Author: Indigo Curnick
Date: 2025-03-08
This article is a WIP - I'll add more notations as I find them
Kalman filtering notation is terrible. Every author uses a different, confusing, notation. I am not able to standardise it, but at least I can provide some of the key references and explain what notation they use.
This article assumes some familiarity with the Kalman filter, as I won't detail every single symbol.
Welch and Bishop
Greg Welch and Gary Bishop published the very popular An Introduction to the Kalman Filter that many people see as a first resource. The equations are
Predict
\[ \hat{x}^-_k = A \hat{x}_{k-1} + Bu_{k-1} \] \[ P^-_k = AP_{k-1}A^T + Q \]
Update
\[ K_k = P^-_k H^T (H P^-_k H^T + R)^{-1} \] \[ \hat{x}_k = \hat{x}^-_k + K_k(z_k - H \hat{x}^-_k) \] \[ P_k = (I - K_k H)P^-_k \]
Wikipedia
Wikipedia uses a more complicated notation
Predict
\[ \hat{\mathbf{x}}_{k\mid k-1} = \mathbf{F}_k \mathbf{x}_{k-1\mid k-1} + \mathbf{B}_k \mathbf{u}_{k} \] \[ \hat{\mathbf{P}}_{k\mid k-1} = \mathbf{F}_k \mathbf{P}_{k-1 \mid k-1} \mathbf{F}_k^\textsf{T} + \mathbf{Q}_k \]
Update
\[ \tilde{\mathbf{y}}_k = \mathbf{z}_k -\mathbf{H}_k\hat{\mathbf{x}}_{k\mid k-1} \] \[ \mathbf{S}_k = \mathbf{H}_k \hat{\mathbf{P}}_{k\mid k-1} \mathbf{H}_k^\textsf{T} + \mathbf{R}_k \] \[ \mathbf{K}_k = \hat{\mathbf{P}}_{k\mid k-1}\mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1} \] \[ \mathbf{x}_{k\mid k} = \hat{\mathbf{x}}_{k\mid k-1} + \mathbf{K}_k\tilde{\mathbf{y}}_k \] \[ \mathbf{P}_{k|k} = \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right) \hat{\mathbf{P}}_{k|k-1} \] \[ \tilde{\mathbf{y}}_{k\mid k} = \mathbf{z}_k - \mathbf{H}_k\mathbf{x}_{k\mid k} \]
Labbe
More recently, Roger Labbe's GitHub textbook Kalman and Bayesian Filters in Python has become one of the go to texts. He uses a very simple notation
Predict
\[ \mathbf{\bar{x}} = \mathbf{Fx} + \mathbf{Bu} \] \[ \mathbf{\bar{P}} = \mathbf{FPF^T} + Q \]
Update
\[ \mathbf{y} = \mathbf{z} - \mathbf{H \bar{x}} \] \[ \mathbf{S} = \mathbf{H \bar{P}} \mathbf{H}^T + R \] \[ \mathbf{K} = \mathbf{\bar{P}} \mathbf{H}^T \mathbf{S}^{-1} \] \[ \mathbf{x} = \mathbf{\bar{x}} + \mathbf{Ky} \] \[ \mathbf{P} = (\mathbf{I} - \mathbf{KH}) \mathbf{\bar{P}} \]
Zarchan and Musoff
Paul Zarchan and Howard Musoff in Fundamnetals of Kalman Filtering: A Practical Approach present an unusual notation. I have mixed feelings about this textbook, since it comes with great examples of actually using the filter and an exploration of how the filter behaves. But, I hate this notation because it doesn't make a clear, distinct separation between predictions and updates.
On the other hand, Zarchan is one of few authors who actually provides enough information to reproduce his Kalman filters. Not only that, but the methods he describes to create an extended Kalman filter seem to be the only reliable way to produce one.
\[ \hat{x}_k = \Phi_k \hat{x}_{k-1} + G_k u_{k-1} + K_k [z_k - H \Phi_k \hat{x}_{k-1} - H G_k u_{k-1}] \] \[ M_k = \Phi_k P_{k-1} \Phi^T_k + Q_k \] \[ K_k = M_k H^T [H M_k H^T + R_k]^{-1} \] \[ P_k = (I - K_k H) M_k \]
Brookner
Eli Brookner in his Tracking and Kalman Filtering Made Easy presents an interesting notation. Again, no separation between predict and update.
\[ X^*_{n+1,n} = \Phi X^*_{n,n} \] \[ X^*_{n,n} = X^*_{n,n-1} + H_n (Y_n - MX^*_{n,n-1}) \] \[ H_n = S^*_{n,n-1} M^T [R_n + MS^*_{n,n-1} M^T]^{-1} \] \[ S^*_{n,n-1} = \Phi S^*_{n-1,n-1} \Phi^T + Q_n \] \[ S^*_{n-1,n-1} = (I - H_{n-1} M) S^*_{n-1,n-2} \]
Gelb
Arthur Gelb et al in Applied Optimal Estimation present another system for notation. Again, no separation between predict and update.
\[ \hat{\underline{x}}_k(-) = \Phi_{k-1} \hat{\underline{x}}_{k-1}(+) \] \[ \hat{\underline{x}}_k(+) = \hat{\underline{x}}_k(-) + K_k[Z_k -H_k ] \hat{\underline{x}}_k(-) \] \[ K_k = P_k(-) H_k^T [H_k P_k(-) H_k^T + R_k]^{-1} \] \[ P_k(+) = \Phi_{k-1} P_{k-1}(+) \Phi^T_{k-1} + Q_{k-1} \] \[ P_k(-) = (I - K_k H_k) P_k(-) \]
Brown and Hwang
Robert Grover Brown and Patrick Y.C. Hwang in Introduction to Random Signals and Applied Kalman Filtering also don't make a distinction between predict and update.
\[ \hat{\mathbf{x}}^-_{k+1} = \phi_k \hat{\mathbf{x}}_k \] \[ \hat{\mathbf{x}}_k = \hat{\mathbf{x}}^-_k + \mathbf{K}_k [\mathbf{z}_k - \mathbf{H}_k \hat{\mathbf{x}}_k] \] \[ \mathbf{K}_k = \mathbf{P}^-_k \mathbf{H}^T_k [\mathbf{H}_k \mathbf{P}^-_k \mathbf{H}_k^T + \mathbf{R}_k]^{-1} \] \[ \mathbf{P}^-_{k+1} = \phi \mathbf{P}_k \phi^T_k + \mathbf{Q}_k \] \[ \mathbf{P}_k = (\mathbf{I} - \mathbf{K}_k \mathbf{H}_k) \mathbf{P}^-_k \]
My Own Notation
In the tradition of other Kalman filter authors, I also make my own notation. I mostly use inspiration from Zarchan, but I introduce back in the concept of the distinct predict and update steps.
Predict
\[ \bar{x}_k = \Phi_k \hat{x}_{k-1} + G_k u_{k-1} \] \[ M_k = \Phi_k P_{k-1} \Phi^T_k + Q_k \]
Update
\[ K_k = M_k H^T [H M_k H^T + R_k]^{-1} \] \[ \tilde{x}_k = z_k - H \Phi_k \hat{x}_{k-1} - H G_k u_{k-1} \] \[ \hat{x}_k = \bar{x}_k + K_k \tilde{x}_k \] \[ P_k = (I - K_k H) M_k \]
References
Zarchan, P., Musoff, H. (2009) Fundamentals of Kalman Filtering: A Practical Approach (3rd Ed.). American Institude of Aeronautics and Astronautics
Curnick, I. (2025) Kalman Filter: How to Solve the Riccati Equations. Available from https://indigocurnick.xyz/blog/2025-01-25/solving-riccati-equations
Welch, G., Bishop, G. (2006) An Introduction to the Kalman Filter. University of North Carolina at Chapel Hill
Wikipedia Kalman Filter. Available from https://en.wikipedia.org/wiki/Kalman_filter
Labb, R. Kalman and Bayesian Filters in Python. Available from https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python
Brookner, E. (1998) Tracking and Kalman Filtering Made Easy. John Wiley & Sons, Inc.
Gelb, A. (1974) Applied Optimal Estimation. MIT Press
Brown, R. G., Hwang, P. Y. C. (2012) Introduction to Random Signals and Applied Kalman Filtering (4th Ed). John Wiley & Sons, Inc.