# -*- coding: utf-8 -*-
# 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"""
Gaussian process regression
===========================
Overview
--------
The Gaussian process regression (GPR) surrogate discipline
expresses the model output as a weighted sum of kernel functions
centered on the learning input data:
.. math::
y = \mu
+ w_1\kappa(\|x-x_1\|;\epsilon)
+ w_2\kappa(\|x-x_2\|;\epsilon)
+ ...
+ w_N\kappa(\|x-x_N\|;\epsilon)
Details
-------
The GPR model relies on the assumption
that the original model :math:`f` to replace
is an instance of a Gaussian process (GP) with mean :math:`\mu`
and covariance :math:`\sigma^2\kappa(\|x-x'|;\epsilon)`.
Then, the GP conditioned by the learning set
:math:`(x_i,y_i)_{1\leq i \leq N}`
is entirely defined by its expectation:
.. math::
\hat{f}(x) = \hat{\mu} + w^T k(x)
and its covariance:
.. math::
\hat{c}(x,x') = \hat{\sigma}^2 - k(x)^T K^{-1} k(x')
where :math:`[\hat{\mu};\hat{w}]=([1_N~K]^T[1_N~K])^{-1}[1_N~K]^TY` with
:math:`K_{ij}=\kappa(\|x_i-x_j\|;\hat{\epsilon})`,
:math:`k_i(x)=\kappa(\|x-x_i\|;\hat{\epsilon})`
and :math:`Y_i=y_i`.
The correlation length vector :math:`\epsilon`
is estimated by numerical non-linear optimization.
Surrogate model
---------------
The expectation :math:`\hat{f}` is the GPR surrogate model of :math:`f`.
Error measure
-------------
The standard deviation :math:`\hat{s}` is a local error measure
of :math:`\hat{f}`:
.. math::
\hat{s}(x):=\sqrt{c(x,x)}
Interpolation or regression
---------------------------
The GPR surrogate discipline can be regressive or interpolative
according to the value of the nugget effect :math:`\\alpha\geq 0`
which is a regularization term
applied to the correlation matrix :math:`K`.
When :math:`\\alpha = 0`,
the surrogate model interpolates the learning data.
Dependence
----------
The GPR model relies on the GaussianProcessRegressor class
of the `scikit-learn library <https://scikit-learn.org/stable/modules/
generated/sklearn.gaussian_process.GaussianProcessRegressor.html>`_.
"""
from __future__ import absolute_import, division, unicode_literals
from future import standard_library
from numpy import atleast_2d
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import Matern
from gemseo.mlearning.regression.regression import MLRegressionAlgo
from gemseo.utils.data_conversion import DataConversion
standard_library.install_aliases()
from gemseo import LOGGER
[docs]class GaussianProcessRegression(MLRegressionAlgo):
""" Gaussian process regression """
LIBRARY = "scikit-learn"
ABBR = "GPR"
def __init__(
self,
data,
transformer=None,
input_names=None,
output_names=None,
kernel=None,
alpha=1e-10,
optimizer="fmin_l_bfgs_b",
n_restarts_optimizer=10,
random_state=None,
):
"""Constructor.
:param data: learning dataset
:type data: Dataset
:param transformer: transformation strategy for data groups.
If None, do not transform data. Default: None.
:type transformer: dict(str)
:param input_names: names of the input variables.
:type input_names: list(str)
:param output_names: names of the output variables.
:type output_names: list(str)
:param kernel: kernel function. If None, use a Matern(2.5).
Default: None.
:type kernel: openturns.Kernel
:param alpha: nugget effect. Default: 1e-10.
:type alpha: float or array
:param optimizer: optimization algorithm. Default: 'fmin_l_bfgs_b'.
:type optimizer: str or callable
:param n_restarts_optimizer: number of restarts of the optimizer.
Default: 10.
:type n_restarts_optimizer: int
:param random_state: the seed used to initialize the centers.
If None, the random number generator is the RandomState instance
used by `np.random`
Default: None.
:type random_state: int
"""
super(GaussianProcessRegression, self).__init__(
data,
transformer=transformer,
input_names=input_names,
output_names=output_names,
kernel=kernel,
alpha=alpha,
optimizer=optimizer,
n_restarts_optimizer=n_restarts_optimizer,
random_state=random_state,
)
if kernel is None:
raw_input_shape, _ = self._get_raw_shapes()
self.kernel = Matern(
(1.0,) * raw_input_shape, [(0.01, 100)] * raw_input_shape, nu=2.5
)
else:
self.kernel = kernel
nro = n_restarts_optimizer
self.algo = GaussianProcessRegressor(
normalize_y=False,
kernel=self.kernel,
copy_X_train=True,
alpha=alpha,
optimizer=optimizer,
n_restarts_optimizer=nro,
random_state=random_state,
)
self.parameters["kernel"] = self.kernel.__class__.__name__
def _fit(self, input_data, output_data):
"""Fit the regression model.
:param ndarray input_data: input data (2D)
:param ndarray output_data: output data (2D)
"""
self.algo.fit(input_data, output_data)
def _predict(self, input_data):
"""Predict output.
:param ndarray input_data: input data (2D).
:return: output prediction (2D).
:rtype: ndarray
"""
output_pred = self.algo.predict(input_data, False)
return output_pred
[docs] def predict_std(self, input_data):
"""Predict standard deviation value for given input data.
:param dict(ndarray) input_data: input data (1D or 2D).
:return: output data (1D or 2D, same as input_data).
:rtype: dict(ndarray)
"""
as_dict = isinstance(input_data, dict)
if as_dict:
input_data = DataConversion.dict_to_array(input_data, self.input_names)
input_data = atleast_2d(input_data)
inputs = self.learning_set.INPUT_GROUP
if inputs in self.transformer:
input_data = self.transformer[inputs].transform(input_data)
_, output_std = self.algo.predict(input_data, True)
return sum(output_std) / len(output_std)
