# The limit of gaussian distribution on curve manifold

I am reading this paper https://arxiv.org/abs/1304.8019. In introduction, it said

In many applications, even simple estimation problems involving angular data are often considered as traditional linear or nonlinear estimation problems and handled with classical techniques such as the Kalman Filter [1], the extended Kalman Filter (EKF), or the unscented Kalman Filter (UKF) [2]. In a circular setting, most traditional approaches to filtering suffer from assuming a Gaussian probability density at a certain point. They fail to take into account the periodic nature of the problem and assume a linear vector space instead of a curved manifold. This shortcoming can cause poor results, in particular when the angular uncertainty is large. In certain cases, the filter may even diverge.

My background is engineering, and I understand what Gaussian distribution in N-dimension means and the role of Gaussian distribution in Kalman filter. But I don't understand why Gaussian distribution fail to estimate angular data in its nature. It will be great if there are mathematical explanation and reference. Thank you.

The second problem is that the traditional approach doesn't reflect the geometry of the problem. Particularly, the periodicity of angles, i.e., when you add 360 degrees to an angle it remains the same. However, $$\theta$$ and $$\theta + 2\pi$$ are different values on the real line. This is what the author means by using a "Linear Geometry", i.e., using a vector space such as the real line. Instead, we must use a "Circular Geometry". For example, instead of looking at the real line, we look at the unit circle. When you walk on the real line you keep finding new values indefinitely; however, when you walk on the unit circle, after some time, you return to where you started. So instead of attaching a probability to $$\theta$$, we attach a probability to $$(\cos \theta, \sin \theta)$$.
• On the other concern, dividing by 2*pi and taking the remainder works for periodicity, but the Gaussian distribution does not do that for you. If you look at a typical bell-shaped curve, then you will see that it streches to infinity on both ends and it doesn't stop to just $2\pi$ that is the problem. You want a distribution that is only defined on the interval $[0,2\pi)$ that respects the geometry of the manifold. Jul 17, 2020 at 10:21