Sensitivity vector

Finally for the last parameter, kp, we have the corresponding sensitivity vector gp... 

Hence the sensitivity of the method is the inverse of the slope of the regression line (which is also the norm of the net sensitivity vector) ... 

The sensitivity vector with respect to the linearization point in the load space is given by [7]... 

The accuracy of the decomposition in equation (5.12) depends on the accuracy of the determination of the period sensitivity vector s. Several methods have been proposed for the accurate calculation of s, including the application of an approximate term (Edelson and Thomas 1981), a more accurate integral method (Larter 1983) and the singular value decomposition (SVD) of the original sensitivity matrix S (Zak et al. 2005). 

This means that if the trajectory of a simulation is close to an -dimensional manifold, and the perturbation of the parameters negligibly shifts the location of the manifold, then the rank of the local sensitivity matrix is not higher than n. Even if the rank of the sensitivity matrix n is lower than the number of species, it does not mean that local similarity is valid for any pairs of the sensitivity vectors. The other extreme case is when all sensitivity vectors are locally similar except for n vectors. 

The direction of vector CC is along the tangent of the manifold at point C and is identical for the perturbation of any parameter x. The direction of this vector is identical to the direction of all sensitivity vectors dYIdxic, and the projections of this vector onto the axes are the sensitivity coefficients (see Fig. 8.4b). If the directions of two vectors are identical, then the ratios of their projections onto the axes are identical, even if the lengths of the vectors are different. This explains the scaling law and also why any sensitivity vector can be obtained by multiplying any other nonzero sensitivity vector belonging to a different parameter by an appropriate scalar. 

If the dimension of the manifold is two, then any row of the sensitivity matrix can be obtained as a linear combination of two independent sensitivity vectors ... 

This means that two independent sensitivity vectors determine the tangent plane of the manifold belonging to point C (see Fig. 8.4d). 

If local similarity exists among the sensitivity vectors, then any sensitivity vector can be obtained by multiplying another nonzero sensitivity vector with a nonzero scalar ... 

The correlation function p defined by Eq. (8.14) is a good measure (Zador et al. 2004) of the similarity of the sensitivity functions. Two sensitivity functions are locally similar, if they point in the same direction (or exactly the opposite direction) in the space of parameters. In this case, the angle of the two vectors is 0° (or 180°) corresponding to the case otpy = +1 (orpy = — 1). If the value ofp,y is not close to 1, then the sensitivity vectors are not locally similar. 

Zador et al. (2004) investigated the local similarity of the sensitivity functions in a model of the adiabatic explosion of hydrogen-air mixtures at several equivalence ratios. The correlation of the sensitivity vector of H2O with all other sensitivity vectors [belonging to temperature (T) and the concentrations of species H, O2, H2O2, H, O, OH, HO2 and N2] was studied. The results, presented in Fig. 8.5, show the almost perfect local similarity of the sensitivity functions. 

Fig. 8.10 The ctmelation of the sensitivity vector of enzyme Cln2 with the sensitivity vectras of all other variables of the cell cycle model. The investigated parameters were the following kasbf, kisbf, esbfnS, BCKO, CLN3MAX, Dn3 and Jn3. It is clear that cos 0 is close to 1 during most of the time period of the simulated cycle and for most variables, indicating that these sensitivity vectors are locally similar (Lovrics et al. 2008)...

ZadOT, J., Zsely, LG., Turanyi, T. Investigation of the correlation of sensitivity vectors of hydrogen combustion models. Int. J. Chem. Kinet. 36, 238-252 (2004)... 

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