Parameter Estimation of Chromatography By Bayesian Inference Using Two Monte Carlo Methods

Chromatography is an important separation technique widely used for many industrial applications such as chemical, pharmaceutical, and petrochemical products. The more complex the design and operation of this process are, the more difficult it can be to optimize them. This can lead suboptimal design and operation, and even failure to meet product requirements such as purity and yield. To avoid such problems, model-based development is a powerful approach.

For reliable model-based development, parameters in the model must be estimated. A common method is regression, but the result obtained by this method is usually a point estimate that cannot quantify the uncertainty of the parameters. There are a large number of model uncertainties in the chromatographic model, which come from many error sources, such as experimental procedures and measurement techniques. Quantifying these uncertainties is crucial for chromatographic model parameter estimation, which should be taken into account in design and operation to ensure sufficient accuracy in predicting the process performance.

To quantify model uncertainty, many statistical techniques have been developed. In particular, Bayesian statistics have been demonstrated to be a powerful approach, where the probability distribution of the target parameters can be estimated utilizing prior information about the model parameters (Gelman et al., 2013). In Bayesian inference, the probability distribution of the target parameters with large parameter space must be computed, which requires substantial computational effort. In this study, two numerical approximation methods, Markov chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC) methods are used. They approximate the posterior parameter distributions by random sampling. In addition, by taking sufficiently wide prior distributions and carrying out a large number of sampling, it is possible to avoid falling into a local optimal solution. Compared to the MCMC method, which relies on a single sampling sequence, the SMC method utilizes multiple sampling sequences and allows parallel computation utilizing multiple cores simultaneously, reducing the computation time.

The purpose of this study is to explore Bayesian estimation methods that can estimate chromatographic model parameters and quantify the uncertainty of the parameters. Two experimental cases with different adsorption behaviors, cyclohexanone and cyclopentanone, as well as phenol and cresol, are considered in this study. Bayesian inference combined with MCMC and SMC methods are applied to estimate the model parameters, and the resulting posterior parameter distributions and computation times of the two methods are compared. The reliability of the estimated parameters is assessed from the probability density distributions. Furthermore, the correlations between the parameters are investigated.

References

[1] Gelman, A., et al. Bayesian data analysis. CRC press, 2013.