Geodesy

Algorithm

Benchmark (surveying)

Pure mathematics

Uncertainty quantification

Statistics

Support vector machine

Field (mathematics)

Probabilistic logic

Computer science

Mathematical optimization

Polynomial

Regression analysis

Polynomial chaos

Regression

Metamodeling

Mathematics

Geography

Pointwise

Function approximation

Mathematical analysis

Artificial intelligence

Monte Carlo method

Programming language

Artificial neural network

Polynomial regression

Machine learning

Journal of Computational Physics

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.

Data-driven polynomial chaos expansion for machine learning regression