The Kalman Filter Explained

A Step-by-Step Derivation

The Kalman Filter cycle: prediction step uses a physical model to advance the state estimate, then the update step corrects the prediction using measurements, producing a refined output estimate

The aim of this tutorial is to derive the filtering equations for the simplest Linear Dynamical System case — the Kalman Filter — outline the filter's implementation, do a similar thing for the smoothing equations and conclude with parameter learning in an LDS (calibrating the Kalman Filter).

What the Tutorial Covers

  • The Linear Dynamical System model
  • Deriving the Kalman filtering equations from first principles
  • The Kalman smoother
  • Parameter learning (EM algorithm for LDS)
  • Practical implementation guidance

Download the full tutorial (PDF)

Written by Dr Tristan Fletcher. See also the companion tutorials on Support Vector Machines and Relevance Vector Machines, or browse all ML tutorials.