DimrothWatson¶
- class halotools.empirical_models.DimrothWatson(momtype=1, a=None, b=None, xtol=1e-14, badvalue=None, name=None, longname=None, shapes=None, seed=None)[source]¶
Bases:
rv_continuous
A Dimroth-Watson distribution of :math:`cos(theta)’
- Parameters:
- kfloat
shape paramater
Notes
The Dimroth-Watson distribution is defined as:
\[p(\cos(\theta)) = B(k)\exp[-k\cos(\theta)^2]\mathrm{d}\cos(\theta)\]where
\[B(k) = \frac{1}{2}int_0^1\exp(-k t^2)\mathrm{d}t\]We assume the ISO convention for spherical coordinates, where \(\theta\) is the polar angle, bounded between \([-\pi, \pi]\), and \(\phi\) is the azimuthal angle, where for a Dimroth-Watson distribution, \(phi' is a uniform random variable between :math:`[0, 2\pi]\): for all
k
.For \(k<0\), the distribution of points on a sphere is bipolar. For \(k=0\), the distribution of points on a sphere is uniform. For \(k>0\), the distribution of points on a sphere is girdle.
Note that as \(k \rarrow \infty\):
\[p(\cos(\theta)) = \frac{1}{2}\left[ \delta(\cos(\theta) + 1) + \delta(\cos(\theta) - 1) \right]\mathrm{d}\cos(\theta)\]and as \(k \rarrow -\infty\):
\[p(\cos(\theta)) = \frac{1}{2}\delta(\cos(\theta))\mathrm{d}\cos(\theta)\]Needless to say, for large \(|k|\), the attributes of this class are approximate and not well tested.
Methods Summary
g1_isf
(y, k)inverse survival function of proposal distribution for pdf for k>0
g1_pdf
(x, k)proposal distribution for pdf for k>0
g2_isf
(y, k)inverse survival function of proposal distribution for pdf for k<0
g2_pdf
(x, k)proposal distribution for pdf for k<0
m1
(k)eneveloping factor for proposal distribution for k>0
m2
(k)eneveloping factor for proposal distribution for pdf for k<0
Methods Documentation