def FieldType(ns):
"""
Construct and return a `Field`:
a tuple of (`DataSource`, `Painter`, `Transfer`)
Notes
-----
* the default Painter is set to `DefaultPainter`
* the default Transfer chain is set to
[`NormalizeDC`, `RemoveDC`, `AnisotropicCIC`]
Parameters
----------
fields_dict : OrderedDict
an ordered dictionary where the keys are Plugin names
and the values are instantiated Plugins
Returns
-------
field : list
list of (DataSource, Painter, Transfer)
"""
# define the default Painter and Transfer
default_painter = Painter.create("DefaultPainter")
default_transfer = [Transfer.create(x) for x in ['NormalizeDC', 'RemoveDC', 'AnisotropicCIC']]
# start with a default option for (DataSource, Painter, Transfer)
field = [None, default_painter, default_transfer]
# set the DataSource
if 'DataSource' not in ns:
raise ValueError("exactly one `DataSource` per field must be specified")
field[0] = getattr(ns, 'DataSource')
# set the Painter
if 'Painter' in ns:
field[1] = getattr(ns, 'Painter')
# set the Transfer
if 'Transfer' in ns:
field[2] = getattr(ns, 'Transfer')
return field
def compute_bianchi_poles(comm,
max_ell,
catalog,
Nmesh,
factor_hexadecapole=False,
paintbrush='cic'):
"""
Use the algorithm detailed in Bianchi et al. 2015 to compute and return the 3D
power spectrum multipoles (`ell = [0, 2, 4]`) from one input field, which contains
non-trivial survey geometry.
The estimator uses the FFT algorithm outlined in Bianchi et al. 2015
(http://adsabs.harvard.edu/abs/2015arXiv150505341B) to compute
the monopole, quadrupole, and hexadecapole
Parameters
----------
comm : MPI.Communicator
the communicator to pass to the ParticleMesh object
max_ell : int, {0, 2, 4}
the maximum multipole number to compute up to (inclusive)
catalog : :class:`~nbodykit.fkp.FKPCatalog`
the FKP catalog object that manipulates the data and randoms DataSource
objects and paints the FKP density
Nmesh : int
the number of cells (per side) in the gridded mesh
factor_hexadecapole : bool, optional
if `True`, use the factored expression for the hexadecapole (ell=4) from
eq. 27 of Scoccimarro 2015 (1506.02729); default is `False`
paintbrush : str, {'cic', 'tsc'}
the density assignment kernel to use when painting, either `cic` (2nd order)
or `tsc` (3rd order); default is `cic`
Returns
-------
pm : ParticleMesh
the mesh object used to do painting, FFTs, etc
result : list of arrays
list of 3D complex arrays holding power spectrum multipoles; respectively,
if `ell_max=0,2,4`, the list holds the monopole only, monopole and quadrupole,
or the monopole, quadrupole, and hexadecapole
stats : dict
dict holding the statistics of the input fields, as returned
by the `FKPPainter` painter
References
----------
* Bianchi, Davide et al., `Measuring line-of-sight-dependent Fourier-space clustering using FFTs`,
MNRAS, 2015
* Scoccimarro, Roman, `Fast estimators for redshift-space clustering`, Phys. Review D, 2015
"""
from pmesh.pm import ParticleMesh, RealField
rank = comm.rank
bianchi_transfers = []
# the painting kernel transfer
if paintbrush == 'cic':
transfer = Transfer.create('AnisotropicCIC')
elif paintbrush == 'tsc':
transfer = Transfer.create('AnisotropicTSC')
else:
raise ValueError("valid `paintbrush` values are: ['cic', 'tsc']")
# determine which multipole values we are computing
if max_ell not in [0, 2, 4]:
raise ValueError(
"valid values for the maximum multipole number are [0, 2, 4]")
ells = numpy.arange(0, max_ell + 1, 2)
# determine kernels needed to compute quadrupole
if max_ell > 0:
# the (i,j) index values for each kernel
k2 = [(0, 0), (1, 1), (2, 2), (0, 1), (0, 2), (1, 2)]
# the amplitude of each kernel term
a2 = [1.] * 3 + [2.] * 3
bianchi_transfers.append((a2, k2))
# determine kernels needed to compute hexadecapole
if max_ell > 2 and not factor_hexadecapole:
# the (i,j,k) index values for each kernel
k4 = [(0, 0, 0), (1, 1, 1), (2, 2, 2), (0, 0, 1), (0, 0, 2), (1, 1, 0),
(1, 1, 2), (2, 2, 0), (2, 2, 1), (0, 1, 1), (0, 2, 2), (1, 2, 2),
(0, 1, 2), (1, 0, 2), (2, 0, 1)]
# the amplitude of each kernel term
a4 = [1.] * 3 + [4.] * 6 + [6.] * 3 + [12.] * 3
bianchi_transfers.append((a4, k4))
# load the data/randoms, setup boxsize, etc from the FKPCatalog
with catalog:
# the mean coordinate offset of the input data
offset = catalog.mean_coordinate_offset
# initialize the particle mesh
pm = ParticleMesh(BoxSize=catalog.BoxSize,
Nmesh=[Nmesh] * 3,
dtype='f4',
comm=comm)
# paint the FKP density field to the mesh (paints: data - randoms, essentially)
real, stats = catalog.paint(pm, paintbrush=paintbrush)
# save the painted density field for later
density = real.copy()
if rank == 0: logger.info('%s painting done' % paintbrush)
# FFT density field and apply the paintbrush window transfer kernel
complex = real.r2c()
transfer(pm, complex)
if rank == 0: logger.info('ell = 0 done; 1 r2c completed')
# monopole A0 is just the FFT of the FKP density field
volume = pm.BoxSize.prod()
A0 = complex[:] * volume # normalize with a factor of volume
# store the A0, A2, A4 arrays here
result = []
result.append(A0)
# the real-space grid points
# the grid is properly offset from [-L/2, L/2] to the original positions in space
# this is the grid used when applying Bianchi kernels
cell_size = pm.BoxSize / pm.Nmesh
xgrid = [(ri + 0.5) * cell_size[i] + offset[i]
for i, ri in enumerate(pm.r)]
# loop over the higher order multipoles (ell > 0)
start = time.time()
for iell, ell in enumerate(ells[1:]):
# temporary array to hold sum of all of the terms in Fourier space
Aell_sum = numpy.zeros_like(complex)
# loop over each kernel term for this multipole
for amp, integers in zip(*bianchi_transfers[iell]):
# reset the realspace mesh to the original FKP density
real[:] = density[:]
# apply the real-space Bianchi kernel
if rank == 0:
logger.debug("applying real-space Bianchi transfer for %s..." %
str(integers))
apply_bianchi_kernel(real, xgrid, *integers)
if rank == 0: logger.debug('...done')
# do the real-to-complex FFT
if rank == 0: logger.debug("performing r2c...")
real.r2c(out=complex)
if rank == 0: logger.debug('...done')
# apply the Fourier-space Bianchi kernel
if rank == 0:
logger.debug(
"applying Fourier-space Bianchi transfer for %s..." %
str(integers))
apply_bianchi_kernel(complex, pm.k, *integers)
if rank == 0: logger.debug('...done')
# and this contribution to the total sum
Aell_sum[:] += amp * complex[:] * volume
# apply the paintbrush window transfer function and save
transfer(pm, Aell_sum)
result.append(Aell_sum)
del Aell_sum # delete temp array since appending to list makes copy
# log the total number of FFTs computed for each ell
if rank == 0:
args = (ell, len(bianchi_transfers[iell][0]))
logger.info('ell = %d done; %s r2c completed' % args)
# density array no longer needed
del density
# summarize how long it took
stop = time.time()
if rank == 0:
logger.info("higher order multipoles computed in elapsed time %s" %
timer(start, stop))
if factor_hexadecapole:
logger.info("using factorized hexadecapole estimator for ell=4")
# proper normalization: same as equation 49 of Scoccimarro et al. 2015
norm = 1.0 / stats['A_ran']
# reuse memory for output arrays
P0 = result[0]
if max_ell > 0:
P2 = result[1]
if max_ell > 2:
P4 = numpy.empty_like(P2) if factor_hexadecapole else result[2]
# calculate the power spectrum multipoles, slab-by-slab to save memory
for islab in range(len(P0)):
# save arrays for reuse
P0_star = (P0[islab]).conj()
if max_ell > 0: P2_star = (P2[islab]).conj()
# hexadecapole
if max_ell > 2:
# see equation 8 of Bianchi et al. 2015
if not factor_hexadecapole:
P4[islab, ...] = norm * 9. / 8. * P0[islab] * (
35. * (P4[islab]).conj() - 30. * P2_star + 3. * P0_star)
# see equation 48 of Scoccimarro et al; 2015
else:
P4[islab, ...] = norm * 9. / 8. * (
35. * P2[islab] * P2_star + 3. * P0[islab] * P0_star -
5. / 3. *
(11. * P0[islab] * P2_star + 7. * P2[islab] * P0_star))
# quadrupole: equation 7 of Bianchi et al. 2015
if max_ell > 0:
P2[islab,
...] = norm * 5. / 2. * P0[islab] * (3. * P2_star - P0_star)
# monopole: equation 6 of Bianchi et al. 2015
P0[islab, ...] = norm * P0[islab] * P0_star
return pm, result, stats
def compute_bianchi_poles(comm, max_ell, catalog, Nmesh, factor_hexadecapole=False, paintbrush='cic'):
"""
Use the algorithm detailed in Bianchi et al. 2015 to compute and return the 3D
power spectrum multipoles (`ell = [0, 2, 4]`) from one input field, which contains
non-trivial survey geometry.
The estimator uses the FFT algorithm outlined in Bianchi et al. 2015
(http://adsabs.harvard.edu/abs/2015arXiv150505341B) to compute
the monopole, quadrupole, and hexadecapole
Parameters
----------
comm : MPI.Communicator
the communicator to pass to the ParticleMesh object
max_ell : int, {0, 2, 4}
the maximum multipole number to compute up to (inclusive)
catalog : :class:`~nbodykit.fkp.FKPCatalog`
the FKP catalog object that manipulates the data and randoms DataSource
objects and paints the FKP density
Nmesh : int
the number of cells (per side) in the gridded mesh
factor_hexadecapole : bool, optional
if `True`, use the factored expression for the hexadecapole (ell=4) from
eq. 27 of Scoccimarro 2015 (1506.02729); default is `False`
paintbrush : str, {'cic', 'tsc'}
the density assignment kernel to use when painting, either `cic` (2nd order)
or `tsc` (3rd order); default is `cic`
Returns
-------
pm : ParticleMesh
the mesh object used to do painting, FFTs, etc
result : list of arrays
list of 3D complex arrays holding power spectrum multipoles; respectively,
if `ell_max=0,2,4`, the list holds the monopole only, monopole and quadrupole,
or the monopole, quadrupole, and hexadecapole
stats : dict
dict holding the statistics of the input fields, as returned
by the `FKPPainter` painter
References
----------
* Bianchi, Davide et al., `Measuring line-of-sight-dependent Fourier-space clustering using FFTs`,
MNRAS, 2015
* Scoccimarro, Roman, `Fast estimators for redshift-space clustering`, Phys. Review D, 2015
"""
from pmesh.pm import ParticleMesh, RealField
rank = comm.rank
bianchi_transfers = []
# the painting kernel transfer
if paintbrush == 'cic':
transfer = Transfer.create('AnisotropicCIC')
elif paintbrush == 'tsc':
transfer = Transfer.create('AnisotropicTSC')
else:
raise ValueError("valid `paintbrush` values are: ['cic', 'tsc']")
# determine which multipole values we are computing
if max_ell not in [0, 2, 4]:
raise ValueError("valid values for the maximum multipole number are [0, 2, 4]")
ells = numpy.arange(0, max_ell+1, 2)
# determine kernels needed to compute quadrupole
if max_ell > 0:
# the (i,j) index values for each kernel
k2 = [(0, 0), (1, 1), (2, 2), (0, 1), (0, 2), (1, 2)]
# the amplitude of each kernel term
a2 = [1.]*3 + [2.]*3
bianchi_transfers.append((a2, k2))
# determine kernels needed to compute hexadecapole
if max_ell > 2 and not factor_hexadecapole:
# the (i,j,k) index values for each kernel
k4 = [(0, 0, 0), (1, 1, 1), (2, 2, 2), (0, 0, 1), (0, 0, 2),
(1, 1, 0), (1, 1, 2), (2, 2, 0), (2, 2, 1), (0, 1, 1),
(0, 2, 2), (1, 2, 2), (0, 1, 2), (1, 0, 2), (2, 0, 1)]
# the amplitude of each kernel term
a4 = [1.]*3 + [4.]*6 + [6.]*3 + [12.]*3
bianchi_transfers.append((a4, k4))
# load the data/randoms, setup boxsize, etc from the FKPCatalog
with catalog:
# the mean coordinate offset of the input data
offset = catalog.mean_coordinate_offset
# initialize the particle mesh
pm = ParticleMesh(BoxSize=catalog.BoxSize, Nmesh=[Nmesh]*3, dtype='f4', comm=comm)
# paint the FKP density field to the mesh (paints: data - randoms, essentially)
real, stats = catalog.paint(pm, paintbrush=paintbrush)
# save the painted density field for later
density = real.copy()
if rank == 0: logger.info('%s painting done' %paintbrush)
# FFT density field and apply the paintbrush window transfer kernel
complex = real.r2c()
transfer(pm, complex)
if rank == 0: logger.info('ell = 0 done; 1 r2c completed')
# monopole A0 is just the FFT of the FKP density field
volume = pm.BoxSize.prod()
A0 = complex[:]*volume # normalize with a factor of volume
# store the A0, A2, A4 arrays here
result = []
result.append(A0)
# the real-space grid points
# the grid is properly offset from [-L/2, L/2] to the original positions in space
# this is the grid used when applying Bianchi kernels
cell_size = pm.BoxSize / pm.Nmesh
xgrid = [(ri+0.5)*cell_size[i] + offset[i] for i, ri in enumerate(pm.r)]
# loop over the higher order multipoles (ell > 0)
start = time.time()
for iell, ell in enumerate(ells[1:]):
# temporary array to hold sum of all of the terms in Fourier space
Aell_sum = numpy.zeros_like(complex)
# loop over each kernel term for this multipole
for amp, integers in zip(*bianchi_transfers[iell]):
# reset the realspace mesh to the original FKP density
real[:] = density[:]
# apply the real-space Bianchi kernel
if rank == 0: logger.debug("applying real-space Bianchi transfer for %s..." %str(integers))
apply_bianchi_kernel(real, xgrid, *integers)
if rank == 0: logger.debug('...done')
# do the real-to-complex FFT
if rank == 0: logger.debug("performing r2c...")
real.r2c(out=complex)
if rank == 0: logger.debug('...done')
# apply the Fourier-space Bianchi kernel
if rank == 0: logger.debug("applying Fourier-space Bianchi transfer for %s..." %str(integers))
apply_bianchi_kernel(complex, pm.k, *integers)
if rank == 0: logger.debug('...done')
# and this contribution to the total sum
Aell_sum[:] += amp*complex[:]*volume
# apply the paintbrush window transfer function and save
transfer(pm, Aell_sum)
result.append(Aell_sum); del Aell_sum # delete temp array since appending to list makes copy
# log the total number of FFTs computed for each ell
if rank == 0:
args = (ell, len(bianchi_transfers[iell][0]))
logger.info('ell = %d done; %s r2c completed' %args)
# density array no longer needed
del density
# summarize how long it took
stop = time.time()
if rank == 0:
logger.info("higher order multipoles computed in elapsed time %s" %timer(start, stop))
if factor_hexadecapole:
logger.info("using factorized hexadecapole estimator for ell=4")
# proper normalization: same as equation 49 of Scoccimarro et al. 2015
norm = 1.0 / stats['A_ran']
# reuse memory for output arrays
P0 = result[0]
if max_ell > 0:
P2 = result[1]
if max_ell > 2:
P4 = numpy.empty_like(P2) if factor_hexadecapole else result[2]
# calculate the power spectrum multipoles, slab-by-slab to save memory
for islab in range(len(P0)):
# save arrays for reuse
P0_star = (P0[islab]).conj()
if max_ell > 0: P2_star = (P2[islab]).conj()
# hexadecapole
if max_ell > 2:
# see equation 8 of Bianchi et al. 2015
if not factor_hexadecapole:
P4[islab, ...] = norm * 9./8. * P0[islab] * (35.*(P4[islab]).conj() - 30.*P2_star + 3.*P0_star)
# see equation 48 of Scoccimarro et al; 2015
else:
P4[islab, ...] = norm * 9./8. * ( 35.*P2[islab]*P2_star + 3.*P0[islab]*P0_star - 5./3.*(11.*P0[islab]*P2_star + 7.*P2[islab]*P0_star) )
# quadrupole: equation 7 of Bianchi et al. 2015
if max_ell > 0:
P2[islab, ...] = norm * 5./2. * P0[islab] * (3.*P2_star - P0_star)
# monopole: equation 6 of Bianchi et al. 2015
P0[islab, ...] = norm * P0[islab] * P0_star
return pm, result, stats