Filtering theory

Assume a dynamical system that is governed by a stochastic difference equation:

\[dx_t = f(x_t, t)+G(x_t,t)d \beta_t\]

for all times, \(t \geq 0\). Observations occur at discrete times:

\[y_k=h(x_k,t_k)+\nu_k\]

where \(k=1,2,...;\) and \(t_{k+1} > t_k \geq t_0\).

The observational error is white in time and is Gaussian (this latter assumption is not essential).

\[\nu_k \rightarrow N(0,R_k)\]

The complete history of observations is:

\[Y_\tau=\{y_l;t_l \leq \tau\}\]

Our goal is to find the probability distribution for the state at time \(t\).

\[p(x,t|Y_t)\]

The state between observation times is obtained from the difference equation. We need to update the state given new observations:

\[p(x,t_k | Y_{t_k}) = p(x,t_k |y_k, Y_{t_{k-1}})\]

We do so by applying Bayes’ rule:

\[p(x,t_k | Y_{t_k}) = \frac{p(y_k |x_k, Y_{t_{k-1}}) p(x,t_k | Y_{t_{k-1}})}{p(y_k, Y_{t_{k-1}})}\]

Since the error is white in time:

\[p(y_k | x_k, Y_{t_{k-1}})=p(y_k|x_k)\]

We integrate the numerator to obtain a normalizing denominator:

\[p(y_k | x_k, Y_{t_{k-1}})= \int p(y_k|x) p(x,t_k |Y_{t_{k-1}})dx\]

This allows us to update the probability after a new observation:

\[p(x,t_k | Y_{t_k}) = \frac{p(y_k|x) p(x,t_k |Y_{t_{k-1}})}{\int p(y_k|\xi) p(\xi,t_k |Y_{t_{k-1}})d\xi}\]