The Dispersion Matrix

The specification of D,j is given here in more detail than in previous descriptions. From Eq. (18), D-,j may be split into far-field term and near-field terms in the same way as c. 

Several empirical forms have been proposed for r ,(z). A typical 

FIGURE 1 The normalised dispersion matrix D, u,. Each profile is a column of the matrix, equal to the concentrations c, - c, at heights Zj (i = 1 to n) produced by a unit source in the layer centered at Zj (j = 1 to m). The number of concentration heights (n) is 20, and the number of source layers (m) is 10. The reference height z is 2h. Turbulence profiles and 

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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. 

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). 

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. 

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... 

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. 

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... 

The "quality" of estimated model parameters is determined by the experimental error variance, tr, and the properties of the dispersion matrix, (X X) , which in turn is determined by the experimental design. Let us have a closer look at the concept "quality". 

The joint probability region can have different orientations and extensions in the parameter space. This will correspond to different "quality" aspects of the estimated values of the model parameters. These quality aspects will depend on the properties of the dispersion matrix (X X) , see Fig. 5.8. 

The "volume" of the joint confidence region is proportional to the experimental error variance and to the square root of the determinant of the dispersion matrix. "Volume" oc I (X X)- ... 

The confidence limits for individual model parameters are proportional to the main axes of the ellipsoids and these are related to the eigenvalues of the dispersion matrix. To obtain the same precision in all model parameters, the confidence region should be spherical. Thus, the experiment should be laid out so that the eigenvalues of (X X)- are equal. 

Independent estimates of the model parameters are obtained when the main axes of the confidence (hyper)ellipsoid are parallel to the parameter axes. This is not the case in Fig. 5.8d. Such situations occur if the covariances of the model parameters are not equal to zero. This will happen if the two or more variables have been varied in a correlated way over the series of experiments. In such cases it will not be possible to unequivocally discern which of these variables is responsible for an observed variation of the response. To obtain independent estimates of the model parameters, the experiments should be laid out in such a way that the dispersion matrix is a diagonal matrix. This implies that the settings of the variables of the model are uncorrelated over the set of experiments, which is equivalent to saying... 

The application of the above principles is illustrated by the previously discussed 2 design. The dispersion matrix (X X) is shown below. 

The determinant (X X) = S which is the minimum value. The eigenvalues of the dispersion matrix are all equal and the variance of all estimated model parameters are / 8. The parameters are independently estimated and the dispersion matrix is a diagonal matrix (the covariances of the models parameters are zero). This means that parameters estimated from a two-level factorial design are independently estimated, with equal and maximum precision. 

The dispersion matrix is not a diagonal matrix. There are correlations between the model parameters. Hence, they are not independently estimated. Through a D-optimal design the parameters are estimated as independently as possible. This is the sacrifice which must be made when the number of experiments does not permit an orthogonal design. The covariances of the model parameters are rather small and the correlations are weak and will, hopefully, not lead to erroneous conclusions as to the influence of the variables. The estimated model parameters are summarized in Table 7.4. 

The variance functions, dj and dj, are obtained from the dispersion matrix of the first design... 

It can be shown that the determinant of the dispersion matrix is the product of its eigenvalues. 

D-optimality. The D-optimality criterion will therefore specify that the product of the dispersion matrix eigenvalues should be as small as possible. 

A-optimality. This criterion refers to the average variance of the estimated model parameters, (A stands for Average variance). An experimental design is A-optimal if the sum of the dispersion matrix eigenvalues... 

E-optimality This criterion is fulfilled when the largest eigenvalue of the dispersion matrix is as small as possible. This minimizes the largest variance of the estimated model parameters. 

The model matrix X in least squares modelling describes the variation of the variables included in the model. The matrix X X is symmetric, and hence also the dispersion matrix, (X X). The eigenvalues of the dispersion matrix are related to the precision of the estimated model parameters. The determinant of the dispersion matrix is the product of its eigenvalues. The "Volume" of the joint confidence region of the estimated model parameters is proportional to the square root of the determinant of the dispersion matrix. 

The possibility of interaction of pyridine with the dispersion matrix material was investigated by collecting spectra of pure KC1 and of pyridine adsorbed on pure KC1 before and after evacuation. All traces of pyridine were removed from the KC1 spectrum following a few minutes of evacuation. 

We will proceed, therefore, with an eigenvector analysis of the 5x5 covariance matrix obtained from zero-centred object data. This is referred to as Q-mode factor analysis and is complementary to the scheme illustrated previously with principal components analysis. In the earlier examples the dispersion matrix was formed between the measured variables, and the technique is sometimes referred to as R-mode analysis. For the current MS data, processing by R-mode analysis would involve the data being scaled along each mjz row (as displayed in Table 3.8) and information about relative peak sizes in any single spectrum would be destroyed. In Q-mode analysis, any scaling is performed within a spectrum and the mass fragmentation pattern for each sample is preserved. 

The dispersion matrix D,j is a discrete form of the transition probability P in Eq. (16) and thus carries all the required information about the velocity field. Its elements have the dimension of aerodynamic resistance (s m ). The elements of column j of D-,j are found by considering (/>jAzj to be a steady unit source, with sources in all other canopy layers set to zero. A theory of turbulent dispersion is used to calculate the concentration field c(z) resulting from this source distribution. The elements of column j of D-,j are then given by... 

Figure 1 illustrates the dispersion matrix by plotting the elements Djj, normalized as D,jU, where u, is the friction velocity. 

Having specified the turbulence properties cr and and thence the dispersion matrix Di, the apparatus is now in place to use Ecp (17) to solve three generic kinds of problem the forward problem of determining the scalar concentration profile c(z) from a specified source density profile < (z), the inverse problem of determining 4> z) from specified or measured information about c(z), and the implicit or coupled problem of determining both c z) and 4> z) together when cf> is a given function of c. 

Figure 3 shows a test of the sensitivity of this method to two key factors, the size of Djj (determined by the number of concentration measurements, , and source layers, m) and the presence or absence of the near-field term in in the dispersion matrix D,j. Figure 3a shows the assumed concentration field, based on forward calculation of the concentration profile q (i = 1 to n) from a specified source profile using Eq. (17), with D j as in Figure 1, and with n = 20, m = 10. This concentration field is then used to reconstruct the source profile 4> z) with Eq. (30), and the flux f(z) from Eq. (20), under three scenarios (1) n,m) = (20,10), with DT" included (2) (n,m) = (10,5), with included and (3) (n,m) = (10,5), with omitted. Figures 3b and 3c respectively show the inferred profiles 0, and Fj at layers centered on heights... 

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