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Wiener Chaos in kernel regression: toward untangling aleatoric and epistemic uncertainty
Publikationstyp
Conference Paper
Date Issued
2025-09
Sprache
English
Start Page
105
End Page
120
Citation
Symposium on Systems Theory in Data and Optimization, SysDO 2024
Contribution to Conference
Publisher DOI
Publisher
Springer
ISBN of container
978-3-031-83191-1
978-3-031-83190-4
978-3-031-83193-5
Gaussian Processes (GPs) are a versatile method that enables different approaches toward learning for dynamics and control. Gaussianity assumptions appear in two dimensions in GPs: The positive semi-definite kernel of the underlying reproducing kernel Hilbert space is used to construct the covariance of a Gaussian distribution over functions, while measurement noise (i.e., data corruption) is usually modeled as i.i.d. additive Gaussians. In this note, we generalize the setting and consider kernel ridge regression with additive i.i.d. non-Gaussian measurement noise. To apply the usual kernel trick, we rely on the representation of the uncertainty via polynomial chaos expansions, which are series expansions for random variables of finite variance introduced by Norbert Wiener. We derive and discuss the analytic solution to the arising Wiener kernel regression. Considering a polynomial dynamic system as a numerical example, we show that our approach allows us to distinguish the uncertainty that stems from the noise in the data samples from the total uncertainty encoded in the GP posterior distribution.
Subjects
Kernel regression
Polynomial chaos expansion
Aleatoric uncertainty
Epistemic uncertainty
Non-Gaussian distribution
DDC Class
600: Technology