Multiple Tensor-on-Tensor Regression: An Approach for Modeling Processes With Heterogeneous Sources of Data

Abstract

In recent years, measurement or collection of heterogeneous sets of data such as those containing scalars, waveform signals, images, and even structured point clouds, has become more common. Statistical models based on such heterogeneous sets of data that represent the behavior of an underlying system can be used in the monitoring, control, and optimization of the system. Unfortunately, available methods mainly focus on the scalars and profiles and do not provide a general framework for integrating different sources of data to construct a model. This article addresses the problem of estimating a process output, measured by a scalar, curve, image, or structured point cloud by a set of heterogeneous process variables such as scalar process setting, profile sensor readings, and images. We introduce a general multiple tensor-on-tensor regression approach in which each set of input data (predictor) and output measurements are represented by tensors. We formulate a linear regression model between the input and output tensors and estimate the parameters by minimizing a least square loss function. To avoid overfitting and reduce the number of parameters to be estimated, we decompose the model parameters using several basis matrices that span the input and output spaces, and provide efficient optimization algorithms for learning the basis and coefficients. Through several simulation and case studies, we evaluate the performance of the proposed method. The results reveal the advantage of the proposed method over some benchmarks in the literature in terms of the mean square prediction error. Supplementary materials for this article are available online.