Dear researchers,
Can anyone help me understand this phenomenon and why does it happen?
Homogeneity of variance on the y scale (e.g. a von Mises distribution y \sim vM(mu, kappa) for constant kappa, as mu varies) implies heterogeneity of variance on the v scale, where v = tan (y/2), and vice versa.
Attached is the plot for such behaviour.
Homogeneity in circular space refers to the property that all points in the space are equivalent or indistinguishable from one another. This means that any point in the circular space can be transformed or mapped onto any other point through a symmetry or transformation operation without changing the overall structure or properties of the space.
Now, let's consider a linear space, such as a line or a higher-dimensional Euclidean space. Homogeneity in a linear space means that all points in the space are also equivalent or indistinguishable from one another. This implies that any point in the linear space can be transformed or mapped onto any other point through a symmetry or transformation operation without altering the overall structure or properties of the space.
The connection between homogeneity in circular space and homogeneity in linear space can be understood by considering the relationship between circles and lines. A circle can be seen as a one-dimensional object embedded in a two-dimensional Euclidean space. By projecting this circle onto the Euclidean space, we obtain a line segment. In this projection, the indistinguishable or equivalent points on the circle become indistinguishable or equivalent points on the line.
Therefore, the homogeneity in the circular space, where all points on the circle are equivalent, translates to homogeneity in the linear space, where all points on the line are also equivalent. The mapping or projection from the circular space to the linear space preserves the property of homogeneity, as the symmetries or transformations in the circular space correspond to symmetries or transformations in the linear space.
In summary, homogeneity in circular space implies homogeneity in linear space through the projection or mapping of the circular space onto the linear space, which preserves the equivalence or indistinguishability of points.
Dear Dr. Ma’Mon,
Thank you very much for your detailed response.
Unfortunately ,the question title was not correct. I edited it now.
I meant how the homogeneity of variance on the y scale (e.g. a von Mises distribution y \sim vM(mu, kappa) for constant kappa, as mu varies) implies heterogeneity of variance on the v scale, where v = tan (y/2), and vice versa.
appreciate your understanding.
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