The joint estimation of the parameters and the states of the hemodynamic model from the blood oxygen level dependent (BOLD) signal is a challenging problem. In the functional magnetic resonance imaging (fMRI) literature, quite interestingly, many proposed algorithms work only as a filtering method. This makes the estimation of hidden states and parameters less reliable compared with the algorithms that use smoothing. In standard implementations, smoothing is performed only once. However, joint state and parameter estimation can be improved substantially by iterating smoothing schemes such as the extended Kalman smoother (IEKS). In the fMRI literature, extended Kalman filtering is thought to be less accurate than standard particle filtering (PF). We compared EKF with PF and observed that the contrary is true. We improved the EKF performance by adding smoother. By iterative scheme joint hemodynamic and parameter estimation is improved substantially. We compared IEKS performance with the square-root cubature Kalman smoother (SCKS) algorithm. We show that its accuracy for the state and the parameter estimation is better and much faster than iterative SCKS. SCKS was found to be a better estimator than the dynamic expectation maximization (DEM), EKF, local linearization filter (LLF) and PF methods. We show in this paper that IEKS is a better estimator than iterative SCKS under different process and measurement noise conditions. As a result, IEKS seems to be the best method we evaluated in all aspects. 
Joint parameter and state estimation of the hemodynamic model by iterative extended Kalman smoother