I am starting to look at particle filtering for a problem that I have. In particular, I would like to reduce the dimensionality of the particles.
The model that I have is able to be partitioned. Let my state vector be $\mathbf{x}=\begin{bmatrix}{\mathbf{x}_1}^T && {\mathbf{x}_2}^T \end{bmatrix}^T$. All are continuous variables. My dynamic model may be expressed as:
$\begin{bmatrix}{\mathbf{x}_1}(k+1) \\ {\mathbf{x}_2}(k+1) \end{bmatrix} = \begin{bmatrix}f_1\left( \begin{bmatrix}{\mathbf{x}_1(k)} \\ {\mathbf{x}_2}(k) \end{bmatrix}\ , \mathbf{u}(k), \mathbf{w}_1(k) \right)\\ \mathbf{F}_2 \mathbf{x}_2(k) + \mathbf{w}_2(k)\end{bmatrix} \\ \mathbf{z}(k) = \mathbf{H}_1 \mathbf{x}_1 + \mathbf{H}_2\left(\mathbf{x}_1\right)\mathbf{x}_2 + \mathbf{v}(k)$
where $\left(\bullet\right)$ is "a function of", upper-case bold are matrices and $\mathbf{u}(k)$ is a control vector. Random vectors $\mathbf{w}_1(k)$, $\mathbf{w}_2(k)$ and $\mathbf{v}(k)$ are mutually independent white Gaussian noise.
My state space model is not quite the conditionally linear structure as suggested in Gustafsson:
$\begin{bmatrix}{\mathbf{x}_1}(k+1) \\ {\mathbf{x}_2}(k+1) \end{bmatrix} = \begin{bmatrix} f_1\left( \mathbf{x}_1 \right) + \mathbf{F}_1\left( \mathbf{x}_1\right) \mathbf{x}_2 + g_1 \left( \mathbf{x}_1 \right) \mathbf{w}_1(k) \\ f_2\left( \mathbf{x}_1 \right) + \mathbf{F}_2\left( \mathbf{x}_1\right) \mathbf{x}_2 + g_2 \left( \mathbf{x}_2 \right) \mathbf{w}_2(k) \end{bmatrix} \\ \mathbf{y}(k) = h_1\left( \mathbf{x}_1 \right) + \mathbf{H}_2\left( \mathbf{x}_1 \right) \mathbf{x}_2 + \mathbf{v}(k)$
I can approximate the state transition with additive noise without too much of an approximation, but I cannot easily split $f_1\left( \begin{bmatrix}{\mathbf{x}_1(k)} \\ {\mathbf{x}_2}(k) \end{bmatrix}\right)$.
Is this model able to Rao-Blackwellised for a particle filter? If so, what papers should I be reading to learn how to do it?
ADDENDUM:
I have managed to split the model in terms of:
$\begin{bmatrix}{\mathbf{x}_1}(k+1) \\ {\mathbf{x}_2}(k+1) \end{bmatrix} = $
$\begin{bmatrix} f_1\left( \mathbf{x}_1(k) , \mathbf{u}(k), \mathbf{w}(k) \right) \\ \mathbf{F}_2 \left( \mathbf{x}_1(k) \right) \mathbf{x}_2(k) + \mathbf{B}\left(\mathbf{x}_1(k)\right) \mathbf{u}(k) + \mathbf{G}\left( \mathbf{x}_1(k)\right) \mathbf{w}(k) \end{bmatrix} \\ \mathbf{z}(k) = \mathbf{H}_2\left(\mathbf{x}_1\right)\mathbf{x}_2 + \mathbf{v}(k)$
The (Gaussian) noise vector $\mathbf{w}(k)$ is the same for both the top and bottom parts of the state transition elements, but I can split them if really required at the expense of accuracy. Measurement noise $\mathbf{v}(k)$ is independent.
This model is still not conditionally linear in the form above. Is either the original form, or the form above able to be marginalised?
Context:
I have previously used an Unscented Kalman filter for this model, which has worked reasonably well, but I would like to extract a little more performance, if at all possible. If I can partition as above, I can split 18 states into 9 states each for the linear and non-linear parts which, as I understand the theory so far, should drastically reduce the number of particles that I need.
