nifty_rg.py 99.2 KB
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# NIFTY (Numerical Information Field Theory) has been developed at the
# Max-Planck-Institute for Astrophysics.
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##
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# Copyright (C) 2015 Max-Planck-Society
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##
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# Author: Marco Selig
# Project homepage: <http://www.mpa-garching.mpg.de/ift/nifty/>
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##
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# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
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##
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# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
# See the GNU General Public License for more details.
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##
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# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
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"""
    ..                  __   ____   __
    ..                /__/ /   _/ /  /_
    ..      __ ___    __  /  /_  /   _/  __   __
    ..    /   _   | /  / /   _/ /  /   /  / /  /
    ..   /  / /  / /  / /  /   /  /_  /  /_/  /
    ..  /__/ /__/ /__/ /__/    \___/  \___   /  rg
    ..                               /______/

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    NIFTY submodule for regular Cartesian grids.
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"""
from __future__ import division
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import os
import numpy as np
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from scipy.special import erf
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import pylab as pl
from matplotlib.colors import LogNorm as ln
from matplotlib.ticker import LogFormatter as lf
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from nifty.nifty_core import point_space,\
                             field
import nifty_fft
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from nifty.keepers import about,\
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                          global_dependency_injector as gdi,\
                          global_configuration as gc
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from nifty.nifty_mpi_data import distributed_data_object
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from nifty.nifty_mpi_data import STRATEGIES as DISTRIBUTION_STRATEGIES
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from nifty.nifty_paradict import rg_space_paradict
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from nifty.nifty_power_indices import rg_power_indices
from nifty.nifty_random import random
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import nifty.nifty_utilities as utilities
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MPI = gdi[gc['mpi_module']]
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RG_DISTRIBUTION_STRATEGIES = DISTRIBUTION_STRATEGIES['global']
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class rg_space(point_space):
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    """
        ..      _____   _______
        ..    /   __/ /   _   /
        ..   /  /    /  /_/  /
        ..  /__/     \____  /  space class
        ..          /______/

        NIFTY subclass for spaces of regular Cartesian grids.

        Parameters
        ----------
        num : {int, numpy.ndarray}
            Number of gridpoints or numbers of gridpoints along each axis.
        naxes : int, *optional*
            Number of axes (default: None).
        zerocenter : {bool, numpy.ndarray}, *optional*
            Whether the Fourier zero-mode is located in the center of the grid
            (or the center of each axis speparately) or not (default: True).
        hermitian : bool, *optional*
            Whether the fields living in the space follow hermitian symmetry or
            not (default: True).
        purelyreal : bool, *optional*
            Whether the field values are purely real (default: True).
        dist : {float, numpy.ndarray}, *optional*
            Distance between two grid points along each axis (default: None).
        fourier : bool, *optional*
            Whether the space represents a Fourier or a position grid
            (default: False).

        Notes
        -----
        Only even numbers of grid points per axis are supported.
        The basis transformations between position `x` and Fourier mode `k`
        rely on (inverse) fast Fourier transformations using the
        :math:`exp(2 \pi i k^\dagger x)`-formulation.

        Attributes
        ----------
        para : numpy.ndarray
            One-dimensional array containing information on the axes of the
            space in the following form: The first entries give the grid-points
            along each axis in reverse order; the next entry is 0 if the
            fields defined on the space are purely real-valued, 1 if they are
            hermitian and complex, and 2 if they are not hermitian, but
            complex-valued; the last entries hold the information on whether
            the axes are centered on zero or not, containing a one for each
            zero-centered axis and a zero for each other one, in reverse order.
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        dtype : numpy.dtype
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            Data type of the field values for a field defined on this space,
            either ``numpy.float64`` or ``numpy.complex128``.
        discrete : bool
            Whether or not the underlying space is discrete, always ``False``
            for regular grids.
        vol : numpy.ndarray
            One-dimensional array containing the distances between two grid
            points along each axis, in reverse order. By default, the total
            length of each axis is assumed to be one.
        fourier : bool
            Whether or not the grid represents a Fourier basis.
    """
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    epsilon = 0.0001  # relative precision for comparisons
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    def __init__(self, num, zerocenter=False, complexity=0, dist=None,
                 harmonic=False, datamodel='fftw', fft_module='pyfftw',
                 comm=gc['default_comm']):
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        """
            Sets the attributes for an rg_space class instance.

            Parameters
            ----------
            num : {int, numpy.ndarray}
                Number of gridpoints or numbers of gridpoints along each axis.
            naxes : int, *optional*
                Number of axes (default: None).
            zerocenter : {bool, numpy.ndarray}, *optional*
                Whether the Fourier zero-mode is located in the center of the
                grid (or the center of each axis speparately) or not
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                (default: False).
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            hermitian : bool, *optional*
                Whether the fields living in the space follow hermitian
                symmetry or not (default: True).
            purelyreal : bool, *optional*
                Whether the field values are purely real (default: True).
            dist : {float, numpy.ndarray}, *optional*
                Distance between two grid points along each axis
                (default: None).
            fourier : bool, *optional*
                Whether the space represents a Fourier or a position grid
                (default: False).

            Returns
            -------
            None
        """
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        self.paradict = rg_space_paradict(num=num,
                                          complexity=complexity,
                                          zerocenter=zerocenter)
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        # set dtype
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        if self.paradict['complexity'] == 0:
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            self.dtype = np.dtype('float64')
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        else:
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            self.dtype = np.dtype('complex128')
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        # set datamodel
        if datamodel not in ['np'] + RG_DISTRIBUTION_STRATEGIES:
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            about.warnings.cprint("WARNING: datamodel set to default.")
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            self.datamodel = \
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                gc['default_distribution_strategy']
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        else:
            self.datamodel = datamodel

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        # set volume/distances
        naxes = len(self.paradict['num'])
        if dist is None:
            dist = 1 / np.array(self.paradict['num'], dtype=np.float)
        elif np.isscalar(dist):
            dist = np.ones(naxes, dtype=np.float) * dist
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        else:
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            dist = np.array(dist, dtype=np.float)
            if np.size(dist) == 1:
                dist = dist * np.ones(naxes, dtype=np.float)
            if np.size(dist) != naxes:
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                raise ValueError(about._errors.cstring(
                    "ERROR: size mismatch ( " + str(np.size(dist)) + " <> " +
                    str(naxes) + " )."))
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        if np.any(dist <= 0):
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            raise ValueError(about._errors.cstring(
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                "ERROR: nonpositive distance(s)."))
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        self.distances = tuple(dist)
        self.harmonic = bool(harmonic)
        self.discrete = False

        self.comm = self._parse_comm(comm)
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        # Initializes the fast-fourier-transform machine, which will be used
        # to transform the space
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        if not gc.validQ('fft_module', fft_module):
            fft_module = gc['fft_module']
        self.fft_machine = nifty_fft.fft_factory(fft_module)
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        # Initialize the power_indices object which takes care of kindex,
        # pindex, rho and the pundex for a given set of parameters
        if self.harmonic:
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            self.power_indices = rg_power_indices(
                    shape=self.get_shape(),
                    dgrid=dist,
                    zerocentered=self.paradict['zerocenter'],
                    comm=self.comm,
                    datamodel=self.datamodel,
                    allowed_distribution_strategies=RG_DISTRIBUTION_STRATEGIES)
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    @property
    def para(self):
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        temp = np.array(self.paradict['num'] +
                        [self.paradict['complexity']] +
                        self.paradict['zerocenter'], dtype=int)
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        return temp
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    @para.setter
    def para(self, x):
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        self.paradict['num'] = x[:(np.size(x) - 1) // 2]
        self.paradict['zerocenter'] = x[(np.size(x) + 1) // 2:]
        self.paradict['complexity'] = x[(np.size(x) - 1) // 2]
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    # __identiftier__ returns an object which contains all information needed
    # to uniquely identify a space. It returns a (immutable) tuple which
    # therefore can be compared.
    # The rg_space version of __identifier__ filters out the vars-information
    # which is describing the rg_space's structure
    def _identifier(self):
        # Extract the identifying parts from the vars(self) dict.
        temp = [(ii[0],
                 ((lambda x: tuple(x) if
                  isinstance(x, np.ndarray) else x)(ii[1])))
                for ii in vars(self).iteritems()
                if ii[0] not in ['fft_machine', 'power_indices', 'comm']]
        temp.append(('comm', self.comm.__hash__()))
        # Return the sorted identifiers as a tuple.
        return tuple(sorted(temp))
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    def copy(self):
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        return rg_space(num=self.paradict['num'],
                        complexity=self.paradict['complexity'],
                        zerocenter=self.paradict['zerocenter'],
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                        dist=self.distances,
                        harmonic=self.harmonic,
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                        datamodel=self.datamodel)
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    def get_shape(self):
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        return tuple(self.paradict['num'])
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    def _cast_to_d2o(self, x, dtype=None, hermitianize=True, **kwargs):
        casted_x = super(rg_space, self)._cast_to_d2o(x=x,
                                                      dtype=dtype,
                                                      **kwargs)
        if hermitianize and self.paradict['complexity'] == 1 and \
           not casted_x.hermitian:
            about.warnings.cflush(
                 "WARNING: Data gets hermitianized. This operation is " +
                 "extremely expensive\n")
            casted_x = utilities.hermitianize(casted_x)
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        return casted_x
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    def _cast_to_np(self, x, dtype=None, hermitianize=True, **kwargs):
        casted_x = super(rg_space, self)._cast_to_np(x=x,
                                                     dtype=dtype,
                                                     **kwargs)
        if hermitianize and self.paradict['complexity'] == 1:
            about.warnings.cflush(
                 "WARNING: Data gets hermitianized. This operation is " +
                 "extremely expensive\n")
            casted_x = utilities.hermitianize(casted_x)
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        return casted_x
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    def enforce_power(self, spec, size=None, kindex=None, codomain=None,
                      log=False, nbin=None, binbounds=None):
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        """
            Provides a valid power spectrum array from a given object.

            Parameters
            ----------
            spec : {float, list, numpy.ndarray, nifty.field, function}
                Fiducial power spectrum from which a valid power spectrum is to
                be calculated. Scalars are interpreted as constant power
                spectra.

            Returns
            -------
            spec : numpy.ndarray
                Valid power spectrum.

            Other parameters
            ----------------
            size : int, *optional*
                Number of bands the power spectrum shall have (default: None).
            kindex : numpy.ndarray, *optional*
                Scale of each band.
            codomain : nifty.space, *optional*
                A compatible codomain for power indexing (default: None).
            log : bool, *optional*
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                Flag specifying if the spectral binning is performed on
                logarithmic scale or not; if set, the number of used bins is
                set automatically (if not given otherwise); by default no
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                binning is done (default: None).
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            nbin : integer, *optional*
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                Number of used spectral bins; if given `log` is set to
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                ``False``; iintegers below the minimum of 3 induce an automatic
                setting; by default no binning is done (default: None).
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            binbounds : {list, array}, *optional*
                User specific inner boundaries of the bins, which are preferred
                over the above parameters; by default no binning is done
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                (default: None).
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        """
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        # Setting up the local variables: kindex
        # The kindex is only necessary if spec is a function or if
        # the size is not set explicitly
        if kindex is None and (size is None or callable(spec)):
            # Determine which space should be used to get the kindex
            if self.harmonic:
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                kindex_supply_space = self
            else:
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                # Check if the given codomain is compatible with the space
                try:
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                    assert(self.check_codomain(codomain))
                    kindex_supply_space = codomain
                except(AssertionError):
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                    about.warnings.cprint("WARNING: Supplied codomain is " +
                                          "incompatible. Generating a " +
                                          "generic codomain. This can " +
                                          "be expensive!")
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                    kindex_supply_space = self.get_codomain()
            kindex = kindex_supply_space.\
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                power_indices.get_index_dict(log=log, nbin=nbin,
                                             binbounds=binbounds)['kindex']

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        return self._enforce_power_helper(spec=spec,
                                          size=size,
                                          kindex=kindex)
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    def check_codomain(self, codomain):
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        """
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            Checks whether a given codomain is compatible to the space or not.
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            Parameters
            ----------
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            codomain : nifty.space
                Space to be checked for compatibility.
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            Returns
            -------
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            check : bool
                Whether or not the given codomain is compatible to the space.
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        """
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        if codomain is None:
            return False
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        if not isinstance(codomain, rg_space):
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            raise TypeError(about._errors.cstring(
                "ERROR: The given codomain must be a nifty rg_space."))
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        if self.datamodel is not codomain.datamodel:
            return False
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        # check number of number and size of axes
        if not np.all(np.array(self.paradict['num']) ==
                      np.array(codomain.paradict['num'])):
            return False
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        # check harmonic flag
        if self.harmonic == codomain.harmonic:
            return False
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        # check complexity-type
        # prepare the shorthands
        dcomp = self.paradict['complexity']
        cocomp = codomain.paradict['complexity']
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        # Case 1: if the domain is copmleteley complex
        # -> the codomain must be complex, too
        if dcomp == 2:
            if cocomp != 2:
                return False
        # Case 2: domain is hermitian
        # -> codmomain can be real. If it is marked as hermitian or even
        # fully complex, a warning is raised
        elif dcomp == 1:
            if cocomp > 0:
                about.warnings.cprint("WARNING: Unrecommended codomain! " +
                                      "The domain is hermitian, hence the " +
                                      "codomain should be restricted to " +
                                      "real values!")
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        # Case 3: domain is real
        # -> codmain should be hermitian
        elif dcomp == 0:
            if cocomp == 2:
                about.warnings.cprint("WARNING: Unrecommended codomain! " +
                                      "The domain is real, hence the " +
                                      "codomain should be restricted to " +
                                      "hermitian configurations!")
            elif cocomp == 0:
                return False
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        # Check if the distances match, i.e. dist'=1/(num*dist)
        if not np.all(
                np.absolute(np.array(self.paradict['num']) *
                            np.array(self.distances) *
                            np.array(codomain.distances) - 1) < self.epsilon):
            return False
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        return True
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    def get_codomain(self, cozerocenter=None, **kwargs):
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        """
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            Generates a compatible codomain to which transformations are
            reasonable, i.e.\  either a shifted grid or a Fourier conjugate
            grid.
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            Parameters
            ----------
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            coname : string, *optional*
                String specifying a desired codomain (default: None).
            cozerocenter : {bool, numpy.ndarray}, *optional*
                Whether or not the grid is zerocentered for each axis or not
                (default: None).
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            Returns
            -------
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            codomain : nifty.rg_space
                A compatible codomain.
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            Notes
            -----
            Possible arguments for `coname` are ``'f'`` in which case the
            codomain arises from a Fourier transformation, ``'i'`` in which
            case it arises from an inverse Fourier transformation.If no
            `coname` is given, the Fourier conjugate grid is produced.
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        """
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        naxes = len(self.get_shape())
        # Parse the cozerocenter input
        if(cozerocenter is None):
            cozerocenter = self.paradict['zerocenter']
        # if the input is something scalar, cast it to a boolean
        elif(np.isscalar(cozerocenter)):
            cozerocenter = bool(cozerocenter)
        # if it is not a scalar...
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        else:
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            # ...cast it to a numpy array of booleans
            cozerocenter = np.array(cozerocenter, dtype=np.bool)
            # if it was a list of length 1, extract the boolean
            if(np.size(cozerocenter) == 1):
                cozerocenter = np.asscalar(cozerocenter)
            # if the length of the input does not match the number of
            # dimensions, raise an exception
            elif(np.size(cozerocenter) != naxes):
                raise ValueError(about._errors.cstring(
                    "ERROR: size mismatch ( " +
                    str(np.size(cozerocenter)) + " <> " + str(naxes) + " )."))
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        # Set up the initialization variables
        num = self.paradict['num']
        dist = 1 / (np.array(self.paradict['num']) * np.array(self.distances))
        datamodel = self.datamodel
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        complexity = {0: 1, 1: 0, 2: 2}[self.paradict['complexity']]
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        harmonic = bool(not self.harmonic)
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        new_space = rg_space(num,
                             zerocenter=cozerocenter,
                             complexity=complexity,
                             dist=dist,
                             harmonic=harmonic,
                             datamodel=datamodel)
        return new_space
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    def get_random_values(self, **kwargs):
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        """
            Generates random field values according to the specifications given
            by the parameters, taking into account possible complex-valuedness
            and hermitian symmetry.

            Returns
            -------
            x : numpy.ndarray
                Valid field values.

            Other parameters
            ----------------
            random : string, *optional*
                Specifies the probability distribution from which the random
                numbers are to be drawn.
                Supported distributions are:

                - "pm1" (uniform distribution over {+1,-1} or {+1,+i,-1,-i}
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                - "gau" (normal distribution with zero-mean and a given
                    standard
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                    deviation or variance)
                - "syn" (synthesizes from a given power spectrum)
                - "uni" (uniform distribution over [vmin,vmax[)

                (default: None).
            dev : float, *optional*
                Standard deviation (default: 1).
            var : float, *optional*
                Variance, overriding `dev` if both are specified
                (default: 1).
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            spec : {scalar, list, numpy.ndarray, nifty.field, function},
                *optional*
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                Power spectrum (default: 1).
            pindex : numpy.ndarray, *optional*
                Indexing array giving the power spectrum index of each band
                (default: None).
            kindex : numpy.ndarray, *optional*
                Scale of each band (default: None).
            codomain : nifty.rg_space, *optional*
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                A compatible codomain (default: None).
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            log : bool, *optional*
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                Flag specifying if the spectral binning is performed on
                    logarithmic
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                scale or not; if set, the number of used bins is set
                automatically (if not given otherwise); by default no binning
                is done (default: None).
            nbin : integer, *optional*
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                Number of used spectral bins; if given `log` is set to
                    ``False``;
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                integers below the minimum of 3 induce an automatic setting;
                by default no binning is done (default: None).
            binbounds : {list, array}, *optional*
                User specific inner boundaries of the bins, which are preferred
                over the above parameters; by default no binning is done
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                (default: None).
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            vmin : float, *optional*
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                Lower limit for a uniform distribution (default: 0).
            vmax : float, *optional*
                Upper limit for a uniform distribution (default: 1).
        """
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        # TODO: Without hermitianization the random-methods from pointspace
        # could be used.

        # Parse the keyword arguments
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        arg = random.parse_arguments(self, **kwargs)
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        # Prepare the empty distributed_data_object
        sample = distributed_data_object(global_shape=self.get_shape(),
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                                         dtype=self.dtype)
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        # Should the output be hermitianized? This does not depend on the
        # hermitianize boolean in about, as it would yield in wrong,
        # not recoverable results
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        hermitianizeQ = self.paradict['complexity']
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        # Case 1: uniform distribution over {-1,+1}/{1,i,-1,-i}
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        if arg[0] == 'pm1' and not hermitianizeQ:
            sample.apply_generator(lambda s: random.pm1(dtype=self.dtype,
                                                        shape=s))
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        elif arg[0] == 'pm1' and hermitianizeQ:
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            sample = self.get_random_values(random='uni', vmin=-1, vmax=1)
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            local_data = sample.get_local_data()
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            if issubclass(sample.dtype.type, np.complexfloating):
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                temp_data = local_data.copy()
                local_data[temp_data.real >= 0.5] = 1
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                local_data[(temp_data.real >= 0) * (temp_data.real < 0.5)] = -1
                local_data[(temp_data.real < 0) * (temp_data.imag >= 0)] = 1j
                local_data[(temp_data.real < 0) * (temp_data.imag < 0)] = -1j
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            else:
                local_data[local_data >= 0] = 1
                local_data[local_data < 0] = -1
            sample.set_local_data(local_data)
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        # Case 2: normal distribution with zero-mean and a given standard
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        #         deviation or variance
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        elif arg[0] == 'gau':
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            var = arg[3]
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            if np.isscalar(var) == True or var is None:
                processed_var = var
            else:
                try:
                    processed_var = sample.distributor.extract_local_data(var)
                except(AttributeError):
                    processed_var = var
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            sample.apply_generator(lambda s: random.gau(dtype=self.dtype,
                                                        shape=s,
                                                        mean=arg[1],
                                                        dev=arg[2],
                                                        var=processed_var))
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            if hermitianizeQ:
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                sample = utilities.hermitianize(sample)
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        # Case 3: uniform distribution
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        elif arg[0] == "uni" and not hermitianizeQ:
            sample.apply_generator(lambda s: random.uni(dtype=self.dtype,
                                                        shape=s,
                                                        vmin=arg[1],
                                                        vmax=arg[2]))

        elif arg[0] == "uni" and hermitianizeQ:
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            # For a hermitian uniform sample, generate a gaussian one
            # and then convert it to a uniform one
            sample = self.get_random_values(random='gau')
            # Use the cummulative of the gaussian, the error function in order
            # to transform it to a uniform distribution.
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            if issubclass(sample.dtype.type, np.complexfloating):
                def temp_func(x):
                    return erf(x.real) + 1j * erf(x.imag)
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            else:
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                def temp_func(x):
                    return erf(x / np.sqrt(2))
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            sample.apply_scalar_function(function=temp_func,
                                         inplace=True)

            # Shift and stretch the uniform distribution into the given limits
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            # sample = (sample + 1)/2 * (vmax-vmin) + vmin
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            vmin = arg[1]
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            vmax = arg[2]
            sample *= (vmax - vmin) / 2.
            sample += 1 / 2. * (vmax + vmin)
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        elif(arg[0] == "syn"):
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            spec = arg[1]
            kpack = arg[2]
            harmonic_domain = arg[3]
            log = arg[4]
            nbin = arg[5]
            binbounds = arg[6]
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            # Check whether there is a kpack available or not.
            # kpack is only used for computing kdict and extracting kindex
            # If not, take kdict and kindex from the fourier_domain
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            if kpack is None:
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                power_indices =\
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                    harmonic_domain.power_indices.get_index_dict(
                                                        log=log,
                                                        nbin=nbin,
                                                        binbounds=binbounds)
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                kindex = power_indices['kindex']
                kdict = power_indices['kdict']
                kpack = [power_indices['pindex'], power_indices['kindex']]
            else:
                kindex = kpack[1]
                kdict = harmonic_domain.power_indices.\
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                    _compute_kdict_from_pindex_kindex(kpack[0], kpack[1])

            # draw the random samples
            # Case 1: self is a harmonic space
            if self.harmonic:
                # subcase 1: self is real
                # -> simply generate a random field in fourier space and
                # weight the entries accordingly to the powerspectrum
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                if self.paradict['complexity'] == 0:
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                    # set up the sample object. Overwrite the default from
                    # above to be sure, that the distribution strategy matches
                    # with the one from kdict
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                    sample = kdict.copy_empty(dtype=self.dtype)
                    # set up and apply the random number generator
                    sample.apply_generator(lambda s: np.random.normal(
                                                       loc=0, scale=1, size=s))
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                # subcase 2: self is hermitian but probably complex
                # -> generate a real field (in position space) and transform
                # it to harmonic space -> field in harmonic space is
                # hermitian. Now weight the modes accordingly to the
                # powerspectrum.
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                elif self.paradict['complexity'] == 1:
                    temp_codomain = self.get_codomain()
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                    # set up the sample object. Overwrite the default from
                    # above to be sure, that the distribution strategy matches
                    # with the one from kdict
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                    sample = kdict.copy_empty(
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                        dtype=temp_codomain.dtype)
                    # set up and apply the random number generator
                    sample.apply_generator(lambda s: np.random.normal(
                                                       loc=0, scale=1, size=s))
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                    # In order to get the normalisation right, the sqrt
                    # of self.dim must be divided out.
                    # Furthermore, the normalisation in the fft routine
                    # must be undone
                    # TODO: Insert explanation
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                    sqrt_of_dim = np.sqrt(self.get_dim())
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                    sample /= sqrt_of_dim
                    sample = temp_codomain.calc_weight(sample, power=-1)

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                    # tronsform the random field to harmonic space
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                    sample = temp_codomain.\
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                        calc_transform(sample, codomain=self)

                    # ensure that the kdict and the harmonic_sample have the
                    # same distribution strategy
                    assert(kdict.distribution_strategy ==
                           sample.distribution_strategy)

                # subcase 3: self is fully complex
                # -> generate a complex random field in harmonic space and
                # weight the modes accordingly to the powerspectrum
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                elif self.paradict['complexity'] == 2:
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                    # set up the sample object. Overwrite the default from
                    # above to be sure, that the distribution strategy matches
                    # with the one from kdict
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                    sample = kdict.copy_empty(dtype=self.dtype)

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                    # set up the random number generator
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                    def temp_gen(s):
                        result = (np.random.normal(loc=0,
                                                   scale=1 / np.sqrt(2),
                                                   size=s) +
                                  np.random.normal(loc=0,
                                                   scale=1 / np.sqrt(2),
                                                   size=s) * 1.j)
                        return result
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                    # apply the random number generator
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                    sample.apply_generator(temp_gen)
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                # apply the powerspectrum renormalization
                # therefore extract the local data from kdict
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                local_kdict = kdict.get_local_data()
                rescaler = np.sqrt(
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                    spec[np.searchsorted(kindex, local_kdict)])
                sample.apply_scalar_function(lambda x: x * rescaler,
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                                             inplace=True)
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            # Case 2: self is a position space
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            else:
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                # get a suitable codomain
                temp_codomain = self.get_codomain()
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                # subcase 1: self is a real space.
                # -> generate a hermitian sample with the codomain in harmonic
                # space and make a fourier transformation.
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                if self.paradict['complexity'] == 0:
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                    # check that the codomain is hermitian
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                    assert(temp_codomain.paradict['complexity'] == 1)
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                # subcase 2: self is hermitian but probably complex
                # -> generate a real-valued random sample in fourier space
                # and transform it to real space
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                elif self.paradict['complexity'] == 1:
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                    # check that the codomain is real
                    assert(temp_codomain.paradict['complexity'] == 0)
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                # subcase 3: self is fully complex
                # -> generate a complex-valued random sample in fourier space
                # and transform it to real space
                elif self.paradict['complexity'] == 2:
                    # check that the codomain is real
                    assert(temp_codomain.paradict['complexity'] == 2)
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                # Get a hermitian/real/complex sample in harmonic space from
                # the codomain
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                sample = temp_codomain.get_random_values(
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                    random='syn',
                    pindex=kpack[0],
                    kindex=kpack[1],
                    spec=spec,
                    codomain=self,
                    log=log,
                    nbin=nbin,
                    binbounds=binbounds
                )

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                # Perform a fourier transform
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                sample = temp_codomain.calc_transform(sample,
                                                      codomain=self)
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            if self.paradict['complexity'] == 1:
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                sample.hermitian = True

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        else:
            raise KeyError(about._errors.cstring(
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                "ERROR: unsupported random key '" + str(arg[0]) + "'."))
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        return sample
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    def calc_weight(self, x, power=1):
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        """
            Weights a given array with the pixel volumes to a given power.

            Parameters
            ----------
            x : numpy.ndarray
                Array to be weighted.
            power : float, *optional*
                Power of the pixel volumes to be used (default: 1).

            Returns
            -------
            y : numpy.ndarray
                Weighted array.
        """
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        # weight
        x = x * self.get_weight(power=power)
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        return x
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    def get_weight(self, power=1):
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        return np.prod(self.distances)**power
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    def calc_dot(self, x, y):
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        """
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            Computes the discrete inner product of two given arrays of field
            values.
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            Parameters
            ----------
            x : numpy.ndarray
                First array
            y : numpy.ndarray
                Second array

            Returns
            -------
            dot : scalar
                Inner product of the two arrays.
        """
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        x = self.cast(x)
        y = self.cast(y)
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        if self.datamodel == 'np':
            result = np.vdot(x, y)
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        elif self.datamodel in RG_DISTRIBUTION_STRATEGIES:
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            result = x.vdot(y)
        else:
            raise NotImplementedError(about._errors.cstring(
                "ERROR: function is not implemented for given datamodel."))

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        if np.isreal(result):
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            result = np.asscalar(np.real(result))
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        if self.paradict['complexity'] != 2:
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            if (np.absolute(result.imag) >
                    self.epsilon**2 * np.absolute(result.real)):
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                about.warnings.cprint(
                    "WARNING: Discarding considerable imaginary part.")
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            result = np.asscalar(np.real(result))
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        return result
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    def calc_transform(self, x, codomain=None, **kwargs):
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        """
            Computes the transform of a given array of field values.

            Parameters
            ----------
            x : numpy.ndarray
                Array to be transformed.
            codomain : nifty.rg_space, *optional*
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                codomain space to which the transformation shall map
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                (default: None).

            Returns
            -------
            Tx : numpy.ndarray
                Transformed array
        """
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        x = self.cast(x)
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        if codomain is None:
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            codomain = self.get_codomain()
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        # Check if the given codomain is suitable for the transformation
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        if not self.check_codomain(codomain):
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            raise ValueError(about._errors.cstring(
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                "ERROR: unsupported codomain."))
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        if codomain.harmonic:
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            # correct for forward fft
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            x = self.calc_weight(x, power=1)
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        # Perform the transformation
        Tx = self.fft_machine.transform(val=x, domain=self, codomain=codomain,
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                                        **kwargs)

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        if not codomain.harmonic:
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            # correct for inverse fft
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            Tx = codomain.calc_weight(Tx, power=-1)

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        # when the codomain space is purely real, the result of the
        # transformation must be corrected accordingly. Using the casting
        # method of codomain is sufficient
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        # TODO: Let .transform  yield the correct dtype
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        Tx = codomain.cast(Tx)
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        return Tx

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    def calc_smooth(self, x, sigma=0, codomain=None):
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        """
            Smoothes an array of field values by convolution with a Gaussian
            kernel.

            Parameters
            ----------
            x : numpy.ndarray
                Array of field values to be smoothed.
            sigma : float, *optional*
                Standard deviation of the Gaussian kernel, specified in units
                of length in position space; for testing: a sigma of -1 will be
                reset to a reasonable value (default: 0).

            Returns
            -------
            Gx : numpy.ndarray
                Smoothed array.
        """

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        # Check sigma
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        if sigma == 0:
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            return x
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        elif sigma == -1:
            about.infos.cprint(
                "INFO: Resetting sigma to sqrt(2)*max(dist).")
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            sigma = np.sqrt(2) * np.max(self.distances)
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        elif(sigma < 0):
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            raise ValueError(about._errors.cstring("ERROR: invalid sigma."))
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        # if a codomain was given...
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        if codomain is not None:
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            # ...check if it was suitable
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            if not isinstance(codomain, rg_space):
                raise ValueError(about._errors.cstring(
                    "ERROR: codomain is not a rg_space instance!"))
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            if not self.harmonic and not codomain.harmonic:
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                raise ValueError(about._errors.cstring(
                    "ERROR: fourier_domain is not a fourier space!"))
            if not self.check_codomain(codomain):
                raise ValueError(about._errors.cstring(
                    "ERROR: fourier_codomain is not a valid codomain!"))
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        elif not self.harmonic:
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            codomain = self.get_codomain()

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        # Case1:
        # If self is a position-space, fourier transform the input and
        # call calc_smooth of the fourier codomain
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        if not self.harmonic:
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            x = self.calc_transform(x, codomain=codomain)
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            x = codomain.calc_smooth(x, sigma)
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            x = codomain.calc_transform(x, codomain=self)
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            return x
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        # Case 2:
        # if self is fourier multiply the gaussian kernel, etc...

        # Cast the input
        x = self.cast(x)

        # if x is hermitian it remains hermitian during smoothing
        if self.datamodel in RG_DISTRIBUTION_STRATEGIES:
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            remeber_hermitianQ = x.hermitian
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        # Define the Gaussian kernel function
        gaussian = lambda x: np.exp(-2. * np.pi**2 * x**2 * sigma**2)

        # Define the variables in the dialect of the legacy smoothing.py
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        nx = np.array(self.get_shape())
        dx = 1 / nx / self.distances
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        # Multiply the data along each axis with suitable the gaussian kernel
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        for i in range(len(nx)):
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            # Prepare the exponent
            dk = 1. / nx[i] / dx[i]
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            nk = nx[i]
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            k = -0.5 * nk * dk + np.arange(nk) * dk
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            if self.paradict['zerocenter'][i] == False:
                k = np.fft.fftshift(k)
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            # compute the actual kernel vector
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            gaussian_kernel_vector = gaussian(k)
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            # blow up the vector to an array of shape (1,.,1,len(nk),1,.,1)
            blown_up_shape = [1, ] * len(nx)
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            blown_up_shape[i] = len(gaussian_kernel_vector)
            gaussian_kernel_vector =\
                gaussian_kernel_vector.reshape(blown_up_shape)
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            # apply the blown-up gaussian_kernel_vector
            x = x*gaussian_kernel_vector
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        if self.datamodel in RG_DISTRIBUTION_STRATEGIES:
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            x.hermitian = remeber_hermitianQ
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        return x
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    def calc_power(self, x, **kwargs):
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        """
            Computes the power of an array of field values.

            Parameters
            ----------
            x : numpy.ndarray
                Array containing the field values of which the power is to be
                calculated.

            Returns
            -------
            spec : numpy.ndarray
                Power contained in the input array.

            Other parameters
            ----------------
            pindex : numpy.ndarray, *optional*
                Indexing array assigning the input array components to
                components of the power spectrum (default: None).
            rho : numpy.ndarray, *optional*
                Number of degrees of freedom per band (default: None).
            codomain : nifty.space, *optional*
                A compatible codomain for power indexing (default: None).
            log : bool, *optional*
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                Flag specifying if the spectral binning is performed on
                logarithmic
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                scale or not; if set, the number of used bins is set
                automatically (if not given otherwise); by default no binning
                is done (default: None).
            nbin : integer, *optional*
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                Number of used spectral bins; if given `log` is set to
                ``False``;
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                integers below the minimum of 3 induce an automatic setting;
                by default no binning is done (default: None).
            binbounds : {list, array}, *optional*
                User specific inner boundaries of the bins, which are preferred
                over the above parameters; by default no binning is done
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                (default: None).
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        """
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        x = self.cast(x)

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        # If self is a position space, delegate calc_power to its codomain.
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        if not self.harmonic:
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            try:
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                codomain = kwargs['codomain']
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            except(KeyError):
                codomain = self.get_codomain()
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            y = self.calc_transform(x, codomain)
            kwargs.update({'codomain': self})
            return codomain.calc_power(y, **kwargs)
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        # If some of the pindex, kindex or rho arrays are given explicitly,
        # favor them over those from the self.power_indices dictionary.
        # As the default value in kwargs.get(key, default) does NOT evaluate
        # lazy, a distinction of cases is necessary. Otherwise the
        # powerindices might be computed, although not necessary
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        if 'pindex' in kwargs and 'kindex' in kwargs and 'rho' in kwargs:
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            pindex = kwargs.get('pindex')
            rho = kwargs.get('rho')
        else:
            log = kwargs.get('log', None)
            nbin = kwargs.get('nbin', None)
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            binbounds = kwargs.get('binbounds', None)
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            power_indices = self.power_indices.get_index_dict(
                                                        log=log,
                                                        nbin=nbin,
                                                        binbounds=binbounds)
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            pindex = kwargs.get('pindex', power_indices['pindex'])
            rho = kwargs.get('rho', power_indices['rho'])
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        fieldabs = abs(x)**2
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        power_spectrum = np.zeros(rho.shape)
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        if self.datamodel == 'np':
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            power_spectrum = np.bincount(pindex.flatten(),
                                         weights=fieldabs.flatten())
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        elif self.datamodel in RG_DISTRIBUTION_STRATEGIES:
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            power_spectrum = pindex.bincount(weights=fieldabs)
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        else:
            raise NotImplementedError(about._errors.cstring(
                "ERROR: function is not implemented for given datamodel."))
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        # Divide out the degeneracy factor
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        power_spectrum /= rho
        return power_spectrum
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    def get_plot(self,x,title="",vmin=None,vmax=None,power=None,unit="",norm=None,cmap=None,cbar=True,other=None,legend=False,mono=True,**kwargs):
        """
            Creates a plot of field values according to the specifications
            given by the parameters.

            Parameters
            ----------
            x : numpy.ndarray
                Array containing the field values.

            Returns
            -------
            None

            Other parameters
            ----------------
            title : string, *optional*
                Title of the plot (default: "").
            vmin : float, *optional*
                Minimum value to be displayed (default: ``min(x)``).
            vmax : float, *optional*
                Maximum value to be displayed (default: ``max(x)``).
            power : bool, *optional*
                Whether to plot the power contained in the field or the field
                values themselves (default: False).
            unit : string, *optional*
                Unit of the field values (default: "").
            norm : string, *optional*
                Scaling of the field values before plotting (default: None).
            cmap : matplotlib.colors.LinearSegmentedColormap, *optional*
                Color map to be used for two-dimensional plots (default: None).
            cbar : bool, *optional*
                Whether to show the color bar or not (default: True).
            other : {single object, tuple of objects}, *optional*
                Object or tuple of objects to be added, where objects can be
                scalars, arrays, or fields (default: None).
            legend : bool, *optional*
                Whether to show the legend or not (default: False).
            mono : bool, *optional*
                Whether to plot the monopole or not (default: True).
            save : string, *optional*
                Valid file name where the figure is to be stored, by default
                the figure is not saved (default: False).
            error : {float, numpy.ndarray, nifty.field}, *optional*
                Object indicating some confidence interval to be plotted
                (default: None).
            kindex : numpy.ndarray, *optional*
                Scale corresponding to each band in the power spectrum
                (default: None).
            codomain : nifty.space, *optional*
                A compatible codomain for power indexing (default: None).
            log : bool, *optional*
                Flag specifying if the spectral binning is performed on logarithmic
                scale or not; if set, the number of used bins is set
                automatically (if not given otherwise); by default no binning
                is done (default: None).
            nbin : integer, *optional*
                Number of used spectral bins; if given `log` is set to ``False``;
                integers below the minimum of 3 induce an automatic setting;
                by default no binning is done (default: None).
            binbounds : {list, array}, *optional*
                User specific inner boundaries of the bins, which are preferred
                over the above parameters; by default no binning is done
                (default: None).            vmin : {scalar, list, ndarray, field}, *optional*
                Lower limit of the uniform distribution if ``random == "uni"``
                (default: 0).

        """
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        try:
            x = x.get_full_data()
        except AttributeError:
            pass
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