Scoring calibration

The application of principal components regression (PCR) to multivariate calibration introduces a new element, viz. data compression through the construction of a small set of new orthogonal components or factors. Henceforth, we will mainly use the term factor rather than component in order to avoid confusion with the chemical components of a mixture. The factors play an intermediary role as regressors in the calibration process. In PCR the factors are obtained as the principal components (PCs) from a principal component analysis (PC A) of the predictor data, i.e. the calibration spectra S (nxp). In Chapters 17 and 31 we saw that any data matrix can be decomposed ( factored ) into a product of (object) score vectors T(nxr) and (variable) loadings P(pxr). The number of columns in T and P is equal to the rank r of the matrix S, usually the smaller of n or p. It is customary and advisable to do this factoring on the data after columncentering. This allows one to write the mean-centered spectra Sq as ... 

The suffix in T (nxA) and P (< xA) indicates that only the first A columns of T and P are used, A being much smaller than n and q. In principal component regression we use the PC scores as regressors for the concentrations. Thus, we apply inverse calibration of the property of interest on the selected set of factor scores ... 

Measurements were made using two types of passive track-etch alpha dosimeters. One of them was the bare detector of CR-39. After exposure these dosimeters were etched by 30 % NaOH at 70°C for 5 hours. The number of pits was scored under a microscope with a television camera in Shiga University of Medical Science. Methods of calibration and adjustment for deposition of radon daughters introduced by Yonehara (Yonehara et al., 1986) were adopted. The second detectors were Terradex type SF (Alter and Price, 1972). These detectors consist of a plastic cup, covered by a filter to allow entry only of gases, with a track-etch detector inside. The reading of results was carried out by Terradex Corp. in Walnut Creek, California, U.S.A.. The measurements of radon concentration were carried out by both methods in each location, except for Hokkaido where the measurements were done only by Terradex. However, the data obtained by CR-39 detectors will be mainly presented in this paper, because the two methods did not give identical results as separately reported in this proceedings by Yonehara et al. (Yonehara et al., 1986). 

En numbers are used when the assigned value has been produced by a reference laboratory, which has provided an estimate of the expanded uncertainty. This scoring method also requires a valid estimate of the expanded uncertainty for each participant s result. A score of En < 1 is considered satisfactory. The acceptability criterion is different from that used for z-, z - or zeta-scores as En numbers are calculated using expanded uncertainties. However, the En number is equal to zeta/2 if a coverage factor of 2 is used to calculate the expanded uncertainties (see Chapter 6, Section 6.3.6). En numbers are not normally used by proficiency testing scheme providers but are often used in calibration studies. 

PLS with a matrix Y instead of a vector y is called PLS2. The purpose of data evaluation can still be to create calibration models for a prediction of the y-variables from the x-variables in PLS2 the models for the various y-variables are connected. In a geometric interpretation (Figure 4.25), the m-dimensional x-space is projected on to a small number of PLS-x-components (summarizing the x-variables), and the -dimensional y-space is projected on to a small number of PLS-y-components (summarizing the y-variables). The x- and the y-components are related pairwise by maximum covariance of the scores, and represent a part of the relationship between X and Y. Scatter plots with the x-scores or the y-scores are projections of... 

FIGURE 4.25 PLS2 works with X- and K-matrix in this scheme both have three dimensions. t and u are linear latent variables with maximum covariance of the scores (inner relation) the corresponding loading vectors are p und q. The second pair of x- and y-components is not shown. A PLS2 calibration model allows a joint prediction of all y-variables from the x-variables via x- and y-scores. 

Table VIII contains rate constants for reactions of tin hydrides with carbon-centered radicals. A striking feature of Table VIII in comparison to other tables in this work is the high percentage of reactions for which Arrhenius parameters were determined by direct LFP or the LFP-clock method. These results are expected to be among the most accurate listed in this work. Scores of radical clocks have been studied with Bu3SnH, but the objectives of those studies were to determine rate constants for the clocks using tin hydride trapping as the calibrated basis reaction.

Like MLR, PCR [63] is an inverse calibration method. However, in PCR, the compressed variables (or PCs) from PCA are used as variables in the multiple linear regression model, rather than selected original X variables. In PCR, PCA is first done on the calibration x data, thus generating PCA scores (T) and loadings (P) (see Section 12.2.5), then a multiple linear regression is carried out according to the following model ... 

When the PCA method is applied to the NIR spectra in Figure 12.16, it is found that three PCs are sufficient to use in the model, and that PCs 1, 2 and 3 explain 52.6%, 20.5%, and 13.6% of the variation in the spectral data, respectively. Figure 12.17 shows a scatter plot of the first two of the three significant PC scores of all 26 foam samples. Note that all of the calibration samples visually agglomerate into four different groups in these first two dimensions of the space. Furthermore, it can be seen that each group corresponds to a single known class. 

Regarding relevance, the spectral miscibility of the data obtained from these two different sources can be readily observed by doing a PCA analysis of the combined spectral data. The scatter plot of the first two PC scores obtained from PCA of such a data set for one of the process analytes is shown in Figure 12.31a. Note that there is considerable common space for the two data sources in the PC1/PC2 space, and there are some regions of this space where only samples from the old calibration strategy lie. A similar pattern is observed in the later PCs of this model. This result indicates that the on-line spectra contain some unique information, but that the on-line and injected-standard spectra are generally quite similar. 

Figure 12.31 Illustration of a hybrid calibration strategy. (A) Scatter plot of first two PCA scores obtained from a process analytical calibration data set containing both synthesized standards (circles) and actual process samples (triangles). (B) Results of a PLS regression fit to the property of Interest, using all of the calibration samples represented In (A).

The body of samples selected is split into two subsets, namely the calibration set and the validation set. The former is used to construct the calibration model and the latter to assess its predictive capacity. A number of procedures for selecting the samples to be included in each subset have been reported. Most have been applied to situations of uncontrolled variability spanning much wider ranges than those typically encountered in the pharmaceutical field. One especially effective procedure is that involving the selection of as many samples as required to span the desired calibration range and encompassing the whole possible spectral variability (i.e. the contribution of physical properties). The choice can be made based on a plot of PCA scores obtained from all the samples. 

For theexample discussed here, the calibration sets for classes A and B are selected gs hically, and for class C are selected as the extremes and centers of each ofdie three levels in the experimental design. The selection results in 15 samples in each of the calibration sets and 12 in each of the validation sets. A score pS>t of all samples in class A is shown in Figure 4.69 with the calibration set samples indicated by X and the validation samples indicated by O. Similarly, SCO plots of clas.es B and C with calibration and validation samples identifiedsre shown in Figures 4.70 and 4.71, respectively. 

Figure 4. Plot of score 1 of all samples in class A after addition of the mislabeled sanies). The samples in the calibration set are Xs and the validation samples are...

This regressaon vector can then be used to predict the concentration in an unknown sample using a two-step process. First, given the measurement of the unknown and the V and S from the calibration, the score vector for the unknown ) is obtained using Equation 531 ... 

Leverage is a measure of the location of a prediction sample in the calibration measurement row space. A high leverage indicates a sample that has an unusual score vector relative to the calibration samples. 

Other strong advantages of PCR over other methods of calibration are that the spectra of the analytes have not to be known, the number of compounds contributing to the signal have not to be known on the beforehand, and the kind and concentration of the interferents should not be known. If interferents are present, e.g. NI, then the principal components analysis of the matrix, D, will reveal that there are NC = NA -I- NI significant eigenvectors. As a consequence the dimension of the factor score matrix A becomes (NS x NC). Although there are NC components present in the samples, one can suffice to relate the concentrations of the NA analytes to the factor score matrix by C = A B and therefore, it is not necessary to know the concentrations of the interferents. 

The concentrations of the analytes in an unknown sample are calculated from the measured spectrum, d , as follows first the factor scores, a , of the spectrum of the unknown are calculated in the eigenvector space, V, of the calibration standards a = d V. The concentrations of the NA analytes are calculated from the relationship found between the factor score and concentration ... 

FIGURE 8 Self-calibrating image comparison for counterfeit identification score images. White higher score. Black lower score. 

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