# Source code for verde.spline

```
# Copyright (c) 2017 The Verde Developers.
# Distributed under the terms of the BSD 3-Clause License.
# SPDX-License-Identifier: BSD-3-Clause
#
# This code is part of the Fatiando a Terra project (https://www.fatiando.org)
#
"""
Biharmonic splines in 2D.
"""
import itertools
import warnings
import numpy as np
from sklearn.utils.validation import check_is_fitted
from .base import BaseGridder, check_fit_input, least_squares, n_1d_arrays
from .coordinates import get_region
from .model_selection import cross_val_score
from .utils import dispatch, parse_engine
try:
import numba
from numba import jit
except ImportError:
numba = None
from .utils import dummy_jit as jit
[docs]class SplineCV(BaseGridder):
r"""
Cross-validated biharmonic spline interpolation.
Similar to :class:`verde.Spline` but automatically chooses the best
*damping* and *mindist* parameters using cross-validation. Tests all
combinations of the given *dampings* and *mindists* and selects the maximum
(or minimum) mean cross-validation score (i.e., a grid search).
This can optionally run in parallel using :mod:`dask`. To do this, use
``delayed=True`` to dispatch computations with :func:`dask.delayed`.
In this case, each fit and score operation of the grid search will be
performed in parallel.
.. note::
When using *delayed*, the ``scores_`` attribute will be
:func:`dask.delayed` objects instead of the actual scores. This is
because the scores are an intermediate step in the computations and
their results are not stored. If you need the scores, run
:func:`dask.compute` on ``scores_`` to calculate them. Be warned that
**this will run the grid search again**. It might still be faster than
serial execution but not necessarily.
.. warning::
The ``client`` parameter is deprecated and will be removed in Verde
v2.0.0. Use ``delayed`` instead.
Other cross-validation generators from :mod:`sklearn.model_selection` can
be used by passing them through the *cv* argument.
Parameters
----------
mindists : iterable or 1d array
List (or other iterable) of *mindist* parameter values to try. Can be
considered a minimum distance between the point forces and data points.
Needed because the Green's functions are singular when forces and data
points coincide. Acts as a fudge factor.
dampings : iterable or 1d array
List (or other iterable) of *damping* parameter values to try. Is the
positive damping regularization parameter. Controls how much smoothness
is imposed on the estimated forces. If None, no regularization is used.
force_coords : None or tuple of arrays
The easting and northing coordinates of the point forces. If None
(default), then will be set to the data coordinates the first time
:meth:`~verde.SplineCV.fit` is called.
engine : str
Computation engine for the Jacobian matrix and prediction. Can be
``'auto'``, ``'numba'``, or ``'numpy'``. If ``'auto'``, will use numba
if it is installed or numpy otherwise. The numba version is
multi-threaded and usually faster, which makes fitting and predicting
faster.
cv : None or cross-validation generator
Any scikit-learn cross-validation generator. If not given, will use the
default set by :func:`verde.cross_val_score`.
client : None or dask.distributed.Client
**DEPRECATED:** This option is deprecated and will be removed in Verde
v2.0.0. If None, then computations are run serially. Otherwise, should
be a dask ``Client`` object. It will be used to dispatch computations
to the dask cluster.
delayed : bool
If True, will use :func:`dask.delayed` to dispatch computations and
allow mod:`dask` to execute the grid search in parallel (see note
above).
Attributes
----------
force_ : array
The estimated forces that fit the observed data.
force_coords_ : tuple of arrays
The easting and northing coordinates of the point forces. Same as
*force_coords* if it is not None. Otherwise, same as the data locations
used to fit the spline.
region_ : tuple
The boundaries (``[W, E, S, N]``) of the data used to fit the
interpolator. Used as the default region for the
:meth:`~verde.SplineCV.grid` and :meth:`~verde.SplineCV.scatter`
methods.
scores_ : array
The mean cross-validation score for each parameter combination. If
``delayed=True``, will be a list of :func:`dask.delayed` objects (see
note above).
mindist_ : float
The optimal value for the *mindist* parameter.
damping_ : float
The optimal value for the *damping* parameter.
spline_ : :class:`verde.Spline`
A fitted :class:`~verde.Spline` with the optimal configuration
parameters.
See also
--------
Spline : The bi-harmonic spline
cross_val_score : Score an estimator/gridder using cross-validation
"""
def __init__(
self,
mindists=(1e3, 10e3, 100e3),
dampings=(1e-10, 1e-5, 1e-1),
force_coords=None,
engine="auto",
cv=None,
client=None,
delayed=False,
):
super().__init__()
self.dampings = dampings
self.mindists = mindists
self.force_coords = force_coords
self.engine = engine
self.cv = cv
self.client = client
self.delayed = delayed
if client is not None:
warnings.warn(
"The 'client' parameter of 'verde.SplineCV' is "
"deprecated and will be removed in Verde 2.0.0. "
"Use the 'delayed' parameter instead.",
FutureWarning,
)
[docs] def fit(self, coordinates, data, weights=None):
"""
Fit the spline to the given data and automatically tune parameters.
For each combination of the parameters given, computes the mean cross
validation score using :func:`verde.cross_val_score` and the given CV
splitting class (the *cv* parameter of this class). The configuration
with the best score is then chosen and used to fit the entire dataset.
The data region is captured and used as default for the
:meth:`~verde.SplineCV.grid` and :meth:`~verde.SplineCV.scatter`
methods.
All input arrays must have the same shape.
Parameters
----------
coordinates : tuple of arrays
Arrays with the coordinates of each data point. Should be in the
following order: (easting, northing, vertical, ...). Only easting
and northing will be used, all subsequent coordinates will be
ignored.
data : array
The data values of each data point.
weights : None or array
If not None, then the weights assigned to each data point.
Typically, this should be 1 over the data uncertainty squared.
Returns
-------
self
Returns this estimator instance for chaining operations.
"""
parameter_sets = [
dict(
mindist=combo[0],
damping=combo[1],
engine=self.engine,
force_coords=self.force_coords,
)
for combo in itertools.product(self.mindists, self.dampings)
]
if self.client is not None:
# Deprecated and will be removed in v2.0.0
scores = []
for params in parameter_sets:
spline = Spline(**params)
scores.append(
self.client.submit(
cross_val_score,
spline,
coordinates=coordinates,
data=data,
weights=weights,
cv=self.cv,
)
)
scores = [np.mean(score.result()) for score in scores]
else:
scores = []
for params in parameter_sets:
spline = Spline(**params)
score = cross_val_score(
spline,
coordinates=coordinates,
data=data,
weights=weights,
cv=self.cv,
delayed=self.delayed,
)
scores.append(dispatch(np.mean, delayed=self.delayed)(score))
best = dispatch(np.argmax, delayed=self.delayed)(scores)
if self.delayed:
best = best.compute()
else:
scores = np.asarray(scores)
self.spline_ = Spline(**parameter_sets[best])
self.spline_.fit(coordinates, data, weights=weights)
self.scores_ = scores
return self
@property
def force_(self):
"The estimated forces that fit the data."
return self.spline_.force_
@property
def region_(self):
"The bounding region of the data used to fit the spline"
return self.spline_.region_
@property
def damping_(self):
"The optimal damping parameter"
return self.spline_.damping
@property
def mindist_(self):
"The optimal mindist parameter"
return self.spline_.mindist
@property
def force_coords_(self):
"The optimal force locations"
return self.spline_.force_coords_
[docs] def predict(self, coordinates):
"""
Evaluate the best estimated spline on the given set of points.
Requires a fitted estimator (see :meth:`~verde.SplineCV.fit`).
Parameters
----------
coordinates : tuple of arrays
Arrays with the coordinates of each data point. Should be in the
following order: (easting, northing, vertical, ...). Only easting
and northing will be used, all subsequent coordinates will be
ignored.
Returns
-------
data : array
The data values evaluated on the given points.
"""
check_is_fitted(self, ["spline_"])
return self.spline_.predict(coordinates)
[docs]class Spline(BaseGridder):
r"""
Biharmonic spline interpolation using Green's functions.
This gridder assumes Cartesian coordinates.
Implements the 2D splines of [Sandwell1987]_. The Green's function for the
spline corresponds to the elastic deflection of a thin sheet subject to a
vertical force. For an observation point at the origin and a force at the
coordinates given by the vector :math:`\mathbf{x}`, the Green's function
is:
.. math::
g(\mathbf{x}) = \|\mathbf{x}\|^2 \left(\log \|\mathbf{x}\| - 1\right)
In practice, this function is not defined for data points that coincide
with a force. To prevent this, a fudge factor is added to
:math:`\|\mathbf{x}\|`.
The interpolation is performed by estimating forces that produce
deflections that fit the observed data (using least-squares). Then, the
interpolated points can be evaluated at any location.
By default, the forces will be placed at the same points as the input data
given to :meth:`~verde.Spline.fit`. This configuration provides an exact
solution on top of the data points. However, this solution can be unstable
for certain configurations of data points.
Approximate (and more stable) solutions can be obtained by applying damping
regularization to smooth the estimated forces (and interpolated values) or
by not using the data coordinates to position the forces (use the
*force_coords* parameter).
Data weights can be used during fitting but only have an any effect when
using the approximate solutions.
Before fitting, the Jacobian (design, sensitivity, feature, etc) matrix for
the spline is normalized using
:class:`sklearn.preprocessing.StandardScaler` without centering the mean so
that the transformation can be undone in the estimated forces.
Parameters
----------
mindist : float
A minimum distance between the point forces and data points. Needed
because the Green's functions are singular when forces and data points
coincide. Acts as a fudge factor.
damping : None or float
The positive damping regularization parameter. Controls how much
smoothness is imposed on the estimated forces. If None, no
regularization is used.
force_coords : None or tuple of arrays
The easting and northing coordinates of the point forces. If None
(default), then will be set to the data coordinates used to fit the
spline.
engine : str
Computation engine for the Jacobian matrix and prediction. Can be
``'auto'``, ``'numba'``, or ``'numpy'``. If ``'auto'``, will use numba
if it is installed or numpy otherwise. The numba version is
multi-threaded and usually faster, which makes fitting and predicting
faster.
Attributes
----------
force_ : array
The estimated forces that fit the observed data.
force_coords_ : tuple of arrays
The easting and northing coordinates of the point forces. Same as
*force_coords* if it is not None. Otherwise, same as the data locations
used to fit the spline.
region_ : tuple
The boundaries (``[W, E, S, N]``) of the data used to fit the
interpolator. Used as the default region for the
:meth:`~verde.Spline.grid` and :meth:`~verde.Spline.scatter` methods.
See also
--------
SplineCV : Cross-validated version of the bi-harmonic spline
"""
def __init__(self, mindist=1e-5, damping=None, force_coords=None, engine="auto"):
super().__init__()
self.mindist = mindist
self.damping = damping
self.force_coords = force_coords
self.engine = engine
[docs] def fit(self, coordinates, data, weights=None):
"""
Fit the biharmonic spline to the given data.
The data region is captured and used as default for the
:meth:`~verde.Spline.grid` and :meth:`~verde.Spline.scatter` methods.
All input arrays must have the same shape.
Parameters
----------
coordinates : tuple of arrays
Arrays with the coordinates of each data point. Should be in the
following order: (easting, northing, vertical, ...). Only easting
and northing will be used, all subsequent coordinates will be
ignored.
data : array
The data values of each data point.
weights : None or array
If not None, then the weights assigned to each data point.
Typically, this should be 1 over the data uncertainty squared.
Returns
-------
self
Returns this estimator instance for chaining operations.
"""
coordinates, data, weights = check_fit_input(coordinates, data, weights)
warn_weighted_exact_solution(self, weights)
# Capture the data region to use as a default when gridding.
self.region_ = get_region(coordinates[:2])
if self.force_coords is None:
self.force_coords_ = tuple(i.copy() for i in n_1d_arrays(coordinates, n=2))
else:
self.force_coords_ = self.force_coords
jacobian = self.jacobian(coordinates[:2], self.force_coords_)
self.force_ = least_squares(jacobian, data, weights, self.damping)
return self
[docs] def predict(self, coordinates):
"""
Evaluate the estimated spline on the given set of points.
Requires a fitted estimator (see :meth:`~verde.Spline.fit`).
Parameters
----------
coordinates : tuple of arrays
Arrays with the coordinates of each data point. Should be in the
following order: (easting, northing, vertical, ...). Only easting
and northing will be used, all subsequent coordinates will be
ignored.
Returns
-------
data : array
The data values evaluated on the given points.
"""
check_is_fitted(self, ["force_"])
shape = np.broadcast(*coordinates[:2]).shape
force_east, force_north = n_1d_arrays(self.force_coords_, n=2)
east, north = n_1d_arrays(coordinates, n=2)
data = np.empty(east.size, dtype=east.dtype)
if parse_engine(self.engine) == "numba":
data = predict_numba(
east, north, force_east, force_north, self.mindist, self.force_, data
)
else:
data = predict_numpy(
east, north, force_east, force_north, self.mindist, self.force_, data
)
return data.reshape(shape)
[docs] def jacobian(self, coordinates, force_coords, dtype="float64"):
"""
Make the Jacobian matrix for the 2D biharmonic spline.
Each column of the Jacobian is the Green's function for a single force
evaluated on all observation points [Sandwell1987]_.
Parameters
----------
coordinates : tuple of arrays
Arrays with the coordinates of each data point. Should be in the
following order: (easting, northing, vertical, ...). Only easting
and northing will be used, all subsequent coordinates will be
ignored.
force_coords : tuple of arrays
Arrays with the coordinates for the forces. Should be in the same
order as the coordinate arrays.
dtype : str or numpy dtype
The type of the Jacobian array.
Returns
-------
jacobian : 2D array
The (n_data, n_forces) Jacobian matrix.
"""
force_east, force_north = n_1d_arrays(force_coords, n=2)
east, north = n_1d_arrays(coordinates, n=2)
jac = np.empty((east.size, force_east.size), dtype=dtype)
if parse_engine(self.engine) == "numba":
jac = jacobian_numba(
east, north, force_east, force_north, self.mindist, jac
)
else:
jac = jacobian_numpy(
east, north, force_east, force_north, self.mindist, jac
)
return jac
def warn_weighted_exact_solution(spline, weights):
"""
Warn the user that a weights doesn't work for the exact solution.
Parameters
----------
spline : estimator
The spline instance that we'll check. Needs to have the ``damping``
attribute.
weights : array or None
The weights given to fit.
"""
# Check if we're using weights without damping and warn the user that it
# might not have any effect.
if weights is not None and spline.damping is None:
warnings.warn(
"Weights might have no effect if no regularization is used. "
"Use damping or specify force positions that are different from the data."
)
def greens_func(east, north, mindist):
"Calculate the Green's function for the Bi-Harmonic Spline"
distance = np.sqrt(east**2 + north**2)
# The mindist factor helps avoid singular matrices when the force and
# computation point are too close
distance += mindist
return (distance**2) * (np.log(distance) - 1)
def predict_numpy(east, north, force_east, force_north, mindist, forces, result):
"Calculate the predicted data using numpy."
result[:] = 0
for j in range(forces.size):
green = greens_func(east - force_east[j], north - force_north[j], mindist)
result += green * forces[j]
return result
def jacobian_numpy(east, north, force_east, force_north, mindist, jac):
"Calculate the Jacobian using numpy broadcasting."
# Reshaping the data to a column vector will automatically build a distance
# matrix between each data point and force.
jac[:] = greens_func(
east.reshape((east.size, 1)) - force_east,
north.reshape((north.size, 1)) - force_north,
mindist,
)
return jac
@jit(nopython=True, fastmath=True, parallel=True)
def predict_numba(east, north, force_east, force_north, mindist, forces, result):
"Calculate the predicted data using numba to speed things up."
for i in numba.prange(east.size):
result[i] = 0
for j in range(forces.size):
green = GREENS_FUNC_JIT(
east[i] - force_east[j], north[i] - force_north[j], mindist
)
result[i] += green * forces[j]
return result
@jit(nopython=True, fastmath=True, parallel=True)
def jacobian_numba(east, north, force_east, force_north, mindist, jac):
"Calculate the Jacobian matrix using numba to speed things up."
for i in numba.prange(east.size):
for j in range(force_east.size):
jac[i, j] = GREENS_FUNC_JIT(
east[i] - force_east[j], north[i] - force_north[j], mindist
)
return jac
# Jit compile the Green's functions for use in the numba functions
GREENS_FUNC_JIT = jit(nopython=True, fastmath=True)(greens_func)
```