Calibration matrices

An illustrative example generates a 2 x 2 calibration matrix from which we can determine the concentrations xi and X2 of dichromate and permanganate ions simultaneously by making spectrophotometric measurements yi and j2 at different wavelengths on an aqueous mixture of the unknowns. The advantage of this simple two-component analytical problem in 3-space is that one can envision the plane representing absorbance A as a linear function of two concentration variables A =f xuX2). 

For many applications, quantitative band shape analysis is difficult to apply. Bands may be numerous or may overlap, the optical transmission properties of the film or host matrix may distort features, and features may be indistinct. If one can prepare samples of known properties and collect the FTIR spectra, then it is possible to produce a calibration matrix that can be used to assist in predicting these properties in unknown samples. Statistical, chemometric techniques, such as PLS (partial least-squares) and PCR (principle components of regression), may be applied to this matrix. Chemometric methods permit much larger segments of the spectra to be comprehended in developing an analysis model than is usually the case for simple band shape analyses. 

Thus, we can predict the concentrations in an unknown by a simple matrix multiplication of a calibration matrix and the unknown spectrum. 

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... 

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. 

But the nonzero intercepts also allow an additional degree of freedom when we calculate the calibration matrix, K, . This provides additional opportunity to adjust to the effects of the extraneous absorbances. 

Recognizing the difficulty satisfying the requirements for successful CLS, you may wonder why anyone would ever use CLS. There are a number of applications where CLS is particularly appropriate. One of the best examples is the case where a library of quantitative spectra is available, and the application requires the analysis of one or more components that suffer little or no interference other than that caused by the components themselves. In such cases, we do not need to use equation [33] to calculate the pure component spectra if we already have them in a library. We can simply construct a K matrix containing the required library spectra and proceed directly to equation [34] to calculate the calibration matrix K., . 

Calculate a calibration matrix using all of the training set samples except for one. 

Now we are ready to solve for the PCR calibration matrix. We do this exactly the same way we solved for the ILS calibration. First, we post-multiply both sides of equation [59] by ATpreJ. 

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. 

More than one analyte can be quantified simultaneously in the presence of interfering compounds. The required measurements are identical to RAFA a data matrix X of the unknown sample and a calibration matrix with the analytes X. ... 

K is called the calibration matrix or the regression matrix. It contains the calibration, or regression, coefficients which are used to predict the concentrations of an unknown from its spectrum. K, will contain one row of coefficients for each component being predicted. Each row will have one coefficient for each spectral wavelength. Thus, K, will have as many columns as there are spectral wavelengths. Substituting equation [38] into equation [37] gives us... 

Next, we examine how well CLS was able to fit the training set data. To do this, we use the CLS calibration matrix K,. to predict (or estimate) the concentrations of the samples with which the calibration was generated. We then examine the differences between these predicted (or estimated) concentrations and the actual concentrations. Notice that "predict" and "estimate" may be used interchangeably in this context. We first substitute K1 and A1 into equation [39], naming the resulting matrix with the predicted concentrations Kl ,. We then repeat the process with K2,., and A2, naming the resulting concentration matrix K2 ,. 

First the responses Rq are measured for the sample. Thereafter K is determined by fitting the changes in the concentrations of the analytes in the sample, brought about by the standard additions, to the changes in the responses. Once all elements in the calibration matrix, K, have been determined, the concentration vector of the analytes in the sample is calculated. The method has been successfully applied to absorption spectrophotometry , anodic stripping voltametry and ICP-atomic emission spectrophotometry Attractive features of the method are that automation is very easy and automatic drift compensation is possible . A drawback is that all interferents should be known and be corrected for. 

PLS is best described in matrix notation where the matrix X represents the calibration matrix (the training set, here physicochemical parameters) and Y represents the test matrix (the validation set, here the coordinates of the odor stimulus space). If there are n stimuli, p physicochemical parameters, and m dimensions of the stimulus space, the equations in Figure 6a apply. The C matrix is an m x p coefficient matrix to be determined and the residuals not explained by the model are contained in E. The X matrix is decomposed as shown in Figure 6b into two small matrices, an n x a matrix T and an a x p matrix B where a << n and a << p. F is the error matrix. The computation of T is such that it both models X and correlates with T and is accomplished with a weight matrix W and a set of latent variables U for Y with a corresponding loading matrix B. ... 

The diligent analyst would develop a robust method with rigorous matrix effect tests on multiple lots, including hemolyzed and lipidemic samples. An initial test would be a spike-recovery evaluation on at least six individual lots. Samples should be spiked at or near the LLOQ, and at a high level near the ULOQ. If matrix interference were indicated by unacceptable relative error (RE) percentage in certain lots, the spiked sample of the unacceptable lots should be diluted with the standard calibrator matrix to estimate the minimum dilution requirement (MDR) at and above which the spike-recovery is acceptable. The spike-recovery test should then be repeated with the test samples diluted at the MDR. Note that this approach will increase the LLOQ for a less sensitive assay. If sensitivity is an issue, then other venues will be required to address the matrix effect problem. For example, the method can be modified to include sample clean-up, antibodies and/or assay conditions may be changed, or the study purpose may be tolerable to acknowledge that the method may not be selective for a few patients (whose data may require special interpretation). 

Internal standards could be used in external calibration, matrix-matched external calibration, and standard addition calibration [2], However, the use of internal standards in LC-MS quantitative methods should not be confused with internal calibration in which an internal standard is employed as a calibrant and the concentration of a unknown sample is calculated from the concentration of this internal standard and its analyte/IS signal ratio, i.e., the concentration of the unknown sample is calculated without the need for a calibration curve [3], The use of internal standards in most LC-MS quantitative methods belongs to signal-ratio calibration or internal standardization [2,4], In fact, the majority of bioanalytical LC-MS methods use matrix-matched signal-ratio external calibration. 

Figure 12.20 shows the loading-vector ( eigenspectrum ) of the first dimension, explaining 74,9% of the variance in the spectral data set, and predicting 82,5% of the variance in the calibration matrix. The most prominent feature to be observed is the inverse relationship between band 1 and 3, which was indeed expected to be the most important source of spectral variance. 

Predict the concentrations of A in the calibration set using MLR and assuming only compound A can be calibrated, as follows, (i) Determine the vector s = c. X/Y c2, where X is the 25 x 22 spectral calibration matrix and c a 25 x 1 vector, (ii) Determine the predicted concentration vector c = X.s / s2 (note that the denominator is simply the sum of squares when only one compound is used). 

Collect not less than 64 spectral scans of the standards. Constmct a calibration matrix containing infrared absorbance values for unsaturation types in the standards and their known concentrations. Confirm the validity of the calibration matrix model as recommended in the software manual. A recommended method is cross-validation for all standards by sequentially excluding one of the standards from the calibration matrix, then using the remaining standards to predict the concentrations. After validation, determine the optimum number of factors, or loading vectors, needed to minimize the deviation between actual and predicted concentrations. This determination is automated in most multicomponent analysis... 

The principle involved is that, provided the analytical signal is proportional to concentration, the initial analyte content is determined through measurement of this signal before and after the addition of a known amount of the analyte to the analyzed sample. The method of standard addition, also denoted as spiking, is used when an analyte is to be quantified inside a matrix, the effects of which are likely to affect the chromatographic peak behavior. In this case, the sample itself is used as the calibration matrix. 

The computer allows more than one method of analysis to be applied to a sample, so two methods are used. In one (7), several different calibration matrices are available to measure hydrocarbons by class, that is, paraffins and cycloparaffins by number of rings (1-6) and 1-ring aromatics. The calibration matrix used in this work is iso-C24- As frequently happens, an unknown sample may have a low aromatic content. For such a sample, results from the above method are accepted as its composition. 

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