"""Functions making use of spectral decompositions.
.. autosummary::
:nosignatures:
make_colored_noise
make_correlated_noise
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from __future__ import annotations
import logging
import warnings
from typing import Callable, Literal
import numpy as np
from .typing import NumberOrArray, NumericArray
try:
from pyfftw.interfaces.numpy_fft import irfftn as np_irfftn
from pyfftw.interfaces.numpy_fft import rfftn as np_rfftn
except ImportError:
from numpy.fft import irfftn as np_irfftn
from numpy.fft import rfftn as np_rfftn
_logger = logging.getLogger(__name__)
""":class:`logging.Logger`: Logger instance."""
def _make_isotropic_correlated_noise(
shape: tuple[int, ...],
corr_spectrum: Callable[[NumericArray], NumericArray],
*,
discretization: NumberOrArray = 1.0,
rng: np.random.Generator | None = None,
) -> Callable[[], NumericArray]:
r"""Return a function creating an array of random values with spatial correlations.
An ensemble average (calling the returned function) multiple times produces normal
distributed variables for each entry in the field. Different entries of the same
field have a correlation function defined by `corr_spectrum`.
Args:
shape (tuple of ints):
Number of supports points in each spatial dimension. The number of the list
defines the spatial dimension.
corr_spectrum (callable):
Implementation of the Fourier transform of the correlation function, i.e.,
the square root of its power spectrum. The arguments of the function are the
squared wave vectors, ensuring isotropy.
discretization (float or list of floats):
Discretization along each dimension. A uniform discretization in each
direction can be indicated by a single number.
rng (:class:`~numpy.random.Generator`):
Random number generator (default: :func:`~numpy.random.default_rng()`)
Returns:
callable: a function returning a random realization
"""
rng = np.random.default_rng(rng)
# extract some information about the grid
dim = len(shape)
dx_arr = np.broadcast_to(discretization, (dim,))
# prepare wave vectors
k2s = np.array(0)
for i in range(dim):
if i == dim - 1:
k = np.fft.rfftfreq(shape[i], dx_arr[i])
else:
k = np.fft.fftfreq(shape[i], dx_arr[i])
k2s = np.add.outer(k2s, k**2)
# scaling of all modes with k != 0
k2s.flat[0] = 1
scaling = np.sqrt(corr_spectrum(k2s))
scaling.flat[0] = 0
scaling /= scaling.mean() # normalize scaling to correct the variance
def noise_corr() -> NumericArray:
"""Return array of correlated noise."""
# initialize uncorrelated random field
arr: NumericArray = rng.normal(size=shape)
# forward transform to frequency space
arr = np_rfftn(arr, s=shape, axes=range(dim))
# scale frequency according to transformed correlation function
arr *= scaling
# backward transform to return to real space
return np_irfftn(arr, s=shape, axes=range(dim)) # type: ignore
return noise_corr
CorrelationType = Literal["none", "gaussian", "power law", "cosine"]
[docs]
def make_colored_noise(
shape: tuple[int, ...],
dx=1.0,
exponent: float = 0,
scale: float = 1,
rng: np.random.Generator | None = None,
) -> Callable[[], NumericArray]:
r"""Return a function creating an array of random values that obey.
.. math::
\langle c(\boldsymbol k) c(\boldsymbol k’) \rangle =
\Gamma^2 |\boldsymbol k|^\nu \delta(\boldsymbol k-\boldsymbol k’)
in spectral space on a Cartesian grid. The special case :math:`\nu = 0` corresponds
to white noise. For simplicity, the correlations respect periodic boundary
conditions.
Args:
shape (tuple of ints):
Number of supports points in each spatial dimension. The number of the list
defines the spatial dimension.
dx (float or list of floats):
Discretization along each dimension. A uniform discretization in each
direction can be indicated by a single number.
exponent:
Exponent :math:`\nu` of the power spectrum
scale:
Scaling factor :math:`\Gamma` determining noise strength
rng (:class:`~numpy.random.Generator`):
Random number generator (default: :func:`~numpy.random.default_rng()`)
Returns:
callable: a function returning a random realization
"""
# deprecated since 2025-04-04
warnings.warn(
"`make_colored_noise` is deprecated. Use `make_correlated_noise` with "
"correlation='power law' instead",
DeprecationWarning,
)
rng = np.random.default_rng(rng)
# extract some information about the grid
dim = len(shape)
dx = np.broadcast_to(dx, (dim,))
if exponent == 0:
# fast case of white noise
def noise_normal():
"""Return array of colored noise."""
return scale * rng.normal(size=shape)
return noise_normal
# deal with colored noise in the following
# prepare wave vectors
k2s = np.array(0)
for i in range(dim):
if i == dim - 1:
k = np.fft.rfftfreq(shape[i], dx[i])
else:
k = np.fft.fftfreq(shape[i], dx[i])
k2s = np.add.outer(k2s, k**2)
# scaling of all modes with k != 0
k2s.flat[0] = 1
scaling = scale * k2s ** (exponent / 4)
scaling.flat[0] = 0
def noise_colored() -> NumericArray:
"""Return array of colored noise."""
# random field
arr: NumericArray = rng.normal(size=shape)
# forward transform
arr = np_rfftn(arr)
# scale according to frequency
arr *= scaling
# backwards transform
return np_irfftn(arr, s=shape, axes=range(dim)) # type: ignore
return noise_colored