The limit of gaussian distribution on curve manifoldI am reading the paper. In introduction, it saidIn 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.

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The limit of gaussian distribution on curve manifoldI am reading the paper. In introduction, it saidIn 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&#039;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.

Carlos Nicholson · Accepted Answer

The traditional approach was that the models produced a probability distribution for all possible values of an angle. In most cases, the distribution is assumed to be Gaussian since it is easier to deal with and reason about, also because the mean of the Gaussian distribution (where the distribution is centered at) always has the highest probability and the probabilities around it diminish symmetrically. So imagine that you have a model that outputs for each possible angle (0-359) a probability (0-1).The problem with this approach is twofold. The first is that the Gaussian distribution is unimodal which means the highest probability only occurs at one point which is the mean as discussed. However, if an object is symmetric (think of a double-ended arrow) then when rotating it 180 degrees it will remain the same. This means that if it was initially placed at 0 degrees or 180 degrees then there wouldn&#039;t be any difference. So the distribution must be multimodal and must have, in this case, two identical and high probabilities at 0 and 180 degrees.The second problem is that the traditional approach doesn&#039;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,   θ and   θ  +  2  π are different values on the real line. This is what the author means by using a &quot;Linear Geometry&quot;, i.e., using a vector space such as the real line. Instead, we must use a &quot;Circular Geometry&quot;. 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 θ, we attach a probability to (cos  θ,sin  θ).

The limit of gaussian distribution on curve manifold I am reading the paper. In introduction, it sa

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