Component calibration coefficients

Principal component regression (PCR) is the algorithm by which PCA is used for quantitative analysis and involves a two-step process. The first is to decompose a calibration data set with PCA to calculate all the significant principal components, and the second step is to regress the concentrations against the scores to produce the component calibration coefficients. Generally, the ILS model is preferred, as it does not require knowledge of the complete composition of all the spectra. Therefore, if we use the ILS model from Eq. 9.16 but rewrite it for scores, S, instead of absorbances. A, we have... 

We now use CLS to generate calibrations from our two training sets, A1 and A2. For each training set, we will get matrices, Kl and K2, respectively, containing the best least-squares estimates for the spectra of pure components 1-3, and matrices, Kl i and K2cnl, each containing 3 rows of calibration coefficients, one row for each of the 3 components we will predict. First, we will compare the estimated pure component spectra to the actual spectra we started with. Next, we will see how well each calibration matrix is able to predict the concentrations of the samples that were used to generate that calibration. Finally, we will see how well each calibration is able to predict the... 

Figure 21. Plots of the CLS calibration coefficients calculated for each component with each training set.

We perform CLS on A6 to produce 2 calibrations. K6 and K6, are the matrices holding the pure component spectra and calibration coefficients, respectively, for CLS with zero intercepts. K6a and K6aMl are the corresponding matrices for CLS with nonzero intercepts. 

The calibration matrix, Fc t has exactly the same format as K, the calibration matrix for CLS. It has one row for each component being predicted. Each row has one calibration coefficient for each wavelength in the spectrum. We can now use F , to predict the concentrations in an unknown sample from its measured spectrum. First, we place the spectrum into a new absorbance matrix, A,. We can now use equation [64] to produce a new concentration matrix, CHah, containing the predicted concentration values for the unknown sample. 

Each column of the matrix P cXn) contains calibration coefficients for each component in a mixture, at each wavelength. 

Calibration of a single chemical constituent by means of full spectrum, or calibration of several components by means of mixture calibration techniques. In mathematical terms, these are problems of parameter estimation where the parameters represent the calibration coefficients. 

A disadvantage of this calibration method is the fact that the calibration coefficients (elements of the P matrix) have no physical meaning, since they do not reflect the spectra of the individual components. The usual assumptions about errorless independent variables (here, the absorbances) and error-prone dependent variables (here, concentrations) are not valid. Therefore, if this method of inverse calibration is used in coimection with OLS for estimating the P coefficients, there is only a slight advantage over the classical /C-matrix approach, due to the fact that a second matrix inversion is avoided. However, in coimection with more soft modeling methods, such as PCR or PLS, the inverse calibration approach is one of the most frequently used calibration tools. 

Thus, the relative calibration coefficient Cs of equivalent sensitivity is obtained for each sensor. In a few cases, absolutely calibrated sensors are available. In this respect, the moment tensor analysis to determine the relative tensor components is preferable in practical applications. 

Then, the distance R and its direction vector r are determined. The amplitudes of the first motions P2 in Fig. 8.7 at more than 6 channels are substituted into Eq. 8.10, and thus the components of the moment tensor are determined from a series of algebraic equations. Since the SiGMA code requires only relative values of the moment tensor components, the relative calibration coefficient Cs of AE sensors is sufficient. The code is already implemented in the AE device. 

To perform prediction using this approach requires retaining not only the calibration coefficients but also the lists of values representing the principal components that were used to create the calibration, so that when unknowns are measured in the future the scores can be calculated. This is necessary because it is the scores that the calibration coefficients operate on. 

The approach used is similar to that of Honigs et al. [18], and which is further discussed in Chapter 16 of this book. The variable in Equation (8.4) of Reference 17 is analogous to the inverse of the coefficients of the Principal Component calibration, expressed in Equation (8.31). This is perhaps most clearly indicated from dimensional analysis as pointed out in Reference 16, the variable Cab has units of absorbance/concentration, while the units of the coefficients of any quantitative... 

It is also interesting to examine the actual regression coefficients that each calibration produces. Recall that we get one row in the calibration matrix, K, for each component that is predicted. Each row contains one coefficient for each wavelength. Thus, we can conveniently plot each row of as if it were a spectrum. Figure 21 contains a set of such plots for each component for Klu, and K2 . We can think of these as plots of the "strategy" of the calibration... 

This will cause CLS to calculate an additional pure component spectrum for the G s. It will also give us an additional row of regression coefficients in our calibration matrix, Kc , which we can, likewise, discard. 

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