Sensitivity, coefficient

To obtain the different values of p, it is only hecessary to produce as many independent equations as there are components in the mixture and, if the mixture has n components, to solve a system of n equations having n unknowns. Individual analysis is now possible for mixtures having a few components but even gasoline has more than 200 It soon becomes unrealistic to have ail the sensitivity coefficients necessary for analysis in this case, 200. ... 

Because and AAi are known, iC can be found using the generalized inverse method. The sensitivity coefficients matrix iCis given by... 

We shall now derive a result first obtained24 by more complicated mathematics than the alternative25 given here. The 1992 Nobel Prize in Chemistry was awarded to R. A. Marcus for developing this work. We construct a family of reaction profiles (see Fig. 10-8) for different members of the series. The horizontal axis is now used to show the relative locations of the transition states. The larger AG is, the closer to product the transition states lies, and the larger AG is. By assuming that the sensitivity coefficient... 

These two contributions can be evaluated separately in terms of the sensitivity coefficients of the result to the measured quantities by using the propagation equation of Kline and McClintock (1953) ... 

Figure 3.6b shows the same sensitivity coefficients expressed as a percentage of their maximum value. From these graphs it is apparent (without much... 

Figure 3.6a Absolute sensitivity coefficients of the Gompertz model. Each curve portrays the time course of the sensitivity of the model to a specific parameter a (+), b (open circles), and Vq ( ).

In our case, it is evident that, independently of the absolute values taken, even qualitatively the shapes of the time courses of the MCCC and of the classic sensitivity coefficients do not seem to agree. 

In our example, for very small times the theoretical influence of b (given by its sensitivity coefficient) grows more slowly than the theoretical influence of a, while the theoretical influence of Vo (initially the only effective one) increases much more slowly than those of either a or b. Assembling these separate effects we have a combined situation in which the practical influence of a (measured by its MCCC) rises quickly while overcoming the influence of Vo, peaks when a is the only effective parameter, then decreases to reach a steady level as the action of b also asserts itself. 

In summary, at each iteration of the estimation method we compute the model output, y(x kw), and the sensitivity coefficients, G for each data point i=l,...,N which are used to set up matrix A and vector b. Subsequent solution of the linear equation yields Akf f 1 and hence k[Pg.53]

The elements of the (2x2)-sensitivity coefficient matrix G are obtained as follows ... 

Equations 4.14 and 4.15 are used to evaluate the model response and the sensitivity coefficients that are required for setting up matrix A and vector b at each iteration of the Gauss-Newton method. 

Equation 6.9 is a matrix differential equation and represents a set of nxp ODEs. Once the sensitivity coefficients are obtained by solving numerically the above ODEs, the output vector, y(tl,k l+I ), can be computed. 

In this case the n-dimemional vector gi represents the sensitivity coefficients of the state variables with respect to parameter k, and satisfies the following ODE,... 

Thus, the error in the solution vector is expected to be large for an ill-conditioned problem and small for a well-conditioned one. In parameter estimation, vector b is comprised of a linear combination of the response variables (measurements) which contain the error terms. Matrix A does not depend explicitly on the response variables, it depends only on the parameter sensitivity coefficients which depend only on the independent variables (assumed to be known precisely) and on the estimated parameter vector k which incorporates the uncertainty in the data. As a result, we expect most of the uncertainty in Equation 8.29 to be present in Ab. 

When the parameters differ by more than one order of magnitude, matrix A may appear to be ill-conditioned even if the parameter estimation problem is well-posed. The best way to overcome this problem is by introducing the reduced sensitivity coefficients, defined as... 

As we have already pointed out in this chapter for systems described by algebraic equations, the introduction of the reduced sensitivity coefficients results in a reduction of cond(A). Therefore, the use of the reduced sensitivity coefficients should also be beneficial to non-stiff systems. 

Finally it is noted that in the above equations we can substitute G(t) with GR(t) and Ak(i+1) with AkRtrM) in case we wish to use the reduced sensitivity coefficient formulation. 

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