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Understanding the Kalman filter with a simple radar example

kalmanfilter.net

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Re: Understanding the Kalman filter with a simple radar example

#11

Post by KellyCriterion »

alex_be wrote: post #2

Author here. I recently updated the homepage of my Kalman Filter tutorial with a new example based on a simple radar tracking problem. The goal was to make the Kalman Filter understandable to anyone with basic knowledge of statistics and linear algebra, without requiring advanced mathematics. The example starts with a radar measuring the distance to a moving object and gradually builds intuition around noisy measurem…

You could do a line extension of your product, like "Kalman Filter in Financial Markets" and sell additional copies :)

Re: Understanding the Kalman filter with a simple radar example

#13

Post by lelandbatey »

Kalman filters are very cool, but when applying them you've got to know that they're not magic. I struggled to apply Kalman Filters for a toy project about ten years ago, because the thing I didn't internalize is that Kalman filters excel at offsetting low-quality data by sampling at a higher rate. You can "retroactively" apply a Kalman filter to a dataset and see some improvement, but you'll only get amazing results if you sample your very-noisy data at a much higher rate than if you were sampling at a "good enough" rate. The higher your sample rate, the better your results will be. In that way, a Kalman filter is something you want to design around, not a "fix all" for data you already have.

Re: Understanding the Kalman filter with a simple radar example

#14

Post by alex_be »

seanhunter wrote: post #9
alex_be wrote: post #2

Author here. I recently updated the homepage of my Kalman Filter tutorial with a new example based on a simple radar tracking problem. The goal was to make the Kalman Filter understandable to anyone with basic knowledge of statistics and linear algebra, without requiring advanced mathematics. The example starts with a radar measuring the distance to a moving object and gradually builds intuition around noisy measurem…

Firstly I think the clarity in general is good. The one piece I think you could do with explaining early on is which pieces of what you are describing are the model of the system and which pieces are the Kalman filter. I was following along as you built the markov model of the state matrix etc and then you called those equations the Kalman filter, but I didn't think we had built a Kalman filter yet. Your early explan…

You’re pointing out a real conceptual issue: where the system model ends and where the Kalman filter begins.

In Kalman filter theory there are two different components:

- The system model

- The Kalman filter (the algorithm)

The state transition and measurement equations belong to the system model. They describe the physics of the system and can vary from one application to another.

The Kalman filter is the algorithm that uses this model to estimate the current state and predict the future state.

I'll consider making that distinction more explicit when introducing the equations. Thanks for pointing this out.

Re: Understanding the Kalman filter with a simple radar example

#15

Post by alex_be »

KellyCriterion wrote: post #11
alex_be wrote: post #2

Author here. I recently updated the homepage of my Kalman Filter tutorial with a new example based on a simple radar tracking problem. The goal was to make the Kalman Filter understandable to anyone with basic knowledge of statistics and linear algebra, without requiring advanced mathematics. The example starts with a radar measuring the distance to a moving object and gradually builds intuition around noisy measurem…

You could do a line extension of your product, like "Kalman Filter in Financial Markets" and sell additional copies :)

That's an interesting idea. The Kalman filter is definitely used in finance, often together with time-series models like ARMA. I've been thinking about writing something, although it's a bit outside my usual engineering focus.

The challenge would be to keep it intuitive and accessible without oversimplifying. Still, it could be an interesting direction to explore.

Re: Understanding the Kalman filter with a simple radar example

#16

Post by alex_be »

joshu wrote: post #8

i liked how https://www.bzarg.com/p/how-a-kalman-filter-works-in-picture... uses color visualization to explain

That's a good article. I also like the visual approach there. My goal here was a bit different. I walk through a concrete radar example step by step, and use multiple examples throughout the tutorial to build intuition and highlight common pitfalls.

Re: Understanding the Kalman filter with a simple radar example

#17

Post by alex_be »

smokel wrote: post #5

This seems to be an ad for a fairly expensive book on a topic that is described in detail in many (free) resources. See for example: https://rlabbe.github.io/Kalman-and-Bayesian-Filters-in-Pyth... Is there something in this particular resource that makes it worth buying?

That's a fair question. My goal with the site was to make as much material available for free as possible, and the core linear Kalman filter content is indeed freely accessible.

The book goes further into topics like tuning, practical design considerations, common pitfalls, and additional examples. But there are definitely many good free resources out there, including the one you linked.

Re: Understanding the Kalman filter with a simple radar example

#18

Post by alex_be »

lelandbatey wrote: post #13

Kalman filters are very cool, but when applying them you've got to know that they're not magic. I struggled to apply Kalman Filters for a toy project about ten years ago, because the thing I didn't internalize is that Kalman filters excel at offsetting low-quality data by sampling at a higher rate. You can "retroactively" apply a Kalman filter to a dataset and see some improvement, but you'll only get amazing results…

I agree that Kalman filters are not magic and that having a reasonable model is essential for good performance.

Higher sampling rates can help in some cases, especially when tracking fast dynamics or reducing measurement noise through repeated updates. However, the main strength of the Kalman filter is combining a model with noisy measurements, not necessarily relying on high sampling rates.

In practice, Kalman filters can work well even with relatively low-rate measurements, as long as the model captures the system dynamics reasonably well.

I also agree that it's often something you design into the system rather than applying as a post-processing step.

Re: Understanding the Kalman filter with a simple radar example

#19

Post by magicalhippo »

alex_be wrote: post #6
magicalhippo wrote: post #3

Earlier quoted context omitted.

I just glossed through for now so might have missed it, but it seemed you pulled the process noise matrix Q out of a hat. I guess it's explained properly in the book but would be nice with some justification for why the entries are what they are.

To keep the example focused and reasonably short, I treated Q matrix as given and concentrated on building intuition around prediction and update. But you're right that this can feel like it appears out of nowhere. The derivation of the Q matrix is a separate topic and requires additional assumptions about the motion model and noise characteristics, which would have made the example significantly longer. I cover this…

Yeah I understand. I do think a brief explanation would help a lot though. As it sits it's not even entirely clear if the presented matrix is general or highly specific. I can easily see someone just use that as their Q matrix because that's what the Q matrix is, says so right there.
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