Source code for gemseo.mlearning.regression.gpr

# Copyright 2021 IRT Saint Exupéry,
# 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
# 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
r"""Gaussian process regression model.


The Gaussian process regression (GPR) model
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)


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} + \hat{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
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 surrogate model of :math:`f`.

Error measure

The standard deviation :math:`\hat{s}` is a local error measure
of :math:`\hat{f}`:

.. math::


Interpolation or regression

The GPR model 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.

The GPR model relies on the GaussianProcessRegressor class
of the `scikit-learn library <
from __future__ import annotations

from typing import Callable
from typing import ClassVar
from typing import Final
from typing import Iterable
from typing import Mapping
from typing import Tuple

import sklearn.gaussian_process
from numpy import atleast_2d
from numpy import ndarray
from numpy import newaxis
from numpy import repeat
from sklearn.gaussian_process.kernels import Kernel

from gemseo.datasets.io_dataset import IODataset
from gemseo.mlearning.core.ml_algo import DataType
from gemseo.mlearning.core.ml_algo import TransformerType
from gemseo.mlearning.regression.regression import MLRegressionAlgo
from gemseo.utils.data_conversion import concatenate_dict_of_arrays_to_array

__Bounds = Tuple[float, float]

[docs]class GaussianProcessRegressor(MLRegressionAlgo): """Gaussian process regression model.""" SHORT_ALGO_NAME: ClassVar[str] = "GPR" LIBRARY: Final[str] = "scikit-learn" __DEFAULT_BOUNDS: Final[tuple[float, float]] = (0.01, 100.0) def __init__( self, data: IODataset, transformer: TransformerType = MLRegressionAlgo.IDENTITY, input_names: Iterable[str] | None = None, output_names: Iterable[str] | None = None, kernel: Kernel | None = None, bounds: __Bounds | Mapping[str, __Bounds] | None = None, alpha: float | ndarray = 1e-10, optimizer: str | Callable = "fmin_l_bfgs_b", n_restarts_optimizer: int = 10, random_state: int | None = None, ) -> None: """ Args: kernel: The kernel specifying the covariance model. If ``None``, use a Matérn(2.5). bounds: The lower and upper bounds of the parameter length scales when ``kernel`` is ``None``. Either a unique lower-upper pair common to all the inputs or lower-upper pairs for some of them. When ``bounds`` is ``None`` or when an input has no pair, the lower bound is 0.01 and the upper bound is 100. alpha: The nugget effect to regularize the model. optimizer: The optimization algorithm to find the parameter length scales. n_restarts_optimizer: The number of restarts of the optimizer. random_state: The seed used to initialize the centers. If ``None``, the random number generator is the RandomState instance used by `numpy.random`. """ super().__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: kernel = sklearn.gaussian_process.kernels.Matern( (1.0,) * self._reduced_dimensions[0], self.__compute_parameter_length_scale_bounds(bounds), nu=2.5, ) self.algo = sklearn.gaussian_process.GaussianProcessRegressor( kernel=kernel, alpha=alpha, optimizer=optimizer, n_restarts_optimizer=n_restarts_optimizer, random_state=random_state, ) self.parameters["kernel"] = kernel.__class__.__name__ @property def kernel(self): # (...) -> Kernel """The kernel used for prediction.""" if self.is_trained: return self.algo.kernel_ else: return self.algo.kernel def __compute_parameter_length_scale_bounds( self, bounds: __Bounds | Mapping[str, __Bounds] | None, ) -> list[tuple[float, float]]: """Return the lower and upper bounds for the parameter length scales. Args: bounds: The lower and upper bounds of the parameter length scales. Either a unique lower-upper pair common to all the inputs or lower-upper pairs for some of them. When an input has no pair, the lower bound is 0.01 and the upper bound is 100. Returns: The lower and upper bounds of the parameter length scales. """ dimension = self._reduced_dimensions[0] if bounds is None: return [self.__DEFAULT_BOUNDS] * dimension if isinstance(bounds, tuple): return [bounds] * dimension bounds_ = [] for name in self.input_names: name_bounds = bounds.get(name, self.__DEFAULT_BOUNDS) bounds_.extend([name_bounds] * self.sizes[name]) return bounds_ def _fit( self, input_data: ndarray, output_data: ndarray, ) -> None:, output_data) def _predict( self, input_data: ndarray, ) -> ndarray: output_data = self.algo.predict(input_data) if output_data.ndim == 1: output_data = output_data[:, newaxis] return output_data
[docs] def predict_std( self, input_data: DataType, ) -> ndarray: """Predict the standard deviation from input data. The user can specify these input data either as a NumPy array, e.g. ``array([1., 2., 3.])`` or as a dictionary of NumPy arrays, e.g. ``{'a': array([1.]), 'b': array([2., 3.])}``. If the NumPy arrays are of dimension 2, their i-th rows represent the input data of the i-th sample; while if the NumPy arrays are of dimension 1, there is a single sample. Args: input_data: The input data. Returns: The standard deviation at the query points. Warnings: If the output variables are transformed before the training stage, then the standard deviation is related to this transformed output space unlike :meth:`.predict` which returns values in the original output space. """ if isinstance(input_data, Mapping): input_data = concatenate_dict_of_arrays_to_array( input_data, self.input_names ) input_data = atleast_2d(input_data) transformer = self.transformer.get(self.learning_set.INPUT_GROUP) if transformer: input_data = transformer.transform(input_data) output_data = self.algo.predict(input_data, return_std=True)[1] if output_data.ndim == 1: output_data = repeat( output_data[:, newaxis], self._reduced_dimensions[1], 1 ) return output_data