Source code for treecorr.gggcorrelation

# Copyright (c) 2003-2024 by Mike Jarvis
#
# TreeCorr is free software: redistribution and use in source and binary forms,
# with or without modification, are permitted provided that the following
# conditions are met:
#
# 1. Redistributions of source code must retain the above copyright notice, this
#    list of conditions, and the disclaimer given in the accompanying LICENSE
#    file.
# 2. Redistributions in binary form must reproduce the above copyright notice,
#    this list of conditions, and the disclaimer given in the documentation
#    and/or other materials provided with the distribution.

"""
.. module:: nnncorrelation
"""

import numpy as np

from . import _treecorr
from .catalog import calculateVarG
from .corr3base import Corr3
from .util import make_writer, make_reader
from .config import make_minimal_config


[docs]class GGGCorrelation(Corr3): r"""This class handles the calculation and storage of a 3-point shear-shear-shear correlation function. We use the "natural components" of the shear 3-point function described by Schneider & Lombardi (2003) [Astron.Astrophys. 397 (2003) 809-818]. In this paradigm, the shears are projected relative to some point defined by the geometry of the triangle. They give several reasonable choices for this point. We choose the triangle's centroid as the "most natural" point, as many simple shear fields have purely real :math:`\Gamma_0` using this definition. It is also a fairly simple point to calculate in the code compared to some of the other options they offer, so projections relative to it are fairly efficient. There are 4 complex-valued 3-point shear corrletion functions defined for triples of shear values projected relative to the line joining the location of the shear to the cenroid of the triangle: .. math:: \Gamma_0 &= \langle \gamma(\mathbf{x1}) \gamma(\mathbf{x2}) \gamma(\mathbf{x3}) \rangle \\ \Gamma_1 &= \langle \gamma(\mathbf{x1})^* \gamma(\mathbf{x2}) \gamma(\mathbf{x3}) \rangle \\ \Gamma_2 &= \langle \gamma(\mathbf{x1}) \gamma(\mathbf{x2})^* \gamma(\mathbf{x3}) \rangle \\ \Gamma_3 &= \langle \gamma(\mathbf{x1}) \gamma(\mathbf{x2}) \gamma(\mathbf{x3})^* \rangle \\ where :math:`\mathbf{x1}, \mathbf{x2}, \mathbf{x3}` are the corners of the triange opposite sides d1, d2, d3 respectively, where d1 > d2 > d3, and :math:`{}^*` indicates complex conjugation. See the doc string of `Corr3` for a description of how the triangles are binned. Ojects of this class holds the following attributes: Attributes: nbins: The number of bins in logr where r = d2. bin_size: The size of the bins in logr. min_sep: The minimum separation being considered. max_sep: The maximum separation being considered. logr1d: The nominal centers of the nbins bins in log(r). If the bin_type is LogRUV, then it will have these attributes: Attributes: nubins: The number of bins in u where u = d3/d2. ubin_size: The size of the bins in u. min_u: The minimum u being considered. max_u: The maximum u being considered. nvbins: The number of bins in v where v = +-(d1-d2)/d3. vbin_size: The size of the bins in v. min_v: The minimum v being considered. max_v: The maximum v being considered. u1d: The nominal centers of the nubins bins in u. v1d: The nominal centers of the nvbins bins in v. If the bin_type is LogSAS, then it will have these attributes: Attributes: nphi_bins: The number of bins in phi. phi_bin_size: The size of the bins in phi. min_phi: The minimum phi being considered. max_phi: The maximum phi being considered. phi1d: The nominal centers of the nphi_bins bins in phi. If the bin_type is LogMultipole, then it will have these attributes: Attributes: max_n: The maximum multipole index n being stored. n1d: The multipole index n in the 2*max_n+1 bins of the third bin direction. In addition, the following attributes are numpy arrays whose shape is: * (nbins, nubins, nvbins) if bin_type is LogRUV * (nbins, nbins, nphi_bins) if bin_type is LogSAS * (nbins, nbins, 2*max_n+1) if bin_type is LogMultipole If bin_type is LogRUV: Attributes: logr: The nominal center of each bin in log(r). rnom: The nominal center of each bin converted to regular distance. i.e. r = exp(logr). u: The nominal center of each bin in u. v: The nominal center of each bin in v. meanu: The (weighted) mean value of u for the triangles in each bin. meanv: The (weighted) mean value of v for the triangles in each bin. If bin_type is LogSAS: Attributes: logd2: The nominal center of each bin in log(d2). d2nom: The nominal center of each bin converted to regular d2 distance. i.e. d2 = exp(logd2). logd3: The nominal center of each bin in log(d3). d3nom: The nominal center of each bin converted to regular d3 distance. i.e. d3 = exp(logd3). phi: The nominal center of each angular bin. meanphi: The (weighted) mean value of phi for the triangles in each bin. If bin_type is LogMultipole: Attributes: logd2: The nominal center of each bin in log(d2). d2nom: The nominal center of each bin converted to regular d2 distance. i.e. d2 = exp(logd2). logd3: The nominal center of each bin in log(d3). d3nom: The nominal center of each bin converted to regular d3 distance. i.e. d3 = exp(logd3). n: The multipole index n for each bin. For any bin_type: Attributes: gam0: The 0th "natural" correlation function, :math:`\Gamma_0`. gam1: The 1st "natural" correlation function, :math:`\Gamma_1`. gam2: The 2nd "natural" correlation function, :math:`\Gamma_2`. gam3: The 3rd "natural" correlation function, :math:`\Gamma_3`. vargam0: The variance of :math:`\Gamma_0`, only including the shot noise propagated into the final correlation. This (and the related values for 1,2,3) does not include sample variance, so it is always an underestimate of the actual variance. vargam1: The variance of :math:`\Gamma_1`. vargam2: The variance of :math:`\Gamma_2`. vargam3: The variance of :math:`\Gamma_3`. meand1: The (weighted) mean value of d1 for the triangles in each bin. meanlogd1: The (weighted) mean value of log(d1) for the triangles in each bin. meand2: The (weighted) mean value of d2 for the triangles in each bin. meanlogd2: The (weighted) mean value of log(d2) for the triangles in each bin. meand3: The (weighted) mean value of d3 for the triangles in each bin. meanlogd3: The (weighted) mean value of log(d3) for the triangles in each bin. weight: The total weight in each bin. ntri: The number of triangles going into each bin (including those where one or more objects have w=0). If ``sep_units`` are given (either in the config dict or as a named kwarg) then the distances will all be in these units. .. note:: If you separate out the steps of the `Corr3.process` command and use `process_auto` and/or `Corr3.process_cross`, then the units will not be applied to ``meanr`` or ``meanlogr`` until the `finalize` function is called. The typical usage pattern is as follows:: >>> ggg = treecorr.GGGCorrelation(config) >>> ggg.process(cat) # For auto-correlation. >>> ggg.process(cat1,cat2,cat3) # For cross-correlation. >>> ggg.write(file_name) # Write out to a file. >>> gam0 = ggg.gam0, etc. # To access gamma values directly. >>> gam0r = ggg.gam0r # You can also access real and imag parts separately. >>> gam0i = ggg.gam0i Parameters: config (dict): A configuration dict that can be used to pass in kwargs if desired. This dict is allowed to have addition entries besides those listed in `Corr3`, which are ignored here. (default: None) logger: If desired, a logger object for logging. (default: None, in which case one will be built according to the config dict's verbose level.) Keyword Arguments: **kwargs: See the documentation for `Corr3` for the list of allowed keyword arguments, which may be passed either directly or in the config dict. """ _cls = 'GGGCorrelation' _letter1 = 'G' _letter2 = 'G' _letter3 = 'G' _letters = 'GGG' _builder = _treecorr.GGGCorr _calculateVar1 = staticmethod(calculateVarG) _calculateVar2 = staticmethod(calculateVarG) _calculateVar3 = staticmethod(calculateVarG) _sig1 = 'sig_sn (per component)' _sig2 = 'sig_sn (per component)' _sig3 = 'sig_sn (per component)'
[docs] def __init__(self, config=None, *, logger=None, **kwargs): """Initialize `GGGCorrelation`. See class doc for details. """ Corr3.__init__(self, config, logger=logger, **kwargs) shape = self.data_shape self._z1 = np.zeros(shape, dtype=float) self._z2 = np.zeros(shape, dtype=float) self._z3 = np.zeros(shape, dtype=float) self._z4 = np.zeros(shape, dtype=float) self._z5 = np.zeros(shape, dtype=float) self._z6 = np.zeros(shape, dtype=float) self._z7 = np.zeros(shape, dtype=float) self._z8 = np.zeros(shape, dtype=float) self._vargam0 = None self._vargam1 = None self._vargam2 = None self._vargam3 = None self.logger.debug('Finished building GGGCorr')
@property def gam0(self): return self._z1 + 1j * self._z2 @property def gam1(self): return self._z3 + 1j * self._z4 @property def gam2(self): return self._z5 + 1j * self._z6 @property def gam3(self): return self._z7 + 1j * self._z8 @property def gam0r(self): return self._z1 @property def gam0i(self): return self._z2 @property def gam1r(self): return self._z3 @property def gam1i(self): return self._z4 @property def gam2r(self): return self._z5 @property def gam2i(self): return self._z6 @property def gam3r(self): return self._z7 @property def gam3i(self): return self._z8
[docs] def process_auto(self, cat, *, metric=None, num_threads=None): """Process a single catalog, accumulating the auto-correlation. This accumulates the auto-correlation for the given catalog. After calling this function as often as desired, the `finalize` command will finish the calculation of meand1, meanlogd1, etc. Parameters: cat (Catalog): The catalog to process metric (str): Which metric to use. See `Metrics` for details. (default: 'Euclidean'; this value can also be given in the constructor in the config dict.) num_threads (int): How many OpenMP threads to use during the calculation. (default: use the number of cpu cores; this value can also be given in the constructor in the config dict.) """ super()._process_auto(cat, metric, num_threads)
[docs] def process_cross12(self, cat1, cat2, *, metric=None, ordered=True, num_threads=None): """Process two catalogs, accumulating the 3pt cross-correlation, where one of the points in each triangle come from the first catalog, and two come from the second. This accumulates the cross-correlation for the given catalogs as part of a larger auto- or cross-correlation calculation. E.g. when splitting up a large catalog into patches, this is appropriate to use for the cross correlation between different patches as part of the complete auto-correlation of the full catalog. Parameters: cat1 (Catalog): The first catalog to process. (1 point in each triangle will come from this catalog.) cat2 (Catalog): The second catalog to process. (2 points in each triangle will come from this catalog.) metric (str): Which metric to use. See `Metrics` for details. (default: 'Euclidean'; this value can also be given in the constructor in the config dict.) ordered (bool): Whether to fix the order of the triangle vertices to match the catalogs. (default: True) num_threads (int): How many OpenMP threads to use during the calculation. (default: use the number of cpu cores; this value can also be given in the constructor in the config dict.) """ super()._process_cross12(cat1, cat2, metric, ordered, num_threads)
[docs] def finalize(self, varg1, varg2, varg3): """Finalize the calculation of the correlation function. The `process_auto`, `process_cross12` and `Corr3.process_cross` commands accumulate values in each bin, so they can be called multiple times if appropriate. Afterwards, this command finishes the calculation by dividing by the total weight. Parameters: varg1 (float): The variance per component of the first shear field. varg2 (float): The variance per component of the second shear field. varg3 (float): The variance per component of the third shear field. """ self._finalize() mask1 = self.weightr != 0 mask2 = self.weightr == 0 self._var_num = 4 * varg1 * varg2 * varg3 # I don't really understand why the variance is coming out 2x larger than the normal # formula for LogSAS. But with just Gaussian noise, I need to multiply the numerator # by two to get the variance estimates to come out right. if self.bin_type in ['LogSAS', 'LogMultipole']: self._var_num *= 2
@property def vargam0(self): if self._vargam0 is None: self._vargam0 = np.zeros(self.data_shape) if self._var_num != 0: self._vargam0.ravel()[:] = self.cov_diag[0:self._nbins].real return self._vargam0 @property def vargam1(self): if self._vargam1 is None: self._vargam1 = np.zeros(self.data_shape) if self._var_num != 0: self._vargam1.ravel()[:] = self.cov_diag[self._nbins:2*self._nbins].real return self._vargam1 @property def vargam2(self): if self._vargam2 is None: self._vargam2 = np.zeros(self.data_shape) if self._var_num != 0: self._vargam2.ravel()[:] = self.cov_diag[2*self._nbins:3*self._nbins].real return self._vargam2 @property def vargam3(self): if self._vargam3 is None: self._vargam3 = np.zeros(self.data_shape) if self._var_num != 0: self._vargam3.ravel()[:] = self.cov_diag[3*self._nbins:4*self._nbins].real return self._vargam3 def _clear(self): super()._clear() self._vargam0 = None self._vargam1 = None self._vargam2 = None self._vargam3 = None def _sum(self, others): # Equivalent to the operation of: # self._clear() # for other in others: # self += other # but no sanity checks and use numpy.sum for faster calculation. np.sum([c.gam0r for c in others], axis=0, out=self.gam0r) np.sum([c.gam0i for c in others], axis=0, out=self.gam0i) np.sum([c.gam1r for c in others], axis=0, out=self.gam1r) np.sum([c.gam1i for c in others], axis=0, out=self.gam1i) np.sum([c.gam2r for c in others], axis=0, out=self.gam2r) np.sum([c.gam2i for c in others], axis=0, out=self.gam2i) np.sum([c.gam3r for c in others], axis=0, out=self.gam3r) np.sum([c.gam3i for c in others], axis=0, out=self.gam3i) np.sum([c.meand1 for c in others], axis=0, out=self.meand1) np.sum([c.meanlogd1 for c in others], axis=0, out=self.meanlogd1) np.sum([c.meand2 for c in others], axis=0, out=self.meand2) np.sum([c.meanlogd2 for c in others], axis=0, out=self.meanlogd2) np.sum([c.meand3 for c in others], axis=0, out=self.meand3) np.sum([c.meanlogd3 for c in others], axis=0, out=self.meanlogd3) np.sum([c.meanu for c in others], axis=0, out=self.meanu) if self.bin_type == 'LogRUV': np.sum([c.meanv for c in others], axis=0, out=self.meanv) np.sum([c.weightr for c in others], axis=0, out=self.weightr) if self.bin_type == 'LogMultipole': np.sum([c.weighti for c in others], axis=0, out=self.weighti) np.sum([c.ntri for c in others], axis=0, out=self.ntri)
[docs] def getStat(self): """The standard statistic for the current correlation object as a 1-d array. In this case, the concatenation of gam0.ravel(), gam1.ravel(), gam2.ravel(), gam3.ravel(). .. note:: This is a complex array, unlike most other statistics. The computed covariance matrix will be complex, although since it is Hermitian the diagonal is real, so the resulting vargam0, etc. will all be real arrays. """ return np.concatenate([self.gam0.ravel(), self.gam1.ravel(), self.gam2.ravel(), self.gam3.ravel()])
[docs] def getWeight(self): """The weight array for the current correlation object as a 1-d array. In this case, 4 copies of self.weight.ravel(). """ return np.concatenate([np.abs(self.weight.ravel())] * 4)
[docs] def toSAS(self, *, target=None, **kwargs): """Convert a multipole-binned correlation to the corresponding SAS binning. This is only valid for bin_type == LogMultipole. Keyword Arguments: target: A target GGGCorrelation object with LogSAS binning to write to. If this is not given, a new object will be created based on the configuration paramters of the current object. (default: None) **kwargs: Any kwargs that you want to use to configure the returned object. Typically, might include min_phi, max_phi, nphi_bins, phi_bin_size. The default phi binning is [0,pi] with nphi_bins = self.max_n. Returns: A GGGCorrelation object with bin_type=LogSAS containing the same information as this object, but with the SAS binning. """ sas = super().toSAS(target=target, **kwargs) sas._var_num = self._var_num # Z(d2,d3,phi) = 1/2pi sum_n Z_n(d2,d3) exp(i n phi) expiphi = np.exp(1j * self.n1d[:,None] * sas.phi1d) gam0 = self.gam0.dot(expiphi) / (2*np.pi) * sas.phi_bin_size gam1 = self.gam1.dot(expiphi) / (2*np.pi) * sas.phi_bin_size gam2 = self.gam2.dot(expiphi) / (2*np.pi) * sas.phi_bin_size gam3 = self.gam3.dot(expiphi) / (2*np.pi) * sas.phi_bin_size # We leave the gam_mu unnormalized in the Multipole class, so after the FT, # we still need to divide by weight. mask = sas.weightr != 0 gam0[mask] /= sas.weightr[mask] gam1[mask] /= sas.weightr[mask] gam2[mask] /= sas.weightr[mask] gam3[mask] /= sas.weightr[mask] # Now fix the projection. # The multipole algorithm uses the Porth et al x projection. # We need to switch that to the canoical centroid projection. # Define some complex "vectors" where p1 is at the origin and # p3 is on the x axis: # s = p3 - p1 # t = p2 - p1 # u = angle bisector of s, t # q1 = (s+t)/3. (this is the centroid) # q2 = q1-t # q3 = q1-s s = sas.meand2 t = sas.meand3 * np.exp(1j * sas.meanphi * sas._phi_units) u = (1 + t/np.abs(t))/2 q1 = (s+t)/3. q2 = q1-t q3 = q1-s # Currently the projection is as follows: # g1 is projected along u # g2 is projected along t # g3 is projected along s # # We want to have # g1 projected along q1 # g2 projected along q2 # g3 projected along q3 # # The phases to multiply by are exp(2iphi_current) * exp(-2iphi_target). I.e. # g1phase = (u conj(q1))**2 / |u conj(q1)|**2 # g2phase = (t conj(q2))**2 / |t conj(q2)|**2 # g3phase = (s conj(q3))**2 / |s conj(q3)|**2 g1phase = (u * np.conj(q1))**2 g2phase = (t * np.conj(q2))**2 g3phase = (s * np.conj(q3))**2 g1phase /= np.abs(g1phase) g2phase /= np.abs(g2phase) g3phase /= np.abs(g3phase) # Now just multiply each gam by the appropriate combination of phases. gam0phase = g1phase * g2phase * g3phase gam1phase = np.conj(g1phase) * g2phase * g3phase gam2phase = g1phase * np.conj(g2phase) * g3phase gam3phase = g1phase * g2phase * np.conj(g3phase) gam0 *= gam0phase gam1 *= gam1phase gam2 *= gam2phase gam3 *= gam3phase sas.gam0r[:] = np.real(gam0) sas.gam0i[:] = np.imag(gam0) sas.gam1r[:] = np.real(gam1) sas.gam1i[:] = np.imag(gam1) sas.gam2r[:] = np.real(gam2) sas.gam2i[:] = np.imag(gam2) sas.gam3r[:] = np.real(gam3) sas.gam3i[:] = np.imag(gam3) for k,v in self.results.items(): temp = sas.results[k] gam0 = v.gam0.dot(expiphi) / (2*np.pi) * sas.phi_bin_size * gam0phase gam1 = v.gam1.dot(expiphi) / (2*np.pi) * sas.phi_bin_size * gam1phase gam2 = v.gam2.dot(expiphi) / (2*np.pi) * sas.phi_bin_size * gam2phase gam3 = v.gam3.dot(expiphi) / (2*np.pi) * sas.phi_bin_size * gam3phase temp.gam0r[:] = np.real(gam0) temp.gam0i[:] = np.imag(gam0) temp.gam1r[:] = np.real(gam1) temp.gam1i[:] = np.imag(gam1) temp.gam2r[:] = np.real(gam2) temp.gam2i[:] = np.imag(gam2) temp.gam3r[:] = np.real(gam3) temp.gam3i[:] = np.imag(gam3) return sas
[docs] def write(self, file_name, *, file_type=None, precision=None, write_patch_results=False, write_cov=False): r"""Write the correlation function to the file, file_name. As described in the doc string for `GGGCorrelation`, we use the "natural components" of the shear 3-point function described by Schneider & Lombardi (2003) using the triangle centroid as the projection point. There are 4 complex-valued natural components, so there are 8 columns in the output file. For bin_type = LogRUV, the output file will include the following columns: ========== ================================================================ Column Description ========== ================================================================ r_nom The nominal center of the bin in r = d2 where d1 > d2 > d3 u_nom The nominal center of the bin in u = d3/d2 v_nom The nominal center of the bin in v = +-(d1-d2)/d3 meanu The mean value :math:`\langle u\rangle` of triangles that fell into each bin meanv The mean value :math:`\langle v\rangle` of triangles that fell into each bin ========== ================================================================ For bin_type = LogSAS, the output file will include the following columns: ========== ================================================================ Column Description ========== ================================================================ d2_nom The nominal center of the bin in d2 d3_nom The nominal center of the bin in d3 phi_nom The nominal center of the bin in phi, the opening angle between d2 and d3 in the counter-clockwise direction meanphi The mean value :math:`\langle phi\rangle` of triangles that fell into each bin ========== ================================================================ For bin_type = LogMultipole, the output file will include the following columns: ========== ================================================================ Column Description ========== ================================================================ d2_nom The nominal center of the bin in d2 d3_nom The nominal center of the bin in d3 n The multipole index n ========== ================================================================ In addition, all bin types include the following columns: ========== ================================================================ Column Description ========== ================================================================ meand1 The mean value :math:`\langle d1\rangle` of triangles that fell into each bin meanlogd1 The mean value :math:`\langle \log(d1)\rangle` of triangles that fell into each bin meand2 The mean value :math:`\langle d2\rangle` of triangles that fell into each bin meanlogd2 The mean value :math:`\langle \log(d2)\rangle` of triangles that fell into each bin meand3 The mean value :math:`\langle d3\rangle` of triangles that fell into each bin meanlogd3 The mean value :math:`\langle \log(d3)\rangle` of triangles that fell into each bin gam0r The real part of the estimator of :math:`\Gamma_0` gam0i The imag part of the estimator of :math:`\Gamma_0` gam1r The real part of the estimator of :math:`\Gamma_1` gam1i The imag part of the estimator of :math:`\Gamma_1` gam2r The real part of the estimator of :math:`\Gamma_2` gam2i The imag part of the estimator of :math:`\Gamma_2` gam3r The real part of the estimator of :math:`\Gamma_3` gam3i The imag part of the estimator of :math:`\Gamma_3` sigma_gam0 The sqrt of the variance estimate of :math:`\Gamma_0` sigma_gam1 The sqrt of the variance estimate of :math:`\Gamma_1` sigma_gam2 The sqrt of the variance estimate of :math:`\Gamma_2` sigma_gam3 The sqrt of the variance estimate of :math:`\Gamma_3` weight The total weight of triangles contributing to each bin. (For LogMultipole, this is split into real and imaginary parts, weight_re and weight_im.) ntri The number of triangles contributing to each bin ========== ================================================================ If ``sep_units`` was given at construction, then the distances will all be in these units. Otherwise, they will be in either the same units as x,y,z (for flat or 3d coordinates) or radians (for spherical coordinates). Parameters: file_name (str): The name of the file to write to. file_type (str): The type of file to write ('ASCII' or 'FITS'). (default: determine the type automatically from the extension of file_name.) precision (int): For ASCII output catalogs, the desired precision. (default: 4; this value can also be given in the constructor in the config dict.) write_patch_results (bool): Whether to write the patch-based results as well. (default: False) write_cov (bool): Whether to write the covariance matrix as well. (default: False) """ self.logger.info('Writing GGG correlations to %s',file_name) precision = self.config.get('precision', 4) if precision is None else precision with make_writer(file_name, precision, file_type, self.logger) as writer: self._write(writer, None, write_patch_results, write_cov=write_cov)
@property def _write_col_names(self): if self.bin_type == 'LogRUV': col_names = ['r_nom', 'u_nom', 'v_nom', 'meand1', 'meanlogd1', 'meand2', 'meanlogd2', 'meand3', 'meanlogd3', 'meanu', 'meanv'] elif self.bin_type == 'LogSAS': col_names = ['d2_nom', 'd3_nom', 'phi_nom', 'meand1', 'meanlogd1', 'meand2', 'meanlogd2', 'meand3', 'meanlogd3', 'meanphi'] else: col_names = ['d2_nom', 'd3_nom', 'n', 'meand1', 'meanlogd1', 'meand2', 'meanlogd2', 'meand3', 'meanlogd3'] col_names += ['gam0r', 'gam0i', 'gam1r', 'gam1i', 'gam2r', 'gam2i', 'gam3r', 'gam3i', 'sigma_gam0', 'sigma_gam1', 'sigma_gam2', 'sigma_gam3'] if self.bin_type == 'LogMultipole': col_names += ['weight_re', 'weight_im', 'ntri'] else: col_names += ['weight', 'ntri'] return col_names @property def _write_data(self): if self.bin_type == 'LogRUV': data = [ self.rnom, self.u, self.v, self.meand1, self.meanlogd1, self.meand2, self.meanlogd2, self.meand3, self.meanlogd3, self.meanu, self.meanv ] elif self.bin_type == 'LogSAS': data = [ self.d2nom, self.d3nom, self.phi, self.meand1, self.meanlogd1, self.meand2, self.meanlogd2, self.meand3, self.meanlogd3, self.meanphi ] else: data = [ self.d2nom, self.d3nom, self.n, self.meand1, self.meanlogd1, self.meand2, self.meanlogd2, self.meand3, self.meanlogd3 ] data += [ self.gam0r, self.gam0i, self.gam1r, self.gam1i, self.gam2r, self.gam2i, self.gam3r, self.gam3i, np.sqrt(self.vargam0), np.sqrt(self.vargam1), np.sqrt(self.vargam2), np.sqrt(self.vargam3) ] if self.bin_type == 'LogMultipole': data += [ self.weightr, self.weighti, self.ntri ] else: data += [ self.weight, self.ntri ] data = [ col.flatten() for col in data ] return data def _read_from_data(self, data, params): super()._read_from_data(data, params) s = self.data_shape self._z1 = data['gam0r'].reshape(s) self._z2 = data['gam0i'].reshape(s) self._z3 = data['gam1r'].reshape(s) self._z4 = data['gam1i'].reshape(s) self._z5 = data['gam2r'].reshape(s) self._z6 = data['gam2i'].reshape(s) self._z7 = data['gam3r'].reshape(s) self._z8 = data['gam3i'].reshape(s) self._vargam0 = data['sigma_gam0'].reshape(s)**2 self._vargam1 = data['sigma_gam1'].reshape(s)**2 self._vargam2 = data['sigma_gam2'].reshape(s)**2 self._vargam3 = data['sigma_gam3'].reshape(s)**2 @classmethod def _calculateT(cls, s, t, k1, k2, k3): # First calculate q values: q1 = (s+t)/3. q2 = q1-t q3 = q1-s # |qi|^2 shows up a lot, so save these. # The a stands for "absolute", and the ^2 part is implicit. a1 = np.abs(q1)**2 a2 = np.abs(q2)**2 a3 = np.abs(q3)**2 a123 = a1*a2*a3 # These combinations also appear multiple times. # The b doesn't stand for anything. It's just the next letter after a. b1 = np.conjugate(q1)**2*q2*q3 b2 = np.conjugate(q2)**2*q1*q3 b3 = np.conjugate(q3)**2*q1*q2 if k1==1 and k2==1 and k3==1: # Some factors we use multiple times expfactor = -np.exp(-(a1 + a2 + a3)/2) # JBJ Equation 51 # Note that we actually accumulate the Gammas with a different choice for # alpha_i. We accumulate the shears relative to the q vectors, not relative to s. # cf. JBJ Equation 41 and footnote 3. The upshot is that we multiply JBJ's formulae # by (q1q2q3)^2 / |q1q2q3|^2 for T0 and (q1*q2q3)^2/|q1q2q3|^2 for T1. # Then T0 becomes # T0 = -(|q1 q2 q3|^2)/24 exp(-(|q1|^2+|q2|^2+|q3|^2)/2) T0 = expfactor * a123 / 24 # JBJ Equation 52 # After the phase adjustment, T1 becomes: # T1 = -[(|q1 q2 q3|^2)/24 # - (q1*^2 q2 q3)/9 # + (q1*^4 q2^2 q3^2 + 2 |q2 q3|^2 q1*^2 q2 q3)/(|q1 q2 q3|^2)/27 # ] exp(-(|q1|^2+|q2|^2+|q3|^2)/2) T1 = expfactor * (a123 / 24 - b1 / 9 + (b1**2 + 2*a2*a3*b1) / (a123 * 27)) T2 = expfactor * (a123 / 24 - b2 / 9 + (b2**2 + 2*a1*a3*b2) / (a123 * 27)) T3 = expfactor * (a123 / 24 - b3 / 9 + (b3**2 + 2*a1*a2*b3) / (a123 * 27)) else: # SKL Equation 63: k1sq = k1*k1 k2sq = k2*k2 k3sq = k3*k3 Theta2 = ((k1sq*k2sq + k1sq*k3sq + k2sq*k3sq)/3.)**0.5 k1sq /= Theta2 # These are now what SKL calls theta_i^2 / Theta^2 k2sq /= Theta2 k3sq /= Theta2 Theta4 = Theta2*Theta2 Theta6 = Theta4*Theta2 S = k1sq * k2sq * k3sq # SKL Equation 64: Z = ((2*k2sq + 2*k3sq - k1sq) * a1 + (2*k3sq + 2*k1sq - k2sq) * a2 + (2*k1sq + 2*k2sq - k3sq) * a3) / (6*Theta2) expfactor = -S * np.exp(-Z) / Theta4 # SKL Equation 65: f1 = (k2sq+k3sq)/2 + (k2sq-k3sq)*(q2-q3)/(6*q1) f2 = (k3sq+k1sq)/2 + (k3sq-k1sq)*(q3-q1)/(6*q2) f3 = (k1sq+k2sq)/2 + (k1sq-k2sq)*(q1-q2)/(6*q3) f1c = np.conjugate(f1) f2c = np.conjugate(f2) f3c = np.conjugate(f3) # SKL Equation 69: g1 = k2sq*k3sq + (k3sq-k2sq)*k1sq*(q2-q3)/(3*q1) g2 = k3sq*k1sq + (k1sq-k3sq)*k2sq*(q3-q1)/(3*q2) g3 = k1sq*k2sq + (k2sq-k1sq)*k3sq*(q1-q2)/(3*q3) g1c = np.conjugate(g1) g2c = np.conjugate(g2) g3c = np.conjugate(g3) # SKL Equation 62: T0 = expfactor * a123 * f1c**2 * f2c**2 * f3c**2 / (24.*Theta6) # SKL Equation 68: T1 = expfactor * ( a123 * f1**2 * f2c**2 * f3c**2 / (24*Theta6) - b1 * f1*f2c*f3c*g1c / (9*Theta4) + (b1**2 * g1c**2 + 2*k2sq*k3sq*a2*a3*b1 * f2c * f3c) / (a123 * 27*Theta2)) T2 = expfactor * ( a123 * f1c**2 * f2**2 * f3c**2 / (24*Theta6) - b2 * f1c*f2*f3c*g2c / (9*Theta4) + (b2**2 * g2c**2 + 2*k1sq*k3sq*a1*a3*b2 * f1c * f3c) / (a123 * 27*Theta2)) T3 = expfactor * ( a123 * f1c**2 * f2c**2 * f3**2 / (24*Theta6) - b3 * f1c*f2c*f3*g3c / (9*Theta4) + (b3**2 * g3c**2 + 2*k1sq*k2sq*a1*a2*b3 * f1c * f2c) / (a123 * 27*Theta2)) return T0, T1, T2, T3
[docs] def calculateMap3(self, *, R=None, k2=1, k3=1): r"""Calculate the skewness of the aperture mass from the correlation function. The equations for this come from Jarvis, Bernstein & Jain (2004, MNRAS, 352). See their section 3, especially equations 51 and 52 for the :math:`T_i` functions, equations 60 and 61 for the calculation of :math:`\langle \cal M^3 \rangle` and :math:`\langle \cal M^2 M^* \rangle`, and equations 55-58 for how to convert these to the return values. If k2 or k3 != 1, then this routine calculates the generalization of the skewness proposed by Schneider, Kilbinger & Lombardi (2005, A&A, 431): :math:`\langle M_{ap}^3(R, k_2 R, k_3 R)\rangle` and related values. If k2 = k3 = 1 (the default), then there are only 4 combinations of Map and Mx that are relevant: - map3 = :math:`\langle M_{ap}^3(R)\rangle` - map2mx = :math:`\langle M_{ap}^2(R) M_\times(R)\rangle`, - mapmx2 = :math:`\langle M_{ap}(R) M_\times(R)\rangle` - mx3 = :math:`\langle M_{\rm \times}^3(R)\rangle` However, if k2 or k3 != 1, then there are 8 combinations: - map3 = :math:`\langle M_{ap}(R) M_{ap}(k_2 R) M_{ap}(k_3 R)\rangle` - mapmapmx = :math:`\langle M_{ap}(R) M_{ap}(k_2 R) M_\times(k_3 R)\rangle` - mapmxmap = :math:`\langle M_{ap}(R) M_\times(k_2 R) M_{ap}(k_3 R)\rangle` - mxmapmap = :math:`\langle M_\times(R) M_{ap}(k_2 R) M_{ap}(k_3 R)\rangle` - mxmxmap = :math:`\langle M_\times(R) M_\times(k_2 R) M_{ap}(k_3 R)\rangle` - mxmapmx = :math:`\langle M_\times(R) M_{ap}(k_2 R) M_\times(k_3 R)\rangle` - mapmxmx = :math:`\langle M_{ap}(R) M_\times(k_2 R) M_\times(k_3 R)\rangle` - mx3 = :math:`\langle M_\times(R) M_\times(k_2 R) M_\times(k_3 R)\rangle` To accommodate this full generality, we always return all 8 values, along with the estimated variance (which is equal for each), even when k2 = k3 = 1. .. note:: The formulae for the ``m2_uform`` = 'Schneider' definition of the aperture mass, described in the documentation of `calculateMapSq`, are not known, so that is not an option here. The calculations here use the definition that corresponds to ``m2_uform`` = 'Crittenden'. Parameters: R (array): The R values at which to calculate the aperture mass statistics. (default: None, which means use self.rnom1d) k2 (float): If given, the ratio R2/R1 in the SKL formulae. (default: 1) k3 (float): If given, the ratio R3/R1 in the SKL formulae. (default: 1) Returns: Tuple containing: - map3 = array of :math:`\langle M_{ap}(R) M_{ap}(k_2 R) M_{ap}(k_3 R)\rangle` - mapmapmx = array of :math:`\langle M_{ap}(R) M_{ap}(k_2 R) M_\times(k_3 R)\rangle` - mapmxmap = array of :math:`\langle M_{ap}(R) M_\times(k_2 R) M_{ap}(k_3 R)\rangle` - mxmapmap = array of :math:`\langle M_\times(R) M_{ap}(k_2 R) M_{ap}(k_3 R)\rangle` - mxmxmap = array of :math:`\langle M_\times(R) M_\times(k_2 R) M_{ap}(k_3 R)\rangle` - mxmapmx = array of :math:`\langle M_\times(R) M_{ap}(k_2 R) M_\times(k_3 R)\rangle` - mapmxmx = array of :math:`\langle M_{ap}(R) M_\times(k_2 R) M_\times(k_3 R)\rangle` - mx3 = array of :math:`\langle M_\times(R) M_\times(k_2 R) M_\times(k_3 R)\rangle` - varmap3 = array of variance estimates of the above values """ # As in the calculateMapSq function, we Make s and t matrices, so we can eventually do the # integral by doing a matrix product. if R is None: R = self.rnom1d else: R = np.asarray(R) # Pick s = d2, so dlogs is bin_size s = d2 = np.outer(1./R, self.meand2.ravel()) if self.bin_type == 'LogRUV': # We take t = d3, but we need the x and y components. (relative to s along x axis) # cf. Figure 1 in JBJ. # d1^2 = d2^2 + d3^2 - 2 d2 d3 cos(theta1) # tx = d3 cos(theta1) = (d2^2 + d3^2 - d1^2)/2d2 # Simplify this using u=d3/d2 and v=(d1-d2)/d3 # = (d3^2 - (d1+d2)(d1-d2)) / 2d2 # = d3 (d3 - (d1+d2)v) / 2d2 # = d3 (u - (2+uv)v)/2 # = d3 (u - 2v - uv^2)/2 # = d3 (u(1-v^2)/2 - v) # Note that v here is really |v|. We'll account for the sign of v in ty. d3 = np.outer(1./R, self.meand3.ravel()) d1 = np.outer(1./R, self.meand1.ravel()) u = self.meanu.ravel() v = self.meanv.ravel() tx = d3*(0.5*u*(1-v**2) - np.abs(v)) # This form tends to be more stable near potentially degenerate triangles # than tx = (d2*d2 + d3*d3 - d1*d1) / (2*d2) # However, add a check to make sure. bad = (tx <= -d3) | (tx >= d3) if np.any(bad): # pragma: no cover self.logger.warning("Warning: Detected some invalid triangles when computing Map^3") self.logger.warning("Excluding these triangles from the integral.") self.logger.debug("N bad points = %s",np.sum(bad)) self.logger.debug("d1[bad] = %s",d1[bad]) self.logger.debug("d2[bad] = %s",d2[bad]) self.logger.debug("d3[bad] = %s",d3[bad]) self.logger.debug("tx[bad] = %s",tx[bad]) bad = np.where(bad) tx[bad] = 0 # for now to avoid nans ty = np.sqrt(d3**2 - tx**2) ty[:,self.meanv.ravel() > 0] *= -1. t = tx + 1j * ty else: d3 = np.outer(1./R, self.meand3.ravel()) t = d3 * np.exp(1j * self.meanphi.ravel() * self._phi_units) # Next we need to construct the T values. T0, T1, T2, T3 = self._calculateT(s,t,1.,k2,k3) if self.bin_type == 'LogRUV': # Finally, account for the Jacobian in d^2t: jac = |J(tx, ty; u, v)|, # since our Gammas are accumulated in s, u, v, not s, tx, ty. # u = d3/d2, v = (d1-d2)/d3 # tx = d3 (u - 2v - uv^2)/2 # = s/2 (u^2 - 2uv - u^2v^2) # dtx/du = s (u - v - uv^2) # dtx/dv = -us (1 + uv) # ty = sqrt(d3^2 - tx^2) = sqrt(u^2 s^2 - tx^2) # dty/du = s^2 u/2ty (1-v^2) (2 + 3uv - u^2 + u^2v^2) # dty/dv = s^2 u^2/2ty (1 + uv) (u - uv^2 - 2uv) # # After some algebra... # # J = s^3 u^2 (1+uv) / ty # = d3^2 d1 / ty # jac = np.abs(d3*d3*d1/ty) jac[bad] = 0. # Exclude any bad triangles from the integral. d2t = jac * self.ubin_size * self.vbin_size / (2.*np.pi) else: # In SAS binning, d2t is easier. # We bin directly in ln(d3) and phi, so # tx = d3 cos(phi) # ty = d3 sin(phi) # dtx/dlnd3 = d3 dtx/dd3 = d3 cos(phi) # dty/dlnd3 = d3 dty/dd3 = d3 sin(phi) # dtx/dphi = -d3 sin(phi) # dty/dphi = d3 cos(phi) # J(tx,ty; lnd3, phi) = d3^2 d2t = d3**2 * self.bin_size * self.phi_bin_size / (2*np.pi) sds = s * s * self.bin_size # Remember bin_size is dln(s) # Note: these are really d2t/2piR^2 and sds/R^2, which are what actually show up # in JBJ equations 45 and 50. T0 *= sds * d2t T1 *= sds * d2t T2 *= sds * d2t T3 *= sds * d2t # Now do the integral by taking the matrix products. gam0 = self.gam0.ravel() gam1 = self.gam1.ravel() gam2 = self.gam2.ravel() gam3 = self.gam3.ravel() vargam0 = self.vargam0.ravel() vargam1 = self.vargam1.ravel() vargam2 = self.vargam2.ravel() vargam3 = self.vargam3.ravel() mmm = T0.dot(gam0) mcmm = T1.dot(gam1) mmcm = T2.dot(gam2) mmmc = T3.dot(gam3) # These accumulate the coefficients that are being dotted to gam0,1,2,3 respectively. # Below, we will take the abs^2 and dot it to gam0,1,2,3 in each case to compute the # total variance. # Note: This assumes that gam0, gam1, gam2, gam3 have no covariance. # This is not technically true, but I think it's approximately ok. var0 = T0.copy() var1 = T1.copy() var2 = T2.copy() var3 = T3.copy() if self.bin_type == 'LogRUV': if k2 == 1 and k3 == 1: mmm *= 6 mcmm += mmcm mcmm += mmmc mcmm *= 2 mmcm = mmmc = mcmm var0 *= 6 var1 *= 6 var2 *= 6 var3 *= 6 else: # Repeat the above for the other permutations for (_k1, _k2, _k3, _mcmm, _mmcm, _mmmc) in [ (1,k3,k2,mcmm,mmmc,mmcm), (k2,1,k3,mmcm,mcmm,mmmc), (k2,k3,1,mmcm,mmmc,mcmm), (k3,1,k2,mmmc,mcmm,mmcm), (k3,k2,1,mmmc,mmcm,mcmm) ]: T0, T1, T2, T3 = self._calculateT(s,t,_k1,_k2,_k3) T0 *= sds * d2t T1 *= sds * d2t T2 *= sds * d2t T3 *= sds * d2t # Relies on numpy array overloading += to actually update in place. mmm += T0.dot(gam0) _mcmm += T1.dot(gam1) _mmcm += T2.dot(gam2) _mmmc += T3.dot(gam3) var0 += T0 var1 += T1 var2 += T2 var3 += T3 else: # SAS binning counts each triangle with each vertex in the c1 position. # Just need to account for the cases where 1-2-3 are clockwise, rather than CCW. if k2 == 1 and k3 == 1: mmm *= 2 mcmm *= 2 mmcm += mmmc mmmc = mmcm var0 *= 2 var1 *= 2 var2 *= 2 var3 *= 2 else: # Repeat the above with 2,3 swapped. T0, T1, T2, T3 = self._calculateT(s,t,1,k3,k2) T0 *= sds * d2t T1 *= sds * d2t T2 *= sds * d2t T3 *= sds * d2t mmm += T0.dot(gam0) mcmm += T1.dot(gam1) mmmc += T2.dot(gam2) mmcm += T3.dot(gam3) var0 += T0 var1 += T1 var2 += T2 var3 += T3 map3 = 0.25 * np.real(mcmm + mmcm + mmmc + mmm) mapmapmx = 0.25 * np.imag(mcmm + mmcm - mmmc + mmm) mapmxmap = 0.25 * np.imag(mcmm - mmcm + mmmc + mmm) mxmapmap = 0.25 * np.imag(-mcmm + mmcm + mmmc + mmm) mxmxmap = 0.25 * np.real(mcmm + mmcm - mmmc - mmm) mxmapmx = 0.25 * np.real(mcmm - mmcm + mmmc - mmm) mapmxmx = 0.25 * np.real(-mcmm + mmcm + mmmc - mmm) mx3 = 0.25 * np.imag(mcmm + mmcm + mmmc - mmm) var0 /= 4 var1 /= 4 var2 /= 4 var3 /= 4 # Now finally add up the coefficient squared times each vargam element. var = np.abs(var0**2).dot(vargam0) var += np.abs(var1**2).dot(vargam1) var += np.abs(var2**2).dot(vargam2) var += np.abs(var3**2).dot(vargam3) return map3, mapmapmx, mapmxmap, mxmapmap, mxmxmap, mxmapmx, mapmxmx, mx3, var
[docs] def writeMap3(self, file_name, *, R=None, file_type=None, precision=None): r"""Write the aperture mass skewness based on the correlation function to the file, file_name. The output file will include the following columns: ========== ========================================================== Column Description ========== ========================================================== R The aperture radius Map3 An estimate of :math:`\langle M_{ap}^3\rangle(R)` (cf. `calculateMap3`) Map2Mx An estimate of :math:`\langle M_{ap}^2 M_\times\rangle(R)` MapMx2 An estimate of :math:`\langle M_{ap} M_\times^2\rangle(R)` Mx3 An estimate of :math:`\langle M_\times^3\rangle(R)` sig_map The sqrt of the variance estimate of each of these ========== ========================================================== Parameters: file_name (str): The name of the file to write to. R (array): The R values at which to calculate the statistics. (default: None, which means use self.rnom) file_type (str): The type of file to write ('ASCII' or 'FITS'). (default: determine the type automatically from the extension of file_name.) precision (int): For ASCII output catalogs, the desired precision. (default: 4; this value can also be given in the constructor in the config dict.) """ self.logger.info('Writing Map^3 from GGG correlations to %s',file_name) if R is None: R = self.rnom1d stats = self.calculateMap3(R=R) if precision is None: precision = self.config.get('precision', 4) col_names = ['R','Map3','Map2Mx', 'MapMx2', 'Mx3','sig_map'] columns = [ R, stats[0], stats[1], stats[4], stats[7], np.sqrt(stats[8]) ] with make_writer(file_name, precision, file_type, logger=self.logger) as writer: writer.write(col_names, columns)