Figure 1.
Flowchart of the proposed method.
The algorithm is a three-stage process, which involves Kalman filtering, a statistical accuracy test and an optimization problem.
Figure 2.
Estimation of 2 parameters in the heat shock model.
The data points (green squares) are obtained by evaluating the true model solution (red dashed curve) at the chosen time points, and then adding white Gaussian noise. The blue solid line shows the reconstructed solution corresponding to the HEKF estimates for the parameters and
. Both the reconstructed measurement signal for
(top) and the one for
(bottom) are very close to the respective true solutions. The graphs are zoomed to highlight the transient response of the heat shock system after a temperature increase.
Figure 3.
Time evolution of the Kalman filter parameter estimates in the heat shock model.
After an initial transient, the estimates of the two parameters (top) and
(bottom), represented by the triangles, keep oscillating around the respective true values (blue dashed line). The last 10 samples (connected by the green line) are averaged to extract a single number from this time-varying signal.
Figure 4.
Discrimination between competing heat shock models.
The models (18) (blue) and (19) (red) are compared in terms of their (top) and
(bottom) outputs. Both signals evolve to the same steady state, but with different transient behavior. The dashed lines represent the ideal model solutions, the triangles are the corresponding Kalman filter estimates.
Table 1.
Discrimination of the heat shock models.
Figure 5.
interval estimates in the case of valid and invalid parameter sets.
The red set of interval estimates corresponds to a parameter set computed with the HEKF only (invalid). The green set corresponds to a parameter set that was obtained with the combination of HEKF and moment matching optimization (valid). The real variances (blue triangles) only lie inside the intervals corresponding to a valid estimation. The top panel is relative to the measurement signal, the bottom panel to the
measurement signal.
Figure 6.
Estimation of 6 parameters in the heat shock model.
The data points (green squares) are obtained by evaluating the true model solution (red dashed curve) at the chosen time points, and then adding white Gaussian noise. The blue solid line shows the reconstructed solution corresponding to the parameters estimates. Both the reconstructed measurement signal for (top) and the one for
(bottom) are very close to the respective true solutions. The graphs are zoomed to highlight the transient response of the heat shock system after a temperature increase.
Table 2.
test results for the estimation of 6 parameters in the heat shock model (moment matching).
Table 3.
test results for the estimation of 6 parameters in the heat shock model (nonlinear least-squares).
Table 4.
test results for the estimation of 6 parameters in the heat shock model (genetic algorithm).
Figure 7.
Estimation of 18 parameters in the repressilator model.
The data points (green squares) are obtained by evaluating the true model solution (red dashed curve) at the chosen time points, and then adding white Gaussian noise. The blue solid line shows the reconstructed solution corresponding to the estimated parameters. Both the reconstructed measurement signal for (top) and the one for
(bottom) are very close to the respective true solutions. The graph for the measurement
is presented in the supporting Figure S1.
Table 5.
test results for the estimation of 18 parameters in the repressilator model (moment matching).
Table 6.
test results for the estimation of 18 parameters in the repressilator model (least-squares).
Table 7.
test results for the estimation of 18 parameters in the repressilator model (genetic algorithm).