Sparse and Structured Function-on-Function Quality Predictive Modeling by Hierarchical Variable Selection and Multitask Learning

Modern manufacturing industries are often featured with a data-rich environment. The real-time behaviors of process variables can be completely recorded as multiple various signal signatures, and the geometric quality of finished products can be thoroughly characterized by their 2-D surface data. Learning the relationship between such signal predictors and surface responses, where the input and output are no longer the conventional scalar variables but are in fact both functions in the time domain and spatial domain, respectively, is critical for quality prediction in many applications nowadays. To this end, this article proposes a novel sparse and structured function-on-function regression (SSF
²
R) model, where a hierarchical variable selection is developed to identify informative signals and further screen significant elements within the selected signals, and a multitask learning is devised to exploit the smoothness nature of surface response and the similarity structure among a series of subregression tasks. Our SSF
²
R model is concisely formulated as a convex problem with an efficient iterative algorithm derived to obtain the global optimum. Moreover, our quality prediction can be performed dynamically during an ongoing manufacturing process when only partial observations of the signal predictors are available. The superiority of our proposed method is validated by numerical simulations and a real case study in the semiconductor industry.