Parameter estimation and prediction uncertainties for multi‐response kinetic models with uncertain inputs

Abstract

Error-in-variables model (EVM) methods are used for parameter estimation when independent variables are uncertain. During EVM parameter estimation, output measurement variances are required as weighting factors in the objective function. These variances can be estimated based on data from replicate experiments. However, conducting replicates is complicated when independent variables are uncertain. Instead, pseudo-replicate runs may be performed where the target values of inputs for repeated runs are the same, but the true input values may be different. Here, we propose a method to estimate output-measurement variances for use in multivariate EVM estimation problems, based on pseudo-replicate data. We also propose a bootstrap technique for quantifying uncertainties in resulting parameter estimates and model predictions. The methods are illustrated using a case study involving n-hexane hydroisomerization in a well-mixed reactor. Case-study results reveal that assumptions about input uncertainties can have important influences on parameter estimates, model predictions and their confidence intervals.

Propagating input uncertainties into parameter uncertainties and model prediction uncertainties—A review

“…Recently, our group extended the available methods so they can be used for characterizing parameter and prediction uncertainties for multi‐response models. [ 37 ] In future, it will be possible to use parameter and prediction uncertainty quantification techniques for complicated dynamic multi‐response models involving ordinary differential equations.…”

“…The uncertainty associated with in Equation () is the uncertainty in the mean response that would be obtained from a very large number of runs with measured values of uncertain inputs given by , whereas the uncertainty associated with in Equation () accounts for both input uncertainty and output measurement uncertainty. To the best of our knowledge, distinguishing between these two types of prediction uncertainties has not been considered in the chemical engineering literature that focuses on the effects of input uncertainties until recently [ 37 ] (even though some modelling papers and popular undergraduate textbooks do talk about two types of prediction uncertainties when model parameters are uncertain but inputs are perfectly known). [ 38–40 ]…”

“…Yousefi et al [93] Triisopropylsilyl acrylate/Methyl methacrylate/Butyl Acrylate terpolymerization Linearization-based JCR* Krieger et al [24] Itaconic acid/N-vinyl-2-pyrrolidone Copolymerization Linearization-based JCR and CIs* Santos et al [23] Styrene/VeoVa-10 copolymerization Linearization-based CIs* Abdi and McAuley [74] N-butyl methacrylate and n-butyl acrylate copolymerization Linearization-based JCR and Bootstrap-based JCR Reischke [94] 2-methylene-1,3-dioxepane/vinyl acetate and methyl methacrylate/ dodecyl methacrylate copolymerization Linearization-based CIs and JCR, and Likelihood-based CIs and JCR Pablo-Morales et al [95] Myrcene and glycidyl methacrylate copolymerization Linearization-based JCR and CIs* Abdi et al [37] N-hexane hydroisomerization Bootstrap-based CIs and boxplots…”

“…Frutiger et al [115] Heat pump Monte Carlo Morgan et al [116] Solvent-based CO 2 capture Monte Carlo Mondejar et al [117] Rankine cycle Monte Carlo Jones et al [118] Heat recovery of organic Rankine cycle Monte Carlo Neba et al [119] Biogas plants Monte Carlo Fenila and Shastri [120] Enzymatic hydrolysis of lignocellulosic biomass Monte Carlo Kimaev et al [121] Chemical vapour deposition Monte Carlo Son and Du [122] Fed-batch penicillin manufacturing process Monte Carlo Abdi et al [37] Hexane hydroisomerization Monte Carlo used to obtain approximate parameter covariance matrices and approximate covariances for the model predictions. Lack of knowledge about degrees of freedom makes it difficult to construct accurate joint confidence regions and confidence intervals for parameters and model predictions.…”

“…To our knowledge, only three research groups have considered this combined EVM parameter estimation and prediction uncertainty quantification problem. [33,37,88] Two groups considered error propagation in simple models with a single response (dependent) variable. Recently, our group extended the available methods so they can be used for characterizing parameter and prediction uncertainties for multi-response models.…”