Kalman Filters

Kalman filters are a sequential algorithm that can be used to generate updated estimates using a the last state of a series of measures. A classic example is to estimate the new position of an object using a new set of observations combined with the previous estimates of position, velocity and acceleration.

Kalman filters are typically expressed as a series of linked linear models:

$$ x_k = A x_{k-1} + \delta_{x} $$

$$ z_k = B x_{k-1} + \delta_{z} $$

$$ \delta_{x} \sim \mathcal{N}(0, Q) $$

$$ \delta_{z} \sim \mathcal{N}(0, R) $$

In this form, \( x_k \) is a state vector at time \( k \) (e.g. the position, velocity and acceleration vector of our example) and \( z_k \) is the observation vector (e.g. a radar echo or other such signal). \( A \) is the forward model. It uses a linear transform to update the state variable. \( B \) is the observation operator that updates the expected observation given the state vector. The linear models have an error term \( \delta \) which we’ve assumed to be normally distributed.

In this