GGGCorrelation: Shear-shear-shear correlations

class treecorr.GGGCorrelation(config=None, *, logger=None, **kwargs)[source]

Bases: Corr3

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 \(\Gamma_0\) 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:

\[\begin{split}\Gamma_0 &= \langle \gamma(\mathbf{x1}) \gamma(\mathbf{x2}) \gamma(\mathbf{x3}) \rangle \\ \Gamma_1 &= \langle \gamma(\mathbf{x1})^* \gamma(\mathbf{x2}) \gamma(\mathbf{x3}) \rangle \\ \Gamma_2 &= \langle \gamma(\mathbf{x1}) \gamma(\mathbf{x2})^* \gamma(\mathbf{x3}) \rangle \\ \Gamma_3 &= \langle \gamma(\mathbf{x1}) \gamma(\mathbf{x2}) \gamma(\mathbf{x3})^* \rangle \\\end{split}\]

where \(\mathbf{x1}, \mathbf{x2}, \mathbf{x3}\) are the corners of the triange opposite sides d1, d2, d3 respectively, where d1 > d2 > d3, and \({}^*\) indicates complex conjugation.

See the doc string of Corr3 for a description of how the triangles are binned along with the attributes related to the different binning options.

In addition to the attributes common to all Corr3 subclasses, objects of this class hold the following attributes:

Attributes:

  • gam0 – The 0th “natural” correlation function, \(\Gamma_0\).

  • gam1 – The 1st “natural” correlation function, \(\Gamma_1\).

  • gam2 – The 2nd “natural” correlation function, \(\Gamma_2\).

  • gam3 – The 3rd “natural” correlation function, \(\Gamma_3\).

  • vargam0 – The variance of \(\Gamma_0\), 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 \(\Gamma_1\).

  • vargam2 – The variance of \(\Gamma_2\).

  • vargam3 – The variance of \(\Gamma_3\).

The typical usage pattern is as follows:

>>> ggg = treecorr.GGGCorrelation(config)
>>> ggg.process(cat)              # Compute auto-correlation.
>>> ggg.process(cat1, cat2, cat3) # Compute cross-correlation.
>>> ggg.write(file_name)          # Write out to a file.
>>> gam0 = ggg.gam0, etc.         # Access gamma values directly.
>>> gam0r = ggg.gam0r             # Or 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.

__init__(config=None, *, logger=None, **kwargs)[source]
calculateMap3(*, R=None, k2=1, k3=1)[source]

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 \(T_i\) functions, equations 60 and 61 for the calculation of \(\langle \cal M^3 \rangle\) and \(\langle \cal M^2 M^* \rangle\), 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): \(\langle M_{ap}^3(R, k_2 R, k_3 R)\rangle\) and related values.

If k2 = k3 = 1 (the default), then there are only 4 combinations of Map and Mx that are relevant:

  • map3 = \(\langle M_{ap}^3(R)\rangle\)

  • map2mx = \(\langle M_{ap}^2(R) M_\times(R)\rangle\),

  • mapmx2 = \(\langle M_{ap}(R) M_\times(R)\rangle\)

  • mx3 = \(\langle M_{\rm \times}^3(R)\rangle\)

However, if k2 or k3 != 1, then there are 8 combinations:

  • map3 = \(\langle M_{ap}(R) M_{ap}(k_2 R) M_{ap}(k_3 R)\rangle\)

  • mapmapmx = \(\langle M_{ap}(R) M_{ap}(k_2 R) M_\times(k_3 R)\rangle\)

  • mapmxmap = \(\langle M_{ap}(R) M_\times(k_2 R) M_{ap}(k_3 R)\rangle\)

  • mxmapmap = \(\langle M_\times(R) M_{ap}(k_2 R) M_{ap}(k_3 R)\rangle\)

  • mxmxmap = \(\langle M_\times(R) M_\times(k_2 R) M_{ap}(k_3 R)\rangle\)

  • mxmapmx = \(\langle M_\times(R) M_{ap}(k_2 R) M_\times(k_3 R)\rangle\)

  • mapmxmx = \(\langle M_{ap}(R) M_\times(k_2 R) M_\times(k_3 R)\rangle\)

  • mx3 = \(\langle M_\times(R) M_\times(k_2 R) M_\times(k_3 R)\rangle\)

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:

  • map3 = array of \(\langle M_{ap}(R) M_{ap}(k_2 R) M_{ap}(k_3 R)\rangle\)

  • mapmapmx = array of \(\langle M_{ap}(R) M_{ap}(k_2 R) M_\times(k_3 R)\rangle\)

  • mapmxmap = array of \(\langle M_{ap}(R) M_\times(k_2 R) M_{ap}(k_3 R)\rangle\)

  • mxmapmap = array of \(\langle M_\times(R) M_{ap}(k_2 R) M_{ap}(k_3 R)\rangle\)

  • mxmxmap = array of \(\langle M_\times(R) M_\times(k_2 R) M_{ap}(k_3 R)\rangle\)

  • mxmapmx = array of \(\langle M_\times(R) M_{ap}(k_2 R) M_\times(k_3 R)\rangle\)

  • mapmxmx = array of \(\langle M_{ap}(R) M_\times(k_2 R) M_\times(k_3 R)\rangle\)

  • mx3 = array of \(\langle M_\times(R) M_\times(k_2 R) M_\times(k_3 R)\rangle\)

  • varmap3 = array of variance estimates of the above values

Return type:

Tuple containing

finalize(varg1, varg2, varg3)[source]

Finalize the calculation of the correlation function.

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.

getStat()[source]

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.

getWeight()[source]

The weight array for the current correlation object as a 1-d array.

In this case, 4 copies of self.weight.ravel().

write(file_name, *, file_type=None, precision=None, write_patch_results=False, write_cov=False)[source]

Write the correlation function to the file, file_name.

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 \(\langle u\rangle\) of triangles that fell into each bin

meanv

The mean value \(\langle v\rangle\) 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 \(\langle phi\rangle\) 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 \(\langle d1\rangle\) of triangles that fell into each bin

meanlogd1

The mean value \(\langle \log(d1)\rangle\) of triangles that fell into each bin

meand2

The mean value \(\langle d2\rangle\) of triangles that fell into each bin

meanlogd2

The mean value \(\langle \log(d2)\rangle\) of triangles that fell into each bin

meand3

The mean value \(\langle d3\rangle\) of triangles that fell into each bin

meanlogd3

The mean value \(\langle \log(d3)\rangle\) of triangles that fell into each bin

gam0r

The real part of the estimator of \(\Gamma_0\)

gam0i

The imag part of the estimator of \(\Gamma_0\)

gam1r

The real part of the estimator of \(\Gamma_1\)

gam1i

The imag part of the estimator of \(\Gamma_1\)

gam2r

The real part of the estimator of \(\Gamma_2\)

gam2i

The imag part of the estimator of \(\Gamma_2\)

gam3r

The real part of the estimator of \(\Gamma_3\)

gam3i

The imag part of the estimator of \(\Gamma_3\)

sigma_gam0

The sqrt of the variance estimate of \(\Gamma_0\)

sigma_gam1

The sqrt of the variance estimate of \(\Gamma_1\)

sigma_gam2

The sqrt of the variance estimate of \(\Gamma_2\)

sigma_gam3

The sqrt of the variance estimate of \(\Gamma_3\)

weight

The total weight of triangles contributing to each bin. (For LogMultipole, this is split into real and imaginary parts, weightr and weighti.)

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)

writeMap3(file_name, *, R=None, file_type=None, precision=None)[source]

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 \(\langle M_{ap}^3\rangle(R)\) (cf. calculateMap3)

Map2Mx

An estimate of \(\langle M_{ap}^2 M_\times\rangle(R)\)

MapMx2

An estimate of \(\langle M_{ap} M_\times^2\rangle(R)\)

Mx3

An estimate of \(\langle M_\times^3\rangle(R)\)

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.)