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Negative correlation can be seen geometrically when two normalized random vectors are viewed as points on a sphere, and the correlation between them is the cosine of the circular arc of separation of the points on a great circle of the sphere. [1] When this arc is more than a quarter-circle (θ > π/2), then the cosine is negative.
For example, a sphere of radius r has Gaussian curvature 1 / r 2 everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus .
Blue portions represent regions where the function is positive, and yellow portions represent where it is negative. The distance of the surface from the origin indicates the absolute value of Y ℓ m ( θ , φ ) {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} in angular direction ( θ , φ ) {\displaystyle (\theta ,\varphi )} .
A positive curvature corresponds to the inverse square radius of curvature; an example is a sphere or hypersphere. An example of negatively curved space is hyperbolic geometry (see also: non-positive curvature). A space or space-time with zero curvature is called flat. For example, Euclidean space is an example of a flat space, and Minkowski ...
The Kent distribution on the two-dimensional sphere. The Kumaraswamy distribution is as versatile as the Beta distribution but has simple closed forms for both the cdf and the pdf. The logit metalog distribution, which is highly shape-flexible, has simple closed forms, and can be parameterized with data using linear least squares.
From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature , and a surface of positive Gaussian curvature . In higher dimensions, a manifold may have different curvatures in different directions, described by the Riemann curvature tensor.
The Bingham distribution is a distribution over axes in N dimensions, or equivalently, over points on the (N − 1)-dimensional sphere with the antipodes identified. [9] For example, if N = 2, the axes are undirected lines through the origin in the plane. In this case, each axis cuts the unit circle in the plane (which is the one-dimensional ...
The name "pseudosphere" comes about because it has a two-dimensional surface of constant negative Gaussian curvature, just as a sphere has a surface with constant positive Gaussian curvature. Just as the sphere has at every point a positively curved geometry of a dome the whole pseudosphere has at every point the negatively curved geometry of a ...