Dual Estimation Experiments

We present results on two time-series to provide a clear illustration of the use of the UKF over the EKF. The first series is again the Mackey-Glass-30 chaotic series with additive noise (SNR $\approx$ 3dB). The second time series (also chaotic) comes from an autoregressive neural network with random weights driven by Gaussian process noise and also corrupted by additive white Gaussian noise (SNR $\approx$ 3dB). A standard 6-4-1 MLP with $tanh$ hidden activation functions and a linear output layer was used for all the filters in the Mackey-Glass problem. A 5-3-1 MLP was used for the second problem. The process and measurement noise variances were assumed to be known. Note that in contrast to the state-estimation example in the previous section, only the noisy time-series is observed. A clean reference is never provided for training. Example training curves for the different dual and joint Kalman based estimation methods are shown in Figure 3. A final estimate for the Mackey-Glass series is also shown for the Dual UKF. The superior performance of the UKF based algorithms are clear. These improvements have been found to be consistent and statistically significant on a number of additional experiments.

Figure: Comparative learning curves and results for the dual estimation experiments.