Kalman filter

Kalman filter is the optimal recursive algorithm for state estimation in linear dynamical systems with Gaussian noise. Developed by Rudolf Kalman in 1960, it provides the mathematical foundation for tracking the state of a system from noisy measurements — from spacecraft navigation to GPS positioning to economic forecasting.

The filter operates in two steps: prediction, where the system's model is used to project the current state estimate forward in time; and update, where the prediction is corrected using the new measurement, weighted by the relative certainty of the model versus the sensor. The result is a minimum-mean-square-error estimate that is computationally efficient and provably optimal under its assumptions.

The Kalman filter's assumptions — linear dynamics, Gaussian noise, known model — are rarely met in practice, yet the filter remains the workhorse of state estimation because it is the first approximation against which all nonlinear alternatives are judged. Its extensions — the extended Kalman filter, the unscented Kalman filter, and particle filters — each relax one assumption at a cost in complexity. The filter is also the formal ancestor of the Bayesian approach to sequential inference, demonstrating that optimal estimation is not merely engineering but a principled theory of how to learn from incomplete information.