Statistics/Data Science: Real world example for a donut destribution? I'm looking for different real world examples for donut distribution (in the statistic meaning).
https://i.stack.imgur.com/VrGq4.jpg
For the most other distributions, I found good examples. But for this the only one is the way to focus a laser beam:
https://www.researchgate.net/figure/Donut-shaped-laser-beam-for-thermocapillary-induced-dewetting-a-Microscopic-pictures_fig2_274400395
But this wouldn´t fit to e.g. a classification problem.
Has anybody a good source oder idea, where donut distributions are existing in real world data? Thanks a lot!
 A: I remember an example of a donut distribution (not a data distribution with a multi-modal radial component if that is what you are looking for) from a course that I took in probabilistic robotics which dealt with Bayes filters, particle filters, Kalman filters, and so on. It might be a bit to specific to this area.
The example is related to a task called localization, which is a setting where you are trying to locate a robot in a certain area. You have access to a map of this area, meaning that you know about specific landmarks on this map. However, you do not know exactly where on the map the robot is so you use the robot's laser to take measurements of distances to landmarks as it moves around in the area. The sensor is a bit noisy and measurements are assumed to have Gaussian noise.
Now, imagine that the robot uses its laser and measures the range $d$ to a known landmark.
You use this information to create a posterior belief of the robot's location $p(x, y, \theta|d)$, where x and y are cartesian coordinates, and theta is the angle of the robot relative to the landmark.
Now, after taking this measurement the only information that the robot has is the distance to the landmark. Therefore, it cannot tell what its angle $\theta$ is relative to the
landmark and any $\theta$ angle between $[-\pi,\pi]$ would be equally probable. Thus, the robot could be located anywhere at the edge of a circle around the landmark.
When it comes to the (x, y) coordinates we can infer more information. At each point of the circle with radius $d$ we get an error with a Gaussian spread centered at $d$. Such that the robot could be within a certain range inside or outside
of the circle. All in all, we get a circle with a Gaussian spread at every single point at its circumference, i.e. a donut.
