Process Modeling and Prediction with Large Number of High-Dimensional Variables Using Functional Regression

Learning the relationship between a response variable (e.g., a quality characteristic) and a set of predictors (e.g., process variables) is of special importance in process modeling, prediction, and optimization. In many applications, not only is the number of these variables large but these variables are also high-dimensional (HD) (e.g., they are represented by waveform signals). This high dimensionality requires a systematic approach to both modeling the relationship between the variables and removing the noninformative input variables. This article proposes a functional regression method in which an HD response is estimated and predicted through a set of informative and noninformative HD covariates. For this purpose, the functional regression coefficients are expanded through a set of low-dimensional smooth basis functions. In order to estimate the low-dimensional set of parameters, a penalized loss function with both smoothing and group lasso penalties is defined. The block coordinate decent (BCD) method is employed to develop a computationally tractable algorithm for minimizing the loss function. Through simulations and case studies, the performance of the proposed method is evaluated and compared with benchmarks. The results illustrate the advantage of the proposed method over the benchmarks. Note to Practitioners - This article proposes a method for efficient and interpretable modeling of processes with high-dimensional (HD) data, such as waveform signals. Especially, the proposed method generates a regression model that predicts a function (e.g., a sensor's readings over time) using several functional inputs. Existing functional regression techniques are mostly limited to a single functional input and are focused on profile data. In many applications, however, a large number of process variables are available for estimating an HD output, such as an image. This article addresses these problems by employing basis functions to reduce the dimension of functions and introducing specific penalties that removes noninformative inputs and improves computational efficiency. A model generated by the proposed approach can be used for process monitoring and optimization. Using simulation and case studies, the performance of the developed method is evaluated and compared with other methods under various scenarios. This can provide practitioners with useful guidelines for selecting an appropriate method for process modeling.