Smoothing for continuous dynamical state space models with sampled system coefficients based on sparse kernel learning

摘要

A new smoother for a continuous dynamical state space model with sampled system coefficients is proposed. This is completely different from conventional approaches, such as Rauch-Tung-Striebel smoother. In the proposed method, the state vector as a continuous function of time is represented by kernel models. The state process model, namely the differential equation, is treated as part of the measurement model at the sampling instants of the system coefficients. Sparse solution of the kernel weights is obtained through a special regularization strategy called the Lasso estimator. The optimization problem appearing in the Lasso estimation is solved by the fast iterative shrinkage threshold algorithm. The hyperparameters involved, namely the kernel widths and the regularization coefficients, are selected objectively through generalized cross-validation or corrected Akaike information criterion tailored to the Lasso estimator. A simple two-dimension example is employed in the simulation to demonstrate the application and also the performance of the proposed method. It is shown that the proposed method could provide state vector estimates with satisfactory accuracy not only at the sampling instants of the observations but also at any other instants. The sparsity of the solution could also be clearly seen in the experiment. The proposed method can be viewed as an alternative smoothing method, rather than a replacement for conventional smoothers, due to the difficult model tuning and increased computation load.