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Introduction to Kalman Filters

A quantity you cannot observe directly evolves over time, and you receive noisy measurements of it. The Kalman filter is the optimal way to track it, and it does so recursively: only the current estimate is carried forward, never the full history.

The model

The state equation
xt=Axt−1+wt,wt∼N(0,Q)x_t = Ax_{t-1} + w_t, \qquad w_t \sim \mathcal{N}(0, Q)

A hidden state evolving under noise, which the filter estimates from noisy observations of it.

zt=Hxt+vt,vt∼N(0,R)z_t = Hx_t + v_t, \qquad v_t \sim \mathcal{N}(0, R)

A hidden state evolving with process noise QQ, and observations corrupted by measurement noise RR.

Predict, then update

Predict: apply the dynamics to get a prior for the current state. Uncertainty grows, since time has passed and the process is noisy.

Update: combine that prediction with the new observation. Uncertainty shrinks, since information has arrived.

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