Truong-Vinh Hoang

Truong-Vinh Hoang, Hermann G. Matthies

A random set is a generalisation of a random variable, i.e. a set-valued random variable. The random set theory allows a unification of other uncertainty descriptions such as interval variable, mass belief function in Dempster-Shafer theory of evidence, possibility theory, and set of probability distributions. The aim of this work is to develop a non-deterministic inference framework, including theory, approximation and sampling method, that deals with the inverse problems in which uncertainty is represented using random sets. The proposed inference method yields the posterior random set based on the intersection of the prior and the measurement induced random sets. That inference method is an extension of Dempster's rule of combination, and a generalisation of Bayesian inference as well. A direct evaluation of the posterior random set might be impractical. We approximate the posterior random set by a random discrete set whose domain is the set of samples generated using a proposed probability distribution. We use the capacity transform density function of the posterior random set for this proposed distribution. This function has a special property: it is the posterior density function yielded by Bayesian inference of the capacity transform density function of the prior random set. The samples of such proposed probability distribution can be directly obtained using the methods developed in the Bayesian inference framework. With this approximation method, the evaluation of the posterior random set becomes tractable.

Truong-Vinh Hoang, Sebastian Krumscheid, Hermann G. Matthies et al.

This paper presents the machine learning-based ensemble conditional mean filter (ML-EnCMF) -- a filtering method based on the conditional mean filter (CMF) previously introduced in the literature. The updated mean of the CMF matches that of the posterior, obtained by applying Bayes' rule on the filter's forecast distribution. Moreover, we show that the CMF's updated covariance coincides with the expected conditional covariance. Implementing the EnCMF requires computing the conditional mean (CM). A likelihood-based estimator is prone to significant errors for small ensemble sizes, causing the filter divergence. We develop a systematical methodology for integrating machine learning into the EnCMF based on the CM's orthogonal projection property. First, we use a combination of an artificial neural network (ANN) and a linear function, obtained based on the ensemble Kalman filter (EnKF), to approximate the CM, enabling the ML-EnCMF to inherit EnKF's advantages. Secondly, we apply a suitable variance reduction technique to reduce statistical errors when estimating loss function. Lastly, we propose a model selection procedure for element-wisely selecting the applied filter, i.e., either the EnKF or ML-EnCMF, at each updating step. We demonstrate the ML-EnCMF performance using the Lorenz-63 and Lorenz-96 systems and show that the ML-EnCMF outperforms the EnKF and the likelihood-based EnCMF.