Function-on-Function Gaussian Process with Application in Robust Parameter Design

As data sensing technology advances, functional data have become increasingly popular in complex systems. Function-on-function regression models, where both input and output variables are functional data, have attracted increasing attention in research. However, all the existing models have limitations that cannot qualify prediction uncertainty. To fill this gap, we propose a novel function-on-function Gaussian process (FFGP). It employs a detachable structure based on the operator-valued kernel to represent the covariance between functional inputs and output. Compared with existing Gaussian process models, FFGP can model functional data directly in the continuous space, and a scalar-valued operator-covariance is defined to qualify the output uncertainty. We further apply FFGP to robust parameter design by proposing an expected loss function to measure the functional output bias and uncertainty given a functional input. Then, an effective and scalable functional gradient descent algorithm (FRGD) is proposed to identify the optimal functional input that minimizes the loss function. Some theoretical properties of FFGP and its corresponding robust parameter optimization via FRGD are discussed.

History: Accepted by Ram Ramesh, Area Editor for Data Science and Machine Learning.

Funding: This work was supported by the National Natural Science Foundation of China [Grant 72271138] and Tsinghua-NUS Joint Funding [Grant 20243080039].

Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information (https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0751) as well as from the IJOC GitHub software repository (https://github.com/INFORMSJoC/2024.0751). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/.