Robust Partial-Learning in Linear Gaussian Systems

This technical note deals with unsupervised and off-line learning of parameters involved in linear Gaussian systems, i.e., the estimation of the transition and the noise covariances matrices of a state-space system from a finite series of observations only. In practice, these systems are the result of a physical problem for which there is a partial knowledge either on the sensors from which the observations are issued or on the state of the studied system. We therefore propose in this work an “Expectation-Maximization” learning type algorithm that takes into account constraints on parameters such as the fact that two identical sensors have the same noise characteristics, and so estimation procedure should exploit this knowledge. The algorithms are designed for the pairwise linear Gaussian system that takes into account supplementary cross-dependences between observations and hidden states w.r.t. the conventional linear system, while still allowing optimal filtering by means of a Kalman-like filter. The algorithm is made robust through QR decompositions and the propagation of a square-root of the covariance matrices instead of the matrices themselves. It is assessed through a series of experiments that compare the algorithms which incorporate or not partial knowledge, for short as well as for long signals.