(6e) Vine Copula-Based Dependence Description for Multivariate Multimode Process Monitoring

For complex chemical processes, process variables with high dimensionality, nonlinearity and non-Gaussian properties are frequently encountered. To maintain process safety and product quality of chemical processes, in recent years, a host of data-based methods have been successfully applied in chemical process monitoring. The existing methods mainly focus on dimensionality reduction or variable decoupling processes, which more or less, will result in information lost or distortion for the process data, thus leading to poor monitoring performance.

As an efficient tool in dependence modeling, vine copula aims to simplify multivariate dependence problem into dealing with optimization problems for a series of bivariate copulas listed in a sparse matrix. It somewhat prevents the problem called “curse of dimensionality” that exists for traditional multivariate copula. In addition, due to its flexibility structure and the rank correlation information preserved, vine copula is capable of handling data reflecting both nonlinear and non-Gaussian properties.

In this work, a novel vine copula-based method is proposed for multimode process monitoring. C-vine copula is initially introduced to establish the joint distribution of normal samples for each mode. The offline modeling procedure includes (a) pair copula construction, (b) conditional distribution calculation, (c) selection and optimization of vine copula models and (d) goodness-of-fit testing. Herein, the statistical information of each mode can be obtained via sampling methods (e.g., Markov Chain Monte Carlo Method). To achieve online monitoring for multimode processes, the global Bayesian Inference-based probability (BIP) index proposed by Yu and Qin is used. Note that Mahalanobis distance is no longer efficient on measuring the distance of monitored data from each non-Gaussian mode, a generalized local probability (GLP) index is then defined. Because the traditional numerical integration method appears rather time-consuming in estimating GLP, density quantile approach is introduced. By analyzing the one-dimensional distribution of probability density function values of each mode, the corresponding density quantile table can be constructed. Therefore, the GLP index as well as the BIP index can immediately be updated in real time by just checking the static table created offline.

The proposed vine copula-based dependence description (VCDD) approach is successfully applied to a numerical example and TE benchmark process. Compared with the FGMM-based approach, the proposed method can achieve higher detection rates, lower false alarm rates and shorter time delays especially to those non-Gaussian processes with tail dependence. Moreover, It is demonstrated that both offline modeling and online monitoring of this approach achieves low computation load, which will make this approach more practical.