noisy_percentile¶

halotools.empirical_models.
noisy_percentile
(percentile, correlation_coeff, seed=None, random_percentile=None)[source] [edit on github]¶ Starting from an input array storing the rankorder percentile of some quantity, add noise to these percentiles to achieve the desired Spearman rankorder correlation coefficient between
percentile
andnoisy_percentile
.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 precomputed random percentiles that will be used to mix with the input
percentile
. Default is None, in which case therandom_percentile
array will be automatically generated as uniform randoms according to the inputseed
.Returns: noisy_percentile : ndarray
Numpy array of shape (ngals, ) storing an array such that the Spearman rankorder correlation coefficient between
percentile
andnoisy_percentile
is equal to the inputcorrelation_coeff
.Notes
The plot below shows how the
noisy_percentile
function adds stochasticity to the inputpercentile
:In the topleft panel, the
correlation_coeff
argument has been set to 0.1, so that there is only a weak correlation between the inputpercentile
and the returned result. Conversely, in the bottomright 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 thehalotools.utils.monte_carlo_from_cdf_lookup
function and thehalotools.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 stellartohalo mass relation. In this particular case, we will use a lognormal PDF for the distribution of \(M_\ast\) 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 rankorder 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 rankorder percentile.Also see Tutorial on Conditional Abundance Matching demonstrating how to use this function in galaxyhalo modeling with several worked examples.
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 lognormal distribution, such that the scatter in \(M_{\ast}\) is correlated with halo concentration. The code below shows how to use thenoisy_percentile
function for this purpose, together with thescipy
implementation of a Gaussian PDF,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 onthefly; note that the API would be the same for any
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
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 \(\log_{10}M_\ast\), with deviations from the mean value of 11 being uncorrelated with anything. To implement a correlation between \(M_\ast  \langle M_{\ast}\rangle\) and concentration, we first calculate the rankorder 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 \(M_\ast\), with lower concentrations receiving lower stellar mass, we would just pass the array
1  percentile
to theisf
function:>>> mstar_maxcorr = norm.isf(1percentile, 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 \(M_\ast\) 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(1result, loc=mean_logmstar, scale=std_logmstar)
In the figure below, we visually demonstrate the results of this calculation by showing the PDF of \(\log_{10}M_\ast\) for our halo sample, colorcoded by the mean concentration of the halos with a given stellar mass:
For each of the different curves, the overall normalization of \(\phi(M_{\ast})\) has been offset for clarity. For the case of a correlation coefficient of unity (the top curve), we see that halos with aboveaverage \(M_\ast\) values for their mass tend to have aboveaverage concentration values for their mass, and conversely for halos with belowaverage \(M_\ast\). 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.