# Kalman filter updating numerical example

The state of the filter is represented by two variables: The Kalman filter has two distinct phases: Predict and Update.

This is justified because, as an optimal estimator, the Kalman filter makes best use of the measurements, therefore the PDF for given the measurements is the Kalman filter estimate.In order to use the Kalman filter to estimate the internal state of a process given only a sequence of noisy observations, one must model the process in accordance with the framework of the Kalman filter.This means specifying the matrices F The initial state, and the noise vectors at each step are all assumed to be mutually independent.The true state is assumed to be an unobserved Markov process, and the measurements are the observed states of a hidden Markov model.However, when the Kalman filter is used to estimate the state x, the probability distribution of interest is that associated with the current states conditioned on the measurements upto the current timestep.The filter was developed in papers by Swerling (1958), Kalman (1960) and Kalman and Bucy (1961).

A wide variety of Kalman filters have now been developed, from Kalman's original formulation, now called the simple Kalman filter, to Schmidt's extended filter, the information filter and a variety of square-root filters developed by Bierman, Thornton and many others.

For example, in a radar application, where one is interested in tracking a target, information about the location, speed, and acceleration of the target is measured with a great deal of corruption by noise at any time instant.

The Kalman filter exploits the dynamics of the target, which govern its time evolution, to remove the effects of the noise and get a good estimate of the location of the target at the present time (filtering), at a future time (prediction), or at a time in the past (interpolation or smoothing). Kalman, though Peter Swerling actually developed a similar algorithm earlier.

In the update phase measurement information from the current timestep is used to refine this prediction to arrive at a new, (hopefully) more accurate estimate. Consider a truck on perfectly frictionless, infinitely long straight rails.

Initially the truck is stationary at position 0, but it is buffeted this way and that by random acceleration.

The Kalman filter is an efficient recursive filter which estimates the state of a dynamic system from a series of incomplete and noisy measurements.