# 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

g1_isf(y, k)[source]

inverse survival function of proposal distribution for pdf for k>0

g1_pdf(x, k)[source]

proposal distribution for pdf for k>0

g2_isf(y, k)[source]

inverse survival function of proposal distribution for pdf for k<0

g2_pdf(x, k)[source]

proposal distribution for pdf for k<0

m1(k)[source]

eneveloping factor for proposal distribution for k>0

m2(k)[source]

eneveloping factor for proposal distribution for pdf for k<0