A New Approach to Linear Filtering and Prediction Problems
Track the truth through the noise by blending each prediction with the next imperfect measurement.
Your phone knows where you are by trusting neither its map nor its GPS completely — it blends a guess about where you should be with each noisy reading of where you seem to be.
The big idea
Suppose you want to know something that keeps changing — where a plane is, how fast a rocket is climbing — but every measurement you get is blurred by noise. You could just believe the latest reading, except it jitters. You could ignore the readings and trust your prediction of where things ought to be, except predictions drift. Kalman's filter does neither and both: at every instant it makes a prediction, takes a measurement, and settles on a weighted blend of the two.
The weight is the clever part. If the sensor is reliable, lean on the measurement; if it is noisy, lean on the prediction — and the filter works out the right balance on its own, from how uncertain it currently is. It then updates that uncertainty and rolls forward. One short loop, run over and over, turns a stream of poor measurements into a steadily good estimate.
How it came about
By the 1950s the mathematics of filtering noise out of a signal already existed: Norbert Wiener and Andrey Kolmogorov had built it during the war, for aiming anti-aircraft guns (wiener-1948). But their method demanded a signal whose statistics never changed, the entire history of the data, and a hard integral to solve. It was beautiful and nearly unusable for a vehicle whose motion changes by the second.
Rudolf Kálmán, a Hungarian-born engineer at a small research institute in Baltimore, reframed the question. Instead of working in frequencies across all time, he described the system by its state — position, velocity, whatever mattered — and let it march forward step by step. Filtering became a simple recursion a computer could run in real time. The control establishment was sceptical and the paper was nearly rejected; then, in 1960, Kálmán described it to Stanley Schmidt at NASA Ames, who saw at once that it could navigate Apollo to the Moon — and the method's future was sealed.
Why it mattered
It made optimal estimation something you could actually compute — on the modest machines of the day, for systems that move and change. That is why it escaped the journals and went everywhere real-time decisions ride on noisy data: spacecraft, aircraft, ships, and eventually the GPS chip and motion sensors in your pocket. Few equations have done so much quiet work.
A way to picture it
Imagine driving through thick fog with an unreliable speedometer. You keep a running sense of where you are and how fast you are going, and you predict where you will be a second from now. Every so often a road sign looms out of the fog — a noisy measurement. You don't fully trust the sign, and you don't fully trust your guess; you split the difference, leaning toward whichever you currently trust more, and drive on with a sharper sense of your position. That moving, self-correcting blend is the Kalman filter.
Where it sits
Kalman's filter is the recursive heir to Wiener's wartime filtering (wiener-1948), and it shares the spirit of Bayes (bayes-1763) — each measurement updating a belief — recast for a moving target and a real-time computer. Downstream it is the engine of modern navigation and a building block of today's probabilistic time-series models and robotic sensor fusion. When a self-driving car fuses many noisy signals into one estimate of the world, it is standing on this 1960 loop.