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Main: KalmanFilter

The Kalman filter is the on-line Bayes algorithm applied to a class of processes with linear dynamics and Gaussian noise. There are two vector sequences that describe the dynamics of the system:

The dynamics is described by:

$$
  X_{t+1} = A X_t + N_t,
  \quad
  Y_t = B X_t + M_t,
$$

where $A$ is a matrix called the dynamics matrix, $B$ is a matrix called the observation matrix, and $N_t$ and $M_t$ are independent samples from multi-dimensional normal distributions. These are called the dynamics noise and the measurement noise, respectively.

In the standard usage, the matrices $A$ and $B$ and the parameters of $N$ and $M$ are all assumed to be known. The algorithm recieves as input the observation sequence $y_1,y_2,\ldots, y_t$ and its goal is to estimate $x_t$. This is done using the online Bayes estimator (which is equivalent to the strong aggregating algorithm for the log loss function). In this case the computation of the prediction can be done in closed form using matrix operations.

The Kalman filter does not operate well when the dynamics is far from linear or the noise far from Gaussian. In these cases the only semi-practical solution is "particle filters", which is a Monte Carlo method that samples from the posterior distribution over the states. It is only semi-practical because one needs to use a very large number for particles (samples) to get reliable results.

References

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Page last modified on July 06, 2008, at 05:10 PM