# Particle Filtering with Nonlinear Observation Equation of 2 set of variables

I am stuck with this problem in my research. I am having a State Space Model like the below mentioned one:

State Equation: $\mathbf{d}_k = \mathbf{d}_{k-1} + \mathbf{u}_k + \boldsymbol{\epsilon}_k$ and

Observation equation: $\mathbf{y}_k = \mathbf{f}(\mathbf{d}_k, \boldsymbol{\theta}_k) + \boldsymbol{\eta}_k$, where $\boldsymbol{\theta}_k$ are independent of $\mathbf{d}_k$. Its a multivariate state space model, i.e. the state variables, observation variables form a vector at each $k$. For this I am using Particle filtering approach as the observation equation is having nonlinear function of the State variables.

To proceed for implementing I am stuck with the additional $\boldsymbol{\theta}_k$ variables. This observation equation is having two sets of variables, state and another non-state variables. I am confused that whether there will be two sets of "weights" $w_k^i$ for two sets of variables i.e. $\boldsymbol{\theta}_k$ and $\mathbf{d}_k$.

Has anybody encountered this type of situation??? I have searched a lot but never found this type of situation. Can anybody help me out??? Thanks in advance...

• you will only have one set of weights and particles. $\theta_k$ is probably just a deterministic constant. This allows you to change your observation equation over time. The only thing this changes is the weight update formulas. The evaluation of your observation density will now be a function of three variables instead of two ($d_k$, $y_k$). – Taylor Jan 31 '17 at 13:28