A two-observation Kalman framework for maximum-likelihood modeling of noisy time series

Modeling a noisy time series requires the dual estimation of both the model parameters and the underlying clean time series. Most approaches estimate the model parameters by minimizing the mean squared prediction error, but estimate the time series by minimizing another cost function. We justify the use of the same maximum-likelihood cost for both parameter and time series estimation, and present a new weight update procedure for recursive minimization of this cost. This learning algorithm uses a two-observation form of the extended Kalman filter, and provides a natural extension of the dual extended Kalman filter procedure previously developed by the authors.