Interactive demo · Bayesian and Probabilistic Machine Learning · Chapter 13

Linear dynamical systems: Kalman filter and smoother

An object moves in the plane with nearly constant velocity (blue); we see only noisy measurements of its position (green). The latent state z_n holds position and velocity, z_n = Az_n−1 + noise of covariance Γ, and x_n = Cz_n + noise of covariance Σ. The Kalman filter (red) predicts each new state and corrects the prediction by the gain times the measurement error, using the data up to time n; the smoother (pink) uses all the data, so it is never less certain. Ellipses: one standard deviation.

Show

Kalman filter p(z_n|x₁..x_n) RTS smoother p(z_n|x₁..x_N) one-step predictions p(z_n|x₁..x_n−1) observations 11 to 20 missing

Process noise q (acceleration)

Measurement noise sd r

Time steps N

Try these

Model (time step 1): A adds the velocity to the position; Γ = q[[⅓I, ½I], [½I, I]] is the covariance of a random acceleration of strength q over one step; C picks the position; Σ = r²I. Prior: position (0, 0) and velocity (1, 0.6), standard deviations 0.71 and 0.32. The filter uses the same q and r as the simulation. The random numbers are seeded, so a class sees the same pictures.

In the lecture notes

Chapter 13 notes: Section 13.3 (linear dynamical systems), Section 13.3.1 (inference: the Kalman filter (13.89)-(13.92) and the Rauch-Tung-Striebel smoother (13.100)-(13.102)) and Section 13.3.2 (learning by EM).
Nonlinear models: the companion demo particle filters.
Python code: code/ch13_sequential_data.py (functions tracking_model, lds_sample, kalman_filter, rts_smoother, lds_em; Figure 13.22) and the checks in code/checks_ch13.py, which compare the filter and smoother with the conditionals of the joint Gaussian. lds_sample(JSRandom(seed), ...) reproduces this page's paths; preset 1 is our Figure 13.22.