Dispersion matrix

A basic assumption of OPA is that the purest spectra are mutually more dissimilar than the corresponding mixture spectra. Therefore, OPA uses a dissimilarity criterion to find the number of components and the corresponding purest spectra. Spectra are sequentially selected, taking into account their dissimilarity. The dissimilarity of spectrum i is defined as the determinant of a dispersion matrix Y,. In general, matrices Y, consist of one or more reference spectra, and the spectrum measured at the /th elution time. 

The set of selected wavelengths (i.e. the experimental design) affects the variance-covariance matrix, and thus the precision of the results. For example, the set 22, 24 and 26 (Table 41.5) gives a less precise result than the set 22, 32 and 24 (Table 41.7). The best set of wavelengths can be derived in the same way as for multiple linear regression, i.e. the determinant of the dispersion matrix (h h) which contains the absorptivities, should be maximized. 

Variances and covariances can be lumped together into the n x n symmetric sample covariance or dispersion matrix S (or ) with current element siU2 such that... 

The matrix of all of either the correlations or covariances or the dispersion matrix can be obtained from the original or transformed data matrices. The data matrices contain the data for the m variables measured over the n samples. The correlation about the mean is given by... 

The interactions observed between the individual components and the target analytes in MSPD are greater and different, in part, from SPE. They appear between the analyte and the solid support, the analyte and the bonded phase, the analyte and the dispersed matrix, the matrix and the solid support, and the matrix and the bonded phase all of the above components interact with the elution solvents, and the dynamic interactions of all of the above occur simultaneously. As a result, both the bonded phase and the solid support are expected to affect the results (99-104). 

S. K. Chandrasekaran and D. R. Paul. Dissolution-controlled transport from dispersed matrixes. J. Pharm. Sci. 71 1399—1402, 1982. 

The term tr(M)-1 designates the trace of the dispersion matrix. Because the diagonal elements of M 1 present the variances of the regression coefficients, the trace (e.g., their sum) is a measure of the overall variance of the regression coefficients. The minimization of this measure ensures better precision in the estimation of the regression coefficients. 

Using the notation of experimental design, F represents the extended design matrix, where the elements of its k x I row-vectors, f, are known functions of x. The matrix (FT) is the Fisher information matrix and its inverse, (FT)-1, is the dispersion matrix of the regression coefficients. 

FIGURE 6.16 A schematic diagram of drug release from a dispersed matrix system. 

Fractional Drug Release and Exhaustion Time for Dispersed Matrix Systems... 

Recall from Eq. (2) that the dispersion matrix depends on velocity. Thus, for the first and second terms on the right-hand-side of Eq. (18), the groundwater average linear velocity vector (which was assumed steady in time) must be determined. This is accomplished in two-steps. In the first step, the distribution of hydraulic heads must be determined in order to calculate the hydraulic gradient, for use in Eq. (1). For steady flow, the head field must satisfy Laplace s equation that is... 

The extraction of the eigenvectors from a symmetric data matrix forms the basis and starting point of many multivariate chemometric procedures. The way in which the data are preprocessed and scaled, and how the resulting vectors are treated, has produced a wide range of related and similar techniques. By far the most common is principal components analysis. As we have seen, PCA provides n eigenvectors derived from a. nx n dispersion matrix of variances and covariances, or correlations. If the data are standardized prior to eigenvector analysis, then the variance-covariance matrix becomes the correlation matrix [see Equation (25) in Chapter 1, with Ji = 52]. Another technique, strongly related to PCA, is factor analysis. ... 

Each of these transformations can be expressed in matrix form as a transform of the data matrix. Y to a new matrix Y followed by calculating the appropriate dispersion matrix, C (the variance-covariance, or correlation matrix). The relevant equations are... 

As the analytical data are all in the same units and cover a similar range of magnitude, standardization is not required either and the variance-covariance matrix will be used as the dispersion matrix. 

Decision limit 32, 33 Degrees of freedom, 8 Dendrogram, 97, 105 Detection limit, 32, 33 Determination limit, 32, 33 Differentiation, 55 Savitsky-Golay, 57 Discriminant function, 124, 130 Discriminant score, 130 Discrimination, 123 Dispersion matrix, 82 Distance measures, 99 Dixon s Q-test, 13... 

Vibration frequencies in the solid phase are generally lower than in the liquid phase. Certain bands of solid samples do not appear in liquid and gaseous samples. Physical and/or chemical interactions of the sample with a dispersing matrix can alter the spectrum and make comparison impossible. 

A least squares estimate is no guarantee whatsoever, that the model parameters will have good properties, i.e. that they will measure what we want them to do, viz. the influence of the variables. The quality of the model parameters is governed by the properties of the dispersion matrix (X X)" and hence it depends ultimately on the experimental design used to determine the model. The requirements for a good design will be discussed in Chapter 5. 

Then, compute the dispersion matrix (X X) which in this case will be... 

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