Batch process modelling with mixtures of Gaussian processes

In recent years Gaussian processes have attracted a significant amount of interest with the particular focus being that of process modelling. This has primarily been a consequence of their good predictive performance and inherent analytical properties. Gaussian processes are a member of the family of non-parametric Bayesian regression models and can be derived from the perspective of neural networks. Their behaviour is controlled through the structure of the covariance function. However, when applied to batch processes, whose data exhibits different variance structures throughout the duration of the batch, a single Gaussian process may not be appropriate for the accurate modelling of its behaviour. Furthermore there are issues with respect to the computational costs of Gaussian processes. The implementation of a Gaussian process model requires the repeated computation of a matrix inverse whose order is the cubic of the number of training data points. This renders the algorithm impractical when dealing with large data sets. To address these two issues, a mixture model of Gaussian processes is proposed. The resulting prediction is attained as a weighted sum of the outputs from each Gaussian process component, with the weights determined by a Gaussian kernel gating network. The model is implemented through a Bayesian approach utilising Markov chain Monte Carlo algorithms. The proposed methodology is applied to data from a bench-mark batch simulation polymerization process, methyl methacrylate (MMA), and the results are compared with those from a single Gaussian process to illustrate the advantages of the proposed mixture model approach.