Quality prediction using functional linear regression with in-situ image and functional sensor data

This article studies a general regression model for a scalar quality response with mixed types of process predictors including process images, functional sensing signals, and scalar process setup attributes. To represent a set of time-dependent process images, a third-order tensor is employed for preserving not only the spatial correlation of pixels within one image but also the temporal dependency among a sequence of images. Although there exist some papers dealing with either tensorial or functional regression, there is little research to thoroughly study a regression model consisting of both tensorial and functional predictors. For simplicity, the presented regression model is called functional linear regression with tensorial and functional predictor (FLR-TFP). The advantage of the presented FLR-TFP model, which is compared to the classical stack-up strategy, is that FLR-TFP can handle both tensorial and functional predictors without destroying the data correlation structure. To estimate an FLR-TFP model, this article presents a new alternating Elastic Net (AEN) estimation algorithm, in which the problem is reformed as three sub-problems by iteratively estimating each group of tensorial, functional, and scalar parameters. To execute the proposed AEN algorithm, a systematic approach is developed to effectively determine the initial running sequence among three sub-problems. The performance of the FLR-TFP model is evaluated using simulations and a real-world case study of friction stir blind riveting process.