Sensitivity matrices

Mechanical Properties. Although wool has a compHcated hierarchical stmcture (see Fig. 1), the mechanical properties of the fiber are largely understood in terms of a two-phase composite model (27—29). In these models, water-impenetrable crystalline regions (generally associated with the intermediate filaments) oriented parallel to the fiber axis are embedded in a water-sensitive matrix to form a semicrystalline biopolymer. The parallel arrangement of these filaments produces a fiber that is highly anisotropic. Whereas the longitudinal modulus of the fiber decreases by a factor of 3 from dry to wet, the torsional modulus, a measure of the matrix stiffness, decreases by a factor of 10 (30). 

The new approach for development of pH sensor with wide acidity range (2.5 M H SO - pH 5.5) based on the use of Congo Red and Benzopurpurin 4B immobilized in polyamido- or arachidic acid nanosized sensitive matrix will be demonstrated. 

In the case of ODE models, the sensitivity matrix G(t() = (3xT/3k)T cannot be obtained by a simple differentiation. However, we can find a differential equation that G(t) satisfies and hence, the sensitivity matrix G(t) can be determined as a function of time by solving simultaneously with the state ODEs another set of differential equations. This set of ODEs is obtained by differentiating both sides of Equation 6.1 (the state equations) with respect to k, namely... 

The parameter sensitivity matrix G(t) can be obtained as shown in the previous section by solving the matrix differential equation,... 

If we consider the limiting case where p=0 and q O, i.e., the case where there are no unknown parameters and only some of the initial states are to be estimated, the previously outlined procedure represents a quadratically convergent method for the solution of two-point boundary value problems. Obviously in this case, we need to compute only the sensitivity matrix P(t). It can be shown that under these conditions the Gauss-Newton method is a typical quadratically convergent "shooting method." As such it can be used to solve optimal control problems using the Boundary Condition Iteration approach (Kalogerakis, 1983). 

The sensitivity matrix, G(t), is a (2x2)-dimemional matrix with elements ... 

The ordinary differential equation that a particular element, G . of the (nxp)-dimensional sensitivity matrix satisfies, can be written directly using the following expression. 

Step 2. Given the current estimate of the parameters, k , compute the parameter sensitivity matrix, G the response variables f(x k[Pg.161]

The output sensitivity matrix (dyT/dk)T is related to the sensitivity coefficient matrix defined as... 

At this point we can summarize the steps required to implement the Gauss-Newton method for PDE models. At each iteration, given the current estimate of the parameters, ky we obtain w(t,z) and G(t,z) by solving numerically the state and sensitivity partial differential equations. Using these values we compute the model output, y(t k(i)), and the output sensitivity matrix, (5yr/5k)T for each data point i=l,...,N. Subsequently, these are used to set up matrix A and vector b. Solution of the linear equation yields Ak(jH) and hence k°M) is obtained. The bisection rule to yield an acceptable step-size at each iteration of the Gauss-Newton method should also be used. 

The solution of Equation 10.28 is obtained in one step by performing a simple matrix multiplication since the inverse of the matrix on the left hand side of Equation 10.28 is already available from the integration of the state equations. Equation 10.28 is solved for r=l,...,p and thus the whole sensitivity matrix G(tr,) is obtained as [gi(tHt), g2(t,+1),- - , gP(t,+i)]. The computational savings that are realized by the above procedure are substantial, especially when the number of unknown parameters is large (Tan and Kalogerakis, 1991). With this modification the computational requirements of the Gauss-Newton method for PDE models become reasonable and hence, the estimation method becomes implementable. 

Analyte dilution sacrifices sensitivity. Matrix matching can only be applied for simple matrices, but is clearly not applicable for complex matrices of varying composition. Accurate correction for matrix effect is possible only if the IS is chosen with a mass number as close as possible to that of the analyte elements). Standard addition of a known amount of the element(s) of interest is a safe method for samples of unknown composition and thus unknown matrix effect. Chemical separations avoid spectral interference and allow preconcentration of the analyte elements. Sampling and sample preparation have recently been reviewed [4]. 

If the Newton-Raphson method is used to solve Eq. (1), the Jacobian matrix (df/3x)u is already available. The computation of the sensitivity matrix amounts to solving the same Eq. (59) with m different right-hand side vectors which form the columns — (3f/<5u)x. Notice that only the partial derivatives with respect to those external variables subject to actual changes in values need be included in the m right-hand sides. 

Instead of the symbol A and the term sensitivity matrix also the symbol K (matrix of calibration coefficients, matrix of linear response constants etc) is used. Because of the direct metrological and analytical meaning of the sensitivities aj - in the A-matrix the term sensitivity matrix is preferred. 

To avoid confusion, A is used here as symbol for the sensitivity matrix Bauer et al. [1991a, b] use S for this purpose and B for the background vector (here y0). 

More extensive, multicomponent system are described by the sensitivity matrix (matrix of partial sensitivities according to Kaiser [1972], also called K-matrix according to Jochum et al. [1981]) ... 

In the ideal case only the diagonal elements of the sensitivity matrix are different from zero. Then no component disturbs any other and the analytical procedure works selectively (see Sect. 7.3). The K-malrix is defined analogously except that the elements are called kij, their definition is the same as the Sij according to Eq. (7.16). 

In case of serious overlappings, multivariate techniques (see Sect. 6.4) are used and p ) > n sensors (measuring points zjt) are measured for n components. From this an overdetermined systems of equations results and, therefore, non-squared sensitivity matrixes. Then the total multicomponent sensitivity is given by... 

Starting from a relationship like Eq. (6.70a), Kaiser [1972] defined sensitivity, partial sensitivities (cross sensitivities, Eq. (7.16)), and the sensitivity matrix Eq. (7.17). From these quantities he derived the following measures ... 

In such cases the sensitivity matrix, Eq. (7.17), will be extended into... 

Table 7.3. Selectivities and specificities in case of various sensitivity matrixes... 

Sect. 7.2 Eq. (3.11) Sensitivity Sensitivity matrix Total sensitivity... 

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