# How to choose the “best” measurment (from a given set) as input for a kalman filter?

Problem: For an object tracking scenario with multiple objects and multiple tracks, I want to choose the "best" assignment object<->track. Therefore, I need a parameter indicating how suitable an object is for a track. (Each object refers to a measurement and each track is a Kalman filter)

Idea: Given the classical Kalman filter with the matrices state $x$ and measurement transition $H$ We can calculate the residual $y_t$ between the predicted measurement $x_{pred}$ and the measurement $z$ by:

$y_t = z-H x_{pred}$

However, this will be a vector. Is there a general approach to reduce this to one parameter? (e.g distance when having just position information). I assume that some kind of normalisation is needed to get all the residuals for each state in the same range? Building the l2-norm would be my first approach.

Question: Which parameter is suitable to indicate how "good" the measurement fits to the Kalman filter?

Is the residual suitable? If so, how to handle different value ranges, and how to "compress" the vector to just one value?

• It sounds like you're referring to the "association" function in a target tracking system. Like you said, when you get a new measurement in, you need to know which of your existing tracks it might be associated with so you can input it to the appropriate tracking filter (or, it could be a new target that you've never seen before!). There are many techniques for doing this; I think this might be too broad of a question. You might try looking in a tracking system text on this topic. – Jason R Dec 8 '16 at 13:19

Question: Which parameter is suitable to indicate how "good" the measurement fits to the Kalman filter?

To estimate a quality of association you can use likelihood function. The likelihood considers not only residual but also uncertainty and represented as scalar value:

$$\mathcal{L} = \frac{1}{\sqrt{2\pi S}}\exp [-\frac{1}{2}\mathbf y^\mathsf T\mathbf S^{-1}\mathbf y]$$

Problem: For an object tracking scenario with multiple objects and multiple tracks, I want to choose the "best" assignment object<->track.

The question which you are interested is called data association and as was said there are a lot of solution and this question is too broad. Here are some of methods (you can google it and find more information):
- Global nearest neighbour (in Euclidean space);
- Global strongest neighbour;
- Joint probability data association;
- Multiple hypothesis tracking (the best choice).

• @Glutton Can you explain what the $S$ in the root is? Should be a scalar value from my understanding. While $\mathbf{S}$ is a Matrix? I found the same notation in rlabbe's script Kalman and Bayesian Filters but no explanation givens as well. – user6522399 Dec 9 '16 at 12:14
• @user6522399, $S$ is determinant of $\mathbf S$. – Gluttton Dec 9 '16 at 12:47
• For those not sure of what the matrix S is, this formula is detailed in An Introduction to the Kalman Filter by Welch & Bishop search "Likelihood of the Measurements Given a Particular Model". S is called "Innovation (or pre-fit residual) covariance" on Wikipedia. – Gabriel Devillers Sep 11 '19 at 13:30

Search for radar plot to track association. There's a lot of algorithms on this subject. To your question: The residual itself will not give you information without its associated covariance matrix

Try a chi-squared test on it. Putting a threshold on this scalar is called gating and it's a first step of plot to track association.