Residuals matrix

This reduces the problem to that of finding the eigenvector V2 associated to k from the residual matrix ... 

It can be shown that the second eigenvector V2 can also be computed directly from the original matrix A, rather than from the residual matrix A - v,, by solving the relation ... 

This follows from the orthogonality of the eigenvectors v, and V2. We have preferred the residual matrix because this approach is used in iterative algorithms for the calculation of eigenvectors, as is explained in Section 31.4. 

The residual matrix E contains the contributions of the remaining second factor to the concentrations of the trace elements at the various wind directions. 

Z is a coefficient which relates the concentration of the analyte in the unknown sample to the concentration in the calibration standard, where = bc. R is a residual matrix which contains the measurement error. Its rows represent null spectra. However, in the presence of other (interfering) compounds, the residual matrix R is not random, but contains structure. Therefore the rank of R is greater than zero. A PCA of R, after retaining the significant PCs, gives ... 

Liquid samples, other than those that are inherently liquid, can arise from the solid sample extraction techniques described above. As mentioned previously, sometimes a simple dilute-and-shoot approach can be utilized, i.e., add solvent to the sample and then inject directly into the instrument. Other times, evaporation of residual liquid can be utilized—then the sample is either directly injected, or if the sample is evaporated to dryness, a new solvent can be added. Often, however, the residual matrix causes interference and the following techniques can be employed for further sample cleanup. 

Y in equation (5.9) is a good representation of the original matrix Y, but not identical. There is a residual matrix of decreasing significance the more eigenvectors are used to compute Y. ... 

The A-matrix can be reconstructed from the PCA scores, T. Usually, only a few PCs are used (the maximum number is the minimum of n and m), corresponding to the main structure of the data. This results in an approximated A-matrix with reduced noise (Figure 3.3). If all possible PCs would be used, the error (residual) matrix E would be zero. 

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. 

After a PC has been calculated the information of this component is peeled off from the currently used X-matrix (Figure 3.12b). This process is called deflation, and is a projection of the object points on to a subspace which is orthogonal to p, the previously calculated loading vector. The obtained X-residual matrix Xres is then used as a new X-matrix for the calculation of the next PC. The process is stopped after the desired number of PCs is calculated or no further PCA components can be calculated because the elements in Xres are very small. 

Calculate the residual matrix of X. Stop if the elements of Xres are very small because no further PCA components are reasonable. 

From the residual matrix, the next PLS component is derived—again with maximum covariance between the scores and y. 

CalibraJwn Measurement Residual Plot (Model Diagnostic) After the pure specta are estimated (S), they are used with the original C matrix to generate esti es of the mixture spectra (R CS). These are then used to calculate a caUbration residual matrix which contains the portion of the mixture spectra that are not fit by the estimated pures (Equation 5.18). 

The residual matrix contains four parameters therefore, four II numbers result ... 

The dimensional matrix consists of a square core matrix and a residual matrix. 

Quantities of the square core matrix may eventually appear in all of the dimensionless numbers as fillers, whereas each element of the residual matrix will appear in only one dimensionless number. For this reason the residual matrix should be loaded with essential variables like the target quantity and the most important physical properties and pro-cess-related parameters. 

After the generation of the matrix of unity, the dimensionless numbers are created as follows Each element of the residual matrix forms the numerator of a fraction, while its denominator consists of the fillers from the matrix of unity with the exponents indicated in the residual matrix. 

The residual matrix consists of only two parameters, so only two pi numbers... 

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