1. Realistic Assessment of Parameter Uncertainty in Dynamic Parameter Estimation
Mordechai Shacham, Dept. of Chem. Engng, Ben Gurion University of the Negev, Beer-Sheva, Israel
Neima Brauner, School of Engineering, Tel-Aviv University, Tel-Aviv, Israel
Abstract
Assessment of the uncertainty in parameter estimation is essential for confidence in subsequent use of the dynamic model with the associated parameters. The assessment of the parameter uncertainty in highly nonlinear kinetic models is often a very difficult task. In this paper a new method for parameter uncertainty assessment is presented and its use is demonstrated for a cellulose hydrolysis kinetic model. The new method involves generation of pseudo-experimental data using a known set of “reference” parameter values. Stepwise regression is used in an attempt to generate alternative sets of parameter values that yield results with precision similar to the reference set. The difference between the individual parameter values in the separate sets represents the uncertainty of these values. High uncertainty level indicates that no physical meaning can be attributed to the predicted parameter values. Application of the proposed method is therefore recommended prior to applying the individual parameter values in other models.
Kinetic closed-system model of enzymatic saccharification of linocellulosic biomass1
Model Re-parametrization to Allow for Stepwise Regression
Data
Model
Part of the model, which is essential for describing the parameter estimation process, is shown in the equation sets above. The model contains three state variables: S, G2 and G, for the substrate, cellobiose and glucose concentration, respectively and 12 kinetic parameters: k1r, K1IG2, K1IG, k2r, K2IG2, K2IG, k3r, K3M and K3IG.
Using the experimental data provided by Kadam et al.1 , Sin et. al2 obtained estimates for the parameter values (including 95% confidence intervals).
Observe that all the confidence intervals are much larger in absolute value than the parameter values. This indicates that for the suggested model (and the available data) the parameters are un-identifiable.
Re-parametrization is carried out in order to prevent singularity of the model when a parameter value is set at zero.
In r1 (original expression) there are divisions by two parameters: K1IG2 and K1IG. In order to prevent singularity when these parameters are set at zero they are replaced by: π1 = 1/ K1IG2 and π2 = 1/ K1IG.
Similarly, the parameters in groups r2 and r3 are replaced as necessary, yielding: π3 = 1/ K2IG2, π4 = 1/ K2IG, π5 = k3r / K3M, π6 = 1/ K3IG and π7 = 1 / K3M.
Inspection of (re-parametrized) group r1 reveals that setting the parameter k1r at zero nullifies the whole group. Thus, k1r cannot be set at zero (assigned H = 1 hierarchy), while the rest of the parameters in group r1 : π1 and π2 , are of H = 2. Similarly for groups r2 and r3, k2r and π5 are of H = 1, and π3, π4, π6 , π7 are of H = 2.
Estimated parameters2
1Source: Kadam et. al, Biotechnol. Prog., 20, (2004).
2Source: Sin et. al, Comp. Chem. Engng., 34, (2010).
Stepwise Regression for the Simulated Data
Identification of different sets of parameter values that represent the data at similar precision levels
Stepwise regression was carried out using the simulated data. All parameter values were bounded: 10-3 lower bound and 100 upper bound.
The parameter values obtained by stepwise regression are significantly different from the “data verification” parameters, however, the sum of squared errors (Ф values) are very similar (7.16*10-5 vs *10-5 ).
The proposed method for assessing parameter uncertainty involve the use of stepwise regression to identify different sets of parameter values that represent the available data at similar precision levels.
For this aim simulated “experimental” data ( 150 data points) were generated using the re-parametrized model and the parameter values provided by Kadam et al1.
To verify the agreement of the stored “experimental” data with the re-parametrized kinetic model and the “Reference” parameter values, parameter identification was carried out using the fmincon function of MATLAB yielding the following results.
The π4 value is very close to the upper bound (set for the stepwise regression solution) and the value changes upon changing the upper bound.
In spite of the large differences in the parameter values, the stepwise regression curves fit well the simulated experimental data.
Observe the close agreement between the “reference” and “data verification” parameter values
Sum of squared differences between simulated “experimental” data and calculated values
1Source: Kadam et. al, Biotechnol. Prog., 20, (2004).
Conclusions
The results demonstrate that the proposed method enables identification of different sets of parameter values which can represent the available data at similar precision levels.
The difference between the individual parameter values in the different sets represents the uncertainty of the parameter values.
The main source of the parameters’ uncertainty is the interaction between the parameters. In such case no physical meaning can be attributed to the parameter values.
Application of the proposed method is therefore recommended prior to borrowing the individual parameter values for application in other models.