Source code for gemseo.uncertainty.sensitivity.sobol.analysis

# 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: Matthias De Lozzo
r"""Class for the estimation of Sobol' indices.

Let us consider the model :math:`Y=f(X_1,\ldots,X_d)`

- :math:`X_1,\ldots,X_d` are independent random variables,
- :math:`E\left[f(X_1,\ldots,X_d)^2\right]<\infty`.

Then, the following decomposition is unique:

.. math::

   Y=f_0 + \sum_{i=1}^df_i(X_i) + \sum_{i,j=1\atop i\neq j}^d f_{i,j}(X_i,X_j)
   + \sum_{i,j,k=1\atop i\neq j\neq k}^d f_{i,j,k}(X_i,X_j,X_k) + \ldots +


- :math:`f_0=E[Y]`,
- :math:`f_i(X_i)=E[Y|X_i]-f_0`,
- :math:`f_{i,j}(X_i,X_j)=E[Y|X_i,X_j]-f_i(X_i)-f_j(X_j)-f_0`
- and so on.

Then, the shift to variance leads to:

.. math::

   V[Y]=\sum_{i=1}^dV\left[f_i(X_i)\right] +
   \sum_{i,j=1\atop j\neq i}^d V\left[f_{i,j}(X_i,X_j)\right] + \ldots +

and the Sobol' indices are obtained by dividing by the variance and sum up to 1:

.. math::

   1=\sum_{i=1}^dS_i + \sum_{i,j=1\atop j\neq i}^d S_{i,j} +
   \sum_{i,j,k=1\atop i\neq j\neq k}^d S_{i,j,k} + \ldots + S_{1,\ldots,d}

A Sobol' index represents the share of output variance explained
by a parameter or a group of parameters. For the parameter :math:`X_i`,

- :math:`S_i` is the first-order Sobol' index
  measuring the individual effect of :math:`X_i`,
- :math:`S_{i,j}` is the second-order Sobol' index
  measuring the joint effect between :math:`X_i` and :math:`X_j`,
- :math:`S_{i,j,k}` is the third-order Sobol' index
  measuring the joint effect between :math:`X_i`, :math:`X_j` and :math:`X_k`,
- and so on.

In practice, we only consider the first-order Sobol' index:

.. math::


and the total-order Sobol' index:

.. math::

   S_i^T=\sum_{u\subset\{1,\ldots,d\}\atop u \ni i}S_u

The latter represents the sum of the individual effect of :math:`X_i` and
the joint effects between :math:`X_i` and any parameter or group of parameters.

This methodology relies on the :class:`.SobolAnalysis` class. Precisely,
:attr:`.SobolAnalysis.indices` contains
both :attr:`.SobolAnalysis.first_order_indices` and
while :attr:`.SobolAnalysis.main_indices` represents total-order Sobol'
Lastly, the :meth:`.SobolAnalysis.plot` method represents
the estimations of both first-order and total-order Sobol' indices along with
their 95% confidence interval.

The user can select the algorithm to estimate the Sobol' indices.
The computation relies on
`OpenTURNS capabilities <>`_.
from __future__ import annotations

import logging
from pathlib import Path
from typing import Any
from typing import Collection
from typing import Iterable
from typing import Mapping
from typing import Sequence

import matplotlib.pyplot as plt
from matplotlib.transforms import Affine2D
from numpy import array
from openturns import JansenSensitivityAlgorithm
from openturns import MartinezSensitivityAlgorithm
from openturns import MauntzKucherenkoSensitivityAlgorithm
from openturns import SaltelliSensitivityAlgorithm
from openturns import Sample

from gemseo.algos.doe.doe_lib import DOELibraryOptionType
from gemseo.algos.doe.lib_openturns import OpenTURNS
from gemseo.algos.parameter_space import ParameterSpace
from gemseo.core.discipline import MDODiscipline
from gemseo.uncertainty.sensitivity.analysis import IndicesType
from gemseo.uncertainty.sensitivity.analysis import SensitivityAnalysis
from gemseo.utils.data_conversion import split_array_to_dict_of_arrays

LOGGER = logging.getLogger(__name__)

[docs]class SobolAnalysis(SensitivityAnalysis): """Sensitivity analysis based on the Sobol' indices. Examples: >>> from numpy import pi >>> from gemseo.api import create_discipline, create_parameter_space >>> from gemseo.uncertainty.sensitivity.sobol.analysis import SobolAnalysis >>> >>> expressions = {"y": "sin(x1)+7*sin(x2)**2+0.1*x3**4*sin(x1)"} >>> discipline = create_discipline( ... "AnalyticDiscipline", expressions=expressions ... ) >>> >>> parameter_space = create_parameter_space() >>> parameter_space.add_random_variable( ... "x1", "OTUniformDistribution", minimum=-pi, maximum=pi ... ) >>> parameter_space.add_random_variable( ... "x2", "OTUniformDistribution", minimum=-pi, maximum=pi ... ) >>> parameter_space.add_random_variable( ... "x3", "OTUniformDistribution", minimum=-pi, maximum=pi ... ) >>> >>> analysis = SobolAnalysis([discipline], parameter_space, n_samples=10000) >>> indices = analysis.compute_indices() """ _ALGOS = { "Saltelli": SaltelliSensitivityAlgorithm, "Jansen": JansenSensitivityAlgorithm, "MauntzKucherenko": MauntzKucherenkoSensitivityAlgorithm, "Martinez": MartinezSensitivityAlgorithm, } _FIRST = "first" _TOTAL = "total" _FIRST_METHOD = f"Sobol({_FIRST})" _TOTAL_METHOD = f"Sobol({_TOTAL})" AVAILABLE_ALGOS = sorted(_ALGOS.keys()) DEFAULT_DRIVER = OpenTURNS.OT_SOBOL_INDICES def __init__( # noqa: D107,D205,D212,D415 self, disciplines: Collection[MDODiscipline], parameter_space: ParameterSpace, n_samples: int, output_names: Iterable[str] | None = None, algo: str | None = None, algo_options: Mapping[str, DOELibraryOptionType] | None = None, formulation: str = "MDF", **formulation_options: Any, ) -> None: self.__sobol = None super().__init__( disciplines, parameter_space, n_samples=n_samples, output_names=output_names, algo=algo, algo_options=algo_options, formulation=formulation, **formulation_options, ) self.main_method = self._FIRST @SensitivityAnalysis.main_method.setter def main_method( # noqa: D102 self, name: str, ) -> None: if name == self._FIRST: self._main_method = self._FIRST_METHOD"Use first order indices as main indices.") elif name == self._TOTAL: self._main_method = self._TOTAL_METHOD"Use total order indices as main indices.") else: raise NotImplementedError( "{} is a bad method name. " "Available ones are {}.".format(name, [self._FIRST, self._TOTAL]) )
[docs] def compute_indices( self, outputs: Sequence[str] | None = None, algo: str = "Saltelli", ) -> dict[str, IndicesType]: """.. # noqa:D205,D212,D415 Args: algo: The name of the algorithm to estimate the Sobol' indices """ try: algo = self._ALGOS[algo] except Exception: raise TypeError( "{} is not an available algorithm " "to compute Sobol indices.".format(algo) ) output_names = outputs or self.default_output if not isinstance(output_names, list): output_names = [output_names] inputs = Sample(self.dataset.get_data_by_group(self.dataset.INPUT_GROUP)) outputs = self.dataset.get_data_by_names(output_names, True) dim = self.dataset.dimension[self.dataset.INPUT_GROUP] n_samples = int(len(self.dataset) / (dim + 2)) self.__sobol = {} for output_name, value in outputs.items(): self.__sobol[output_name] = [] for index in range(value.shape[1]): sub_outputs = Sample(value[:, index][:, None]) self.__sobol[output_name].append(algo(inputs, sub_outputs, n_samples)) return self.indices
def __get_indices( self, first_order: bool = True, ) -> IndicesType: """Get the indices, either first-order or total order. Args: first_order: The type of indices. If True, return first-order Sobol' indices. Otherwise, return total-order Sobol' indices. Returns: The Sobol' indices, either first-order or total order. """ if first_order: method = "getFirstOrderIndices" else: method = "getTotalOrderIndices" inputs_names = self.dataset.get_names(self.dataset.INPUT_GROUP) sizes = self.dataset.sizes indices = {} for name in self.__sobol: indices[name] = [ split_array_to_dict_of_arrays( array(getattr(sobol, method)()), sizes, inputs_names ) for sobol in self.__sobol[name] ] return indices @property def first_order_indices(self) -> IndicesType: """dict: The first-order Sobol' indices. With the following structure: .. code-block:: python { "output_name": [ { "input_name": data_array, } ] } """ return self.__get_indices() @property def total_order_indices(self) -> IndicesType: """dict: The total-order Sobol' indices. With the following structure: .. code-block:: python { "output_name": [ { "input_name": data_array, } ] } """ return self.__get_indices(False)
[docs] def get_intervals( self, first_order: bool = True, ) -> IndicesType: """Get the confidence interval for Sobol' indices. Args: first_order: The type of indices. If True, returns the intervals for the first-order indices. Otherwise, for the total-order indices. Returns: dict: The confidence intervals for the Sobol' indices. With the following structure: .. code-block:: python { "output_name": [ { "input_name": data_array, } ] } """ inputs_names = self.dataset.get_names(self.dataset.INPUT_GROUP) sizes = self.dataset.sizes intervals = {} for output_name in self.__sobol: intervals[output_name] = [] for sobol in self.__sobol[output_name]: if first_order: interval = sobol.getFirstOrderIndicesInterval() else: interval = sobol.getTotalOrderIndicesInterval() lower_bound = array(interval.getLowerBound()) upper_bound = array(interval.getUpperBound()) lower_bound = split_array_to_dict_of_arrays( lower_bound, sizes, inputs_names ) upper_bound = split_array_to_dict_of_arrays( upper_bound, sizes, inputs_names ) intervals[output_name].append( { name: array([lower_bound[name][0], upper_bound[name][0]]) for name in lower_bound } ) return intervals
@property def indices( # noqa: D102 self, ) -> dict[str, IndicesType]: # noqa: D102 return {"first": self.first_order_indices, "total": self.total_order_indices} @property def main_indices(self) -> IndicesType: # noqa: D102 if self.main_method == self._TOTAL_METHOD: return self.total_order_indices else: return self.first_order_indices
[docs] def plot( self, output: str | tuple[str, int], inputs: Iterable[str] | None = None, title: str | None = None, save: bool = True, show: bool = False, file_path: str | Path | None = None, directory_path: str | Path | None = None, file_name: str | None = None, file_format: str | None = None, sort: bool = True, sort_by_total: bool = True, ): r""".. # noqa: D415,D417 Plot the first- and total-order Sobol' indices. For :math:`i\in\{1,\ldots,d\}`, plot :math:`S_i^{1}` and :math:`S_T^{1}` with their confidence intervals. Args: sort: The sorting option. If True, sort variables before display. sort_by_total: The type of sorting. If True, sort variables according to total-order Sobol' indices. Otherwise, use first-order Sobol' indices. """ if not isinstance(output, tuple): output = (output, 0) fig, ax = plt.subplots() if sort_by_total: indices = self.total_order_indices else: indices = self.first_order_indices intervals = self.get_intervals() indices = indices[output[0]][output[1]] intervals = intervals[output[0]][output[1]] first_order_indices = self.first_order_indices[output[0]][output[1]] total_order_indices = self.total_order_indices[output[0]][output[1]] if sort: names = [ name for name, _ in sorted( list(indices.items()), key=lambda item: item[1], reverse=True ) ] else: names = list(indices.keys()) names = self._filter_names(names, inputs) errorbar_options = {"marker": "o", "linestyle": "", "markersize": 7} trans1 = Affine2D().translate(-0.01, 0.0) + ax.transData trans2 = Affine2D().translate(+0.01, 0.0) + ax.transData values = [first_order_indices[name][0] for name in names] yerr = array( [ [ first_order_indices[name][0] - intervals[name][0], intervals[name][1] - first_order_indices[name][0], ] for name in names ] ).T ax.errorbar( names, values, yerr=yerr, label="First order", transform=trans2, **errorbar_options, ) intervals = self.get_intervals(False) intervals = intervals[output[0]][output[1]] values = [total_order_indices[name][0] for name in names] yerr = array( [ [ total_order_indices[name][0] - intervals[name][0], intervals[name][1] - total_order_indices[name][0], ] for name in names ] ).T ax.errorbar( names, values, yerr, label="Total order", transform=trans1, **errorbar_options, ) ax.legend( loc="lower left", bbox_to_anchor=(0.0, 1.01), ncol=2, borderaxespad=0, frameon=False, ) ax.set_title(title or f"Sobol indices for the output {output[0]}({output[1]})") self._save_show_plot( fig, save=save, show=show, file_path=file_path, file_name=file_name, file_format=file_format, directory_path=directory_path, )