State Space Model & Particle Filter

State Space Model (SSM)

State Space Model(SSM) is widely used in the field requiring the sequential estimation or online learning. This model is effective if you consider a system having two different variables; one completely represents the actual state but cannot be observed and the other partially represents the actual state but can be observed. Here, I call the former $x$ (state variable) and the latter $y$ (observation variable).

In SSM, we intruduce the following equations $F, H$ (or $f, h$) and identify them by observed data sample $[y_1, \dots, y_t]$.

• Equation of each state $x_t$ :

  $$ \begin{aligned} x_{t+1} &= F(x_t) ~~ (\text{Deterministic process}) \\ x_{t+1} &\sim f(\cdot\vert x_t) ~~ (\text{Stochastic process}) \end{aligned} $$

• Equation of each observation $y_t$ :

  $$ \begin{aligned} y_t &= H(x_t) ~~ (\text{Deterministic process}) \\ y_t &\sim h(\cdot \vert x_t) ~~ (\text{Stochastic process}) \end{aligned} $$

Perticle filter

1. For each $i$ in $[1 \dots M]$

   1. (Prediction)

      Derive prediction distribution $f(x_t \vert \cdot)$ depends on particles $\hat{x}_{t-1}$.

      Sample $x^{i}_{t \vert t-1} ~~~ (i = 1, \dots, M)$ following $f(x_t \vert \cdot)$.

      $$ \begin{align} x^{i}_{t \vert t-1} \sim f(x_t \vert \hat{x}_{t-1}) \end{align} $$

2. (Likelihood)

   Derive the likelihood of $x^i_{t \vert t-1}$ from given sample data $y_t$ based on $h(\cdot)$

   $$ w^i_t \sim h(y_t \vert x^i_{t \vert t-1}) $$

3. (Resampling)

   Resampe $\hat{x}^i_{t \vert t-1}$ based on the likelihood $w^i_t ~~~ (i=1,\dots,M)$ .

   Derive the filter distribution $p(x_t \vert y_{1:t})$ for any $x_t$: $$ \begin{aligned} p(x_t \vert y_{1:t}) &\approx \frac{1}{M} \sum_{i=1}^{M} \delta(x_t - \hat{x}^i_{t \vert t-1}) \\ &\approx \sum_{i=1}^{M} \frac{}{\sum_{i=1}^{M} } \delta(x_t - \hat{x}^i_{t \vert t-1}) \end{aligned} $$