Analytical results for uncertainty propagation through trained machine
  learning regression models
Andrew Thompson
Machine learning (ML) models are increasingly being used in metrology
applications. However, for ML models to be credible in a metrology context they
should be accompanied by principled uncertainty quantification. This paper
addresses the challenge of uncertainty propagation through trained/fixed
machine learning (ML) regression models. Analytical expressions for the mean
and variance of the model output are obtained/presented for certain input data
distributions and for a variety of ML models. Our results cover several popular
ML models including linear regression, penalised linear regression, kernel
ridge regression, Gaussian Processes (GPs), support vector machines (SVMs) and
relevance vector machines (RVMs). We present numerical experiments in which we
validate our methods and compare them with a Monte Carlo approach from a
computational efficiency point of view. We also illustrate our methods in the
context of a metrology application, namely modelling the state-of-health of
lithium-ion cells based upon Electrical Impedance Spectroscopy (EIS) data