(241c) Regularization and the Effect of Uncertainty in Parameter Estimation for NRTL Models
AIChE Annual Meeting
2023
2023 AIChE Annual Meeting
Computing and Systems Technology Division
Advances in Computational Methods and Numerical Analysis - II
Wednesday, November 8, 2023 - 4:13pm to 4:31pm
Typically, least squares estimators are applied to obtain parameter estimates using real-life experimental data. The uncertainty of the parameter estimates is commonly calculated using successive linearizations, which result in the confidence ellipsoids frequently published alongside parameter estimates.
The linearization is also used for the analysis of model uncertainty and consequently for stochastic optimization. The assumptions underlying these linearizations do not in general hold for highly nonlinear models as, e.g., in chemical engineering (cf. Fig. 1a). Hence, parameter estimates of nonlinear models should be handled carefully, until a new way of estimation of parameter errors is established. At the same time, the combination of nonlinearity of these models, measurement uncertainty, and small number of measurements available frequently causes ill-conditioning of parameter estimation (PE) problems.
While engineers use rigorous or semi-rigorous equations to describe phenomena, they end up with models which are âoverparametrizedâ for a system at hand: for example, they lack experiments at broader range of temperatures to describe the temperature dependence of a reaction rate. In such a case, estimation of parameters of an Arrhenius-type equation is impossible without regularization techniques, i.e., techniques ensuring conditioning and identifiability of the PE problems. In cases, where interdependence or correlation of parameters cannot be seen directly, regularization methods are essential.
Many regularization methods exist to solve these least squares problems. Among them are subset selection (Burth et al., 1999; Quaiser & Mönnigmann, 2009), Tikhonov regularization (Tikhonov & Arsenin, 1977), TSVD (Xu, 1998), and the orthogonalizations by Gram-Schmidt (Chu & Hahn, 2007) and Householder (Chen et al., 1989). All these regularizations imply a bias-error tradeoff through their hyperparameters.
In this contribution, we investigate the importance of proper experimental design and regularization and their effect on the solution of PE problems and uncertainty quantification of parameter estimates.
As a case study, the vapor liquid equilibrium (VLE) of 1-propanol and propyl-acetate is taken. The activity coefficients are described by the NRTL model, with five unknown parameters: two parameters for each component and the non-randomness parameter α.
In this example, the NRTL model cannot perfectly describe the VLE performance of the 1-propanol / propyl-acetate mixture, which is a frequent occurrence for complex nonlinear models in real-life engineering applications. Plant / model or data / model mismatch is inherent.
Measurement data both from a full factorial design â with three levels of granularity of 9, 15, and 27 experimental points â as well as from an optimal experimental design (OED), cf. (Vanaret et al., 2021), are applied to investigate the influence of the amount and quality of experimental data as well as the regularizations on the quality of the PE and the resulting prediction uncertainty.
Usually, analysis of the regularization techniques is done using artificially generated data. Here we base the entire analysis on actual experimental data, which is obtained in our own laboratory.
To analyze the effects on the performance of the least squares estimator, the amount of measurement data is varied, and different values are chosen for hyperparameters of the regularization methods mentioned above. At the same time, uncertainty estimates obtained by the Cramér-Rao bound (Fréchet, 1943) are compared against the ones calculated by Monte Carlo simulations.
In Fig. 1, preliminary results hereof are shown. On the right-hand side (b), the influence of selecting different subsets of identifiable parameters can be observed: For the 1-propanol / propyl-acetate mixture three parameters of the NRTL model were estimated: one binary interaction parameter for each component and α. Using regularization by subset selection (Quaiser, 2009), parameter is removed from the PE problem. This runs contrary to what is usually done in thermodynamics, where is typically fixed instead. Fig.1 (b) depicts both cases. Here, fixing leads to a larger confidence ellipsoid, i.e., larger uncertainty, compared to the solution suggested by the regularization.
Regarding PE in general, the results show that regularization is required to solve the ill-conditioned least squares formulations. Unidentifiable parameters lead to more solver iterations and badly conditioned Hessians near the solution. At the same time, the regularization leads to more accurate results and more reliable estimates regarding parameter uncertainty quantification. The identifiable subset of parameters has in general a lower quantifiable uncertainty than the larger set.
Regarding OED, the regularizations in PE problems may help in reducing the parameter space for the OED problems, which can lower computational and experimental cost.
In future work, we will more closely examine how these regularizations affect the iterative cycle of PE, OED, and real-life experiments.
References
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Figure 1: (a) comparison of a 95% confidence ellipses estimated by linearization and actual uncertainty of parameters calculated using Monte Carlo simulations (left); (b) comparison of 95% confidence ellipses that depend on a set of selected parameters (right).