alex_be

I've been using Firefox for almost 20 years as my default browser. Thank you for your work!

I've been and I've seen. It is a fence. A very long fence. You can't cross it with bare hands, and if you try, you will be shot. This is not the case when dozens of bulldozers simultaneously cross the fence and many thousands of trained terrorists coming out of tunnels and cross the fence through the openings. If you don't have prior intelligence, you can't stop it.

"In addition the Moon has no atmosphere and is constantly bombarded by radiation from the Sun that causes the soil to become electrostatically charged." - You can use a magnetic or electric field to push the soil away

486 was my dream. Unfortunately, my parents didn't have money for it. I bought my first PC in 1999 - a Pentium 2. I invested a lot of money in the monitor; computers become obsolete very quickly, while a monitor can serve for many years. Surprisingly, flat monitors appeared soon after...

Thanks a lot for your detailed and valuable comments. I will definitely include them in the tutorial. If you have additional comments, I would be happy to hear them.

True. It's about managing the risk rather than eliminating it. If you remove an outlier, you get a missing measurement and, as a result, higher uncertainty (error). But it is still better than keeping the outlier.

Thanks a lot for this comment, Ted! This probably deserves its own example, not just a brief mention. I will definitely do that.

Interesting. It sounds like you ended up with a data-driven estimator. Did you have a chance to compare the data-driven and model-based approaches?

It's a free accessibility widget by Sienna. I tweaked the CSS to adapt it to the https://kalmanfilter.net/ style. You can find it here: https://accessibility-widget.pages.dev/

Classic :)

I have a chapter in my book that introduces sensor fusion as a concept. If you want to dive deeper into the sensor fusion topic, I would recommend Bar-Shalom's or Blackman's book.

Thanks for your feedback. I am thinking of writing a second volume with more advanced and less introductory topics, but I haven't decided yet. It is a serious commitment and it will take years to complete. If I take this decision, I will consider a chapter on LQG.

Small clarification: nonlinear Kalman filters are suboptimal. EKF relies on linear approximations, and UKF uses heuristic approximations.

Kalman filter is about combining uncertain measurements, and human observations could be viewed as noisy sensors. On the other hand, the standard KF assumes unbiased sensors with Gaussian noise, and I don't know if those assumptions hold for human witnesses.

Yeah. Building things step by step often makes complex topics much easier to understand.

Thanks for your feedback! Actually the KF concept is generic, but as mentioned above: "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."

So it is right to say that the implementation of the KF is tightly coupled to the system. Getting that part right is usually the hardest step.

It is always a good idea to include outliers treatment in KF algorithm to filter out weird measurements.

The tutorial actually predates ChatGPT by quite a few years (first published in 2017). Today, I do sometimes use ChatGPT to fix grammar, but I am responsible for the content and it is always mine.

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.

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.

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.

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.

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.

That's a good point. "Optimal" in this context means that, under the standard assumptions (linear system, Gaussian noise, correct model), the Kalman Filter minimizes the estimation error covariance. In other words, it provides the minimum-variance estimate among all linear unbiased estimators.

You're right that the term can feel vague without that context. I’ll consider adding a short clarification earlier in the introduction to make this clearer before diving into the math. Thanks for the suggestion.

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 topic in detail in the book.

I'll consider adding a brief explanation or reference to make that step clearer. Thanks for pointing this out.

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 measurements, prediction using a motion model, and how the Kalman Filter combines both. I also tried to keep the math minimal while still showing where the equations come from.

I would really appreciate feedback on clarity. Which parts are intuitive? Which parts are confusing? Is the math level appropriate?

If you have used Kalman Filters in practice, I would also be interested to hear whether this explanation aligns with your intuition.

Big step toward AI-assisted mathematical research

Very interesting and so actual. Isaac Asimov is one of my favorite writers.

Hello mojomark, My name is Alex Becker, and I am the author of the "Kalman Filter from the Ground Up book." First, I would like to thank you for your feedback. As an author, it is important for me to receive feedback from the readers. In the book's second edition, I made many changes based on the feedback from the book readers. I've elaborated explanations of the topics some readers failed to understand and added several appendixes. You are right, book authors don't get paid to respond. However, the KalmanFilter.Net is a project, not just a book. I've worked on the project for 6 years, and you will see more in the future. The goal of this project is to help people to understand and implement the Kalman Filter. The website includes a contact page: https://www.kalmanfilter.net/contact.aspx, and the book purchasers receive my email. I do my best to answer emails and help my readers if they fail to understand specific points. Regarding the issues that you and other responders raised. Scientists use terminology that looks weird to non-scientists. But I can't write an engineering book without addressing scientific terms. I agree that "Random Variable" sounds unintuitive, but this is the term. Almost any physical quantity is a random variable. For example, the temperature of your body is a random variable. You can measure the temperature using a thermometer, but it doesn't mean that its reading is your actual temperature. Your temperature is a random variable that can be described by two parameters – the thermometer reading and the thermometer precision. Understanding the "Random Variables" concept is essential for understanding the Kalman Filter. If you make multiple measurements, the temperature is described by mean and variance. You can read again the https://www.kalmanfilter.net/background.html page, where this concept is explained. Mathematicians call the mean and the variance – statistical moments. "Moments" also sounds weird. I mention this term in the introduction, but I don't use it again in the book. If the term "moments" causes confusion, I can consider removing it from the introduction. Regarding the Linear Algebra. You don't need any prior Linear Algebra knowledge to understand the concept of the Kalman Filter and implement the Kalman Filter in one dimension. However, if you want to tackle the multi-dimensional Kalman Filter, you need some basic Linear Algebra knowledge, at least matrix multiplication. Thanks again for your post, Alex Becker.