Kalman filters are insanely popular in many engineering fields, especially those involve sensors and motion tracking. Consider how to design a radar system to track military aircrafts (or warships, submarines, … for that matter), how to track people or vehicles in a video stream, how to predict location of a vehicle carrying a GPS sensor… In all these cases, some (advanced) variation of Kalman filter is probably what you would need.
Learning and teaching Kalman filters is therefore quite challenging, not only because of the mere complexity of the algorithms, but also because there are many variations of them.
With a Computer Science background, I encountered Kalman filters several years ago, but never managed to put them into the global context of the field. I had chances to look at them again recently, and rediscovered yet another way to present and explain Kalman filters. It made a lot of sense to me, and hopefully it does to you too.
Note that there are a lot of details missing from this post (if you are building a radar system to track military aircrafts, look somewhere else!). I was just unhappy to see many introductory material on Kalman filters are either too engineering or too simplified. I want something more Machine Learning-friendly, so this is my attempt.
Let’s say you want to track an enemy’s aircraft. All you have is a lame radar (bought from Russia probably) which, when oriented properly, will give you a 3-tuple of range, angle and angular velocity ![[r \;\phi\;\dot{\phi}]^{T}](http://s0.wp.com/latex.php?latex=%5Br+%5C%3B%5Cphi%5C%3B%5Cdot%7B%5Cphi%7D%5D%5E%7BT%7D&bg=ffffff&fg=333333&s=0 "[r \;\phi\;\dot{\phi}]^{T}")
![\mathbf{x}_k = [x_1\;x_2\;x_3]^{T}](http://s0.wp.com/latex.php?latex=%5Cmathbf%7Bx%7D_k+%3D+%5Bx_1%5C%3Bx_2%5C%3Bx_3%5D%5E%7BT%7D&bg=ffffff&fg=333333&s=0 "\mathbf{x}_k = [x_1\;x_2\;x_3]^{T}")
Visualised as a Bayesian network, it looks like this:
With all the Markov properties hold, i.e. 
Yes, indeed. This looks a lot like the beloved Hidden Markov Models, where 
However, in HMMs, the states
1. The tracking problem
Before we see how we can solve this problem, let’s first formally define it. Let 
The whole point of tracking is to infer the value of the state at the current timestep 
Let’s assume somehow we know the distribution 
Inspired by some similar idea in the field (e.g. the EM algorithm), we can do this in 2 step. The first step is to predict the value of 
The last equation was based on the Markov assumption that 
Assume that is doable (although maybe not trivial), we now have the prior distribution 
Here we use the observation model and the prior computed in the previous step to estimate the posterior 
Unfortunately, both the predict and update equations are intractable in the general case. We therefore need to find some ways to approximate them.
2. Kalman filter
Enter Kalman filter. In Kalman filter, we make the following assumption:
- The prior distribution is Gaussian:
, parameterized by the state mean vector
and covariance matrix
.
Predict and update the states now mean we need to keep track the value ofand
over time.
- The transition model is linear:
whereis the control vector (zero for many application),
is the process noise vector drawn from a zero-mean Gaussian distribution with covariance
.
Practically, this means the transition distribution is Gaussian:
Note that althoughare all subscripted t, but they don’t need to be kept track over the course of tracking. Often they only depend on the time difference between the latest timestep and the previous one.
- The observation model is linear:
whereis the observation noise.
Similarly, this means the observation distribution is Gaussian:
, parameterized by the state mean vector 
and covariance matrix 
.
is the control vector (zero for many application), 
is the process noise vector drawn from a zero-mean Gaussian distribution with covariance 
.
are all subscripted t, but they don’t need to be kept track over the course of tracking. Often they only depend on the time difference between the latest timestep and the previous one.
is the observation noise.
Those assumption are based on the fact that a linear transformation of a Gaussian is also a Gaussian.
Now since everything is Gaussian, the equations become tractable and have closed-form solutions. The derivation of Kalman filter can be found everywhere on the Internet, and only involves a few matrix operations.
In fact this is why it is so popular. At the end of the day, if the state vector is low-dimensional, it can be implemented very efficiently on low-power devices, while giving good solutions once the assumptions above are satisfied.
3. Extended Kalman Filter and Unscented Kalman Filter
Unfortunately, life is not linear. For instance, the location of an aircraft is almost certainly not a linear function of its location in previous timestep (unless it is flying at a constant velocity, which is the best way to be shot down). For those applications, the assumption of Kalman filter doesn’t hold anymore.
There are a few way to deal with this scenario, where the transition function 
First, Extended Kalman Filter uses the Taylor Series expansion to approximate the function 
However, computing the Jacobian matrix is messy, and the Taylor Series expansion might not very well-behaved for complicated process. In those case, Unscented Kalman Filter (UKF) might be used.
UKF is based on the observation that we only need to track the mean vector 
It looks like this:
The details on how to choose the sigma points and how to compute the transformed mean and variance can be found on the Internet. We are not going to repeat them here because they are boring, but the idea now gets closer to sampling methods.
We will have another post for Particle Filtering, with all the sampling methods. This is where things get serious.
