Environmental Models
2.2 SENSITIVITY ANALYSIS AS AN IDENTIFIABILITY TEST
2.2.2 d ynamIc (t rajectory ) S enSItIvIty
A different approach is now considered to show the influence of each parameter along the system evolution. Deriving the definition of sensitivity (2.15) with respect to time and exchanging the order of the derivatives yields
d
producing the dynamic sensitivity system
S S
The sensitivity dynamical equation (2.24) is to be solved in conjunction with the system equations, because the matrix Jn is similar to the Jacobian, but it is not computed at the equilibrium because it follows the nominal trajectory xnom obtained with nominal parameter values. The importance of this new sensitivity definition lies in its relation to system evolution, because it reveals the influence of each parameter along the system trajectory, hence its name of trajectory sensitivity. In other words, it provides information on the parts of the trajectory in which the parameter influence is greatest.
The data collected during this time are therefore the most effective in parameter estimation. Thus, trajectory sensitivity can be used to plan a data acquisition campaign, concentrating the sampling effort at the times when the data are most effective for estimating a given parameter.
This concept will be considered again in due course, when discussing the validity of the esti-mates and how their accuracy can be maximized. For now, let us elaborate on Equation 2.24. To obtain the sensitivity system, calculus must be applied to obtain the Jacobian matrix Jn and the other term ∂ ∂f pi, which can be regarded as an input. Of course, ∂ ∂f pi is obtained by evaluating the derivative of the system function with respect to the parameter of interest pi. Figure 2.12 shows a
Nominal
FIGURE 2.12 Trajectory sensitivity system, including the model evolving along its nominal trajectory and np sensitivity systems fed with the nominal state trajectory, incorporated in Jn.
general trajectory sensitivity system composed of the system model, evolving with nominal param-eters, and a bank of np sensitivity systems, one for each parameter of interest. Notice that the latter systems are not autonomous, because the pseudo-Jacobian matrix Jn must be fed with the nominal state trajectory xnom along the system evolution. The last term in each sensitivity system, ∂ ∂f pi, is the derivative of the system equations with respect to each parameter and only the column of the relevant parameter pi should be used.
As an example, consider again the Monod model (2.18), and suppose that we want to set up the sensitivity system for its four parameters. Using the notation of Equation 2.24, the sensitivity terms take the form
These equations generate four sensitivity systems, one for each parameter, which can be composed as follows:
Sensitivity to μmax:
Sensitivity to Ks:
Sensitivity to bh:
Apart from the considerable calculus involved, the exact computation of sensitivities with the above method requires the joint simulation of the original model, to generate the nominal trajec-tory, and of the sensitivity systems for the parameters of interest, all arranged as in Figure 2.12. If the model consists of a considerable number of equations and/or parameters, this may represent a formidable challenge. Fortunately there is a way to obtain equally accurate sensitivities in a much simpler way.
2.2.2.1 Approximate Trajectory Sensitivity
If we relax the basic hypothesis under which the sensitivity system (2.24) was derived, that
is, Spy y p y p
=∆limp→0
(
∆ ∆)
= ∂ ∂( )
, and instead assume that each parameter is perturbed by a finite increment Δp, then a handy way of computing an approximate trajectory sensitivity is available (Petersen, 2000; Petersen et al., 2003; De Pauw, 2005). Consider perturbing one parameter p by a very small (but finite) amount ∆p = ±δ, and perform the simulations with perturbed parameters, in addition to the nominal simulation with reference parameter value pnom. Then, the sensitivity Spy is computed as the average of the two side sensitivities S+ and S−:p y
If the incremental perturbation ∆p = ±δ is small enough, then Equation 2.31 is an accurate approxi-mation of the trajectory sensitivity system (2.24) without having to go through all the involved calculus. A word of caution is due here regarding very complex models, for which the choice of δ may become critical, and its optimal value should be determined (De Pauw, 2005; Iacopozzi et al., 2007). As an example, Figure 2.13 compares the exact (2.27) and approximate (2.31) sensitivities of the Monod kinetics to the maximum growth rate μmax. Though for δ = 0 05. there is still some discrepancy between the two sensitivities, for a smaller value (δ =0 0001. ) the two trajectories coin-cide for all practical purposes. Figure 2.14 shows the trajectory sensitivities for the complete set of parameters in the Monod kinetics.
The trajectory sensitivity approach can be used for planning the data collection with a view towards parameter estimation. In fact, the instants in which the magnitude of the sensitivity to a parameter is large are those in which that parameter has the highest influence on the system response, and hence the data collected in those time lapses will carry the highest amount of infor-mation for its estiinfor-mation. Figure 2.15 shows the most effective time brackets for collecting substrate and biomass data for μmax estimation.
In a similar way, the sensitivities of the Streeter & Phelps (S&P) model
BOD
with respect to its parameters Kb and Kc, can be obtained, as shown in Figure 2.16, for the single reach case with upstream point source. The shaded areas along the BOD and DO curves indicate the most favourable portion of the system evolution for gathering data used to estimate the two model parameters Kb and Kc. The sensitivity of BOD to Kc was not traced, being identically zero, because this parameter does not appear in the BOD model equation.
2.2.2.2 Sensitivity Ranking
The graphical representation of trajectory sensitivities yields a visual appraisal of the role played by each parameter along a certain system evolution. This information can be used as a basis for plan-ning a data collection campaign, but when it comes to ranking the parameters according to their sensitivity, a numerical figure of merit is certainly preferable. This numerical information can be extracted from the sensitivity trajectories by sampling them at regular intervals and computing their root-mean-square (RMS) value, that is,
ξi py
k N
j q
N S kij i np
= =
=
=
∑
∑
1 ( )2 1, ,
1 1
(2.33)
Extending the computation of Equation 2.33 to all of the np parameters in the model, they can be ranked according to their scores ξ, so that we can decide which are the most sensitive and limit the estimation to that critical subset. The N sampling times in Equation 2.33 can be freely selected and are not required to coincide with the actual sampling times.
Time (h)
0 20 40 60 80 100 120 140 160 180 200
−30
−20
−10 0 10
Substrate (mg/l)
Substrate Exact sensitivity to μmax
Approx. sensitivity to μmax for δ = 0.05
Biomass
Exact sensitivity to μmax
Approx. sensitivity to μmax for δ = 0.05
0 20 40 60 80 100 120 140 160 180 200
−5 0 5 10
Time (h)
Biomass (mg/l)
FIGURE 2.13 Comparison of exact and approximate trajectory sensitivities of the Monod kinetics to μmax. The approximate sensitivities for δ = 10−4 are superimposed to the exact ones, while some deviation is discern-ible for δ = 0.05.