Data-driven polynomial chaos expansions for machine learning regression
Résumé
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.
In classical machine learning (ML) regression settings, however, the system is only known
through observations of its inputs and output, and the interest lies in obtaining accurate
pointwise predictions of the latter. Here, 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. In the presence of
statistically dependent inputs, we investigate two ways to build the PCE, and show through
simulations that one approach is superior to the other in the stated settings.
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