Column-standardized matrix

The matrix Cp contains the variances of the columns of X on the main diagonal and the covariances between the columns in the off-diagonal positions (see also Section 9.3.2.4.4). The correlation matrix Rp is derived from the column-standardized matrix Zp ... 

Nevertheless, if (3.78) is known to be violated, a further issue is to find the variable that is primarily responsible for the violation. The ratio of the absolute value of the correction to the corresponding standard deviation provides some information but may be misleading (ref. 31). The analysis proposed by Almdsy and Sztand (ref. 32) is based on geometric ideas. If exactly one observation is corrupted by gross error then the corresponding column of matrix W and the vector f of equation errors are nearly collinear. Useful measures of collinearity are 7 = cos, where is the... 

Figure 8.3. ID analysis of B1O3" in drinking water by ion chromatography with ICP-MS detection study of polyatomic ions produced at mass 79 and 81 from a 300 p,g g-1 sulfate matrix a) and a 100 p,g g-1 phosphate matrix (b). Std denotes the peak produced by a 100 p.L post-column standard injection of BrOj". Reprinted from [395] with permission Copyright 1999 American Chemical Society.

If any of the internal standard ion abundance ratios as specified in Table 6 are outside the contract specified control limits, the Contractor must reanalyze the sample extract on a second GC column with different elution characteristics, as discussed in Section 16. No reextraction is required for such an analysis. This reanalysis is only billable if the same internal standard ion abundance ratios are outside the control limits on the second column, indicating matrix effects may have occurred. 

Calibration is realized by recording the spectra at -wavelengths of m standard mixtures, of known composition of c components. The spectra (absorbance or emission) are arranged into the columns of matrix Y (dimensions n X m), with the composition of each mixture forming the columns of concentration matrix X (c X m). 

The principal components are calculated using standard matrix techniques [Chatfield and Collins 1980]. The first step is to calculate the variance-covariance matrix. If there are s observations, each of which contains v values, then the data set can be represented as a matrix D with v rows and s columns. The variance-covariance matrix Z is ... 

To improve the accuracy of LC-MS quantitative results, matrix effects can be compensated for by means of isotopically labeled internal standards, matrix-matched standard calibration curves, standard addition, echo-peak technique, post-column infusion, extrapolative dilution, and so on. Isotopically labeled internal standards and/or matrix-matched standard calibration curves are two common approaches that have been widely used. Table 6.1 lists some commercially available isotopically labeled internal standards. Although this method provides the most accurate result, sometimes it is not realistic to have isotopically labeled standards for each individual analyte. Therefore, matrix-matched standard calibration curves, with or without... 

Imagine a hypothetical case when all the standard deviations of measurement errors are 4-times smaller. From the theory it follows that the adjustments, thus the final estimates remain unaltered but increases 16-times and its value 6.47 exceeds now the critical value. Searching for a possible source of a gross error, the (pseudojstandardized adjustments Zj can be computed by (10.5.4) with (10.5.3). Here, aj is -th column of matrix A obtained in the final step by elimination. The greatest Zj are found for the variables and y in both cases approximately... 

Standardization means to divide each centered matrix element with the column standard deviations ... 

To compute the variance, we first find the mean concentration for that component over all of the samples. We then subtract this mean value from the concentration value of this component for each sample and square this difference. We then sum all of these squares and divide by the degrees of freedom (number of samples minus 1). The square root of the variance is the standard deviation. We adjust the variance to unity by dividing the concentration value of this component for each sample by the standard deviation. Finally, if we do not wish mean-centered data, we add back the mean concentrations that were initially subtracted. Equations [Cl] and [C2] show this procedure algebraically for component, k, held in a column-wise data matrix. 

Usually, the raw data in a matrix are preprocessed before being submitted to multivariate analysis. A common operation is reduction by the mean or centering. Centering is a standard transformation of the data which is applied in principal components analysis (Section 31.3). Subtraction of the column-means from the elements in the corresponding columns of an nxp matrix X produces the matrix of... 

From now on, we adopt a notation that reflects the chemical nature of the data, rather than the statistical nature. Let us assume one attempts to analyze a solution containing p components using UV-VIS transmission spectroscopy. There are n calibration samples ( standards ), hence n spectra. The spectra are recorded at q wavelengths ( sensors ), digitized and collected in an nx.q matrix S. The information on the known concentrations of the chemical constituents in the calibration set is stored in an nxp matrix C. Each column of C contains the concentrations of one of the p analytes, each row the concentrations of the analytes for a particular calibration standard. 

Note that eq. (36.5) is a collection of many univariate multiple regression models for each wavelength j the multiple regression of the corresponding spectral channel , i.e. Sj, on the concentration matrix C yields a vector of regression coefficients, ky (the yth column of K). For K to be estimable C C must be invertible, i.e. the number of calibration standards should at least be as large as the number of analytes. It is clearly not possible to obtain, directly or indirectly, say 3 pure spectra from recording the spectra of just 1 or 2 standards of known composition. In practice, the condition n>p, or more precisely rank(C)=p, is hardly a restriction. 

Optimizing the GC instrument is crucial for the quantitation of sulfentrazone and its metabolites. Before actual analysis, the temperatures, gas flow rates, and the glass insert liner should be optimized. The injection standards must have a low relative standard deviation (<15%) and the calibration standards must have a correlation coefficient of at least 0.99. Before injection of the analysis set, the column should be conditioned with a sample matrix. This can be done by injecting a matrix sample extract several times before the standard, repeating this conditioning until the injection standard gives a reproducible response and provides adequate sensitivity. 

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