118.1 Kalman Filter - circa 1960
The KF requires a good understanding of linear algebra, matrix operations, and probability theory. If you are not familiar with these concepts, it is recommended to review them before diving into the Kalman filter.
- Linear algebra:
- Vectors and matrices are used to represent the state of a dynamic system.
- Eigenvalue decomposition is used to analyze the dynamics of the system.
- The matrix exponential is used to solve the state space model.
- Jordan form may be needed if the dynamics matrix is not diagonalizable.
- The Toeplitz matrices and Vandermonde matrices are used in some derivations of the Kalman filter.
- State space vector representation of dynamic systems.
- Laplace transforms are used for the state space model.
- Convolution operation
The course instructor points out that you can learn the Kalman filter in a couple of lessons and that it is not a difficult topic.
This is true, but he goes into the details of the derivation and the mathematical background, which is important for a deep understanding of the topic. Also there are many variations of the Kalman filter, and understanding the derivation can help you understand how to modify the filter for different applications.
One fascinating connection is that of the Kalman filter to the Bellman equation in Reinforcement Learning. The Kalman filter can be seen as a special case of the Bellman equation, where the state transition and observation models are linear and Gaussian.