UKF dual estimation

Recall that the dual estimation problem consists of simultaneously estimating the clean state ${\bf x}_k$ and the model parameters ${\bf w}$ from the noisy data $y_k$ (see Equation 7). As expressed earlier, a number of algorithmic approaches exist for this problem. We present results for the Dual UKF and Joint UKF. Development of a Unscented Smoother for an EM approach [9] was presented in [2]. As in the prior state-estimation example, we utilize a noisy time-series application modeled with neural networks for illustration of the approaches.

In the the dual extended Kalman filter [10], a separate state-space representation is used for the signal and the weights. The state-space representation for the state ${\bf x_k}$ is the same as in Equation 20. In the context of a time-series, the state-space representation for the weights is given by

$${{\bf w}_k} = {\bf w}_{k-1} + {\bf u}_k$$ (21)

$${y_k} = f({{\bf x}_{k-1}, {\bf w}_k}) + {v_k} + {n_k} .$$ (22)

where we set the innovations covariance ${\bf P}_{\bf u}$ equal to $\mu {\bf P}_{\mathbf{w}}$ ³. Two EKFs can now be run simultaneously for signal and weight estimation. At every time-step, the current estimate of the weights is used in the signal-filter, and the current estimate of the signal-state is used in the weight-filter. In the new dual UKF algorithm, both state- and weight-estimation are done with the UKF. Note that the state-transition is linear in the weight filter, so the nonlinearity is restricted to the measurement equation.

In the joint extended Kalman filter [11], the signal-state and weight vectors are concatenated into a single, joint state vector: ${[ {\bf x}_k^T \, {\bf w}_k^T]}^T$. Estimation is done recursively by writing the state-space equations for the joint state as:

$$\left [ \begin{matrix}{\bf x}_{k}\\ {\bf w}_{k} \end{matrix} \right ] = \left [ \begin{matrix}F( {\bf x}_{k-1}, {\bf w}_{k-1})\\ {\bf I......} \right ] + \left [\begin{matrix}B\cdot v_k \\ {\bf u}_k \end{matrix} \right ]$$ (23)

$$y_k = \left [ \begin{matrix}1 & 0 &\cdots&0 \end{matrix} \right] \left[ \begin{matrix}{\bf x}_{k}\\ {\bf w}_{k} \end{matrix} \right ] + n_k ,$$ (24)

and running an EKF on the joint state-space⁴ to produce simultaneous estimates of the states ${\bf x}_k$ and ${\bf w}$. Again, our approach is to use the UKF instead of the EKF.