Editing Kalman filter (section)

== Overview of the calculation ==
Kalman filtering uses a system's dynamic model (e.g., physical laws of motion), known control inputs to that system, and multiple sequential measurements (such as from sensors) to form an estimate of the system's varying quantities (its [[State space (controls)|state]]) that is better than the estimate obtained by using only one measurement alone. As such, it is a common [[sensor fusion]] and [[data fusion]] algorithm.

Noisy sensor data, approximations in the equations that describe the system evolution, and external factors that are not accounted for, all limit how well it is possible to determine the system's state. The Kalman filter deals effectively with the uncertainty due to noisy sensor data and, to some extent, with random external factors. The Kalman filter produces an estimate of the state of the system as an average of the system's predicted state and of the new measurement using a [[Weighted mean|weighted average]]. The purpose of the weights is that values with better (i.e., smaller) estimated uncertainty are "trusted" more. The weights are calculated from the [[covariance]], a measure of the estimated uncertainty of the prediction of the system's state. The result of the weighted average is a new state estimate that lies between the predicted and measured state, and has a better estimated uncertainty than either alone. This process is repeated at every time step, with the new estimate and its covariance informing the prediction used in the following iteration. This means that Kalman filter works [[recursive filter|recursively]] and requires only the last "best guess", rather than the entire history, of a system's state to calculate a new state.

The measurements' certainty-grading and current-state estimate are important considerations. It is common to discuss the filter's response in terms of the Kalman filter's ''[[gain (electronics)|gain]]''. The Kalman gain is the weight given to the measurements and current-state estimate, and can be "tuned" to achieve a particular performance. With a high gain, the filter places more weight on the most recent measurements, and thus conforms to them more responsively. With a low gain, the filter conforms to the model predictions more closely. At the extremes, a high gain (close to one) will result in a more jumpy estimated trajectory, while a low gain (close to zero) will smooth out noise but decrease the responsiveness.

When performing the actual calculations for the filter (as discussed below), the state estimate and covariances are coded into [[Matrix (mathematics)|matrices]] because of the multiple dimensions involved in a single set of calculations. This allows for a representation of linear relationships between different state variables (such as position, velocity, and acceleration) in any of the transition models or covariances.