# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
#
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU Lesser General Public
# License version 3 as published by the Free Software Foundation.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with this program; if not, write to the Free Software Foundation,
# Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
# Contributors:
# INITIAL AUTHORS - initial API and implementation and/or initial
# documentation
# :author: Francois Gallard, Matthias De Lozzo
# OTHER AUTHORS - MACROSCOPIC CHANGES
r"""The RBF network for regression.
The radial basis function surrogate discipline expresses the model output
as a weighted sum of kernel functions centered on the learning input data:
.. math::
y = w_1K(\|x-x_1\|;\epsilon) + w_2K(\|x-x_2\|;\epsilon) + \ldots
+ w_nK(\|x-x_n\|;\epsilon)
and the coefficients :math:`(w_1, w_2, \ldots, w_n)` are estimated
by least squares minimization.
Dependence
----------
The RBF model relies on the Rbf class of the
`scipy library
<https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.Rbf.html>`_.
"""
from __future__ import annotations
import logging
from typing import Callable
from typing import ClassVar
from typing import Iterable
from typing import Mapping
from typing import Union
from numpy import average
from numpy import exp
from numpy import finfo
from numpy import log
from numpy import ndarray
from numpy import sqrt
from numpy.linalg import norm
from scipy.interpolate import Rbf
from gemseo.core.dataset import Dataset
from gemseo.mlearning.core.ml_algo import TransformerType
from gemseo.mlearning.core.supervised import SavedObjectType
from gemseo.mlearning.regression.regression import MLRegressionAlgo
from gemseo.utils.python_compatibility import Final
LOGGER = logging.getLogger(__name__)
SavedObjectType = Union[SavedObjectType, float, Callable]
[docs]class RBFRegressor(MLRegressionAlgo):
r"""Regression based on radial basis functions (RBFs).
This model relies on the SciPy class :class:`scipy.interpolate.Rbf`.
"""
der_function: Callable[[ndarray], ndarray]
"""The derivative of the radial basis function."""
y_average: ndarray
"""The mean of the learning output data."""
SHORT_ALGO_NAME: ClassVar[str] = "RBF"
LIBRARY: Final[str] = "SciPy"
EUCLIDEAN: Final[str] = "euclidean"
MULTIQUADRIC: Final[str] = "multiquadric"
INVERSE_MULTIQUADRIC: Final[str] = "inverse_multiquadric"
GAUSSIAN: Final[str] = "gaussian"
LINEAR: Final[str] = "linear"
CUBIC: Final[str] = "cubic"
QUINTIC: Final[str] = "quintic"
THIN_PLATE: Final[str] = "thin_plate"
AVAILABLE_FUNCTIONS: list[str] = [
MULTIQUADRIC,
INVERSE_MULTIQUADRIC,
GAUSSIAN,
LINEAR,
CUBIC,
QUINTIC,
THIN_PLATE,
]
def __init__(
self,
data: Dataset,
transformer: Mapping[str, TransformerType] | None = None,
input_names: Iterable[str] | None = None,
output_names: Iterable[str] | None = None,
function: str | Callable[[float, float], float] = MULTIQUADRIC,
der_function: Callable[[ndarray], ndarray] | None = None,
epsilon: float | None = None,
smooth: float = 0.0,
norm: str | Callable[[ndarray, ndarray], float] = "euclidean",
) -> None:
r"""
Args:
function: The radial basis function taking a radius ``r`` as input,
representing a distance between two points.
If it is a string,
then it must be one of the following:
.. code::
'multiquadric': sqrt((r/self.epsilon)**2 + 1)
'inverse': 1.0/sqrt((r/self.epsilon)**2 + 1)
'gaussian': exp(-(r/self.epsilon)**2)
'linear': r
'cubic': r**3
'quintic': r**5
'thin_plate': r**2 * log(r)
If it is a callable,
then it must take two arguments ``(self, r)``,
e.g. ``lambda self, r: return sqrt((r/self.epsilon)**2 + 1)``
for the multiquadric function.
The epsilon parameter will be available as ``self.epsilon``.
Other keyword arguments passed in will be available as well.
der_function: The derivative of the radial basis function,
only to be provided if ``function`` is a callable
and if the use of the model with its derivative is required.
If ``None`` and if ``function`` is a callable,
an error will be raised.
If ``None`` and if ``function`` is a string,
the class will look for its internal implementation
and will raise an error if it is missing.
The ``der_function`` shall take three arguments
(``input_data``, ``norm_input_data``, ``eps``).
For an RBF of the form function(:math:`r`),
der_function(:math:`x`, :math:`|x|`, :math:`\epsilon`) shall
return :math:`\epsilon^{-1} x/|x| f'(|x|/\epsilon)`.
epsilon: An adjustable constant for Gaussian or multiquadric functions.
If ``None``, use the average distance between input data.
smooth: The degree of smoothness,
``0`` involving an interpolation of the learning points.
norm: The distance metric to be used,
either a distance function name `known by SciPy
<https://docs.scipy.org/doc/scipy/reference/generated/
scipy.spatial.distance.cdist.html>`_
or a function that computes the distance between two points.
"""
if isinstance(function, str):
function = str(function)
super().__init__(
data,
transformer=transformer,
input_names=input_names,
output_names=output_names,
function=function,
epsilon=epsilon,
smooth=smooth,
norm=norm,
)
self.y_average = 0.0
self.der_function = der_function
[docs] class RBFDerivatives:
r"""Derivatives of functions used in :class:`.RBFRegressor`.
For an RBF of the form :math:`f(r)`, :math:`r` scalar,
the derivative functions are defined by :math:`d(f(r))/dx`,
with :math:`r=|x|/\epsilon`. The functions are thus defined
by :math:`df/dx = \epsilon^{-1} x/|x| f'(|x|/\epsilon)`.
This convention is chosen to avoid division by :math:`|x|` when
the terms may be cancelled out, as :math:`f'(r)` often has a term
in :math:`r`.
"""
TOL = finfo(float).eps
[docs] @classmethod
def der_multiquadric(
cls,
input_data: ndarray,
norm_input_data: float,
eps: float,
) -> ndarray:
r"""Compute derivative of :math:`f(r) = \sqrt{r^2 + 1}` wrt :math:`x`.
Args:
input_data: The 1D input data.
norm_input_data: The norm of the input variable.
eps: The correlation length.
Returns:
The derivative of the function.
"""
return input_data / eps**2 / sqrt((norm_input_data / eps) ** 2 + 1)
[docs] @classmethod
def der_inverse_multiquadric(
cls,
input_data: ndarray,
norm_input_data: float,
eps: float,
) -> ndarray:
r"""Compute derivative w.r.t. :math:`x` of the function
:math:`f(r) = 1/\sqrt{r^2 + 1}`.
Args:
input_data: The 1D input data.
norm_input_data: The norm of the input variable.
eps: The correlation length.
Returns:
The derivative of the function.
"""
return -input_data / eps**2 / ((norm_input_data / eps) ** 2 + 1) ** 1.5
[docs] @classmethod
def der_gaussian(
cls,
input_data: ndarray,
norm_input_data: float,
eps: float,
) -> ndarray:
r"""Compute derivative w.r.t. :math:`x` of the function
:math:`f(r) = \exp(-r^2)`.
Args:
input_data: The 1D input data.
norm_input_data: The norm of the input variable.
eps: The correlation length.
Returns:
The derivative of the function.
"""
return -2 * input_data / eps**2 * exp(-((norm_input_data / eps) ** 2))
[docs] @classmethod
def der_linear(
cls,
input_data: ndarray,
norm_input_data: float,
eps: float,
) -> ndarray:
"""Compute derivative w.r.t. :math:`x` of the function
:math:`f(r) = r`.
If :math:`x=0`, return 0 (determined up to a tolerance).
Args:
input_data: The 1D input data.
norm_input_data: The norm of the input variable.
eps: The correlation length.
Returns:
The derivative of the function.
"""
return (
(norm_input_data > cls.TOL)
* input_data
/ eps
/ (norm_input_data + cls.TOL)
)
[docs] @classmethod
def der_cubic(
cls,
input_data: ndarray,
norm_input_data: float,
eps: float,
) -> ndarray:
"""Compute derivative w.r.t. :math:`x` of the function
:math:`f(r) = r^3`.
Args:
input_data: The 1D input data.
norm_input_data: The norm of the input variable.
eps: The correlation length.
Returns:
The derivative of the function.
"""
return 3 * norm_input_data * input_data / eps**3
[docs] @classmethod
def der_quintic(
cls,
input_data: ndarray,
norm_input_data: float,
eps: float,
) -> ndarray:
"""Compute derivative w.r.t. :math:`x` of the function
:math:`f(r) = r^5`.
Args:
input_data: The 1D input data.
norm_input_data : The norm of the input variable.
eps: The correlation length.
Returns:
The derivative of the function.
"""
return 5 * norm_input_data**3 * input_data / eps**5
[docs] @classmethod
def der_thin_plate(
cls,
input_data: ndarray,
norm_input_data: float,
eps: float,
) -> ndarray:
r"""Compute derivative w.r.t. :math:`x` of the function
:math:`f(r) = r^2 \log(r)`.
If :math:`x=0`, return 0 (determined up to a tolerance).
Args:
input_data: The 1D input data.
norm_input_data: The norm of the input variable.
eps: The correlation length.
Returns:
The derivative of the function.
"""
return (
(norm_input_data > cls.TOL)
* input_data
/ eps**2
* (1 + 2 * log(norm_input_data / eps + cls.TOL))
)
def _fit(
self,
input_data: ndarray,
output_data: ndarray,
) -> None:
self.y_average = average(output_data, axis=0)
output_data -= self.y_average
args = list(input_data.T) + [output_data]
self.algo = Rbf(
*args,
mode="N-D",
function=self.parameters["function"],
epsilon=self.parameters["epsilon"],
smooth=self.parameters["smooth"],
norm=self.parameters["norm"],
)
def _predict(
self,
input_data: ndarray,
) -> ndarray:
output_data = self.algo(*input_data.T)
if len(output_data.shape) == 1:
output_data = output_data[:, None] # n_outputs=1, rbf reduces
return output_data + self.y_average
def _predict_jacobian(
self,
input_data: ndarray,
) -> ndarray:
self._check_available_jacobian()
der_func = self.der_function or getattr(
self.RBFDerivatives, f"der_{self.function}"
)
# predict_samples learn_samples
# Dimensions : ( n_samples , n_outputs , n_inputs , n_learn_samples )
# input_data : ( n_samples , , n_inputs , )
# ref_points : ( , , n_inputs , n_learn_samples )
# nodes : ( , n_outputs , , n_learn_samples )
# jacobians : ( n_samples , n_outputs , n_inputs , )
eps = self.algo.epsilon
ref_points = self.algo.xi[None, None]
nodes = self.algo.nodes.T[None, :, None]
input_data = input_data[:, None, :, None]
diffs = input_data - ref_points
dists = norm(diffs, axis=2)[:, :, None]
contributions = nodes * der_func(diffs, dists, eps=eps)
return contributions.sum(-1)
def _check_available_jacobian(self) -> None:
"""Check if the Jacobian is available for the given setup.
Raises:
NotImplementedError: Either if the Jacobian computation is not implemented
or if the derivative of the radial basis function is missing.
"""
if self.algo.norm != self.EUCLIDEAN:
raise NotImplementedError(
"Jacobian is only implemented for Euclidean norm."
)
if callable(self.function) and self.der_function is None:
raise NotImplementedError(
"No der_function is provided."
"Add der_function in RBFRegressor constructor."
)
def _get_objects_to_save(self) -> dict[str, SavedObjectType]:
objects = super()._get_objects_to_save()
objects["y_average"] = self.y_average
objects["der_function"] = self.der_function
return objects
@property
def function(self) -> str:
"""The name of the kernel function.
The name is possibly different from self.parameters['function'], as it
is mapped (scipy). Examples:
'inverse' -> 'inverse_multiquadric'
'InverSE MULtiQuadRIC' -> 'inverse_multiquadric'
"""
return self.algo.function
