Department of Biomedical Engineering and Computational Science

UNGM-model estimation with extended and unscented Kalman filters, 'ungm_demo'

In this demonstration we use the extended Kalman filter as well as the Unscented Kalman filter to estimate the Univariate Nonstationary Growth (UNGM) model, which is a bimodal and highly nonlinear time series. The purpose of this demonstration is to show the possible advantage of augmented form UKF over nonaugmented one. We also compare the performances of different Kalman filters to a Bootstrap filter, which is a simple particle filter.

In figure 1 we have plotted the estimates produced by augmented UKF, first order EKF and Bootstrap filter. The bimodality of the signal can be seen from the figure. For example, during samples 10-25 the EKF clearly follows the wrong mode. The same can be observed from figure 2, which contains the absolute errors of the previous estimates and the their credible intervals. Figure also shows that EKF is clearly more over confident than UKF and Bootstrap-filter. However, the difference between the two latter methods can't be seen clearly from these two figures. By calculating the mean square errors (see figure 4 for MSEs over 100 Monte Carlo runs) it becomes apparent, that the Bootstrap-filter is clearly superior over other tested methods. This is most likely to the bimodality of the model, as the Gaussian approximation made in EKF and UKF is known not to work well in multi-modal cases.

Another interesting observation can be made from figure 3, in which we have plotted the filtering results of a non-augmented form UKF (UKF1) and a augmented one (UKF2). The phenomenon is also most likely due to the bimodality as it appears that UKF1 can't decide which mode to follow and guesses between the two.

Files used in this example:

UNGM_F
UNGM_DF_DX
UNGM_D2F_DX2
UNGM_H
UNGM_DH_DX
UNGM_D2H_DX2
UNGM_DEMO
Dynamic model function
Jacobian of the dynamic model
Hessian of the dynamic model
Measurement model function
Jacobian of the measurement model
Hessian of the measurement model
UNGM model demonstration
Figure 1

Figure 1. Estimates for 100 first samples of UNGM-model produced by a) augmented UKF, b) first order EKF and c) Bootstrap filter.

Figure 1

Figure 2. Absolute errors and credible intervals of previous estimates.

Figure 1

Figure 3. Estimates for position and velocity produced by Kalman smoother.

Figure 1

Figure 4. Estimates for position and velocity produced by Kalman smoother.