def __init__(self, y, d, sigma=0.0, phi='phs3', eps=1.0, order=None):
y, d, sigma, phi, eps, order, _ = _sanitize_arguments(
y, d, sigma, phi, eps, order, None)
# For improved numerical stability, shift the observations so that
# their centroid is at zero
center = y.mean(axis=0)
y = y - center
# Build the system of equations and solve for the RBF and mononomial
# coefficients
Kyy = phi(y, y, eps=eps)
if sp.issparse(Kyy):
Kyy = sp.csc_matrix(Kyy + sp.diags(sigma**2))
else:
Kyy[range(y.shape[0]), range(y.shape[0])] += sigma**2
Py = mvmonos(y, order)
phi_coeff, poly_coeff = PartitionedSolver(Kyy, Py).solve(d)
self.y = y
self.phi = phi
self.eps = eps
self.order = order
self.center = center
self.phi_coeff = phi_coeff
self.poly_coeff = poly_coeff
def __init__(self, y, d, sigma=0.0, phi='phs3', eps=1.0, order=None):
y = np.asarray(y, dtype=float)
assert_shape(y, (None, None), 'y')
ny, ndim = y.shape
d = np.asarray(d, dtype=float)
assert_shape(d, (ny, ), 'd')
if np.isscalar(sigma):
sigma = np.full(ny, sigma, dtype=float)
else:
sigma = np.asarray(sigma, dtype=float)
assert_shape(sigma, (ny, ), 'sigma')
phi = get_rbf(phi)
if not np.isscalar(eps):
raise ValueError('The shape parameter should be a float')
# If `phi` is not in `_MIN_ORDER`, then the RBF is either positive definite
# (no minimum polynomial order) or user-defined
min_order = _MIN_ORDER.get(phi, -1)
if order is None:
order = max(min_order, 0)
elif order < min_order:
logger.warning(
'The polynomial order should not be below %d for %s in order for the '
'interpolant to be well-posed' % (min_order, phi))
order = int(order)
# For improved numerical stability, shift the observations so that their
# centroid is at zero
center = y.mean(axis=0)
y = y - center
# Build the system of equations and solve for the RBF and mononomial
# coefficients
Kyy = phi(y, y, eps=eps)
S = scipy.sparse.diags(sigma**2)
Py = mvmonos(y, order)
nmonos = Py.shape[1]
if nmonos > ny:
raise ValueError(
'The polynomial order is too high. The number of monomials, %d, '
'exceeds the number of observations, %d' % (nmonos, ny))
z = np.zeros(nmonos, dtype=float)
phi_coeff, poly_coeff = PartitionedSolver(Kyy + S, Py).solve(d, z)
self.y = y
self.phi = phi
self.eps = eps
self.order = order
self.center = center
self.phi_coeff = phi_coeff
self.poly_coeff = poly_coeff
def __init__(self,
y,
d,
sigma=None,
eps=1.0,
phi=phs3,
order=1,
extrapolate=True):
y = np.asarray(y)
assert_shape(y, (None, None), 'y')
nobs, dim = y.shape
d = np.asarray(d)
assert_shape(d, (nobs, ), 'd')
if sigma is None:
# if sigma is not specified then it is zeros
sigma = np.zeros(nobs)
elif np.isscalar(sigma):
# if a float is specified then use it as the uncertainties for
# all observations
sigma = np.repeat(sigma, nobs)
else:
sigma = np.asarray(sigma)
assert_shape(sigma, (nobs, ), 'sigma')
phi = get_rbf(phi)
# form block consisting of the RBF and uncertainties on the
# diagonal
K = phi(y, y, eps=eps)
Cd = scipy.sparse.diags(sigma**2)
# form the block consisting of the monomials
pwr = powers(order, dim)
P = mvmonos(y, pwr)
# create zeros vector for the right-hand-side
z = np.zeros((pwr.shape[0], ))
# solve for the RBF and mononomial coefficients
phi_coeff, poly_coeff = PartitionedSolver(K + Cd, P).solve(d, z)
self._y = y
self._phi = phi
self._order = order
self._eps = eps
self._phi_coeff = phi_coeff
self._poly_coeff = poly_coeff
self._pwr = pwr
self.extrapolate = extrapolate
def __call__(self, x, diff=None, chunk_size=1000):
'''
Evaluates the interpolant at `x`
Parameters
----------
x : (N, D) array
Target points.
diff : (D,) int array, optional
Derivative order for each spatial dimension.
chunk_size : int, optional
Break `x` into chunks with this size and evaluate the
interpolant for each chunk. Smaller values result in decreased
memory usage but also decreased speed.
Returns
-------
out : (N,) array
Values of the interpolant at `x`
'''
x = np.asarray(x, dtype=float)
assert_shape(x, (None, self._y.shape[1]), 'x')
xlen = x.shape[0]
# allocate output array
out = np.zeros(xlen, dtype=float)
count = 0
while count < xlen:
start, stop = count, count + chunk_size
K = self._phi(x[start:stop], self._y, eps=self._eps, diff=diff)
P = mvmonos(x[start:stop], self._pwr, diff=diff)
out[start:stop] = (K.dot(self._phi_coeff) +
P.dot(self._poly_coeff))
count += chunk_size
# return zero for points outside of the convex hull if
# extrapolation is not allowed
if not self.extrapolate:
out[~_in_hull(x, self._y)] = np.nan
return out
def __call__(self, x, diff=None, chunk_size=1000):
'''
Evaluates the interpolant at `x`
Parameters
----------
x : (N, D) float array
Target points
diff : (D,) int array, optional
Derivative order for each spatial dimension
chunk_size : int, optional
Break `x` into chunks with this size and evaluate the interpolant for
each chunk
Returns
-------
(N,) float array
'''
x = np.asarray(x, dtype=float)
assert_shape(x, (None, self.y.shape[1]), 'x')
nx = x.shape[0]
if chunk_size is not None:
out = np.zeros(nx, dtype=float)
for start in range(0, nx, chunk_size):
stop = start + chunk_size
out[start:stop] = self(x[start:stop],
diff=diff,
chunk_size=None)
return out
x = x - self.center
Kxy = self.phi(x, self.y, eps=self.eps, diff=diff)
Px = mvmonos(x, self.order, diff=diff)
out = Kxy.dot(self.phi_coeff) + Px.dot(self.poly_coeff)
return out
def __call__(self, x, diff=None, chunk_size=100):
'''
Evaluates the interpolant at `x`
Parameters
----------
x : (N, D) float array
Target points
diff : (D,) int array, optional
Derivative order for each spatial dimension
chunk_size : int, optional
Break `x` into chunks with this size and evaluate the interpolant for
each chunk
Returns
-------
(N,) float array
'''
x = np.asarray(x, dtype=float)
assert_shape(x, (None, self.y.shape[1]), 'x')
nx = x.shape[0]
if chunk_size is not None:
out = np.zeros(nx, dtype=float)
for start in range(0, nx, chunk_size):
stop = start + chunk_size
out[start:stop] = self(x[start:stop],
diff=diff,
chunk_size=None)
return out
# get the indices of the k-nearest observations for each interpolation
# point
_, nbr = self.tree.query(x, self.k)
# multiple interpolation points may have the same neighborhood. Make the
# neighborhoods unique so that we only compute the interpolation
# coefficients once for each neighborhood
nbr, inv = np.unique(np.sort(nbr, axis=1), return_inverse=True, axis=0)
nnbr = nbr.shape[0]
# Get the observation data for each neighborhood
y, d, sigma = self.y[nbr], self.d[nbr], self.sigma[nbr]
# shift the centers of each neighborhood to zero for numerical stability
centers = y.mean(axis=1)
y = y - centers[:, None]
# build the left-hand-side interpolation matrix consisting of the RBF
# and monomials evaluated at each neighborhood
Kyy = self.phi(y, y, eps=self.eps)
Kyy[:, range(self.k), range(self.k)] += sigma**2
Py = mvmonos(y, self.order)
PyT = np.transpose(Py, (0, 2, 1))
nmonos = Py.shape[2]
Z = np.zeros((nnbr, nmonos, nmonos), dtype=float)
LHS = np.block([[Kyy, Py], [PyT, Z]])
# build the right-hand-side data vector consisting of the observations for
# each neighborhood and extra zeros
z = np.zeros((nnbr, nmonos), dtype=float)
rhs = np.hstack((d, z))
# solve for the RBF and polynomial coefficients for each neighborhood
coeff = np.linalg.solve(LHS, rhs)
# expand the arrays from having one entry per neighborhood to one entry per
# interpolation point
coeff = coeff[inv]
y = y[inv]
centers = centers[inv]
# evaluate at the interpolation points
x = x - centers
phi_coeff = coeff[:, :self.k]
poly_coeff = coeff[:, self.k:]
Kxy = self.phi(x[:, None], y, eps=self.eps, diff=diff)[:, 0]
Px = mvmonos(x, self.order, diff=diff)
out = (Kxy * phi_coeff).sum(axis=1) + (Px * poly_coeff).sum(axis=1)
return out
def weights(x, s, diffs, coeffs=None, phi=phs3, order=None, eps=1.0):
'''
Returns the weights which map a functions values at `s` to an approximation
of that functions derivative at `x`. The weights are computed using the
RBF-FD method described in [1]. In this function `x` is a single point in
D-dimensional space. Use `weight_matrix` to compute the weights for multiple
point.
Parameters
----------
x : (D,) array
Target point. The weights will approximate the derivative at this point.
s : (N, D) array
Stencil points. The derivative will be approximated with a weighted sum of
values at this point.
diffs : (D,) int array or (K, D) int array
Derivative orders for each spatial dimension. For example `[2, 0]`
indicates that the weights should approximate the second derivative with
respect to the first spatial dimension in two-dimensional space. diffs can
also be a (K, D) array, where each (D,) sub-array is a term in a
differential operator. For example the two-dimensional Laplacian can be
represented as `[[2, 0], [0, 2]]`.
coeffs : (K,) array, optional
Coefficients for each term in the differential operator specified with
`diffs`. Defaults to an array of ones. If `diffs` was specified as a (D,)
array then `coeffs` should be a length 1 array.
phi : rbf.basis.RBF instance or str, optional
Type of RBF. Select from those available in `rbf.basis` or create your own.
order : int, optional
Order of the added polynomial. This defaults to the highest derivative
order. For example, if `diffs` is `[[2, 0], [0, 1]]`, then order is set to
2.
eps : float or (N,) array, optional
Shape parameter for each RBF, which have centers `s`. This only makes a
difference when using RBFs that are not scale invariant. All the predefined
RBFs except for the odd order polyharmonic splines are not scale invariant.
Returns
-------
out : (N,) array
RBF-FD weights
Examples
--------
Calculate the weights for a one-dimensional second order derivative.
>>> x = np.array([1.0])
>>> s = np.array([[0.0], [1.0], [2.0]])
>>> diff = (2,)
>>> weights(x, s, diff)
array([ 1., -2., 1.])
Calculate the weights for estimating an x derivative from three points in a
two-dimensional plane
>>> x = np.array([0.25, 0.25])
>>> s = np.array([[0.0, 0.0],
[1.0, 0.0],
[0.0, 1.0]])
>>> diff = (1, 0)
>>> weights(x, s, diff)
array([ -1., 1., 0.])
References
----------
[1] Fornberg, B. and N. Flyer. A Primer on Radial Basis
Functions with Applications to the Geosciences. SIAM, 2015.
'''
x = np.asarray(x, dtype=float)
assert_shape(x, (None, ), 'x')
s = np.asarray(s, dtype=float)
assert_shape(s, (None, x.shape[0]), 's')
diffs = np.asarray(diffs, dtype=int)
diffs = _reshape_diffs(diffs)
if coeffs is None:
coeffs = np.ones(diffs.shape[0], dtype=float)
else:
coeffs = np.asarray(coeffs, dtype=float)
assert_shape(coeffs, (diffs.shape[0], ), 'coeffs')
phi = get_rbf(phi)
# stencil size and number of dimensions
size, dim = s.shape
# get the maximum polynomial order allowed for this stencil size
max_order = _max_poly_order(size, dim)
if order is None:
# If the polynomial order is not specified, make it equal to the derivative
# order, provided that the stencil size is large enough.
order = _default_poly_order(diffs)
order = min(order, max_order)
if order > max_order:
raise ValueError('Polynomial order is too high for the stencil size')
# center the stencil on `x` for improved numerical stability
s = s - x
x = np.zeros_like(x)
# get the powers for the added monomials
pwr = powers(order, dim)
# evaluate the RBF and monomials at each point in the stencil. This becomes
# the left-hand-side
A = phi(s, s, eps=eps)
P = mvmonos(s, pwr)
# Evaluate the RBF and monomials for each term in the differential operator.
# This becomes the right-hand-side.
a = np.zeros((1, size), dtype=float)
p = np.zeros((1, pwr.shape[0]), dtype=float)
for c, d in zip(coeffs, diffs):
a += c * phi(x[None, :], s, eps=eps, diff=d)
p += c * mvmonos(x[None, :], pwr, diff=d)
# squeeze `a` and `p` into 1d arrays. `a` is ran through as_array because it
# may be sparse.
a = as_array(a)[0]
p = p[0]
# attempt to compute the RBF-FD weights
try:
w = PartitionedSolver(A, P).solve(a, p)[0]
return w
except np.linalg.LinAlgError:
raise np.linalg.LinAlgError(
'An error was raised while computing the RBF-FD weights at point %s '
'with the RBF %s and the polynomial order %s. This may be due to a '
'stencil with duplicate or collinear points. The stencil contains the '
'following points:\n%s' % (x, phi, order, s))
def weights(x, s, diffs,
coeffs=None,
phi=phs3,
order=None,
eps=1.0):
'''
Returns the weights which map a function's values at `s` to an approximation
of that function's derivative at `x`. The weights are computed using the
RBF-FD method described in [1]
Parameters
----------
x : (..., D) float array
Target points where the derivative is being approximated
s : (..., M, D) float array
Stencils for each target point
diffs : (D,) int array or (K, D) int array
Derivative orders for each spatial dimension. For example `[2, 0]`
indicates that the weights should approximate the second derivative with
respect to the first spatial dimension in two-dimensional space. `diffs`
can also be a (K, D) array, where each (D,) sub-array is a term in a
differential operator. For example, the two-dimensional Laplacian can be
represented as `[[2, 0], [0, 2]]`.
coeffs : (K, ...) float array, optional
Coefficients for each term in the differential operator specified with
`diffs`. The coefficients can vary between target points. Defaults to an
array of ones.
phi : rbf.basis.RBF instance or str, optional
Type of RBF. See `rbf.basis` for the available options.
order : int, optional
Order of the added polynomial. This defaults to the highest derivative
order. For example, if `diffs` is `[[2, 0], [0, 1]]`, then this is set to
2.
eps : float or float array, optional
Shape parameter for each RBF
Returns
-------
(..., M) float array
RBF-FD weights for each target point
Examples
--------
Calculate the weights for a one-dimensional second order derivative.
>>> x = np.array([1.0])
>>> s = np.array([[0.0], [1.0], [2.0]])
>>> diff = (2,)
>>> weights(x, s, diff)
array([ 1., -2., 1.])
Calculate the weights for estimating an x derivative from three points in a
two-dimensional plane
>>> x = np.array([0.25, 0.25])
>>> s = np.array([[0.0, 0.0],
[1.0, 0.0],
[0.0, 1.0]])
>>> diff = (1, 0)
>>> weights(x, s, diff)
array([ -1., 1., 0.])
References
----------
[1] Fornberg, B. and N. Flyer. A Primer on Radial Basis Functions with
Applications to the Geosciences. SIAM, 2015.
'''
x = np.asarray(x, dtype=float)
assert_shape(x, (..., None), 'x')
bcast = x.shape[:-1]
ndim = x.shape[-1]
s = np.asarray(s, dtype=float)
assert_shape(s, (..., None, ndim), 's')
# broadcast leading dimensions of `s` to match leading dimensions of `x`
s = np.broadcast_to(s, bcast + s.shape[-2:])
ssize = s.shape[-2]
diffs = np.asarray(diffs, dtype=int)
diffs = np.atleast_2d(diffs)
assert_shape(diffs, (None, ndim), 'diffs')
if coeffs is None:
coeffs = np.ones(len(diffs), dtype=float)
else:
coeffs = np.asarray(coeffs, dtype=float)
assert_shape(coeffs, (len(diffs), ...), 'coeffs')
# broadcast each element in `coeffs` to match leading dimensions of `x`
coeffs = [np.broadcast_to(c, bcast) for c in coeffs]
phi = get_rbf(phi)
# get the maximum polynomial order allowed for this stencil size
max_order = _max_poly_order(ssize, ndim)
if order is None:
# If the polynomial order is not specified, make it equal to the derivative
# order, provided that the stencil size is large enough.
order = diffs.sum(axis=1).max()
order = min(order, max_order)
if order > max_order:
raise ValueError('Polynomial order is too high for the stencil size')
# center the stencil on `x` for improved numerical stability
x = x[..., None, :]
s = s - x
x = np.zeros_like(x)
# get the powers for the added monomials
pwr = monomial_powers(order, ndim)
# evaluate the RBF and monomials at each point in the stencil. This becomes
# the left-hand-side
A = phi(s, s, eps=eps)
P = mvmonos(s, pwr)
Pt = np.einsum('...ij->...ji', P)
Z = np.zeros(bcast + (len(pwr), len(pwr)), dtype=float)
LHS = np.concatenate(
(np.concatenate((A, P), axis=-1),
np.concatenate((Pt, Z), axis=-1)),
axis=-2)
# Evaluate the RBF and monomials at the target points for each term in the
# differential operator. This becomes the right-hand-side.
a, p = 0.0, 0.0
for c, d in zip(coeffs, diffs):
a += c[..., None, None]*phi(x, s, eps=eps, diff=d)
p += c[..., None, None]*mvmonos(x, pwr, diff=d)
# convert `a` to an array because phi may be a sparse RBF
a = as_array(a)[..., 0, :]
p = p[..., 0, :]
rhs = np.concatenate((a, p), axis=-1)
w = np.linalg.solve(LHS, rhs)[..., :ssize]
return w
