Partial derivative-based dynamic sensitivity analysis expression for non-linear auto regressive with exogenous (NARX) model – case studies on distillation columns and model's interpretation investigation

Constructing the reliable dynamic sensitivity profile for the output variable using the machine learning model is a challenging task; however, the dynamic sensitivity trends are helpful to understand the impact of the input variables on the system's performance. In this paper, we have derived the partial-derivative approach-based sensitivity analysis expression for the non-linear auto regressive with exogenous (NARX) model for the first time. The engineering systems-based case studies, i.e., two distillation columns with five and ten stages, respectively are taken which are commonly found in the chemical processing plants. Two output variables, i.e., liquid composition in tray 2 and tray 4 (Y2 and Y4) of a five-stage distillation column, and liquid composition in tray 7 (Y7) of a ten-stage (higher) distillation column are modelled by NARX with respect to time, feed concentration (Xf) and feed flow rate (Lf). The dynamic sensitivity profiles of the output variables with respect to Xf and Lf for the two distillation columns are plotted by the derived partial derivative-based sensitivity expression on the NARX model. Furthermore, the forward difference method of sensitivity analysis (first principle method) is also applied on the ordinary differential equations of the distillation columns to compute the sensitivity values of the output variables. A good agreement in the dynamic sensitivity values of the output variables with respect to the input variables is found for the two sensitivity analysis techniques thereby demonstrating the effectiveness of the partial-derivative approach for the improved NARX's interpretability performance. This research presents the explicit partial-derivative based sensitivity analysis expression for the NARX model which can be utilized for time-series applications and can provide the insights about the model's interpretation performance.