Data-driven polynomial chaos expansion for machine learning regression

TL;DR

This paper introduces a data-driven polynomial chaos expansion method for machine learning regression that provides accurate pointwise predictions, uncertainty quantification, and robustness to noise, especially effective with small training datasets.

Contribution

It demonstrates that PCE can be used as a purely data-driven regression model with competitive accuracy and added benefits like uncertainty quantification and simplicity.

Findings

PCE-based regression achieves accuracy comparable to neural networks and SVMs.

The method effectively quantifies output uncertainties.

It performs well with small training datasets and is robust to noise.

Abstract

We present a regression technique for data-driven problems based on polynomial chaos expansion (PCE). PCE is a popular technique in the field of uncertainty quantification (UQ), where it is typically used to replace a runnable but expensive computational model subject to random inputs with an inexpensive-to-evaluate polynomial function. The metamodel obtained enables a reliable estimation of the statistics of the output, provided that a suitable probabilistic model of the input is available. Machine learning (ML) regression is a research field that focuses on providing purely data-driven input-output maps, with the focus on pointwise prediction accuracy. We show that a PCE metamodel purely trained on data can yield pointwise predictions whose accuracy is comparable to that of other ML regression models, such as neural networks and support vector machines. The comparisons are performed on benchmark datasets available from the literature. The methodology also enables the quantification of the output uncertainties, and is robust to noise. Furthermore, it enjoys additional desirable properties, such as good performance for small training sets and simplicity of construction, with only little parameter tuning required.