# Source code for halotools.empirical_models.abunmatch.noisy_percentile

```
"""Module storing the `noisy_percentile` function
"""
import numpy as np
from scipy.special import erf, erfinv
from astropy.utils.misc import NumpyRNGContext
__all__ = ('noisy_percentile', )
[docs]
def noisy_percentile(percentile, correlation_coeff, seed=None, random_percentile=None):
r"""
Starting from an input array storing the rank-order percentile of some quantity,
add noise to these percentiles to achieve the desired
Spearman rank-order correlation coefficient
between ``percentile`` and ``noisy_percentile``.
Notes
-----
The plot below shows how the `noisy_percentile`
function adds stochasticity to the input ``percentile``:
.. image:: /_static/noisy_percentile_demo.png
In the top-left panel, the ``correlation_coeff`` argument has been set to 0.1,
so that there is only a weak correlation between the input ``percentile``
and the returned result. Conversely, in the bottom-right panel, the
correlation is very tight.
Because the `noisy_percentile` function is so general,
there are many variations on how you can use it
to model correlations between galaxy and halo properties. Many such applications
are based on the method of inverse transformation sampling to generate
Monte Carlo realizations of galaxy properties, and so the
`halotools.utils.monte_carlo_from_cdf_lookup` function and the
`halotools.utils.build_cdf_lookup` function may come in handy.
In the Examples section below, we demonstrate how you can
implement a correlation between halo concentration
and scatter in the stellar-to-halo mass relation.
In this particular case, we will use a log-normal PDF for the distribution of
:math:`M_\ast` at fixed halo mass. Note, however, that the
`noisy_percentile` function does not require that
the statistical distribution of the galaxy property being modeled necessarily
have any particular functional form. So long as you have knowledge of the
rank-order percentile of your galaxy property, `noisy_percentile` allows
you to introduce correlations of arbitrary strength with any other variable
for which you also know the rank-order percentile.
Also see :ref:`cam_tutorial` demonstrating how to use this
function in galaxy-halo modeling with several worked examples.
Parameters
----------
percentile : ndarray
Numpy array of shape (npts, ) storing values between 0 and 1, exclusive.
correlation_coeff : float or ndarray
Float or ndarray of shape (npts, ) storing values between 0 and 1, inclusive.
seed : int, optional
Random number seed used to introduce noise
random_percentile: ndarray, optional
Numpy array of shape (npts, ) storing pre-computed random percentiles
that will be used to mix with the input ``percentile``. Default is None,
in which case the ``random_percentile`` array will be automatically
generated as uniform randoms according to the input ``seed``.
Returns
-------
noisy_percentile : ndarray
Numpy array of shape (npts, ) storing an array such that
the Spearman rank-order correlation coefficient between
``percentile`` and ``noisy_percentile`` is equal to the input
``correlation_coeff``.
Examples
--------
The `noisy_percentile` function is useful as the kernel of a calculation
in which you are modeling a correlation between a galaxy property and some
halo property.
For example, suppose you have a sample of halos at fixed mass,
and you want to map stellar mass onto the halos according to a log-normal
distribution, such that the scatter in :math:`M_{\ast}` is correlated with
halo concentration. The code below shows how to use the
`noisy_percentile` function for this purpose, together with
the `scipy` implementation of a Gaussian PDF, `~scipy.stats.norm`.
In the demo below, we'll start out by selecting a sample of halos
at fixed mass using a fake halo catalog that is generated on-the-fly;
note that the API would be the same for any `~halotools.sim_manager.CachedHaloCatalog`.
>>> from halotools.sim_manager import FakeSim
>>> halocat = FakeSim()
>>> mask = (halocat.halo_table['halo_mpeak'] > 10**11.9)
>>> mask *= (halocat.halo_table['halo_mpeak'] < 10**12.1)
>>> halo_sample = halocat.halo_table[mask]
>>> num_sample = len(halo_sample)
If we just wanted random uncorrelated scatter in stellar mass,
we can pass the `~scipy.stats.norm.isf` function a set of random uniform
numbers:
>>> from scipy.stats import norm
>>> mean_logmstar, std_logmstar = 11, 0.1
>>> uran = np.random.rand(num_sample)
>>> mstar_random = norm.isf(uran, loc=mean_logmstar, scale=std_logmstar)
The ``mstar_random`` array is just a normal distribution in :math:`\log_{10}M_\ast`,
with deviations from the mean value of 11 being uncorrelated with anything.
To implement a correlation between :math:`M_\ast - \langle M_{\ast}\rangle` and
concentration, we first calculate the rank-order percentile
of the concentrations of our halo sample,
simply by sorting and normalizing by the number of objects:
>>> from halotools.utils import rank_order_percentile
>>> percentile = rank_order_percentile(halo_sample['halo_nfw_conc'])
If we wanted to implement a perfect correlation between concentration and
scatter in :math:`M_\ast`, with lower concentrations receiving
lower stellar mass, we would just pass the array ``1 - percentile`` to
the `~scipy.stats.norm.isf` function:
>>> mstar_maxcorr = norm.isf(1-percentile, loc=mean_logmstar, scale=std_logmstar)
The `noisy_percentile` function allows you to build correlations of a strength
that is intermediate between these two extremes. If you want :math:`M_\ast` and
concentration to have a Pearson correlation coefficient of 0.5:
>>> correlation_coeff = 0.5
>>> result = noisy_percentile(percentile, correlation_coeff)
>>> mstar_0p5 = norm.isf(1-result, loc=mean_logmstar, scale=std_logmstar)
In the figure below, we visually demonstrate the results of this calculation
by showing the PDF of :math:`\log_{10}M_\ast` for our halo sample, color-coded by
the mean concentration of the halos with a given stellar mass:
.. image:: /_static/color_correlation_pdf.png
For each of the different curves,
the overall normalization of :math:`\phi(M_{\ast})` has been offset for clarity.
For the case of a correlation coefficient of unity (the top curve),
we see that halos with above-average :math:`M_\ast` values for their mass
tend to have above-average concentration values for their mass, and conversely
for halos with below-average :math:`M_\ast`.
For the case of zero correlation (the bottom curve), there is no trend at all.
Correlation strengths between zero and unity span the intermediary cases.
"""
if np.all(correlation_coeff == 1):
return percentile
elif np.all(correlation_coeff == -1):
return percentile[::-1]
percentile = np.atleast_1d(percentile)
correlation_coeff = np.atleast_1d(correlation_coeff)
if random_percentile is None:
with NumpyRNGContext(seed):
random_percentile = np.random.uniform(0, 1, len(percentile))
ztrue = _z_score_from_percentile(percentile)
zran = _z_score_from_percentile(random_percentile)
znoisy = _weighted_mixture_of_two_gaussians(ztrue, zran, correlation_coeff)
return _percentile_from_z_score(znoisy)
def _percentile_from_z_score(z_score):
return 0.5*(1 + erf(z_score/np.sqrt(2)))
def _z_score_from_percentile(percentile):
return np.sqrt(2)*erfinv(2*percentile-1)
def _weighted_mixture_of_two_gaussians(g1, g2, r):
return r*g1 + np.sqrt(1-r*r)*g2
```