# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""
This module provides tools for interpolating data.
"""
import numpy as np
__all__ = ['ShepardIDWInterpolator']
__doctest_requires__ = {('ShepardIDWInterpolator'): ['scipy']}
[docs]
class ShepardIDWInterpolator:
"""
Class to perform Inverse Distance Weighted (IDW) interpolation.
This interpolator uses a modified version of `Shepard's method
<https://en.wikipedia.org/wiki/Inverse_distance_weighting>`_ (see
the Notes section for details).
Parameters
----------
coordinates : float, 1D array_like, or NxM array_like
Coordinates of the known data points. In general, it is expected
that these coordinates are in a form of a NxM-like array where N
is the number of points and M is dimension of the coordinate
space. When M=1 (1D space), then the ``coordinates`` parameter
may be entered as a 1D array or, if only one data point is
available, ``coordinates`` can be a scalar number representing
the 1D coordinate of the data point.
.. note::
If the dimensionality of ``coordinates`` is larger than 2,
e.g., if it is of the form N1 x N2 x N3 x ... x Nn x M, then
it will be flattened to form an array of size NxM where N =
N1 * N2 * ... * Nn.
values : float or 1D array_like
Values of the data points corresponding to each coordinate
provided in ``coordinates``. In general a 1D array is expected.
When a single data point is available, then ``values`` can be a
scalar number.
.. note::
If the dimensionality of ``values`` is larger than 1 then it
will be flattened.
weights : float or 1D array_like, optional
Weights to be associated with each data value. These weights, if
provided, will be combined with inverse distance weights (see
the Notes section for details). When ``weights`` is `None`
(default), then only inverse distance weights will be used. When
provided, this input parameter must have the same form as
``values``.
leafsize : float, optional
The number of points at which the k-d tree algorithm switches
over to brute-force. ``leafsize`` must be positive. See
`scipy.spatial.cKDTree` for further information.
Notes
-----
This interpolator uses a slightly modified version of `Shepard's
method <https://en.wikipedia.org/wiki/Inverse_distance_weighting>`_.
The essential difference is the introduction of a "regularization"
parameter (``reg``) that is used when computing the inverse distance
weights:
.. math::
w_i = 1 / (d(x, x_i)^{power} + r)
By supplying a positive regularization parameter one can avoid
singularities at the locations of the data points as well as control
the "smoothness" of the interpolation (e.g., make the weights of the
neighbors less varied). The "smoothness" of interpolation can also
be controlled by the power parameter (``power``).
Examples
--------
This class can can be instantiated using the following syntax::
>>> from photutils.utils import ShepardIDWInterpolator as idw
Example of interpolating 1D data::
>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> x = rng.random(100) # 100 random values
>>> y = np.sin(x)
>>> f = idw(x, y)
>>> f(0.4) # doctest: +FLOAT_CMP
0.38937843420912366
>>> np.sin(0.4) # doctest: +FLOAT_CMP
0.3894183423086505
>>> xi = rng.random(4) # 4 random values
>>> xi # doctest: +FLOAT_CMP
array([0.47998792, 0.23237292, 0.80188058, 0.92353016])
>>> f(xi) # doctest: +FLOAT_CMP
array([0.46577097, 0.22837422, 0.71856662, 0.80125391])
>>> np.sin(xi) # doctest: +FLOAT_CMP
array([0.46176846, 0.23028731, 0.71866503, 0.7977353 ])
NOTE: In the last example, ``xi`` may be a ``Nx1`` array instead of
a 1D vector.
Example of interpolating 2D data::
>>> rng = np.random.default_rng(0)
>>> pos = rng.random((1000, 2))
>>> val = np.sin(pos[:, 0] + pos[:, 1])
>>> f = idw(pos, val)
>>> f([0.5, 0.6]) # doctest: +FLOAT_CMP
0.8948257014687874
>>> np.sin(0.5 + 0.6) # doctest: +FLOAT_CMP
0.8912073600614354
"""
def __init__(self, coordinates, values, weights=None, leafsize=10):
from scipy.spatial import cKDTree
coordinates = np.asarray(coordinates)
if coordinates.ndim == 0: # scalar coordinate
coordinates = np.atleast_2d(coordinates)
if coordinates.ndim == 1:
coordinates = np.transpose(np.atleast_2d(coordinates))
if coordinates.ndim > 2:
coordinates = np.reshape(coordinates, (-1, coordinates.shape[-1]))
values = np.asanyarray(values).ravel()
ncoords = coordinates.shape[0]
if ncoords < 1:
raise ValueError('You must enter at least one data point.')
if values.shape[0] != ncoords:
raise ValueError('The number of values must match the number '
'of coordinates.')
if weights is not None:
weights = np.asanyarray(weights).ravel()
if weights.shape[0] != ncoords:
raise ValueError('The number of weights must match the '
'number of coordinates.')
if np.any(weights < 0.0):
raise ValueError('All weight values must be non-negative '
'numbers.')
self.coordinates = coordinates
self.ncoords = ncoords
self.coords_ndim = coordinates.shape[1]
self.values = values
self.weights = weights
self.kdtree = cKDTree(coordinates, leafsize=leafsize)
[docs]
def __call__(self, positions, n_neighbors=8, eps=0.0, power=1.0, reg=0.0,
conf_dist=1.0e-12, dtype=float):
"""
Evaluate the interpolator at the given positions.
Parameters
----------
positions : float, 1D array_like, or NxM array_like
Coordinates of the position(s) at which the interpolator
should be evaluated. In general, it is expected that these
coordinates are in a form of a NxM-like array where N is the
number of points and M is dimension of the coordinate space.
When M=1 (1D space), then the ``positions`` parameter may be
input as a 1D-like array or, if only one data point is
available, ``positions`` can be a scalar number representing
the 1D coordinate of the data point.
.. note::
If the dimensionality of the ``positions`` argument is
larger than 2, e.g., if it is of the form N1 x N2 x N3 x
... x Nn x M, then it will be flattened to form an array
of size NxM where N = N1 * N2 * ... * Nn.
.. warning::
The dimensionality of ``positions`` must match the
dimensionality of the ``coordinates`` used during the
initialization of the interpolator.
n_neighbors : int, optional
The maximum number of nearest neighbors to use during the
interpolation.
eps : float, optional
Set to use approximate nearest neighbors; the kth neighbor
is guaranteed to be no further than (1 + ``eps``) times the
distance to the real *k*-th nearest neighbor. See
`scipy.spatial.cKDTree.query` for further information.
power : float, optional
The power of the inverse distance used for the interpolation
weights. See the Notes section for more details.
reg : float, optional
The regularization parameter. It may be used to control the
smoothness of the interpolator. See the Notes section for
more details.
conf_dist : float, optional
The confusion distance below which the interpolator should
use the value of the closest data point instead of
attempting to interpolate. This is used to avoid
singularities at the known data points, especially if
``reg`` is 0.0.
dtype : data-type, optional
The data type of the output interpolated values. If `None`
then the type will be inferred from the type of the
``values`` parameter used during the initialization of the
interpolator.
"""
n_neighbors = int(n_neighbors)
if n_neighbors < 1:
raise ValueError('n_neighbors must be a positive integer')
if conf_dist is not None and conf_dist <= 0.0:
conf_dist = None
positions = np.asanyarray(positions)
if positions.ndim == 0:
# assume we have a single 1D coordinate
if self.coords_ndim != 1:
raise ValueError('The dimensionality of the input position '
'does not match the dimensionality of the '
'coordinates used to initialize the '
'interpolator.')
elif positions.ndim == 1:
# assume we have a single point
if self.coords_ndim not in (1, positions.shape[-1]):
raise ValueError('The input position was provided as a 1D '
'array, but its length does not match the '
'dimensionality of the coordinates used '
'to initialize the interpolator.')
elif positions.ndim != 2:
raise ValueError('The input positions must be an array-like '
'object of dimensionality no larger than 2.')
positions = np.reshape(positions, (-1, self.coords_ndim))
npositions = positions.shape[0]
distances, idx = self.kdtree.query(positions, k=n_neighbors, eps=eps)
if n_neighbors == 1:
return self.values[idx]
if dtype is None:
dtype = self.values.dtype
interp_values = np.zeros(npositions, dtype=dtype)
for k in range(npositions):
valid_idx = np.isfinite(distances[k])
idk = idx[k][valid_idx]
dk = distances[k][valid_idx]
if dk.shape[0] == 0:
interp_values[k] = np.nan
continue
if conf_dist is not None:
# check if we are close to a known data point
confused = (dk <= conf_dist)
if np.any(confused):
interp_values[k] = self.values[idk[confused][0]]
continue
w = 1.0 / ((dk**power) + reg)
if self.weights is not None:
w *= self.weights[idk]
wtot = np.sum(w)
if wtot > 0.0:
interp_values[k] = np.dot(w, self.values[idk]) / wtot
else:
interp_values[k] = np.nan
if len(interp_values) == 1:
return interp_values[0]
else:
return interp_values