# Bayesian and subset-selection methods for parameter estimation in mechanistic models with limited data: A review and comparison

Jakob Straznicky, Lauren A Gibson, Benoit Celse, Kimberley B Mcauley

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Jakob Straznicky, Lauren A Gibson, Benoit Celse, Kimberley B Mcauley. Bayesian and subset-selection methods for parameter estimation in mechanistic models with limited data: A review and comparison. Chemical Engineering Research and Design, 2025, 216, pp.293-311. 10.1016/j.cherd.2025.02.037. hal-05079265

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Submitted on 22 May 2025

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# Bayesian and subset-selection methods for parameter estimation in mechanistic models with limited data: A review and comparison

Jakob I. Straznicky<sup>a</sup>, Lauren A. Gibson<sup>a</sup>, Benoit Celse<sup>b, c</sup>, Kimberley B. McAuley<sup>a, *</sup>

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

### Keywords:

Fundamental models

Parameter estimation

Subset selection

Estimability analysis

Bayesian statistics

## ABSTRACT

Parameters in mathematical models require accurate estimation for the model to give reliable predictions. When data are limited, weighted least-squares methods sometimes result in unreliable parameter estimates. Two popular approaches to combat this issue are subset-selection and Bayesian estimation. Subset-selection ranks model parameters from most- to least-estimable based on prior parameter knowledge and available data. The ranked list is used to determine which parameters should be estimated, and which should be fixed at initial guesses to avoid overfitting. Bayesian estimation methods summarize prior knowledge about parameters using probability distributions. Simple Bayesian methods result in objective functions with penalty terms that keep parameter estimates near their initial guesses unless there is considerable information in the data. Subset-selection and Bayesian methods result in different parameter estimates using the same data and similar prior information. A hydroisomerization case study is presented comparing the merits and shortcoming of each approach. Bayesian estimation is preferred if prior parameter knowledge is reliable, but provides misleading results when the modeler is overly confident about poor parameter guesses. Subset-selection methods are more computationally expensive, less susceptible to problems arising from poor initial guesses, and provide additional information about influences of model parameters and opportunities for model simplification.

## 1. Introduction

Mathematical models are used to simulate chemical and biochemical systems, scale-up new processes and select optimal operating conditions (Maria, 2004; Dobre and Marcano, 2007). When modelling chemical reactor systems, the resulting models often contain many kinetic rate constants and activation energies along with thermodynamic and transport parameters. These parameters are sometimes poorly known and appropriate estimates of their values are required for models to give reliable predictions. Common methods to estimate model parameters rely on optimizers to adjust the parameter values to ensure that model predictions are close to experimental data (de Levie, 1986). Chemical process models often contain many different types of response variables each with different units and different typical measurement errors. A popular method to estimate parameters when different types of measurements are available is weighted least-square (WLS), which corresponds to selecting values for model parameters in the vector $\theta$ that minimize the objective function:

$$J_{\text{WLS}} = \sum_{i=1}^{N_y} \frac{(y_i - g(\mathbf{x}_i, \theta))^2}{s_{y_i}^2} \quad (1)$$

where $N_y$ is the total number of measured values available, $y_i$ is the value of the $i^{th}$ measured response, $g(\mathbf{x}_i, \theta)$ is the corresponding model prediction and $s_{y_i}^2$ is the estimated variance of the $i^{th}$ measurement. In Eq. (1), $\mathbf{x}_i$ is a vector of experimental settings for the $i^{th}$ response. WLS objective functions arise naturally from maximum-likelihood estimation (MLE) when normally-distributed measurement errors are assumed (Flannery et al., 1992). Problems with WLS estimation may arise when there are many model parameters, there is a limited number of data values available for estimation, or when the data are collected in a small operating range (Chen et al., 2022; McLean et al., 2011). These problems include numerical difficulties during optimization and overfitting of the data, which leads to unreliable model predictions. Two popular statistical approaches to combat these issues are parameter subset selection and Bayesian estimation (BE) of the model parameters.

When subset selection techniques are used, the most influential

<footnote><sup>a</sup>Queen's University, Department of Chemical Engineering, Kingston, Canada</footnote>

<footnote><sup>b</sup>IFP Energies Nouvelles, Rond-point de l'échangeur de Solaize, Solaize, France</footnote>

<footnote>* Correspondence to: Queen's University, Department of Chemical Engineering, Kingston, ON, Canada. E-mail address: kim.mcauley@queensu.ca (K.B. McAuley).</footnote>
parameters are selected for estimation while less-important or highly-correlated parameters are fixed at their initial guesses or removed from the estimation problem via simplifying assumptions. Several different subset selection methods have been developed including those relying on orthogonalization, eigenvalues and eigenvectors, and correlation/collinearity measures (Brun et al., 2002; Quaiser and Mönnigmann, 2009; Chu and Hahn, 2009; Degenring et al., 2004; Vajda et al., 1989; Yao et al., 2003; Kou et al., 2005; Thompson et al., 2009). Table 1 summarizes studies (since 2020) involving real experimental data and fundamental models (also called mechanistic models) of chemical processes where the researchers relied on parameter subset selection methods to aid parameter estimation. In Table 1, the column with heading $n_0$ indicates the number of parameters in the model prior to subset selection (or model simplification) and the column with heading $n_{est}$ indicates the number of parameters that were selected and estimated. As indicated in Table 1, 22 studies used orthogonalization-based methods (i.e., those with $O$ in the right-most column) to select which parameters should be estimated, three studies used correlation/collinearity methods (i.e., those with $C$ in the right-most column), one used eigenvalue/principal component methods (i.e., with $E/P$ in the right-most column) and one removed parameters from the estimation that had low overall influence on the model predictions (i.e., with $L$ in the rightmost column). A larger table summarizing earlier (prior to 2020) parameter estimation studies involving subset selection is provided in Table S1 of the Supporting Information (SI) (Brun et al., 2002; Yao et al., 2003; Kou et al., 2005; Thompson et al., 2009, 2007; Dueñas Díez et al., 2006; Schittkowski, 2007; Brockmann et al., 2008; Machado et al., 2009; Freni et al., 2009; Jiménez-Hornero et al., 2009; Schmidt et al., 2010; Kim et al., 2010; Benyahia et al., 2011; Woloszyn and McAuley, 2011; Heitzig et al., 2011; Cosenza et al., 2013; Karimi et al., 2012; López et al., 2013; Lei et al., 2013; Pontes Freitas Alberton et al., 2015; Price et al., 2014; Baker et al., 2015; Aydın et al., 2015; Sildir et al., 2015; Salehi et al., 2016a; Yang et al., 2017; Casas-Orozco et al., 2018; Wierschem et al., 2017; Villegas et al., 2017; Karimi et al., 2018; Chen et al., 2018; Bui and Bhan, 2018; Yoon et al., 2019; Lim et al., 2019; Ho et al., 2019; Fysikopoulos et al., 2019). Additional information about subset-selection methods and how they have been used is provided in Section 2 below.

When Bayesian parameter estimation is used, the entire set of model parameters is estimated. In addition to kinetic and transport parameters, this parameter set may also include reaction rate orders and measurement noise variances. Prior knowledge about plausible parameter values, which is provided by the modeler, is used to ensure that parameters do not move very far from their initial guesses when information in the available data is limited (Gelman et al., 2014; Dochain and Vanrolleghem, 2001; Beck and Arnold, 1977). Table 2 shows studies since 2020 in which Bayesian methods have been employed for parameter estimation in fundamental chemical process models using experimental data. Notice that the models in Table 2 tend to contain many fewer parameters than those in Table 1 (i.e., an average of 9.2 parameters per model compared with an average of 33 parameters). This trend is also evident in Table S2 in the SI, which summarizes earlier BE studies involving real data in the chemical engineering literature (Olomon and Reilly, 1987; Anderson et al., 1992; Durán and White, 1995; Bois et al., 1997; Biglari et al., 2006; Coleman and Block, 2006; Hermanto et al., 2008; Su et al., 2014; Mathew and Duever, 2015; Ferrari et al., 2016; Wang et al., 2018a; Boosari et al., 2017; Matera et al., 2019). Some of the reasons why BE is less popular than subset selection for complicated models are described below in Section 3.

Both subset-selection and Bayesian methods require some prior information provided by the modeler when parameters are being estimated. For example, both approaches require parameter initial guesses and information about the uncertainties associated with different types of measurements. The most popular subset-selection method and Bayesian methods also require information about the modeler's uncertainty in their initial parameter guesses. Subset selection and Bayesian

<caption>**Table 1**</caption>

Recent studies that have used subset-selection methods in chemical process models with experimental data.

<table><thead><tr><th>Refs.</th><th>Process</th><th>Type of Data</th><th>n<sub>0</sub></th><th>n<sub>est</sub></th><th>M</th></tr></thead><tbody><tr><td>Cheng et al.<br>(2020) Aiello<br>et al. (2021)</td><td>Ethylene/α-olefin<br>copolymerization</td><td>MWD, CCD,<br>SLD</td><td>20<br>36</td><td>9<br>22</td><td>O</td></tr><tr><td>Gibson et al.<br>(2022)</td><td>using multi-site<br>catalysts</td><td>Polymerization<br>rate, MWD,<br>CCD</td><td>60<br>61</td><td>53<br>34</td><td></td></tr><tr><td>Gibson et al.<br>(2023)</td><td></td><td>Polymerization<br>rate, MWD,<br>CCD</td><td>27<br>27</td><td>8<br>9</td><td></td></tr><tr><td>Feng et al.<br>(2022)</td><td>Straznicky et al.<br>(2023)</td><td>Polymerization<br>rate, MWD,<br>CCD</td><td></td><td></td><td></td></tr><tr><td>Feng et al.<br>(2023)</td><td></td><td>MWD and short<br>chain<br>branching<br>distribution</td><td></td><td></td><td></td></tr><tr><td>Liu and McAuley<br>(2020)</td><td>Batch<br>copolymerization of<br>nylon 6/6,6</td><td>Polymerization<br>rate, MWD,<br>CCD, triad<br>sequences</td><td></td><td></td><td></td></tr><tr><td>Dieu Vo et al.<br>(2020)</td><td>Polyether<br>production from 1,3-<br>propanediol</td><td>MWD and short<br>chain<br>branching<br>distribution</td><td></td><td></td><td></td></tr><tr><td>Vo et al. (2020)</td><td>Polycondensation of<br>1,3-propanediol in a<br>batch reactor</td><td>Dynamic data<br>on<br>concentrations<br>of end groups,<br>linear monomer<br>and cyclic<br>dimer</td><td>23</td><td>21</td><td>O</td></tr><tr><td>Schenk et al.<br>(2021)</td><td>Multi-phase urea<br>compound synthesis<br>process</td><td>Dynamic data<br>on<br>concentrations<br>of monomer,<br>dimer, trimer,<br>tetramer,<br>pentamer,<br>heptamer and<br>water in the<br>liquid phase of<br>the reactor,<br>water in the<br>condensate<br>leaving the<br>condenser and<br>the mass of<br>liquid<br>accumulated in<br>the condensate<br>collector</td><td>23</td><td>18</td><td>O</td></tr><tr><td>Rodman and<br>Gerogiorgis<br>(2020)</td><td>Beer fermentation</td><td>Cyclic and<br>linear oligomer<br>data<br>Concentration<br>profiles on<br>substrate,<br>products and<br>impurities over<br>the reaction<br>time</td><td>36</td><td>36</td><td>O</td></tr><tr><td>Bouchkira et al.<br>(2021)</td><td>Pitzer's<br>thermodynamic<br>model</td><td>Glucose,<br>maltose and<br>maltotriose<br>concentrations<br>over 120 h at<br>various<br>temperatures</td><td>7</td><td>4</td><td>E/<br>P</td></tr><tr><td>Vo et al. (2021)</td><td>Biobased poly<br>(trimethylene) ether</td><td>Water activity<br>and speciation<br>versus molarity<br>of components<br>Concentrations<br>of water,</td><td>21</td><td>17</td><td>O</td></tr></tbody></table>

(continued on next page)
<table><caption>**Table 1 (continued)**</caption><thead><tr><th>Refs.</th><th>Process</th><th>Type of Data</th><th>n<sub>0</sub></th><th>n<sub>est</sub></th><th>M</th></tr></thead><tbody><tr><td></td><td>glycol production in reactor/condenser system</td><td>monomer, linear dimer, linear tetramer, cyclic dimer and cyclic tetramer over time</td><td></td><td></td><td></td></tr><tr><td>Abdi et al. (2022)</td><td rowspan="3">Free-radical photopolymerization</td><td>Vinyl group conversion</td><td>40</td><td>39</td><td rowspan="3">O</td></tr><tr><td>Vo et al. (2023)</td><td>using different light intensities, film thicknesses and initiator amounts</td><td>35</td><td>35</td></tr><tr><td>El Halabi et al. (2024)</td><td>Cell, substrate, product and by-product concentrations over five 80-hour fermentations</td><td>39</td><td>6</td></tr><tr><td>Chaleewong et al. (2022)</td><td>Succinate production by metabolically engineered e-coli</td><td>Water vapour absorbed as a function of equilibrium water vapour concentration at various temperatures</td><td>12</td><td>4</td><td rowspan="2">C</td></tr><tr><td>Cardenas et al. (2022)</td><td>Water vapor adsorption isotherms on alumina and hopcalite</td><td>Dynamic data on concentrations of viable cells, dead cells, ethanol and acetic acid</td><td>9</td><td>8</td></tr><tr><td>Jiménez-Hornero et al. (2022)</td><td>Wine vinegar acetification bioreactor</td><td>Speciation data on sulfur and phosphate, mineral solubility, and activity of water with 8 binary systems</td><td>9</td><td>8</td><td>L</td></tr><tr><td>Bouchkira et al. (2023)</td><td>Thermodynamic model on commercial phosphoric acid production from phosphate rock</td><td>Coke and benzene mass fractions, paraffin, naphthenic and aromatic mole fractions, temperature and pressure drops</td><td>21</td><td>17</td><td>O</td></tr><tr><td>Atarianshandiz et al. (2023)</td><td>Continuous catalytic naphtha reforming</td><td>Time varying data on starting material, intermediate and impurity concentrations</td><td>180</td><td>8</td><td>O</td></tr><tr><td>Moshiritabrizi et al. (2024)</td><td>Batch liquid phase production of precursor for HIV treatment API</td><td>Data on gas temperature at various reactor locations, and gas flow rate and composition at reactor outlet</td><td>8</td><td>5</td><td>O</td></tr><tr><td>Haidar et al. (2024)</td><td>Biomass downdraft gasifier</td><td>Biomass, glycerol, CO<sub>2</sub>, succinic acid, acetic acid and formic acid concentrations over 140 hours</td><td>40</td><td>27</td><td>O</td></tr><tr><td>Vigato et al. (2024)</td><td>Biosuccinic acid production via continuous fermentation</td><td></td><td>13</td><td>5</td><td>C</td></tr></tbody></table>

<table><caption>**Table 1 (continued)**</caption><thead><tr><th>Refs.</th><th>Process</th><th>Type of Data</th><th>n<sub>0</sub></th><th>n<sub>est</sub></th><th>M</th></tr></thead><tbody><tr><td>Méndez-Blanco and Özkan (2024)</td><td>Batch reactive distillation</td><td>Dynamic data on reactor temperature, pressure and viscosity</td><td>23</td><td>14</td><td>O</td></tr><tr><td>Elmisaoui et al. (2024)</td><td>Dissolution of phosphate ore in phosphoric acid</td><td>Dynamic data on fraction of dissolution at various temperatures</td><td>4</td><td>2</td><td>O</td></tr></tbody></table>

methods use this prior information in different ways, resulting in different parameter estimates from the same data. To our knowledge, no review article has focused on parameter subset-selection and Bayesian methods and how they have been used for parameter estimation in chemical process models. The objective of the current article is to provide such a review and to use a case study to explore the relative merits and shortcomings of the two approaches.

The remainder of this article is organized as follows. First, detailed background information is provided on subset-selection methods, paying closest attention to the popular orthogonalization-based approach. Next, BE methods are described. A case study involving a n-hexane hydroisomerization model with data from the University of Ghent is then presented. Parameter estimation using both subset-selection and Bayesian methods is conducted and the relative merits of the two approaches are assessed. Finally, advice is provided concerning which approach is better to use in different practical modelling situations.

## 2. Subset selection methods

As shown in Table 1 and Table S1, the subset-selection or “leave-out” (LO) approach has been used to aid parameter estimation in models for polymerization (Cheng et al., 2020; Aiello et al., 2021; Straznicky et al., 2023; Liu and McAuley, 2020), biochemical, pharmaceutical, and mineral processes (Schenk et al., 2021; Chaleewong et al., 2022; Bouchkira et al., 2023; Moshiritabrizi et al., 2024; Vigato et al., 2024; Elmisaoui et al., 2024) as well as other types of chemical processes (Schmidt et al., 2010; Aydın et al., 2015; Salehi et al., 2016a; Haidar et al., 2024). The most popular parameter subset-selection method for chemical process models is the orthogonalization-based method developed by Yao et al. and refined by Thompson et al., which was used in 22 of the 27 studies in Table 1 (Yao et al., 2003; Thompson et al., 2009). Details about this algorithm are provided in Table 3, which is discussed in detail below. In addition to chemical process models, the orthogonalization method has been applied to models in many other disciplines including metabolic cellular networks (Pontes Freitas Alberton et al., 2015; Villegas et al., 2017; Yue et al., 2006; Lee et al., 2018; Meshram et al., 2013), agricultural and soil processes (Wu et al., 2024; Wirsching et al., 2023; Agarwal and Sharma, 2022; Chavez Rodriguez et al., 2020; Ngo et al., 2015), aquatic toxicity (Quevedo et al., 2023), human movement and migration (Alsulami et al., 2022), pandemic forecasting (Krivorotko et al., 2021), environmental systems (Amato et al., 2020), mathematical biology (Baker et al., 2015; Olsen et al., 2019) and physiology (Balakrishnan et al., 2014; Maheshwari et al., 2011). As such, the algorithm has been used to select parameters for estimation in models described by algebraic equations (AEs) (Gibson et al., 2023; Feng et al., 2022), ordinary differential equations (ODEs)e.g., (Cheng et al., 2020; Aiello et al., 2021; Gibson et al., 2022; Straznicky et al., 2023; Liu and McAuley, 2020; Dieu Vo et al., 2020; Bouchkira et al., 2023; Méndez-Blanco and Özkan, 2024), differential algebraic equations (DAEs) (Fysikopoulos et al., 2019) and partial differential equations (PDEs) (Lei et al., 2013; El Halabi et al., 2024; Elmisaoui et al., 2024).

The algorithm in Table 3 ranks the model parameters from most-
<table><caption>**Table 2**
Summary of studies that use BE for chemical process models.</caption><thead><tr><th>Refs.</th><th>Process</th><th>Data</th><th>n<sub>0</sub></th><th>Bayesian Estimation</th></tr></thead><tbody><tr><td>Eisenkolb et al. (2020)</td><td>Batch enzyme catalyzed two-substrate reaction</td><td>Dynamic data on substrate concentration</td><td>6</td><td>Uniform prior on the log-transformed parameters<br>Parameter estimation using MCMC</td></tr><tr><td>Hernandez Lalinde et al. (2020)</td><td>CO<sub>2</sub> methanation and reverse water gas shift over mesoporous catalyst</td><td>Gas compositions and temperature difference along reactor axis</td><td>12</td><td>Prior distribution not specified<br>Parameter estimates obtained by minimizing objective function</td></tr><tr><td>Kalyanaraman et al. (2020)</td><td>Adsorption isotherm model for CO<sub>2</sub> sorption on UiO-66</td><td>CO<sub>2</sub> adsorption from NIST database</td><td>5</td><td>Normal prior for parameters<br>Parameter estimation using MC method</td></tr><tr><td>Shih et al. (2020)</td><td>Langmuir adsorption model to describe CO<sub>2</sub> adsorption on UiO-66</td><td>Literature data on CO<sub>2</sub> partial pressure</td><td>3</td><td>Uniform priors for all parameters<br>Parameter estimation using MCMC</td></tr><tr><td>Capecchi et al. (2021)</td><td>Dynamic model of n-butyl levulinate hydrogenation</td><td>Intermediate and product concentrations</td><td>7</td><td>Prior distribution not specified<br>Parameters estimated by minimizing objective function</td></tr><tr><td>Yamamoto et al. (2021)</td><td>Pulse and step injection of glucose and fructose by column chromatography</td><td>Dynamic glucose and fructose concentrations at column outlet</td><td>6</td><td>Uniform and normal priors for parameters<br>Parameter estimation using sequential MC</td></tr><tr><td>Hungenberg et al. (2022)</td><td>Dynamic model of batch copolymerization of acrylic acid and β-acroyloxypropionic acid</td><td>Monomer concentrations</td><td>2</td><td>Uniform priors for reactivity ratios<br>Parameter estimates obtained using grid search</td></tr><tr><td>Ostace et al. (2022)</td><td>Kinetic models for reduction of Fe-based oxygen carrier with CH<sub>4</sub>, CO, and H<sub>2</sub></td><td>Oxygen carrier conversion, and weight of oxidized and reduced samples</td><td>11</td><td>Uniform priors for all parameters<br>Parameter estimation using MCMC</td></tr><tr><td>Kojima et al. (2022)</td><td>Carbon dioxide adsorption isotherm on zeolite catalyst</td><td>CO<sub>2</sub> adsorption from NIST database</td><td>5</td><td>Uniform and normal priors used for parameters<br>Parameter estimation using MCMC</td></tr><tr><td>Hughes et al. (2022)</td><td>Kinetic and mass-transfer model for CO<sub>2</sub> capture</td><td>Wet wall column and pilot plant data on CO<sub>2</sub> outlet flow</td><td>3</td><td>Truncated normal prior distributions for all parameters<br>Parameter estimation using MCMC</td></tr><tr><td>Ferreira et al. (2023)</td><td>PDE model of hydrodynamic characterization of an aerobic fixed-bed bioreactor</td><td>Residence time distribution data with high and low flow using tracers</td><td>14</td><td>Uniform priors for all parameters<br>Parameter estimation using MCMC</td></tr><tr><td>Yuyama et al. (2024)</td><td>Equilibrium and mass-transfer model for CO<sub>2</sub> absorption</td><td>CO<sub>2</sub> gas concentrations at various absorber heights</td><td>15</td><td>Uniform and normal distributions for parameters<br>Parameter estimation using MCMC</td></tr><tr><td>Sugiyama et al. (2024)</td><td>Catalyzed esterification of acetic acid and methanol by cation exchange resin</td><td>Data from reactive and non-reactive chromatography experiments</td><td>10</td><td>Uniform priors for all parameters<br>Parameter estimating using sequential MC methods</td></tr></tbody></table>

<table><caption>**Table 3**
Yao's orthogonalization-based parameter ranking algorithm, adapted from Yao et al. (2003), McLean and McAuley (2012).</caption><thead><tr><th>Step</th><th>Process</th></tr></thead><tbody><tr><td>1.</td><td>Calculate the magnitude of each column of Z using the Euclidian norm.</td></tr><tr><td>2.</td><td>The parameter corresponding to the column of Z with the largest magnitude is set at the most estimable parameter. Set k = 1 for the first iteration.</td></tr><tr><td>3.</td><td>Construct a matrix X<sub>k</sub> that includes the parameters from Z that have been ranked</td></tr><tr><td>4.</td><td>Calculate Ẑ<sub>k</sub>, the prediction of the matrix Z using the subset of k columns that have been ranked. Ẑ<sub>k</sub> = X<sub>k</sub>(X<sub>k</sub><sup>T</sup>X<sub>k</sub>)<sup>-1</sup>X<sub>k</sub><sup>T</sup>Z</td></tr><tr><td>5.</td><td>Calculate the residual matrix R<sub>k</sub> = Z - Ẑ<sub>k</sub></td></tr><tr><td>6.</td><td>Calculate the magnitude of each column of R<sub>k</sub>. The (k+1)th-most estimable parameter will correspond to the column with the largest magnitude</td></tr><tr><td>7.</td><td>Construct a new matrix X<sub>k+1</sub> to include a column for this new ranked parameter.</td></tr><tr><td>8.</td><td>Increase the iteration count, k, by one each time and repeat steps 3–7 until all the parameters have been ranked.</td></tr></tbody></table>

estimable to least-estimable according to their influences on predictions of available data while accounting for correlated effects of parameters on model predictions. This method relies on a scaled sensitivity matrix of the model predictions with respect to the parameters (Yao et al., 2003):

$$Z = \begin{bmatrix} \frac{\partial g_1}{\partial \theta_1} & \frac{s_{\theta_0}}{s_{y_1}} & \dots & \frac{\partial g_1}{\partial \theta_p} & \frac{s_{\theta_0 p}}{s_{y_1}} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \frac{\partial g_n}{\partial \theta_1} & \frac{s_{\theta_0 1}}{s_{y_n}} & \dots & \frac{\partial g_n}{\partial \theta_p} & \frac{s_{\theta_0 p}}{s_n} \end{bmatrix}_{n \times p} \quad (2)$$

Matrix Z has one column for each model parameter and one row for the prediction of each data value that will be used for parameter estimation. In Eq. (2), $g_i$ (where $i = 1...n$) is the model prediction of the ith data value and $\theta_j$ (where $j = 1...p$) is the jth model parameter. In situations involving dynamic models with multiple experimental runs and multiple responses measured at several times in each run, the value of n can be large (e.g., the product of the number of runs, the number of response variables, and the number of sampling times per run). Notice that sensitivity coefficients $\partial g_i / \partial \theta_j$ contained in Z are multiplied by scaling factor $s_{\theta_0 j}$, which reflects the modeler's uncertainty in the initial value of the jth parameter, and by scaling factor $s_{y_i}$, which reflects the typical uncertainty for the ith measured response. The value of $s_{\theta_0 j}$ can come from estimation results from previous experiments or from the modeler's confidence in each initial parameter guess. The value of $s_{y_i}$ can come from replicate experiments or from accuracy information specified by the instrument manufacturer. These scaling factors ensure fair comparisons among sensitivity coefficients involving different types of parameters and different types of measurements which may have a variety of magnitudes and units. The majority of studies in Table 1 obtained the sensitivity coefficients in Z using perturbations around the initial parameter guesses (Aiello et al., 2021; Gibson et al., 2022; Straznicky et al., 2023; El Halabi et al., 2024; Moshiritabrizi et al., 2024). For example, some studies used:

$$\frac{\partial g_1}{\partial \theta_1} \Big|_{\theta_0} = \frac{g_1(1.05\theta_0, \theta_0, \dots, \theta_p) - g_1(\theta_0, \dots, \theta_p)}{0.05\theta_0} \quad (3)$$

to obtain the first element in Z (Aiello et al., 2021; Gibson et al., 2022; Straznicky et al., 2023; El Halabi et al., 2024; Moshiritabrizi et al., 2024). Perturbations of 5% (and also 2%) of the initial parameter
guesses are popular because they are large enough to produce significant changes in the model predictions compared to errors in numerical solutions for ODEs, DAEs and PDEs, (Saltelli et al., 2004) and are small enough to approximate the required derivatives. An alternative is to obtain sensitivity coefficients by solving sensitivity differential equations along with the model equations (Gibson et al., 2024; Leis and Kramer, 1998; Pauw and Vanrolleghem, 2006). Notice that in Eqs. (2) and (3) the sensitivities are evaluated around the initial parameter guesses in $\theta_0 = [\theta_{01}, \theta_{02}, \dots, \theta_{0j}]^T$. The numerical values of the sensitivities in Eq. (3) depend on the initial parameter guesses that are specified. To better account for nonlinearity of the model equations, some researchers have used global sensitivity analysis techniques (e.g., Sobol) when computing sensitivities (Brockmann et al., 2008; Bouchkira et al., 2021; Cardenas et al., 2022; Jiménez-Hornero et al., 2022). Sobol considers variability in sensitivities across the parameter space of interest rather than relying on local sensitivities obtained using the initial parameter guesses (Saltelli et al., 2008; Celse et al., 2015). The downside of using global sensitivities instead of local sensitivities is the high computational expense, because the model equations must be solved a large number of times to evaluate the global sensitivity coefficients for each parameter (Fysikopoulos et al., 2019).

After numerical values for the elements of Z have been calculated, the algorithm in Table 3 is used to rank the parameters from most estimable to least estimable. As indicated in Step 1, the column with the largest magnitude (i.e., Euclidean norm) is selected as most estimable. This column contains large (positive and negative) values compared to other columns, indicating that the corresponding parameter has large influence on model predictions. Note that, a large uncertainty $s_{\theta_{0j}}$ for initial parameter value $\theta_{0j}$ will contribute to a large magnitude for the jth column so that parameters with large initial uncertainties may be selected instead of other important parameters that are well known. Steps 3–6 in Table 3 are used to account for the correlated influences of model parameters. The ranking procedure continues until all model parameters are ranked, or until it becomes impossible to invert $X_k^T X_k$ due to singularity. Any unranked parameters should be excluded from parameter estimation to avoid numerical difficulties. Multiple sets of initial parameter guesses can be used by modelers who are concerned that their parameter ranking results may be influenced by their choices for initial parameter values (McLean and McAuley, 2012).

The algorithm in Table 3 is easier to use and is less computationally expensive than collinearity methods, which consider related influences of all possible pairs of parameters using a combinatorial approach. A further benefit of the method in Table 3 is that it automatically accounts for complex relationships among three or more parameters via Steps 4 and 5. In Step 4, columns in Z are fitted as linear combinations of the columns for parameters that have already been ranked. Step 5 computes a residual matrix $R_k$, which provides information about the marginal influence of the unranked model parameters. Using the residual matrix to select the next parameter on the list ensures that complex interactions among unranked parameters and previously ranked parameters are considered.

The algorithm in Table 3 is a forward selection method because it starts by finding the most important parameter and then adds parameters to the ranked list one-at-a-time, assuming that all previously-ranked parameters will be estimated. Backward selection algorithms have also been developed where the least-estimable parameter is determined first and then the remaining parameters are ranked from least-estimable to most-estimable, assuming that the previously selected parameters will not be estimated (Sutter and Kalivas, 1993). Like the orthogonalization-based method in Table 3, methods based on eigenvalues or principal components also account for complex interactions among parameters (Degenring et al., 2004; Vajda et al., 1989; Schittkowski, 2007; Kim and Lee, 2019). These methods, denoted by E/P in Table 1 and Table S1, find linear combinations of the model parameters that have large influences on the model predictions. These linear

combinations are found using eigenvalue/eigenvector decomposition of $Z^T Z$, which is a Fisher Information Matrix (FIM), or an unscaled analogue of $Z^T Z$ (Jolliffe, 2002). The E/P method which was introduced by Vajda et al. (1989) and improved by Degenring et al. (2004) and Schittkowski (2007) is a backward-selection method. The method begins by determining the eigenvalues of the FIM and ranking them from highest to lowest. The least-estimable parameter is selected by examining the eigenvector corresponding to the smallest eigenvalue. The parameter with the largest weight in this eigenvector is selected as the least-estimable parameter. The second parameter on the ranked list is selected because it has a large weight in the eigenvector for the second-smallest eigenvalue. A downside of this method is that unimportant parameters with relatively high weights in eigenvectors for the small eigenvalues may be ignored multiple times and are included in the parameter estimation problem as a result. A modification by Schittkowski is helpful for addressing this issue. Schittkowski's variant of the E/P algorithm removes the column corresponding to the least-important parameter from Z prior to eigenvalue/eigenvector decomposition to determine the second-least-estimable parameter. In this way, some interaction effects of parameters on eigenvectors is removed, which helps in construction of a reliable ranked list of parameters.

When using E/P methods, researchers have used several approaches for scaling their sensitivity matrices. Some authors have scaled the model predictions using measurement uncertainties $s_{y_i}$, (Schittkowski, 2007; Schmidt et al., 2010; Sildir et al., 2015; Kim and Lee, 2019) as shown in Eq. (2), but others have scaled using typical values of the measured responses (Degenring et al., 2004; Salehi et al., 2016a). Scaling by the measurement uncertainties is preferable because it leads to parameters being selected for estimation that have large influence on predictions for reliable measurements. By contrast, parameters whose main influence is on predictions of unreliable measurements may not be selected due to lack of information in the data. None of the users of the E/P methods scaled their sensitivities by the initial parameter uncertainties $s_{\theta_{0j}}$, as shown in Eq. (2). Some E/P users did not use any scaling for the model parameters, which is problematic when parameters have different units and/or different magnitudes (Schittkowski, 2007; Schmidt et al., 2010; Sildir et al., 2015). Others scaled their sensitivities using the initial parameter guesses, which ensures dimensional consistency in the E/P calculations (Degenring et al., 2004; Salehi et al., 2016b). We recommend that, in future, E/P users should scale based on the modeler's level of uncertainty in each initial parameter guess. One way to obtain suitable uncertainty information for each parameter is from the lower and upper bounds that will be employed during WLS parameter estimation. Some users of the orthogonalization-based method in Table 3 obtained their parameter uncertainty scaling factors from (Aiello et al., 2021; Straznicky et al., 2023):

$$s_{\theta_{0j}} = \frac{1}{2} (\theta_{UB_j} - \theta_{LB_j}) \quad (4)$$

where $\theta_{UB_j}$ is the lower bound on the estimate of $\theta_j$ and $\theta_{LB_j}$ is the upper bound. Parameters that have a small value of $s_{\theta_{0j}}$ will tend to be removed from the estimation because the modeler already has considerable knowledge about plausible values for these parameters.

Three research groups denoted by L in Table 1 and Table S1 used global sensitivity techniques to obtain their parameter sensitivity coefficients and then removed parameters from the estimation problem if they had low sensitivities (Fysikopoulos et al., 2019; Ferrari et al., 2016; Jiménez-Hornero et al., 2022). We do not recommend this simple approach. Removing parameters from the estimation problem because they have small influence on predictions of the available data is often a good idea. However, the L method is not able to detect problems associated with two or more influential parameters that have highly correlated effects. For example, important parameters $\theta_1$ and $\theta_2$ could always appear together as a product in the model equations. Each could have a large global sensitivity, but it would be impossible to estimate these two
parameters uniquely. Unlike the *L* approach, the orthogonalization, collinearity and E/P approaches would detect this problem so that only one of θ₁ or θ₂ would be selected for estimation. Note that Ferrari et al. used the *L* approach to remove parameters with low sensitivities from their estimation problem and then used a Bayesian approach to estimate the influential parameters (Ferrari et al., 2016). As a result, they did not experience problems related to overfitting or parameter correlation. Information about BE and why it is effective is provided in Section 3.

After the parameters have been ranked using *O*, *C* or *E/P* methods, several different techniques have been used to determine a suitable parameter subset for estimation. For example, early researchers who used *O* methods examined the columns of **R**<sub>k</sub> after each parameter was added to the ranked list (Yao et al., 2003; Kou et al., 2005; Jayasankar et al., 2009). If the magnitude of the largest column in **R**<sub>k</sub> was below an arbitrary cut-off value, all remaining unranked parameters were deemed unestimable. The problem with this approach is that a suitable cut-off value was difficult to specify. Thompson et al. adopted a brute force cross-validation method to determine the number of ranked parameters to estimate in a polyethylene reactor model (Thompson et al., 2009). For each candidate parameter subset, parameter estimation was performed multiple times with a different run from the dataset left out. Predictions for the left-out runs were used to compute an overall weighted sum of squared residuals (WSSR). The parameter subset with the smallest WSSR was selected as the number of parameters from the ranked list that should be estimated. This subset-selection method is reliable because it selects the parameter subset that gives the best predictions for data that were not included in parameter estimation. Unfortunately, this cross-validation method requires many repeated parameter estimations, making it computationally expensive.

Littlejohns et al. proposed a less-computationally-expensive option involving a series of parameter estimations (using all the data) as shown in Fig. 1 (Littlejohns et al., 2010a, 2010b). Using this method, the objective function value ($J_{WLS}$) was first computed using the initial parameter guesses. Then, the first parameter from their ranked list was estimated, with all others fixed at their initial guesses. Next the top two parameters were estimated together, with the remaining $p-2$ parameters fixed, and so on. Using this figure, Littlejohns et al. (2010b) decided that 25 out of 27 model parameters should be estimated in their bioscrubber model, because the decrease in $J_{WLS}$ was negligible when the two bottom-ranked parameters were included in the estimation, as shown in the inset. The downside of this approach is that it is often difficult to judge how many parameters should be estimated based on the plot. For example, another modeler could reasonably argue that only 22 parameters should be selected for estimation, because the decrease in

$J_{WLS}$ is small when additional parameters are included.

Wu et al. derived a mean-squared-error (MSE) criterion $r_{cc}$ that can be used to select the appropriate number of parameters from ranked parameter lists (Wu et al., 2011a, 2011b). This criterion is a ratio that focuses on the trade-off between decreases in bias and increases in variance of model predictions when more parameters are included in the estimation. Table S4 in the SI shows how $r_{cc}$ is computed using values of $J_{WLS}$ when $k$ parameters are estimated. Fig. 2 shows a plot of $r_{cc}$ versus the number of estimated parameters for a batch reactor case study which shows that $r_{cc}$ reaches its minimum value when $k = 5$ parameters are estimated (Biegler et al., 1986). Wu et al. used several case studies to confirm that selecting parameters based on the minimum value of $r_{cc}$ gave the same results as the computationally-expensive cross-validation approach of Thompson et al. (2009), Wu et al. (2011b). A benefit of Wu's technique is that it does not require any arbitrary cutoff value or judgments about the shape of a plot to determine the appropriate parameter subset. One complication that can arise when using $r_{cc}$ is that it is computed using $J_p$, which is the value of the WLS objective function obtained when all $p$ parameters are estimated. In situations where it is impossible to estimate all $p$ parameters simultaneously (due to ill-conditioning), $r_{cc}$ can be calculated by assuming that $J_p$ is equal to the objective function value obtained when the largest possible number of parameters is estimated (i.e., by assuming that estimating additional parameters would not lead to a better model fit) (Woloszyn and McAuley, 2011).

Wu's criterion has been used to aid parameter estimation for a variety of chemical process models (Feng et al., 2023; Liu and McAuley, 2020; El Halabi et al., 2024; Bouchkira et al., 2023; Moshiritabrizi et al., 2024; Elmisaoui et al., 2024). The main downside of Wu's MSE-based method is that its derivation focuses on minimizing the MSE of model predictions at conditions where data are available. Chemical engineers are often more interested in using their models to make predictions at new operating conditions where they do not have data. To address this issue, Eghtesadi and McAuley extended the MSE-based criterion to consider the accuracy of predictions at new operating conditions where accurate predictions are desired (Eghtesadi and McAuley, 2014). This method has been demonstrated using case studies with synthetic data, but has not been used to aid parameter estimation in chemical process models using real data. The main impediment to its use is that, during the early stages of model development and parameter estimation, modelers may lack information or confidence about specific operating conditions where their model will be used in the future. Since Wu's method is easier to use than Eghtesadi's, our research group tends to use Wu's method instead.

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** $J_{WLS}$ versus number of parameters estimated for a bioscrubber model, adapted from Littlejohns et al. (2010b).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Plot of Wu's MSE-based criterion $r_{cc}$ versus $k$ the number of parameters estimated from the ranked list, adapted from McLean et al. (2011).</figcaption></figure>
A benefit of using orthogonalization-based ranking combined with the $r_{cc}$ criterion for subset selection is that this approach provides information about why specific parameters have not been selected for estimation. Liu et al. developed a diagnosis method where magnitudes of columns in $R_k$ are used to determine if a parameter was not selected for estimation due to lack of influence on model predictions or to correlated effects with other parameters (Liu et al., 2021). This type of diagnosis is helpful when considering future experiments to aid parameter estimation. Instead of keeping some parameters fixed at their initial guesses or collecting additional data, it is often beneficial to simplify model equations to remove difficult-to-estimate parameters. For example, our research group has sometimes assumed that activities of multiple types of catalyst sites are the same when there was insufficient information in the data to estimate unique values for different sites (Thompson et al., 2009, 2007; Aiello et al., 2021; Gibson et al., 2022, 2023; Straznicky et al., 2023). We have also chosen to ignore temperature effects for some reactions when there is insufficient information to estimate activation energies (Thompson et al., 2009; Moshiritabrizi et al., 2024).

To date, all research groups in Table 1 and Table S1 who used collinearity methods for parameter ranking selected an arbitrary cutoff value for their collinearity measure and used it to determine the number of parameters that should be estimated. Similarly, most researchers who used E/P methods for parameter ranking chose an arbitrary cutoff value for the smallest eigenvalue that should be permitted when selecting a subset of parameters to estimate. Much like the cut-off for columns in $R_k$ used by Yao et al., these similar arbitrary specifications are difficult to choose based on information about the model and the available data. As a result, we recommend that Wu's statistically motivated MSE criterion should be used instead, regardless of the method employed to rank the parameters.

### 3. Bayesian methods

Bayesian parameter-estimation methods are based on the idea that previous knowledge about reasonable values for parameters can be summarized using prior probability distributions and then used to aid parameter estimation (Gelman et al., 2014). This prior knowledge may arise from information about previous parameter estimates from similar systems, or from the modeler's knowledge about the range of parameter values that make physical sense. Fig. 3 shows three popular prior distributions that can be used to summarize prior knowledge on a parameter $\theta$. Fig. 3a shows a normal prior distribution centered at an initial guess of $\theta_0 = 2$ with a standard deviation of $\sigma_{\theta_0} = 1.5$. Fig. 3b shows a truncated normal prior distribution also centered at an initial guess of $\theta_0 = 2$ with a standard deviation of $\sigma_{\theta_0} = 1.5$, but this time truncated between 0 and 4 so that there is zero probability of the parameter value lying outside these bounds. Lastly, Fig. 3c shows a uniform prior distribution between 0 and 4.

As shown in Table 2 and Table S2, Bayesian methods have been used to estimate parameters in fundamental models for polymerization (Mathew and Duever, 2015; Hungenberg et al., 2022), crystallization

(Hermanto et al., 2008; Su et al., 2014), sorption processes (Matera et al., 2019; Kalyanaraman et al., 2020; Shih et al., 2020; Kojima et al., 2022; Yuyama et al., 2024), biochemical processes (Bois et al., 1997; Coleman and Block, 2006; Capecci et al., 2021; Ferreira et al., 2023), and other chemical processes (Olomon and Reilly, 1987; Anderson et al., 1992; Durán and White, 1995; Boosari et al., 2017; Sugiyama et al., 2024). Of the studies in Table 2 and Table S2, 13 used uniform priors (Anderson et al., 1992; Biglari et al., 2006; Coleman and Block, 2006; Su et al., 2014; Mathew and Duever, 2015; Shih et al., 2020; Yamamoto et al., 2021; Hungenberg et al., 2022; Ostace et al., 2022; Kojima et al., 2022; Ferreira et al., 2023; Yuyama et al., 2024; Sugiyama et al., 2024), 10 used normal priors (Olomon and Reilly, 1987; Hermanto et al., 2008; Mathew and Duever, 2015; Ferrari et al., 2016; Wang et al., 2018a; Boosari et al., 2017; Matera et al., 2019; Kalyanaraman et al., 2020; Yamamoto et al., 2021; Kojima et al., 2022) and 4 used truncated normal priors (Bois et al., 1997; Coleman and Block, 2006; Hermanto et al., 2008; Hughes et al., 2022). Like the subset selection techniques described in Section 2, Bayesian methods have been used to aid parameter estimation in models described by AEs (Boosari et al., 2017), ODEs (Olomon and Reilly, 1987; Anderson et al., 1992; Bois et al., 1997; Biglari et al., 2006; Coleman and Block, 2006; Mathew and Duever, 2015; Matera et al., 2019; Kalyanaraman et al., 2020; Shih et al., 2020; Capecci et al., 2021; Hungenberg et al., 2022; Ostace et al., 2022; Kojima et al., 2022) and PDEs (Durán and White, 1995; Hermanto et al., 2008; Su et al., 2014; Ferreira et al., 2023). All of the studies in Table 2 and Table S2 are concerned with parameter estimation using real process data. Other studies that used synthetic data to test BE methods for chemical process models are not included (Reischke, 2023; Fan et al., 2024; Lu et al., 2002, 2004; Hsu et al., 2009; Masoumi et al., 2013; Galagali and Marzouk, 2015; Costa and Naveira-Cotta, 2019; Overstall et al., 2019).

BE works by combining information about parameter values from the prior distributions with information in the available data. When normal priors are specified, simple closed-form Bayesian objective functions can be derived (Hsu et al., 2009; Gelman et al., 2014; Shah-mohammadi and McAuley, 2020):

$$J_B = \sum_{i=1}^{N_y} \frac{(y_i - g(x_i, \theta))^2}{\sigma_{y_i}^2} + (\theta - \theta_0)^T \Sigma_{\theta_0}^{-1} (\theta - \theta_0) \quad (5)$$

where $\sigma_{y_i}^2$ is the variance of the error in the $i^{th}$ measured response, which may be assumed known or may be estimated from replicate experiments, and $\Sigma_{\theta_0}$ is the covariance matrix of the initial parameter guesses, which must be specified by the modeler. If the off-diagonal elements of $\Sigma_{\theta_0}$ are zero, which is the usual case, Eq. (5) simplifies to:

$$J_B = \sum_{i=1}^{N_y} \frac{(y_i - g(x_i, \theta))^2}{\sigma_{y_i}^2} + \sum_{j=1}^{p} \frac{(\theta_{0j} - \theta_j)^2}{\sigma_{\theta_{0j}}^2} \quad (6)$$

where $\theta_{0j}$ is the mean of the prior normal distribution for the $j^{th}$ parameter and $\sigma_{\theta_{0j}}$ is the corresponding standard deviation. This form of

<figure><img src="image_5.png" /><figcaption>Fig. 3. Prior knowledge about parameter $\theta$ summarized using a) a normal distribution, b) a truncated normal distribution and c) a uniform distribution.</figcaption></figure>
the Bayesian objective function is frequently used because modelers often specify independent normal prior distributions for the parameters (Shahmohammadi and McAuley, 2020). To our knowledge, none of the studies in Table 2 and Table S2 involved non-zero off-diagonal elements in Σθ₀ because modelers rarely have access to this type of prior knowledge.

Notice that Eq. (6) is the same as the WLS objective function in Eq. (1), except that there are added quadratic penalty terms. These penalty terms ensure that parameter estimates do not move away from their initial guesses, unless there is sufficient information in the experimental data. The degree to which the modeler believes the initial guess on the jth parameter is reflected in the choice of σ²θ₀ⱼ. If σ²θ₀ⱼ is small, the modeler is confident in the initial parameter guess, leading to a greater penalty when the optimizer moves the parameter estimate away from this value. Alternatively, if σ²θ₀ⱼ is large, the modeler has low confidence in the initial parameter guess, and the optimizer has more freedom to move the parameter estimate away from the initial guess. In the studies in Table 2 and Table S2 that relied on normal priors, the modelers specified prior distributions for all of the model parameters. However, it would also be possible to specify prior distributions for only some of the parameters, resulting in a smaller number of parameter deviation terms in Eq. (6). If a parameter has no specified prior distribution, the optimizer adjusts the parameter value to fit the experimental data, but no additional penalty term ensures that the parameter remains near its initial guess. Modelers may prefer to not impose priors for some parameters whose initial guesses are highly uncertain.

Modelers sometimes feel uncomfortable when specifying their prior knowledge about model parameters using normal distributions because normal distributions permit parameter values that range from negative infinity to positive infinity, albeit with low probabilities for very small and very large parameter values. When estimating model parameters, modelers routinely specify bounds on the parameters to prevent the optimizer from selecting physically unrealistic parameter values. Unfortunately, normal priors are not consistent with these modeler-specified bounds. To address this issue, modelers sometimes specify prior distributions with a finite parameter range, such as uniform and truncated normal distributions. In the past, using these priors forced modelers to resort to complex sampling algorithms to carry out parameter estimation, most commonly Markov Chain Monte Carlo (MCMC) methods and related sampling-based techniques. e.g. Biglari et al., 2006; Coleman and Block, 2006; Hermanto et al., 2008; Mathew and Duever, 2015; Shih et al., 2020; Kojima et al., 2022, These methods were used because modelers did not have a closed-form objective function for parameter estimation (like Eq. (6)). Using MCMC methods (such as the Metropolis-Hastings algorithm) involves thousands of simulations (sometimes millions) with different candidate parameter values to provide a full joint probability distribution of the parameter estimates (known as the *posterior distribution*) (Gelman et al., 2014; Gamerman and Lopes, 2014). As shown in Table 2 and Table S2, these methods tend to be used for chemical process models with only a small number of parameters (i.e., 20 studies with 2 ≤ n₀ ≤ 10 and only 3 study with n₀ ≥ 15). These sampling-based methods tend to be computationally expensive and are more difficult to implement and understand than BE using Eq. (6). In our experience, implementation issues are sometimes encountered using MCMC and in-depth knowledge about how to tune the algorithms is required to obtain accurate results, even when open-source or commercial codes are used (Gelman et al., 2014; Brooks, 1998; Chib, 1995). An important issue is the specification of tuning parameters, such as the choice of the proposal distribution for the Markov chain, the length of the burn-in period, the length of the Markov chain, and the number of chains to run (Brooks, 1998; Chib, 1995). It is particularly difficult to find an efficient proposal distribution for problems with many parameters, making MCMC methods ill-suited for large dynamic chemical process models (Bardsley et al., 2014). As a result, there is a growing interest in using other optimization-based methods

for BE (Bardsley et al., 2014; Wang et al., 2018b; Gibson and McAuley, 2024).

Recently, Gibson and McAuley showed that the closed-form objective function in Eq. (6) arises naturally when truncated normal distributions are specified, as long as parameter bounds of the form (Gibson and McAuley, 2024):

$$ \theta_{LB_j} \le \theta_j \le \theta_{UB_j} \qquad (7) $$

are specified during parameter estimation. They also showed that BE with uniform prior distributions results in the WLS objective function in Eq. (1) when parameter bounds are enforced during estimation. In addition to these proofs, they proposed a parametric bootstrapping approach (see Table S5 for details) to determine confidence intervals (CIs) for parameters while accounting for truncated normal priors and model nonlinearity. They showed that, as expected, their bootstrap CIs and parameter estimates obtained using Eqs. (6) and (7) are nearly identical to results obtained using MCMC for a pharmaceutical case study. This finding is significant because it reveals that traditional gradient-based optimization tools can be readily used to obtain accurate Bayesian CIs and parameter estimates. The resulting computational effort and tuning requirements for this approach should be much more modest than those required for MCMC methods, especially when larger-scale models with many equations and parameters are considered.

Based on the entries in Tables 1, 2, S1 and S2, Bayesian estimation in chemical process models is used less frequently than subset-selection methods, especially for larger-scale problems. This tendency is due to several reasons, including: (i) difficulties in understanding and tuning MCMC and related sampling-based algorithms and (ii) misgivings about specifying prior parameter distributions when there is considerable uncertainty in some or all parameter values. Modelers who are uncomfortable specifying prior distributions for parameters may decide to use subset-selection methodologies as an alternative. However, it should be noted that prior information must also be specified when subset-selection methods are used (i.e., initial parameter guesses, parameter uncertainties for use in scaling and parameter bounds to be enforced during estimation). Employing a truncated normal distribution or a uniform distribution based on this information requires specification of only limited additional knowledge about the shape of the distribution. When there is high uncertainty in some or all parameter values, modelers may choose to specify less-informative priors, such as uniform priors, rather than truncated normal priors. Unfortunately, uniform priors do not help with ill-conditioned parameter-estimation problems and overfitting, because the resulting objective function does not include the second term in Eq. (6). As a result, we recommend that a truncated normal distribution with a large value of σ²θ₀ⱼ should be used instead. In this way, the second term will appear in the objective function (albeit with low weights) so that the optimizer will keep some poorly known parameters near their initial guesses when some parameters have highly correlated influences and/or there is limited information in the dataset.

A benefit of BE is that all parameters in θ are estimated, so that repeated parameter estimations (e.g., like those required to produce Figs. 1 and 2) are not required. As a result, BE using a closed-form objective function is faster and easier to use than subset-selection methods. When Bayesian estimation is used along with bootstrapping to obtain CIs, some repeated parameter estimation is required (Gibson and McAuley, 2024). A benefit of this approach is that CIs (or joint confidence regions) are automatically obtained for all parameters. By contrast, when subset-selection methods are used to obtain parameter CIs, these CIs are conditional on the values of the fixed parameters that were not included in the estimation. The main downsides of using BE are: (i) it can mislead modelers into believing they have estimated all model parameters using their available data, (ii) poor/misleading parameter estimates can be obtained when misinformed prior information is specified and (iii) to our knowledge, there are no good tools for
diagnosing which parameters have been estimated reliably from the data and which parameters are estimated mainly due to unreliable prior information specified by the modeler. One method that has been used to investigate whether or not there was significant information in the experimental data to estimate a particular parameter is by comparing the corresponding prior and posterior distributions (Hughes et al., 2022; Yuyama et al., 2024). If there is significant overlap between the prior and posterior distributions, then the modeler knows that the data had little influence on the estimate. In the next section, we propose a simple method that can be used to quantify this overlap.

## 4. Proposed methodology

We propose that the Jaccard index (also called the Jaccard similarity coefficient, JSC) should be used to quantify overlap between 95 % CIs obtained from prior and posterior distributions for parameters estimated using BE. The JSC is defined as the ratio of the intersection of two sets to the union of two sets (Jaccard, 1901):

$$JSC(A, B) = \frac{|A \cap B|}{|A \cup B|} \qquad (8)$$

Based on this definition, the JSC is always between zero and one. It is zero when there is no overlap between two sets, and unity when two sets are identical. The JSC has been applied in a variety of pattern recognition and genomics applications, but we have not seen any uses of the JSC for diagnosis of the effectiveness of BE (El-Rashidy et al., 2022; Camgoz et al., 2016; Chung et al., 2019).

If the prior and posterior distributions for a model parameter are normally distributed, as is the case in some Bayesian linear regression problems, the overlap in area between the prior and posterior distributions can be readily calculated (details are provided elsewhere) (Straznicky, 2024). Fig. 4 shows 3 examples where JSCs have been computed to compare two normal distributions. In Fig. 4a, the prior and posterior distributions have the same mean, but the posterior is narrower due to information in the data resulting in a JSC of 0.19. In Fig. 4b, the prior and posterior also have the same mean, but the posterior is only slightly narrower, resulting in a JSC of 0.92 which indicates that there was little information about the parameter in the data. In Fig. 4c there is only a small overlap between the prior and the posterior resulting in a JSC of 0.0097. This small value of the JSC indicates that the estimated parameter and its uncertainty arises almost entirely from information in the experimental data. When a small JSC value is obtained for a parameter, the modeler may want to reconsider the prior information that they provided about that parameter. For example, they should plot the prior and posterior distributions to investigate whether the JSC is small due to a very narrow posterior (as in Fig. 4a) or due to a large shift in the mean (as in Fig. 4c). If a situation arises like the one in Fig. 4c, the modeler might increase the standard deviation of the prior distribution to indicate greater initial uncertainty in the initial parameter guess. Also, the modeler should look carefully at the data to investigate whether one or more outliers may be giving final parameter estimates that are inconsistent with prior knowledge. If outliers are

identified, they can be removed so that reliable parameter estimates can be obtained.

Although Fig. 4 focuses on the uncertainty of individual parameter estimates, it would be possible to also compute JSCs to compare multivariate normal distributions that correspond to joint probability distributions for model parameters.

Unfortunately, when BE is conducted for nonlinear chemical process models, the posterior distributions for the model parameters have no analytical Gaussian solution (Gelman et al., 2014; Ghasemi et al., 2011). As a result, we propose that JSCs should be used to compare 95 % CIs obtained via bootstrapping with 95 % CIs that correspond to the prior distribution. If there is significant overlap for these types of intervals for a particular parameter, the JSC will be close to 1, indicating that the data provided little information about the parameter. This proposed methodology will be illustrated using the case study.

## 5. Hydroisomerization case study

The model considered in this report describes n-hexane hydroisomerization in a continuous well-mixed Berty reactor using a bifunctional zeolite catalyst (Toch et al., 2014). In this process, gaseous n-hexane (nC6) and hydrogen (H₂) are fed to the reactor to produce 2-methyl pentane (2MP), 3-methyl pentane (3MP) and propane (C3) as shown in Fig. 5. To model this process, the simplified reaction scheme developed by Toch et al. is used (see Fig. 6) (Toch et al., 2014). Reactions pcp(1) and pcp(2) are protonated cyclopropyl branching reactions, and reaction cr is a cracking reaction. n-Hexane isomerizes to 2MP and 3MP at rates of $r_{pcp(1)}$ and $r_{pcp(2)}$, respectively. The 2MP product is cracked to C3 at a rate of $r_{cr}$.

<figure><img src="image_6.png" /><figcaption>Fig. 5. Process flow diagram for a Berty reactor converting n-hexane to 3MP, 2MP and C3 in the presence of hydrogen. Grey circles represent the zeolite catalyst.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. Overlaps of prior and posterior distributions with JSCs of a) 0.19 b) 0.92 and c) 0.0097.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 6. Simplified reaction network for hydroisomerization of n-hexane adapted from Toch et al. (2014).</figcaption></figure>

The system is modelled assuming an ideal continuous stirred tank reactor (CSTR) operating at steady-state. Outflow rates $F_{2MP}$, $F_{3MP}$ and $F_{C3}$ for the products can be predicted using the model equations in Table 4 (Abdi et al., 2023). These equations are a system of three implicit equations and three unknowns ($F_{2MP}$, $F_{3MP}$, $F_{C3}$), which can be calculated based on the process inputs (T, P, W and $F_{H_2}^0/F_{nC6}^0$) where T is the temperature, P is pressure, W is the mass of catalyst in the reactor, and $F_{H_2}^0/F_{nC6}^0$ is the inlet ratio of hydrogen to n-hexane. Kinetic and equilibrium parameters in the model equations depend on temperature according to the Arrhenius expressions:

$$k_i = k_{ref,i} e^{-\frac{E_{a,i}}{R} \left( \frac{1}{T} \frac{1}{T_{ref}} \right)} \quad (9)$$

$$K_{phys} = K_{ref} e^{-\frac{\Delta H^0}{R} \left( \frac{1}{T} \frac{1}{T_{ref}} \right)} \quad (10)$$

where $k_{ref,i}$ ($i = pcp(1)$, $pcp(2)$, cr) is a kinetic rate constant at the temperature $T_{ref} = 531.48$ K, which was selected because it is the average operating temperature for a series of experiments conducted at the University of Ghent (Toch et al., 2014). Similarly, $K_{ref}$ is value of the equilibrium constant $K_{phys}$ at $T_{ref}$. In Eq. (9), $E_{a,i}$ is the activation energy for the ith reaction. Similarly, $\Delta H^0$ in Eq. (10) is the enthalpy of phys-isorption. Model parameters that require estimation are: $\theta = [k_{ref,pcp(1)}, k_{ref,pcp(2)}, k_{ref,cr}, K_{ref}, E_{a,pcp(1)}, E_{a,pcp(2)}, E_{a,cr}, \Delta H^0]^T$.

Experimental settings used to conduct 36 steady-state experimental runs at the University of Ghent are provided in Table S5. These settings are used to generate synthetic data for the current case study. Synthetic data are used instead of real data to ensure that the model equations accurately reflect the behaviour of the process, and so that the true parameter values are known and can be compared with parameter estimates obtained using subset selection and Bayesian methods. The true parameter values used in the current study are provided in Table 5. They are physically-realistic parameter values that were estimated by Toch et al. using real data. A further benefit of using synthetic data is that

<table><caption>**Table 4**
System of model equations for n-hexane hydroisomerization at steady-state.</caption><tr><td>$F_{2MP}$</td><td>$$ -\left( \frac{k_{pcp(1)}(F_{nC6}^0 - F_{2MP} - F_{3MP} - 0.5F_{C3}) - k_{cr}F_{2MP}}{1 + K_{ref}P\left(\frac{F_{nC6}^0 - 0.5F_{C3}}{F_{H_2}^0 + F_{nC6}^0}\right)} \right) $$</td><td>W = 0</td><td>(4.1)</td></tr><tr><td>$F_{3MP}$</td><td>$$ -\left( \frac{k_{pcp(2)}(F_{nC6}^0 - F_{2MP} - F_{3MP} - 0.5F_{C3})}{1 + K_{ref}P\left(\frac{F_{nC6}^0 - 0.5F_{C3}}{F_{H_2}^0 + F_{nC6}^0}\right)} \right) $$</td><td>W = 0</td><td>(4.2)</td></tr><tr><td>$F_{C3}$</td><td>$$ -\left( \frac{2k_{cr}P\frac{F_{2MP}}{F_{H_2}^0 - 0.5F_{C3}}}{1 + K_{ref}P\left(\frac{F_{nC6}^0 - 0.5F_{C3}}{F_{H_2}^0 + F_{nC6}^0}\right)} \right) $$</td><td>W = 0</td><td>(4.3)</td></tr></table>

<table><caption>**Table 5**
True values of the kinetic rate constants and equilibrium constants $\theta_{true,j}$ and physically realistic lower and upper bounds.</caption><thead><tr><th>Rate Constant</th><th>Units</th><th>Value</th><th>Lower Bound ($\theta_{LBJ}$)</th><th>Upper Bound ($\theta_{UBj}$)</th></tr></thead><tbody><tr><td>$k_{ref,pcp(1)}$</td><td>$\times 10^{-6}$ mol s<sup>-1</sup> g<sub>cat</sub><sup>-1</sup></td><td>200.0</td><td>50</td><td>500</td></tr><tr><td>$k_{ref,pcp(2)}$</td><td>$\times 10^{-6}$ mol s<sup>-1</sup> g<sub>cat</sub><sup>-1</sup></td><td>114.5</td><td>20</td><td>500</td></tr><tr><td>$k_{ref,cr}$</td><td>$\times 10^{-6}$ mol s<sup>-1</sup> g<sub>cat</sub><sup>-1</sup></td><td>14.6</td><td>0</td><td>50</td></tr><tr><td>$K_{ref}$</td><td>$\times 10^{-6}$ Pa<sup>-1</sup></td><td>10</td><td>0</td><td>30</td></tr><tr><td>$E_{a,pcp(1)}$</td><td>kJ mol<sup>-1</sup></td><td>53.5</td><td>0</td><td>200</td></tr><tr><td>$E_{a,pcp(2)}$</td><td>kJ mol<sup>-1</sup></td><td>61.4</td><td>0</td><td>200</td></tr><tr><td>$E_{a,cr}$</td><td>kJ mol<sup>-1</sup></td><td>68.0</td><td>0</td><td>200</td></tr><tr><td>$\Delta H^0$</td><td>kJ mol<sup>-1</sup></td><td>-88.6</td><td>-200</td><td>0</td></tr></tbody></table>

many different experimental runs can be simulated using different random values for measurement errors. As a result, the effectiveness of different parameter estimation techniques can be compared when either a large number of experiments or a small number of experiments are available.

This case study considers several scenarios that a modeler might face when estimating parameters for a new catalyst. The first scenario has only four experimental runs conducted at mild operating conditions (i. e., at the lowest two temperatures from Table S5) as shown in Fig. 7a. A modeler might select these mild conditions for initial experiments to make sure the experimental system is operating properly. As shown in Fig. 7b, the second scenario involves two additional experiments conducted at a higher temperature, which should lead to more-reliable estimation of activation energies. The third scenario shown in Fig. 7c contains 9 runs from a 2³ factorial design with an added center-point run. A fourth scenario that includes experiments conducted at all 36 sets of operating conditions from Table S5 is also considered, but is not shown in Fig. 7.

Data sets for the four scenarios were synthesized using the detailed procedure shown in Table S6. True values of measurement noise variances (i.e., $\sigma_{F_{2MP}}^2 = 1.44 \times 10^{-13}$, $\sigma_{F_{3MP}}^2 = 6.84 \times 10^{-14}$ and $\sigma_{F_{C3}}^2 = 1.94 \times 10^{-14}$ mol² s⁻²) were used to obtain normally-distributed random measurement errors, which were added to simulation results obtained using the true parameter values. These measurement errors were assumed to have constant variance, with different variance estimates obtained for each type of response. In some systems, this constant variance assumption would be inappropriate (e.g., if the relative errors in a response tended to be of similar size). This type of situation could be handled using logarithmic or other transformations on the response variables (Box and Hill, 1974).

## 5.1. Parameter estimation using subset selection and Bayesian methods

Table 6 contains a summary of the findings for the n-hexane hydroisomerization case study performed using the model in Table 4, which contains 8 unknown parameters $\theta = [k_{ref,pcp(1)}, k_{ref,pcp(2)}, k_{ref,cr}, K_{ref}, E_{a,pcp(1)}, E_{a,pcp(2)}, E_{a,cr}, \Delta H^0]^T$. The individual findings and how they were obtained are discussed below. When comparing WLS, LO and BE,
<figure><img src="image_9.png" /><figcaption>Fig. 7. Locations within the operating space for three scenarios where data are collected for a new catalyst.</figcaption></figure>

we generated 100 sets of simulated data for each case in Fig. 7. Each modeler used the same physically-realistic parameter bounds (Table 5) and known measurement variances $\sigma_{F2MP}^2$, $\sigma_{F3MP}^2$ and $\sigma_{FC3}^2$. We assume that the modelers know these measurement noise variances based on a large number of replicate experiments obtained using a previous catalyst. Each modeler uses their own synthetic data set and initial parameter guesses (IG) when estimating the parameters. The unique synthetic data sets for each modeler were synthesized with different random number seeds so that the effects of random measurement noise are considered. Details about how these reliable initial guesses were obtained are provided in Section 7 of the SI. Results for the first case (low-temperature experiments in Fig. 7a) are shown using box plots in Fig. 8. The left-most box plots in Figs. 8a, 8b,...8h show the initial parameter guesses selected by the 100 modelers.

Box plots obtained using WLS, LO and BE approaches are also shown in Fig. 8 for each of the model parameters. WLS estimation was performed using Eq. (1) as the objective function. Notice that the box plots for the 100 sets of WLS parameter estimates indicate more variability in the resulting parameter estimates than in the initial parameter guesses (except perhaps for parameter $k_{ref,pcp(2)}$). LO parameter estimation was performed after ranking the model parameters using the orthogonalization-based algorithm in Table 3 and then selecting the appropriate number of parameters for estimation using the method in Table S3. Depending on the initial parameter guesses and the noise sequences in the synthetic data, different numbers of parameters were selected by each modeler. The most popular subset, selected 15 % of the time, contained only two parameters (i.e., $k_{ref,pcp(1)}$ and $k_{ref,pcp(2)}$). The second-most popular subset selected 13 % of the time contained 3 parameters (i.e., $k_{ref,pcp(1)}$, $k_{ref,pcp(2)}$ and $\Delta H$). Only one parameter (i.e., either $k_{ref,pcp(1)}$ or $k_{ref,pcp(2)}$) was selected for estimation 15 % of the time. Four or more parameters were selected 40 % of the time. Notice from Fig. 8 that the resulting parameter estimates tend to be more precise than those obtained using WLS. Also notice that estimates for $k_{ref,pcp(1)}$ and $k_{ref,pcp(2)}$, which were estimated most often, are more precise than

<table><caption>**Table 6**
Summary of findings from n-hexane hydroisomerization case study.</caption><thead><tr><td>No.</td><td>Highlight</td></tr></thead><tbody><tr><td>1.</td><td>Both LO and BE provide superior parameter estimates and model predictions compared to WLS when limited data are available</td></tr><tr><td>2.</td><td>When the prior information is reliable, BE provides better parameter estimates and model predictions compared to the LO method</td></tr><tr><td>3.</td><td>When the prior information is misinformed, BE can provide very unreliable parameter estimates compared to the LO parameter estimates</td></tr><tr><td>4.</td><td>When the prior information is misinformed, model predictions using LO method tend to be more accurate compared to predictions using BE</td></tr><tr><td>5.</td><td>Unestimable parameters can be identified using both Bayesian and LO methods, but LO methods provide a more detailed diagnosis of parameter estimability difficulties</td></tr><tr><td>6.</td><td>Confidence intervals for estimated parameters are narrower for Bayesian parameter estimates</td></tr></tbody></table>

their initial parameter guesses. When BE was performed using the objective function in Eq. (6), along with the good initial guesses (see Fig. 8) and corresponding good values for $\sigma_{\theta_{0j}}^2$, all of the estimates are more precise than those obtained using WLS and LO methods. Each modeler used their respective WLS, LO and BE parameter values to predict new data corresponding to the full range of experimental conditions from the University of Ghent, shown in Table S5. Root-mean-squared errors (RMSEs) between these model predictions and synthetic data are 5.65, 3.68 and 2.60, respectively, using WLS, LO and BE parameter values. These results and parity plots in Figs. S8-S11 confirm that the BE method gave the most reliable predictions. The results in Fig. 8 and Figs. S4-S11 are consistent with Finding 1 and 2 in Table 6.

Fig. 9a, Fig. 9b and Fig. 9c show box plots for estimates of $k_{ref,pcp(1)}$ obtained using 100 sets of simulated data from 4, 9 and 36 experimental runs, respectively. Box plots for other parameters are provided in Figs. S5-S7. As expected, data sets with more experiments have lower variability for the parameter estimates, regardless of the method used for parameter estimation. Using data from 36 experimental runs results in LO parameter estimates that are the same as the corresponding WLS estimates. In this situation, the LO method selects all 8 parameters for estimation because there is sufficient information in the data. The improved parameter estimates result in improved model predictions (see parity plots in Figs. S8-S11). Because the BE method benefits from additional accurate prior information specified by the modeler, the BE parameter estimates and predictions are always better than those obtained using WLS and LO. For example, when data from 36 runs are used for parameter estimation, the RMSE value for predicting new data for the 36 runs is 0.80 for BE, which is slightly smaller than 0.81 for both WLS and LO. The results in Fig. 8 and Fig. 9 were obtained when reliable initial values for the parameters and their uncertainties are specified by the modelers. It is important to consider situations where unreliable prior information is specified. One problem with BE methods is that modelers have difficulty in providing accurate variances $\sigma_{\theta_{0j}}^2$ for their initial parameter guesses.

Fig. 10 shows three choices of truncated normal prior distributions for $k_{ref,pcp(1)}$ that could be specified by the first two out of 100 modelers. All these truncated distributions ensure that $k_{ref,pcp(1)}$ (in $mol s^{-1} g_{cat}^{-1}$) is between the specified lower and upper bounds (i.e., $0.5 \times 10^{-4} \le k_{ref,pcp(1)} \le 5 \times 10^{-4}$). Fig. 10a shows reliable prior distributions for $k_{ref,pcp(1)}$. The corresponding true parameter value, which is unknown to these modelers, is indicated by the red dashed line. The modes of the truncated normal distributions correspond to initial parameter guesses specified by the modelers. Fig. 10b shows alternative misinformed prior distributions that could be specified instead. These distributions have initial guesses that are larger and variances that are smaller than in Fig. 10a. Finally, Fig. 10c shows prior distributions that have the same initial guesses as in Fig. 10b, but a larger variance so that the prior distributions are more consistent with the true parameter value.

Fig. 11 shows estimates for $k_{ref,pcp(1)}$ from 100 modelers, obtained
<figure><img src="image_10.png" /><figcaption>**Fig. 8.** Box plots showing the initial parameter guesses (IG) and estimates obtained using WLS parameter estimation, LO parameter estimation and BE methods using four runs at mild conditions. Dashed horizontal lines represent the true parameter values.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Parameter estimates of $k_{ref,pcp(1)}$ for 100 modelers when four (a), nine (b) and 36 (c) experimental runs are available.</figcaption></figure>

using WLS, LO and BE methods, using data from 9 experiments when the prior parameter information is accurate (Fig. 11a), larger than the true value with a small uncertainty (Fig. 11b), and larger than the true value with a large uncertainty (Fig. 11c). Box plots for other parameters are provided in the SI, Fig. 11a, which is identical to Fig. 9b, confirms that a good prior distribution is helpful for obtaining accurate parameter estimates. Large initial guesses in Fig. 11b (corresponding to prior

distributions like those in Fig. 10b) cause noticeable bias in the BE results for $k_{ref,pcp(1)}$. The corresponding LO estimates are less prone to this problem because $k_{ref,pcp(1)}$ is selected for estimation by all 100 modelers, so initial guesses for this parameter do not influence the LO estimates. These results are consistent with Finding 3 in Table 6. Fig. 11c shows a similar box plot where the 100 modelers specified prior distributions like those in Fig. 10c. The larger specified variance causes the BE results
<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Truncated normal prior distributions specified by the first two modelers (first modeler shown in blue, second shown in black) corresponding to (a) accurate initial guesses, (b) inaccurate initial guesses with small uncertainty, and (c) inaccurate initial guesses with large uncertainty.</figcaption></figure>

to have less bias compared to Fig. 11b.

The misleading prior in Fig. 10b results in larger objective function values than the priors in Figs. 10a and 10c, as expected. When the misleading prior distribution in Fig. 10b is specified, the average objective function value is 110, which is much larger than 26 obtained using the accurate prior in Fig. 10a. This result makes sense because the good prior enables the optimizer to find parameter estimates that fit the data well while keeping the parameters near their initial guesses. When the prior in Fig. 10c is used, the final objective function value is 32. The larger standard deviation in Fig. 10c results in a smaller penalty term in Eq. (5), and in parameter values that can better match the data, in comparison with parameters obtained using the prior in Fig. 10b.

The prior distributions have different relative influences when different amounts of data are available for parameter estimation. For example, the parameter penalty term contributes 40 % of the final objective function value for BE when only 4 experimental runs are used (see Fig. 7a). As more data are included, the parameter penalty term becomes relatively less important (e.g., only 21 % of the objective function value when 9 runs are used (see Fig. 7c)).

Parity plots for predictions of new F2MP data using the LO and BE parameters are shown in Fig. 12. Additional parity plots for other responses are provided in the SI. As shown in Fig. 12b, the bias in the BE parameter estimates caused by inaccurate prior degrades the model predictions. Corresponding model predictions using the corresponding LO estimates are more accurate. Examination of this and other parity plots in the SI leads to Finding 4 in Table 6. Fig. 12c reveals that specifying a larger uncertainty in the initial parameter guesses results in BE model-prediction results with less bias. In practice, it is difficult for modelers to specify appropriate values of $\sigma_{0j}^2$, which can make BE a riskier parameter estimation approach than LO.

Table 7 shows a comparison of parameter estimates obtained via LO and BE approaches for the first modeler using data from 4 experiments at low temperatures. As shown in the 4th column, the LO approach determines that $k_{ref,pcp(1)}$ is the most-important parameter for estimation from the data. This result makes sense because the pcp(1) reaction

generates the largest portion of the product mixture. The bottom-ranked parameter using the LO approach is the activation energy $E_{a,cr}$ for cracking, which also makes sense because there is limited information in the data about temperature effects for the cracking side reaction. The top 6 parameters from the ranked list are selected for estimation using the MSE-based algorithm in Table S3. The bottom 2 parameters $E_{a,cr}$ and $\Delta H^0$ remain fixed at their initial guesses when the LO approach is used. Parameter $E_{a,cr}$ is not estimable because it has very little influence on predictions of the available data, whereas $\Delta H^0$ is not estimable because it has only a small influence on the predictions and this influence is highly correlated with that of $K_{ref}$. Details are provided in Section 6 of the SI. The values of the parameter estimates $\hat{\theta}_{LOj}$ and their corresponding 95 % bootstrap CIs are shown in the 5th column of Table 7. These CIs account for model nonlinearity and the parameter bounds enforced during estimation. The main downside associated with these CIs is that they are conditional on fixed values of $E_{a,cr}$ and $\Delta H^0$. If these two parameters were fixed at different initial guesses, then different values for the parameter estimates and CIs would be obtained. As a result, the CIs do not account for all the uncertainty in the model parameters.

When BE is used, all 8 parameters are estimated to obtain $\hat{\theta}_B$ based on the available data and the prior knowledge (see Fig. 10). Posterior distributions for the parameters obtained by the first modeler were generated using the parametric bootstrapping method in Table S4. Fig. 13 shows the corresponding posterior and prior distributions for parameters $k_{ref,pcp(1)}$ and $E_{a,cr}$. The posterior distribution for $k_{ref,pcp(1)}$ is much narrower than the corresponding prior distribution, indicating that there is substantial information in the data about this parameter. By contrast, the posterior distribution of $E_{a,cr}$ is similar to its prior. The overlap of these prior and posterior distributions can be summarized by calculating the JSC of the corresponding 95 % CIs, which are 0.245 for $k_{ref,pcp(1)}$ and 0.76 for $E_{a,cr}$. These JSCs and those for the other six parameters, provided in Table 7, are helpful for assessing which CIs were influenced most by the data, and which are influenced most by the prior distributions. Examination of the plot in Fig. 13b and the corresponding large value of JSC suggests that the estimate of $E_{a,cr}$ is influenced mostly

<figure><img src="image_13.png" /><figcaption>**Fig. 11.** Parameter estimates of $k_{ref,pcp(1)}$ using 9 experiments with (a) accurate initial guesses, (b) inaccurate initial guesses with small uncertainty, and (c) inaccurate initial guesses with large uncertainty.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Fig. 12.** Model predictions of F2MP using parameter estimates from 9 experiments and (a) accurate initial guesses, (b) inaccurate initial guesses with small uncertainty, and (c) inaccurate initial guesses with large uncertainty.</figcaption></figure>

<table><caption>**Table 7**
Comparison of LO and BE parameter estimates and confidence intervals obtained by parametric bootstrapping using data from 4 experiments shown in Fig. 6a.</caption><thead><tr><th>Parameter θ<sub>j</sub></th><th>True Values<br>θ<sub>true,j</sub></th><th>Initial Guess θ<sub>0j</sub></th><th>σ<sub>θ0j</sub></th><th>Rank</th><th>LO Estimate and 95 % CI</th><th>Bayesian Estimate, 95 % CI and JSC Value</th></tr></thead><tbody><tr><td>k<sub>ref,pcp(1)</sub></td><td>200.0</td><td>199.3</td><td>66.7</td><td>1</td><td>228.3 [156.7, 291.2]</td><td>181.7 [155.1, 216.0] 0.245</td></tr><tr><td>k<sub>ref,pcp(2)</sub></td><td>114.5</td><td>143.6</td><td>38.2</td><td>3</td><td>102.5 [70.9, 136.4]</td><td>92.0 [78.6, 110.2] 0.182</td></tr><tr><td>k<sub>ref,cr</sub></td><td>14.6</td><td>14.6</td><td>4.9</td><td>5</td><td>21.4 [10.8, 38.2]</td><td>18.4 [16.4, 20.2] 0.201</td></tr><tr><td>K<sub>ref</sub></td><td>10</td><td>3.8</td><td>3.3</td><td>2</td><td>15.7 [1.8, 42.0]</td><td>11.1 [5.1, 20.2] 0.063</td></tr><tr><td>E<sub>a,pcp(1)</sub></td><td>53.5</td><td>39.8</td><td>17.8</td><td>6</td><td>57.9 [25.9, 125.8]</td><td>56.1 [48.3, 64.3] 0.305</td></tr><tr><td>E<sub>a,pcp(2)</sub></td><td>61.4</td><td>48.7</td><td>20.5</td><td>4</td><td>85.4 [3.7, 109.4]</td><td>49.5 [42.9, 57.3] 0.223</td></tr><tr><td>E<sub>a,cr</sub></td><td>68.0</td><td>57.0</td><td>22.7</td><td>8</td><td></td><td>63.2 [31.8, 130.4] 0.760</td></tr><tr><td>ΔH<sup>0</sup></td><td>-88.6</td><td>-155.2</td><td>29.5</td><td>7</td><td></td><td>-151.3 [-192.7, -111.0] 0.564</td></tr></tbody></table>

by the prior, and not by the experimental data. Unfortunately, the BE method does not provide as much detailed diagnostic information about why $E_{a,cr}$ is difficult to estimate from the data (i.e., about whether the large overlap between the prior and posterior distributions is due to low sensitivity or to highly correlated influences of parameters on model predictions). Notice that the Bayesian CIs in Table 7 are all narrower than the corresponding LO CIs, which makes sense because the Bayesian estimates rely on additional prior information provided by the modeler. This result is summarized as Finding 6 in Table 6. When reliable prior information is available, BE results in greater accuracy of the parameter estimates than LO estimation. Unfortunately, when initial parameter guesses are poor, or the modeler is overly confident about the accuracy of the initial guesses, substantial bias in the parameter estimates tends to arise for BE parameter estimation (see Fig. 11), resulting in misleading narrow 95 % CIs. When LO parameter estimation is used, poor initial guesses for parameters that are not selected for estimation also leads to bias in the estimates of other parameters. The size of this bias tends to be smaller than for BE parameter estimation (see, for example, Fig. S20h in

the SI).

## 6. Conclusions and advice

Modelers and applied statisticians have developed subset-selection and Bayesian methods to aid parameter estimation in chemical process models when there is insufficient information in the data to obtain reliable estimates for all of the model parameters. Both methods make use of parameter initial guesses and information supplied by the modeler about the accuracy of these guesses. In addition, modelers who use Bayesian techniques provide information about the structure of the prior distribution for the model parameters. This literature review shows how subset selection and Bayesian methods have been used to estimate parameters in a wide variety of chemical process models. We find that subset-selection methods are more popular than Bayesian techniques, especially when the number of model parameters is large or the model equations are complex. Subset selection usually consists of two parts. First, the model parameters are ranked from most important to estimate

<figure><img src="image_15.png" /><figcaption>**Fig. 13.** Overlap of the prior and posterior distribution for (a) kref,pcp(1) and (b) Ea,cr.</figcaption></figure>
to least important based on sensitivity and uncertainty information. Second, the appropriate number of parameters to estimate is determined. In recent years, chemical process modelers with experimental data have used four different methods to rank their model parameters. The orthogonalization-based method has been used most often, because it is relatively easy to use and it simultaneously accounts for parameter influence and correlated effects of parameters on model predictions (Yao et al., 2003; Thompson et al., 2009). Seven different methods have been used to determine a suitable parameter subset from a ranked parameter list. Five of these methods involve specification of cut-off values (i.e., for collinearity measures, singular values, global sensitivities, residual sensitivities or changes in the objective function when additional parameters are estimated). All of these cut-off values are somewhat arbitrary and difficult to specify appropriately. As a result, we recommend the other two methods, which do not require cut-off values (i.e., cross validation and a MSE-based approach) (Wu et al., 2011a, 2011b). These techniques select the number of parameters from the ranked list that gives the best predictions of new data. Unfortunately, subset-selection methodologies can be time consuming because a large number of repeated parameter estimations with different numbers of parameters are required.

Bayesian estimation is easier to use than subset selection if normal prior distributions are specified for the model parameters. In this situation, simple closed-form objective functions arise so that a simple gradient-based optimization can be performed (only once) to obtain the parameter estimates. One problem with this approach is that modelers may not want to specify normal prior distributions for parameters because tails of these distributions are inconsistent with bounds that are often enforced to ensure that parameters values are physically realistic. The majority of chemical process modelers who used Bayesian parameter estimation techniques specified prior parameter distributions with finite ranges for the parameters. Because closed-form objective functions were not available, these modelers relied on MCMC and related sampling-based algorithms to obtain their parameter estimates. Unfortunately, MCMC is computationally expensive and can be difficult to implement and tune. Recently, closed-form objective functions for Bayesian estimation were derived for situations involving truncated normal priors and uniform priors between lower and upper bounds. We believe that this approach to Bayesian parameter estimation will become popular for chemical process models in the future because it is less computationally expensive and easier to implement than MCMC and other sampling-based methods.

The recommended subset-selection and Bayesian parameter estimation techniques are illustrated and compared using a n-hexane hydroisomerization case study, resulting in the six important findings in Table 6. Based on the results of this case study, we make five recommendations: (1) Bayesian parameter estimation with truncated normal priors should be used instead of subset-selection methods if quick estimation results are desired; (2) Subset-selection methods should be used if detailed analysis of some parameter estimates is desired or if model simplifications are contemplated; (3) Relatively-large standard deviations should be specified for normal or truncated normal priors to reduce biases in parameter estimates and model predictions, which can become large when modelers are overly confident about the accuracy of their initial parameter guesses; (4) Uniform prior distributions are not recommended for Bayesian parameter estimation when it is difficult or impossible to uniquely estimate all model parameters using WLS (i.e., when the FIM is ill-conditioned), because the resulting closed-form objective function does not contain parameter penalty terms that help stabilize the optimization problem; (5) To assess whether a Bayesian parameter estimate has been influenced more by the data or by the prior distribution, we recommend using the Jaccard Similarity Coefficient to quantify the relative overlap between the posterior and prior distributions. In fortunate situations where there is sufficient information in the data to obtain unique and precise estimates for all model parameters without relying on prior information, we recommend that modelers

should use simple WLS parameter estimation, because it will result in reliable parameter estimates. In this type of situation, parameter subset selection and Bayesian estimation are not required.

We hope that the discussion and results in this article will help researchers in academia and industry determine which parameters they should estimate from their limited data and how they should make the best use of prior knowledge about physically-plausible parameter values. We also hope that it will help modelers diagnose whether their parameter estimates and model predictions are reliable.

## CRediT authorship contribution statement

**Benoit Celse:** Writing – review & editing, Conceptualization. **Laurien Alexandra Gibson:** Writing – review & editing, Validation, Software. **Jakob Ivan Straznicky:** Writing – original draft, Software, Methodology, Investigation, Formal analysis, Conceptualization. **Kimberley B. McAuley:** Writing – review & editing, Supervision, Project administration, Methodology, Investigation, Funding acquisition, Conceptualization.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

Funding from the Natural Sciences and Engineering Research Council of Canada and Eurokin is gratefully acknowledged. We would like to thank Liam Flemming for helpful technical discussions.

## Appendix A. Supporting information

Supplementary data associated with this article can be found in the online version at doi:10.1016/j.cherd.2025.02.037.

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