have been obtained via the operator equivalent technique, those for any of the J sub-levels thereof may easily be derived. This procedure could of course be extended to evaluate matrix elements between states of different J, arising from the same LS state, but for calculations, whether in the LSMlMs) or the LSJMj) basis, in which it is required to incorporate mixing of different LS states the resulting cross-product matrix elements cannot be found by the operator equivalent method but must be determined directly from the wave functions. 

Another parametric routine implements a discriminant function by the method commonly called linear discriminant function analysis. It is nearly identical to the linear Bayesian discriminant, except that instead of using the covariance matrix, the sum of cross-products matrix is used. Results obtained with the routine are ordinarily very similar to those obtained using the linear Bayes routine. The routine implemented as LDFA is a highly modified version of program BMD04M taken from the Biomedical Computer Programs Package (47). 

The mean-centred spectral and concentration data are presented in Table 6.17 and the cross-product matrix in Table 6.18. The eigenvalues and eigenvectors of this matrix are shown in Table 6.19. Table 6.20 provides the matrix of principal components and the correlation of each with the tryptophan concentration. 

Table 6.18 Cross-product matrix (Yq.Y ) of data from Table 6.17...

Table 6.19 Decomposition of the cross —product matrix (Table 6.18) provides a matrix of eigenvectors and a vector of eigenvalues...

The leverage expressed by an object provides information on its importance in contributing to the calibration model by relating the position of its independent variables relative to others. It is related to the Mahalanobis distance and is derived from the cross product matrix of factors used in the calibration model. 

Q-mode factor analysis A factor analysis method based on eigen analysis of the cross-product matrix. 

The principal components are the eigenvectors of the sum of cross-products matrix we shall neither prove this result nor discuss how eigenvectors are obtained. The former requires more advanced mathematics than is appropriate for our level of discussion the latter is somewhat farther removed from NIRA than would warrant discussion here. Both topics are well-described elsewhere, the former in texts of mathematical statistics [10] and the latter in texts about numerical analysis [12]. 

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