utils.py

# -*- coding: utf-8 -*-
# Copyright (C) 2013 Centre de données Astrophysiques de Marseille
# Copyright (C) 2013-2014 Institute of Astronomy
# Copyright (C) 2014 Yannick Roehlly <yannick@iaora.eu>
# Licensed under the CeCILL-v2 licence - see Licence_CeCILL_V2-en.txt
# Author: Yannick Roehlly & Médéric Boquien

from functools import lru_cache

from astropy import log
from astropy.cosmology import WMAP7 as cosmo
import numpy as np
from scipy import optimize
from scipy.special import erf

log.setLevel('ERROR')

def save_chi2(obs, variable, models, chi2, values):
    """Save the chi² and the associated physocal properties

    """
    fname = 'out/{}_{}_chi2-block-{}.npy'.format(obs['id'], variable,
                                                 models.iblock)
    data = np.memmap(fname, dtype=np.float64, mode='w+',
                     shape=(2, chi2.size))
    data[0, :] = chi2
    data[1, :] = values


@lru_cache(maxsize=None)
def compute_corr_dz(model_z, obs_z):
    """The mass-dependent physical properties are computed assuming the
    redshift of the model. However because we round the observed redshifts to
    two decimals, there can be a difference of 0.005 in redshift between the
    models and the actual observation. At low redshift, this can cause a
    discrepancy in the mass-dependent physical properties: ~0.35 dex at z=0.010
    vs 0.015 for instance. Therefore we correct these physical quantities by
    multiplying them by corr_dz.

    Parameters
    ----------
    model_z: float
        Redshift of the model.
    obs_z: float
        Redshift of the observed object.

    """
    if model_z == obs_z:
        return 1.
    if model_z > 0.:
        return (cosmo.luminosity_distance(obs_z).value /
                cosmo.luminosity_distance(model_z).value)**2.
    return (cosmo.luminosity_distance(obs_z).value * 1e5)**2.


def dchi2_over_ds2(s, obs_fluxes, obs_errors, mod_fluxes):
    """Function used to estimate the normalization factor in the SED fitting
    process when upper limits are included in the dataset to fit (from Eq. A11
    in Sawicki M. 2012, PASA, 124, 1008).

    Parameters
    ----------
    s: Float
        Contains value onto which we perform minimization = normalization
        factor
    obs_fluxes: RawArray
        Contains observed fluxes for each filter.
    obs_errors: RawArray
        Contains observed errors for each filter.
    model_fluxes: RawArray
        Contains modeled fluxes for each filter.
    lim_flag: Boolean
        Tell whether we use upper limits (True) or not (False).

   Returns
    -------
    func: Float
        Eq. A11 in Sawicki M. 2012, PASA, 124, 1008).

    """
    # We enter into this function if lim_flag = True.

    # The mask "data" selects the filter(s) for which measured fluxes are given
    # i.e., when obs_fluxes is >=0. and obs_errors >=0.
    # The mask "lim" selects the filter(s) for which upper limits are given
    # i.e., when obs_errors < 0

    wlim = np.where(np.isfinite(obs_errors) & (obs_errors < 0.))
    wdata = np.where(obs_errors >= 0.)

    mod_fluxes_data = mod_fluxes[wdata]
    mod_fluxes_lim = mod_fluxes[wlim]

    obs_fluxes_data = obs_fluxes[wdata]
    obs_fluxes_lim = obs_fluxes[wlim]

    obs_errors_data = obs_errors[wdata]
    obs_errors_lim = -obs_errors[wlim]

    dchi2_over_ds_data = np.sum(
        (obs_fluxes_data-s*mod_fluxes_data) *
        mod_fluxes_data/(obs_errors_data*obs_errors_data))

    dchi2_over_ds_lim = np.sqrt(2./np.pi)*np.sum(
        mod_fluxes_lim*np.exp(
            -np.square(
                (obs_fluxes_lim-s*mod_fluxes_lim)/(np.sqrt(2)*obs_errors_lim)
                      )
                             )/(
            obs_errors_lim*(
                1.+erf(
                   (obs_fluxes_lim-s*mod_fluxes_lim)/(np.sqrt(2)*obs_errors_lim)
                      )
                           )
                               )
                                                )

    func = dchi2_over_ds_data - dchi2_over_ds_lim

    return func


def _compute_scaling(model_fluxes, obs_fluxes, obs_errors):
    """Compute the scaling factor to be applied to the model fluxes to best fit
    the observations. Note that we look over the bands to avoid the creation of
    an array of the same size as the model_fluxes array. Because we loop on the
    bands and not on the models, the impact on the performance should be small.

    Parameters
    ----------
    model_fluxes: array
        Fluxes of the models
    obs_fluxes: array
        Observed fluxes
    obs_errors: array
        Observed errors

    Returns
    -------
    scaling: array
        Scaling factors minimising the χ²
    """
    num = np.zeros(model_fluxes.shape[1])
    denom = np.zeros(model_fluxes.shape[1])
    for i in range(obs_fluxes.size):
        if np.isfinite(obs_fluxes[i]) and obs_errors[i] > 0.:
            num += model_fluxes[i, :] * (obs_fluxes[i] / (obs_errors[i] *
                                                          obs_errors[i]))
            denom += np.square(model_fluxes[i, :] * (1./obs_errors[i]))

    return num/denom


def compute_chi2(model_fluxes, obs_fluxes, obs_errors, lim_flag):
    """Compute the χ² of observed fluxes with respect to the grid of models. We
    take into account upper limits if need be. Note that we look over the bands
    to avoid the creation of an array of the same size as the model_fluxes
    array. Because we loop on the bands and not on the models, the impact on
    the performance should be small.

    Parameters
    ----------
    model_fluxes: array
        2D grid containing the fluxes of the models
    obs_fluxes: array
        Fluxes of the observed object
    obs_errors: array
        Uncertainties on the fluxes of the observed object
    lim_flag: boolean
        Boolean indicating whether upper limits should be treated (True) or
        discarded (False)

    Returns
    -------
    chi2: array
        χ² for all the models in the grid
    scaling: array
        scaling of the models to obtain the minimum χ²
    """
    limits = lim_flag and np.any(obs_errors <= 0.)

    scaling = _compute_scaling(model_fluxes, obs_fluxes, obs_errors)

    # Some observations may not have flux values in some filter(s), but
    # they can have upper limit(s).
    if limits == True:
        for imod in range(scaling.size):
            scaling[imod] = optimize.root(dchi2_over_ds2, scaling[imod],
                                          args=(obs_fluxes, obs_errors,
                                                model_fluxes[:, imod])).x

    # χ² of the comparison of each model to each observation.
    chi2 = np.zeros(model_fluxes.shape[1])
    for i in range(obs_fluxes.size):
        if np.isfinite(obs_fluxes[i]) and obs_errors[i] > 0.:
            chi2 += np.square(
                (obs_fluxes[i] - model_fluxes[i, :] * scaling) * (1./obs_errors[i]))

    # Some observations may not have flux values in some filter(s), but
    # they can have upper limit(s).
    if limits == True:
        for i, obs_error in enumerate(obs_errors):
            if obs_error < 0.:
                chi2 -= 2. * np.log(.5 *
                                    (1. + erf(((obs_fluxes[i] -
                                                model_fluxes[i, :] * scaling) /
                                               (-np.sqrt(2.)*obs_errors[i])))))

    return chi2, scaling


def weighted_param(param, weights):
    """Compute the weighted mean and standard deviation of an array of data.

    Parameters
    ----------
    param: array
        Values of the parameters for the entire grid of models
    weights: array
        Weights by which to weight the parameter values

    Returns
    -------
    mean: float
        Weighted mean of the parameter values
    std: float
        Weighted standard deviation of the parameter values

    """

    mean = np.average(param, weights=weights)
    std = np.sqrt(np.average((param-mean)**2, weights=weights))

    return (mean, std)