Asymmetric von Mises Scaling

This website introduces asymmetric von Mises scaling (AMISESCAL). AMISESCAL is a kind of asymmetric multidimensional scaling (AMDS) using a technique deriving from directional statistics. This method is a statistical model analyzing an asymmetric matrix to visualize the structure underlying the matrix.

Quick Introduction PPT

Directional Statistics

Directional statistics is a branch of statistics that deals with angles, courses, and directions as data. This branch of statistics is used in magnetic field analysis, animal migration, disease transmission route, etc. For example, as shown in the figure below, we obtain the directions to which migratory birds flew away as data. And, our concern is which angle the birds prefers to migrate. Then, in the histogram of the directional statistics, frequencies are summarized on the circle.

von Mises Distribution

The normal distribution is the most important distribution in statistics. The von Mises distribution can be considered to be the normal distribution of directional statistics. The functional form of this distribution is as follows:

 

μ: Mean Direction Parameter

      κ: Concentration Parameter

 

The domain of the definition of θ is [0, 2π]. The von Mises distribution has two parameters: mean direction (μ) and concentration (κ). Mean direction is the parameter regulating the highest density direction of the distribution, and concentration is the parameter for adjusting the depth around the mean direction.

The left graph in the figure below shows four von Mises distributions with varying parameters. On the other hand, the right graph shows the four distributions on the unit circle. It is often easy to understand the angle of the mean direction (μ) and the size of the concentration (κ) of a von Mises distribution by showing it on a circle.

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