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

# 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
#
# 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: Matthias De Lozzo
#    OTHER AUTHORS   - MACROSCOPIC CHANGES
r"""Class for the estimation of Sobol' indices.

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

- :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 +
f_{1,\ldots,d}(X_1,\ldots,X_d)

where:

- :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 +
V\left[f_{1,\ldots,d}(X_1,\ldots,X_d)\right]

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::

S_i=\frac{V[E[Y|X_i]]}{V[Y]}

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
:attr:.SobolAnalysis.total_order_indices
while :attr:.SobolAnalysis.main_indices represents total-order Sobol'
indices.
Lastly, the :meth:.SobolAnalysis.plot method represents
the estimations of both first-order and total-order Sobol' indices along with
their confidence intervals whose default level is 95%.

The user can select the algorithm to estimate the Sobol' indices.
The computation relies on
OpenTURNS capabilities <https://openturns.github.io/www/>_.
"""
from __future__ import annotations

import logging
from pathlib import Path
from typing import Any
from typing import ClassVar
from typing import Collection
from typing import Final
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 numpy import newaxis
from numpy.typing import NDArray
from openturns import JansenSensitivityAlgorithm
from openturns import MartinezSensitivityAlgorithm
from openturns import MauntzKucherenkoSensitivityAlgorithm
from openturns import SaltelliSensitivityAlgorithm
from openturns import Sample
from strenum import PascalCaseStrEnum
from strenum import StrEnum

from gemseo.algos.doe.doe_library 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.post.dataset.dataset_plot import VariableType
from gemseo.uncertainty.sensitivity.analysis import FirstOrderIndicesType
from gemseo.uncertainty.sensitivity.analysis import SecondOrderIndicesType
from gemseo.uncertainty.sensitivity.analysis import SensitivityAnalysis
from gemseo.utils.data_conversion import split_array_to_dict_of_arrays
from gemseo.utils.string_tools import repr_variable

LOGGER = logging.getLogger(__name__)

[docs]class SobolAnalysis(SensitivityAnalysis): """Sensitivity analysis based on the Sobol' indices. Examples: >>> from numpy import pi >>> from gemseo 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() """
[docs] class Algorithm(PascalCaseStrEnum): """The algorithms to estimate the Sobol' indices.""" SALTELLI = "Saltelli" JANSEN = "Jansen" MAUNTZ_KUCHERENKO = "MauntzKucherenko" MARTINEZ = "Martinez"
__ALGO_NAME_TO_CLASS: dict[Algorithm, type] = { Algorithm.SALTELLI: SaltelliSensitivityAlgorithm, Algorithm.JANSEN: JansenSensitivityAlgorithm, Algorithm.MAUNTZ_KUCHERENKO: MauntzKucherenkoSensitivityAlgorithm, Algorithm.MARTINEZ: MartinezSensitivityAlgorithm, }
[docs] class Method(StrEnum): """The names of the sensitivity methods.""" FIRST = "first" """The first-order Sobol' index.""" TOTAL = "total" """The total-order Sobol' index."""
__SECOND: Final[str] = "second" __GET_FIRST_ORDER_INDICES: Final[str] = "getFirstOrderIndices" __GET_SECOND_ORDER_INDICES: Final[str] = "getSecondOrderIndices" __GET_TOTAL_ORDER_INDICES: Final[str] = "getTotalOrderIndices" DEFAULT_DRIVER: ClassVar[str] = OpenTURNS.OT_SOBOL_INDICES output_variances: dict[str, NDArray[float]] """The variances of the output variables.""" output_standard_deviations: dict[str, NDArray[float]] """The standard deviations of the output variables.""" def __init__( 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", compute_second_order: bool = True, use_asymptotic_distributions: bool = True, **formulation_options: Any, ) -> None: r""" Args: compute_second_order: Whether to compute the second-order indices. use_asymptotic_distributions: Whether to estimate the confidence intervals of the first- and total-order Sobol' indices with the asymptotic distributions; otherwise, use bootstrap. Notes: The estimators of Sobol' indices rely on the same DOE algorithm. This algorithm starts with two independent input datasets composed of :math:N independent samples and this number :math:N is the usual sampling size for Sobol' analysis. When compute_second_order=False or when the input dimension :math:d is equal to 2, :math:N=\frac{n_\text{samples}}{2+d}. Otherwise, :math:N=\frac{n_\text{samples}}{2+2d}. The larger :math:N, the more accurate the estimators of Sobol' indices are. Therefore, for a small budget n_samples, the user can choose to set compute_second_order to False to ensure a better estimation of the first- and second-order indices. """ # noqa: D205, D212, D415 self.__output_names_to_sobol_algos = {} if algo_options is None: algo_options = {} algo_options["eval_second_order"] = compute_second_order super().__init__( disciplines, parameter_space, n_samples=n_samples, output_names=output_names, algo=algo, algo_options=algo_options, formulation=formulation, **formulation_options, ) self.__eval_second_order = compute_second_order self.__use_asymptotic_distributions = use_asymptotic_distributions self._main_method = self.Method.FIRST dataset = self.dataset input_dimension = parameter_space.dimension sample_size = len(dataset) // ( 2 + input_dimension * (1 + (compute_second_order and input_dimension > 2)) ) # Variance computation. _output_variances = ( dataset.get_view(group_names=dataset.OUTPUT_GROUP) .to_numpy()[:sample_size] .var(0) ) self.output_variances = { column[1]: [] for column in dataset.get_view(group_names=dataset.OUTPUT_GROUP).columns } for i, column in enumerate( dataset.get_view(group_names=dataset.OUTPUT_GROUP).columns ): self.output_variances[column[1]].extend([_output_variances[i]]) self.output_variances = {k: array(v) for k, v in self.output_variances.items()} self.output_standard_deviations = { k: v**0.5 for k, v in self.output_variances.items() }
[docs] def compute_indices( self, outputs: str | Sequence[str] | None = None, algo: Algorithm = Algorithm.SALTELLI, confidence_level: float = 0.95, ) -> dict[str, FirstOrderIndicesType]: """ Args: algo: The name of the algorithm to estimate the Sobol' indices. confidence_level: The level of the confidence intervals. """ # noqa:D205,D212,D415 output_names = outputs or self.default_output if isinstance(output_names, str): output_names = [output_names] inputs = Sample( self.dataset.get_view(group_names=self.dataset.INPUT_GROUP).to_numpy() ) input_dimension = self.dataset.group_names_to_n_components[ self.dataset.INPUT_GROUP ] # If eval_second_order is set to False, the input design is of size N(2+n_X). # If eval_second_order is set to False, # if n_X = 2, the input design is of size N(2+n_X). # if n_X != 2, the input design is of size N(2+2n_X). # Ref: https://openturns.github.io/openturns/latest/user_manual/_generated/ # openturns.SobolIndicesExperiment.html#openturns.SobolIndicesExperiment n_samples = len(self.dataset) if self.__eval_second_order and input_dimension > 2: sub_sample_size = int(n_samples / (2 * input_dimension + 2)) else: sub_sample_size = int(n_samples / (input_dimension + 2)) self.__output_names_to_sobol_algos = {} for output_name in output_names: output_data = self.dataset.get_view( group_names=self.dataset.OUTPUT_GROUP, variable_names=output_name ).to_numpy() algos = self.__output_names_to_sobol_algos[output_name] = [] for sub_output_data in output_data.T: algos.append( self.__ALGO_NAME_TO_CLASS[algo]( inputs, Sample(sub_output_data[:, newaxis]), sub_sample_size ) ) algos[-1].setUseAsymptoticDistribution( self.__use_asymptotic_distributions ) algos[-1].setConfidenceLevel(confidence_level) return self.indices
def __get_indices( self, method_name: str ) -> FirstOrderIndicesType | SecondOrderIndicesType: """Get the first-, second- or total-order indices. Args: method_name: The name of the OpenTURNS method to compute the indices. Returns: The first-, second- or total-order indices. """ input_names = self.dataset.get_variable_names(self.dataset.INPUT_GROUP) names_to_sizes = self.dataset.variable_names_to_n_components indices = { output_name: [ split_array_to_dict_of_arrays( array(getattr(ot_algorithm, method_name)()), names_to_sizes, input_names, ) for ot_algorithm in self.__output_names_to_sobol_algos[output_name] ] for output_name in self.__output_names_to_sobol_algos } if method_name == self.__GET_SECOND_ORDER_INDICES: return { output_name: [ { k: split_array_to_dict_of_arrays( v.T, names_to_sizes, input_names ) for k, v in output_component_indices.items() } for output_component_indices in output_indices ] for output_name, output_indices in indices.items() } return indices @property def first_order_indices(self) -> FirstOrderIndicesType: """The first-order Sobol' indices. With the following structure: .. code-block:: python { "output_name": [ { "input_name": data_array, } ] } """ return self.__get_indices(self.__GET_FIRST_ORDER_INDICES) @property def second_order_indices(self) -> SecondOrderIndicesType: """The second-order Sobol' indices. With the following structure: .. code-block:: python { "output_name": [ { {"input_name": {"other_input_name": data_array}, } ] } """ if not self.__eval_second_order: return {} return self.__get_indices(self.__GET_SECOND_ORDER_INDICES) @property def total_order_indices(self) -> FirstOrderIndicesType: """The total-order Sobol' indices. With the following structure: .. code-block:: python { "output_name": [ { "input_name": data_array, } ] } """ return self.__get_indices(self.__GET_TOTAL_ORDER_INDICES) def __unscale_index( self, sobol_index: NDArray[float] | Mapping[str, NDArray[float]], output_name: str, output_index: int, use_variance: bool, ) -> NDArray[float] | dict[str, NDArray[float]]: """Unscaled a Sobol' index. Args: sobol_index: The Sobol' index to unscale. output_name: The name of the related output. output_index: The index of the related output. use_variance: Whether to use the variance of the outputs; otherwise, use their standard deviation. Returns: The unscaled Sobol' index. """ factor = self.output_variances[output_name][output_index] if isinstance(sobol_index, Mapping): unscaled_data = {k: v * factor for k, v in sobol_index.items()} if not use_variance: return {k: v**0.5 for k, v in unscaled_data.items()} else: unscaled_data = sobol_index * factor if not use_variance: return unscaled_data**0.5 return unscaled_data
[docs] def unscale_indices( self, indices: FirstOrderIndicesType | SecondOrderIndicesType, use_variance: bool = True, ) -> FirstOrderIndicesType | SecondOrderIndicesType: """Unscale the Sobol' indices. Args: indices: The Sobol' indices. use_variance: Whether to express an unscaled Sobol' index as a share of output variance; otherwise, express it as the square root of this part and therefore with the same unit as the output. Returns: The unscaled Sobol' indices. """ return { output_name: [ { input_name: self.__unscale_index( sensitivity_indices, output_name, i, use_variance ) for input_name, sensitivity_indices in output_value.items() } for i, output_value in enumerate(output_sensitivity_indices) ] for output_name, output_sensitivity_indices in indices.items() }
[docs] def get_intervals( self, first_order: bool = True, ) -> FirstOrderIndicesType: """Get the confidence intervals for the Sobol' indices. Warnings: You must first call :meth:.compute_indices. Args: first_order: If True, compute the intervals for the first-order indices. Otherwise, for the total-order indices. Returns: The confidence intervals for the Sobol' indices. With the following structure: .. code-block:: python { "output_name": [ { "input_name": data_array, } ] } """ input_names = self.dataset.get_variable_names(self.dataset.INPUT_GROUP) names_to_sizes = self.dataset.variable_names_to_n_components intervals = {} for output_name, sobol_algos in self.__output_names_to_sobol_algos.items(): intervals[output_name] = [] for sobol_algorithm in sobol_algos: if first_order: interval = sobol_algorithm.getFirstOrderIndicesInterval() else: interval = sobol_algorithm.getTotalOrderIndicesInterval() names_to_lower_bounds = split_array_to_dict_of_arrays( array(interval.getLowerBound()), names_to_sizes, input_names ) names_to_upper_bounds = split_array_to_dict_of_arrays( array(interval.getUpperBound()), names_to_sizes, input_names ) intervals[output_name].append( { input_name: ( names_to_lower_bounds[input_name], names_to_upper_bounds[input_name], ) for input_name in input_names } ) return intervals
@property def indices( # noqa: D102 self, ) -> dict[str, FirstOrderIndicesType | SecondOrderIndicesType]: return { self.Method.FIRST: self.first_order_indices, self.__SECOND: self.second_order_indices, self.Method.TOTAL: self.total_order_indices, }
[docs] def plot( self, output: VariableType, 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, ) -> None: r"""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. """ # noqa: D415 D417 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() output_name, output_component = output indices = indices[output_name][output_component] intervals = intervals[output_name][output_component] first_order_indices = self.first_order_indices[output_name][output_component] total_order_indices = self.total_order_indices[output_name][output_component] if sort: names = [ name for name, _ in sorted( indices.items(), key=lambda item: item[1].sum(), reverse=True ) ] else: names = 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 names_to_sizes = { name: value.size for name, value in first_order_indices.items() } values = [ first_order_indices[name][index] for name in names for index in range(names_to_sizes[name]) ] yerr = array( [ [ first_order_indices[name][index] - intervals[name][0][index], intervals[name][1][index] - first_order_indices[name][index], ] for name in names for index in range(names_to_sizes[name]) ] ).T x_labels = [] for name in names: if names_to_sizes[name] == 1: x_labels.append(name) else: size = names_to_sizes[name] x_labels.extend( [repr_variable(name, index, size) for index in range(size)] ) ax.errorbar( x_labels, values, yerr=yerr, label="First order", transform=trans2, **errorbar_options, ) intervals = self.get_intervals(False) intervals = intervals[output_name][output_component] values = [ total_order_indices[name][index] for name in names for index in range(names_to_sizes[name]) ] yerr = array( [ [ total_order_indices[name][index] - intervals[name][0][index], intervals[name][1][index] - total_order_indices[name][index], ] for name in names for index in range(names_to_sizes[name]) ] ).T ax.errorbar( x_labels, values, yerr, label="Total order", transform=trans1, **errorbar_options, ) ax.legend(loc="lower left") pretty_output_name = repr_variable( output_name, output_component, len(self.total_order_indices[output_name]), ) if title is None: title = f"Sobol' indices for the output {pretty_output_name}" variance = self.output_variances[output_name][output_component] ax.set_title(f"{title}\nVar[{pretty_output_name}]={variance:.1e}") ax.set_axisbelow(True) ax.grid() 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, )