Sensitivity Calculation

Note

In-depth information about the theoretical underlying and the calculation methods will be described in a book chapter releasing in the near future.

The Fisher information matrix (FIM) can be easily calculated via the sensitivity matrix \(S\):

\[\begin{alignat}{3} F = S^T C^{-1} S, \end{alignat}\]

where \(C\) is the covariance matrix of measurement error.

As an example, the mentioned sensitivity matrix for two observables \(y = (y_1, y_2)\), two different inputs \(u = (u_1, u_2)\), \(N\) different time and \(N_p\) parameterscan be built in the following way:

\[\begin{split}S = \begin{bmatrix} s_{11} (t_1, u_1) & ... & s_{1 N_p}(t_1, u_1) \\ \vdots & & \vdots \\ s_{11} (t_{N}, u_1) & ... & s_{1 N_p} (t_{N}, u_1)\\ s_{11} (t_1, u_2) & ... & s_{1 N_p}(t_1, u_2) \\ \vdots & & \vdots \\ s_{11} (t_N, u_2) & ... & s_{1 N_p} (t_N, u_2)\\ s_{21} (t_1, u_1) & ... & s_{2 N_p}(t_1, u_1) \\ \vdots & & \vdots \\ s_{21} (t_{N}, u_1) & ... & s_{2 N_p} (t_{N}, u_1)\\ s_{21} (t_1, u_2) & ... & s_{2 N_p}(t_1, u_2) \\ \vdots & & \vdots \\ s_{21} (t_N, u_2) & ... & s_{2 N_p} (t_N, u_2) \end{bmatrix}\end{split}\]

Here the elements of this matrix are the local sensitivity coefficients

\[\begin{alignat}{3} s_{ij} (t_m, u_n) = \frac{\mathrm{d} y_i}{\mathrm{d} p_j} \end{alignat}\]