def test_schechter():
from skypy.utils.random import schechter
from skypy.utils.special import gammaincc
def schechter_cdf_gen(alpha, x_min, x_max):
a = gammaincc(alpha + 1, x_min)
b = gammaincc(alpha + 1, x_max)
return lambda x: (a - gammaincc(alpha + 1, x)) / (a - b)
# Test the schechter function, sampling dimensionless x values
alpha = -1.3
x_min = 1e-10
x_max = 1e2
x = schechter(alpha, x_min, x_max, size=1000)
assert np.all(x >= x_min)
assert np.all(x <= x_max)
# Test the distribution of galaxy properties follows the right distribution
cdf = schechter_cdf_gen(alpha, x_min, x_max)
_, p = kstest(x, cdf)
assert p > 0.01
# Test output shape when scale is a scalar
scale = 5
samples = schechter(alpha, x_min, x_max, scale=scale)
assert np.shape(samples) == np.shape(scale)
# Test output shape when scale is an array
scale = np.random.uniform(size=10)
samples = schechter(alpha, x_min, x_max, scale=scale)
assert np.shape(samples) == np.shape(scale)
def subhalo_mass_sampler(halo_mass, nsubhalos, alpha, beta,
x, m_min, resolution=100):
r'''Subhalo mass sampler.
This function samples the original, unstriped masses of subhaloes from the
subhalo mass function of their parent halo with a constant mass stripping
factor given by equation (1) in [1]_ and the upper subhalo mass limit from
[2]_.
Parameters
-----------
halo_mass : (nm, ) array_like
The mass of the halo parent, in units of solar mass.
nsubhalos: (nm, ) array_like
Array of the number of subhalos assigned to parent halos with mass `halo_mass`.
alpha, beta : float
Parameters that determines the subhalo Schechter function. Its the amplitude
is defined by equation 2 in [1].
x : float
Parameter that accounts for the added mass of the original, unstripped
subhalos.
m_min : float
Original mass of the least massive subhalo, in units of solar mass.
Current stripped mass is given by :math:`m_{min} / x`.
resolution: int, optional
Resolution of the inverse transform sampling spline. Default is 100.
Returns
--------
sample: (nh, ) array_like
List of original masses drawn from the subhalo mass function for each
parent halo, in units of solar mass. The length corresponds to the total
number of subhalos for all parent halos, i.e. `np.sum(nsubhalos)`.
Examples
---------
>>> import numpy as np
>>> from skypy.halo import mass
This example samples 100 subhalos for a parent halo of mass 1.0E12 Msun:
>>> halo, min_sh = 1.0e12, 1.0e6
>>> alpha, beta, gamma_M = 1.9, 1.0, 0.3
>>> x = 3
>>> nsh = 100
>>> sh = mass.subhalo_mass_sampler(halo, nsh, alpha, beta, x, min_sh)
References
----------
.. [1] Vale, A. and Ostriker, J.P. (2004), arXiv: astro-ph/0402500.
.. [2] Vale, A. and Ostriker, J.P. (2004), arXiv: astro-ph/0511816.
'''
halo_mass = np.atleast_1d(halo_mass)
nsubhalos = np.atleast_1d(nsubhalos)
subhalo_list = []
for M, n in zip(halo_mass, nsubhalos):
x_min = m_min / (x * beta * M)
x_max = 0.5 / (x * beta)
subhalo_mass = schechter(alpha, x_min, x_max, resolution, size=n, scale=x * beta * M)
subhalo_list.append(subhalo_mass)
return np.concatenate(subhalo_list)
def test_schechter_gamma():
from skypy.utils.random import schechter
# when alpha > 0, x_min ≈ 0, x_max ≈ ∞, distribution is gamma
# n.b. if alpha < 0 the distribution becomes too steep to resolve accurately
alpha = np.random.uniform(0, 2)
x_min = 1e-20
x_max = 1e+20
scale = 2.5
x = schechter(alpha, x_min, x_max, resolution=100000, size=1000, scale=scale)
_, p = kstest(x, 'gamma', args=(alpha+1, 0, scale))
assert p > 0.01
def press_schechter(n,
m_star,
size=None,
x_min=0.00305,
x_max=1100.0,
resolution=100):
"""Sampling from Press-Schechter mass function (1974).
Masses following the Press-Schechter mass function following the
Press et al. [1]_ formalism.
Parameters
----------
n : float or int
The n parameter in the Press-Schechter mass function.
m_star : float or int
Factors parameterising the characteristic mass.
size: int, optional
Output shape of luminosity samples.
x_min, x_max : float or int, optional
Lower and upper bounds in units of M*.
resolution : int, optional
Resolution of the inverse transform sampling spline. Default is 100.
Returns
-------
mass : array_like
Drawn masses from the Press-Schechter mass function.
Examples
--------
>>> import skypy.halo.mass as mass
>>> n, m_star = 1, 1e9
>>> sample = mass.press_schechter(n, m_star, size=1000, x_min=1e-10,
... x_max=1e2, resolution=1000)
References
----------
.. [1] Press, W. H. and Schechter, P., APJ, (1974).
"""
alpha = -0.5 * (n + 9.0) / (n + 3.0)
x_sample = schechter(alpha, x_min, x_max, resolution=resolution, size=size)
return m_star * np.power(x_sample, 3.0 / (n + 3.0))
def test_schechter():
# Test the schechter function, sampling dimensionless x values
alpha = -1.3
x_min = 1e-10
x_max = 1e2
def calc_cdf(x):
a = special.upper_incomplete_gamma(alpha + 1, x_min)
b = special.upper_incomplete_gamma(alpha + 1, x)
c = special.upper_incomplete_gamma(alpha + 1, x_min)
d = special.upper_incomplete_gamma(alpha + 1, x_max)
return (a - b) / (c - d)
sample = random.schechter(alpha,
x_min=x_min,
x_max=x_max,
resolution=100,
size=1000)
# Test the distribution of galaxy properties follows the right distribution
p_value = scipy.stats.kstest(sample, calc_cdf)[1]
assert p_value > 0.01
def herbel_luminosities(redshift,
alpha,
a_m,
b_m,
size=None,
x_min=0.00305,
x_max=1100.0,
resolution=100):
r"""Model of Herbel et al (2017)
Luminosities following the Schechter luminosity function following the
Herbel et al. [1]_ model.
Parameters
----------
redshift : (nz,) array-like
The redshift values at which to sample luminosities.
alpha : float or int
The alpha parameter in the Schechter luminosity function.
a_m, b_m : float or int
Factors parameterising the characteristic absolute magnitude M_* as
a linear function of redshift according to Equation 3.3 in [1]_.
size: int, optional
Output shape of luminosity samples. If size is None and redshift
is a scalar, a single sample is returned. If size is None and
redshift is an array, an array of samples is returned with the same
shape as redshift.
x_min, x_max : float or int, optional
Lower and upper luminosity bounds in units of L*.
resolution : int, optional
Resolution of the inverse transform sampling spline. Default is 100.
Returns
-------
luminosity : array_like
Drawn luminosities from the Schechter luminosity function.
Notes
-----
The Schechter luminosity function is given as
.. math::
\Phi(L, z) = \frac{\Phi_\star(z)}{L_\star(z)}
\left(\frac{L}{L_\star(z)}\right)^\alpha
/exp\left(-\frac{L}{L_\star(z)}\right) \;.
Here the luminosity is defined as
.. math::
L = 10^{-0.4M} \;,
with absolute magnitude :math:`M`. Furthermore, Herbel et al. [1]_
introduced
.. math::
\Phi_\star(z) = b_\phi \exp(a_\phi z) \;,
M_\star(z) = a_M z + b_M \;.
Now we have to rescale the Schechter function by the comoving element and
get
.. math::
\phi(L,z) = \frac{d_H d_M^2}{E(z)} \Phi(L,z)\;.
Examples
--------
>>> import skypy.galaxy.luminosity as lum
Sample 100 luminosity values at redshift z = 1.0 with
a_m = -0.9408582, b_m = -20.40492365, alpha = -1.3.
>>> luminosities = lum.herbel_luminosities(1.0, -1.3, -0.9408582,
... -20.40492365, size=100)
Sample a luminosity value for every redshift in an array z with
a_m = -0.9408582, b_m = -20.40492365, alpha = -1.3.
>>> z = np.linspace(0,2, 100)
>>> luminosities = lum.herbel_luminosities(z, -1.3, -0.9408582,
... -20.40492365)
References
----------
.. [1] Herbel J., Kacprzak T., Amara A. et al., 2017, Journal of Cosmology
and Astroparticle Physics, Issue 08, article id. 035 (2017)
"""
if size is None and np.shape(redshift):
size = np.shape(redshift)
luminosity_star = _calculate_luminosity_star(redshift, a_m, b_m)
x_sample = schechter(alpha, x_min, x_max, resolution=100, size=size)
return luminosity_star * x_sample
def schechter_smf(alpha, m_star, x_min, x_max, resolution=100, size=None):
r""" Stellar masses following the Schechter mass function [1]_.
Parameters
----------
alpha : float
The alpha parameter in the Schechter stellar mass function.
m_star : (nm,) array-like
Characteristic stellar mass M_*.
size: int, optional
Output shape of stellar mass samples. If size is None and m_star
is a scalar, a single sample is returned. If size is None and
m_star is an array, an array of samples is returned with the same
shape as m_star.
x_min, x_max : float
Lower and upper bounds for the random variable x in units of M_*.
resolution : int, optional
Resolution of the inverse transform sampling spline. Default is 100.
Returns
-------
stellar mass : (nm,) array_like
Drawn stellar masses from the Schechter stellar mass function in units
of the solar mass.
Notes
-----
The stellar mass probability distribution (pdf) follows a Schechter
profile of the form
.. math::
\Phi(M) = \frac{1}{M_*} \left(\frac{M}{M_*}\right)^\alpha
\exp\left(-\frac{M}{M_*}\right) \;.
From this pdf one can sample the stellar masses.
Examples
--------
>>> from skypy.galaxy import stellar_mass
Sample 100 stellar masses values at redshift z = 1.0 with alpha = -1.4,
m_star = 10**10.67, x_min = 0.0002138 and x_max = 213.8
>>> masses = stellar_mass.schechter_smf(-1.4, 10**10.67, 0.0002138,
... 213.8, size=100)
References
----------
.. [1] Mo, H., Van den Bosch, F., & White, S. (2010). Galaxy Formation and
Evolution. Cambridge: Cambridge University Press.
doi:10.1017/CBO9780511807244
"""
if size is None and np.shape(m_star):
size = np.shape(m_star)
x_sample = schechter(alpha, x_min, x_max, resolution=resolution, size=size)
return m_star * x_sample
