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

# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
#
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# Lesser General Public License for more details.
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# 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 Morris indices.

OAT technique
-------------

The purpose of the One-At-a-Time (OAT) methodology is to quantify the elementary effect

.. math::

   df_i = f(X_1+dX_1,\ldots,X_{i-1}+dX_{i-1},X_i+dX_i,\ldots,X_d)
          -
          f(X_1+dX_1,\ldots,X_{i-1}+dX_{i-1},X_i,\ldots,X_d)

associated with a small variation :math:`dX_i` of :math:`X_i` with

.. math::

   df_1 = f(X_1+dX_1,\ldots,X_d)-f(X_1,\ldots,X_d)

The elementary effects :math:`df_1,\ldots,df_d` are computed sequentially
from an initial point

.. math::

   X=(X_1,\ldots,X_d)

From these elementary effects, we can compare their absolute values
:math:`|df_1|,\ldots,|df_d|` and sort :math:`X_1,\ldots,X_d` accordingly.

Morris technique
----------------

Then, the purpose of the Morris' methodology is to repeat the OAT method
from different initial points :math:`X^{(1)},\ldots,X^{(r)}`
and compare the parameters in terms of mean

.. math::

   \mu_i^* = \frac{1}{r}\sum_{j=1}^r|df_i^{(j)}|

and standard deviation

.. math::

   \sigma_i = \frac{1}{r}\sum_{j=1}^r\left(|df_i^{(j)}|-\mu_i\right)^2

where :math:`\mu_i = \frac{1}{r}\sum_{j=1}^rdf_i^{(j)}`.

This methodology relies on the :class:`.MorrisAnalysis` class.
"""

from __future__ import annotations

from types import MappingProxyType
from typing import TYPE_CHECKING
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 numpy import abs as np_abs
from numpy import array
from numpy import ndarray
from strenum import StrEnum

from gemseo.algos.doe.lib_pydoe import PyDOE
from gemseo.disciplines.utils import get_all_outputs
from gemseo.uncertainty.sensitivity.analysis import FirstOrderIndicesType
from gemseo.uncertainty.sensitivity.analysis import SensitivityAnalysis
from gemseo.uncertainty.sensitivity.morris.oat import _OATSensitivity
from gemseo.utils.string_tools import repr_variable

if TYPE_CHECKING:
    from pathlib import Path

    from gemseo.algos.doe.doe_library import DOELibraryOptionType
    from gemseo.algos.parameter_space import ParameterSpace
    from gemseo.core.discipline import MDODiscipline
    from gemseo.post.dataset.dataset_plot import VariableType


[docs]class MorrisAnalysis(SensitivityAnalysis): r"""Sensitivity analysis based on the Morris' indices. :attr:`.MorrisAnalysis.indices` contains both :math:`\mu^*`, :math:`\mu` and :math:`\sigma` while :attr:`.MorrisAnalysis.main_indices` represents :math:`\mu^*`. Lastly, the :meth:`.MorrisAnalysis.plot` method represents the parameters as a scatter plot where :math:`X_i` has as coordinates :math:`(\mu_i^*,\sigma_i)`. The bigger :math:`\mu_i^*` is, the more significant :math:`X_i` is. Concerning :math:`\sigma_i`, it highlights non-linear effects along :math:`X_i` or cross-effects between :math:`X_i` and other parameter(s). The user can specify the DOE algorithm name to select the initial points, as well as the number of replicates and the relative step for the input variations. Examples: >>> from numpy import pi >>> from gemseo import create_discipline, create_parameter_space >>> from gemseo.uncertainty.sensitivity.morris.analysis import MorrisAnalysis >>> >>> 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 = MorrisAnalysis([discipline], parameter_space, n_samples=None) >>> indices = analysis.compute_indices() """ mu_: dict[str, dict[str, ndarray]] """The mean effects with the following structure: .. code-block:: python { "output_name": [ { "input_name": data_array, } ] } """ mu_star: dict[str, dict[str, ndarray]] """The mean absolute effects with the following structure: .. code-block:: python { "output_name": [ { "input_name": data_array, } ] } """ sigma: dict[str, dict[str, ndarray]] """The variability of the effects with the following structure: .. code-block:: python { "output_name": [ { "input_name": data_array, } ] } """ relative_sigma: dict[str, dict[str, ndarray]] """The relative variability of the effects with the following structure: .. code-block:: python { "output_name": [ { "input_name": data_array, } ] } """ min: dict[str, dict[str, ndarray]] # noqa: A003 """The minimum effect with the following structure: .. code-block:: python { "output_name": [ { "input_name": data_array, } ] } """ max: dict[str, dict[str, ndarray]] # noqa: A003 """The maximum effect with the following structure: .. code-block:: python { "output_name": [ { "input_name": data_array, } ] } """ DEFAULT_DRIVER = PyDOE.PYDOE_LHS
[docs] class Method(StrEnum): """The names of the sensitivity methods.""" MU_STAR = "MU_STAR" """The mean of the absolute finite difference.""" SIGMA = "SIGMA" """The standard deviation of the absolute finite difference."""
def __init__( self, disciplines: Collection[MDODiscipline], parameter_space: ParameterSpace, n_samples: int | None, output_names: Iterable[str] = (), algo: str = "", algo_options: Mapping[str, DOELibraryOptionType] = MappingProxyType({}), n_replicates: int = 5, step: float = 0.05, formulation: str = "MDF", **formulation_options: Any, ) -> None: r""" Args: n_replicates: The number of times the OAT method is repeated. Used only if ``n_samples`` is None. Otherwise, this number is the greater integer :math:`r` such that :math:`r(d+1)\leq` ``n_samples`` and :math:`r(d+1)` is the number of samples actually carried out. step: The finite difference step of the OAT method. Raises: ValueError: If at least one input dimension is not equal to 1. """ # noqa: D205, D212, D415 if parameter_space.dimension != len(parameter_space.variable_names): raise ValueError("Each input dimension must be equal to 1.") self.mu_ = {} self.mu_star = {} self.sigma = {} self.relative_sigma = {} self.min = {} self.max = {} self.__step = step if n_samples is None: self.__n_replicates = n_replicates else: self.__n_replicates = n_samples // (parameter_space.dimension + 1) if self.__n_replicates == 0: raise ValueError( f"The number of samples ({n_samples}) must be " "at least equal to the dimension of the input space plus one " f"({parameter_space.dimension}+1={parameter_space.dimension + 1})." ) disciplines = list(disciplines) if not output_names: output_names = get_all_outputs(disciplines) scenario = self._create_scenario( disciplines, output_names, formulation, formulation_options, parameter_space, ) discipline = _OATSensitivity(scenario, parameter_space, step) super().__init__( [discipline], parameter_space, n_samples=self.__n_replicates, algo=algo, algo_options=algo_options, ) self._main_method = self.Method.MU_STAR self.__outputs_bounds = discipline.output_range self.default_output = output_names @property def outputs_bounds(self) -> dict[str, list[float]]: """The empirical bounds of the outputs.""" return self.__outputs_bounds @property def n_replicates(self) -> int: """The number of OAT replicates.""" return self.__n_replicates
[docs] def compute_indices( self, outputs: str | Sequence[str] = (), normalize: bool = False, ) -> dict[str, FirstOrderIndicesType]: """ Args: normalize: Whether to normalize the indices with the empirical bounds of the outputs. """ # noqa: D205 D212 D415 fd_data = self.dataset.get_view(group_names=self.dataset.OUTPUT_GROUP).to_dict( orient="list" ) output_names = outputs or self.default_output if isinstance(output_names, str): output_names = [output_names] self.mu_ = {name: {} for name in output_names} self.mu_star = {name: {} for name in output_names} self.sigma = {name: {} for name in output_names} self.relative_sigma = {name: {} for name in output_names} self.min = {name: {} for name in output_names} self.max = {name: {} for name in output_names} for fd_name, value in fd_data.items(): value = array([value]).T output_name, input_name = _OATSensitivity.get_io_names(fd_name[1]) if output_name in output_names: lower = self.outputs_bounds[output_name][0] upper = self.outputs_bounds[output_name][1] self.mu_[output_name][input_name] = value.mean(0) self.mu_star[output_name][input_name] = np_abs(value).mean(0) self.sigma[output_name][input_name] = value.std(0) self.min[output_name][input_name] = np_abs(value).min(0) self.max[output_name][input_name] = np_abs(value).max(0) if normalize: self.mu_[output_name][input_name] /= upper - lower self.mu_star[output_name][input_name] /= max(abs(upper), abs(lower)) self.sigma[output_name][input_name] /= upper - lower self.min[output_name][input_name] /= upper - lower self.max[output_name][input_name] /= upper - lower self.relative_sigma[output_name][input_name] = ( self.sigma[output_name][input_name] / self.mu_star[output_name][input_name] ) for output_name in output_names: length = len(next(iter(self.sigma[output_name].values()))) for func in [ self.mu_, self.mu_star, self.sigma, self.relative_sigma, self.min, self.max, ]: func[output_name] = [ {name: array([val[idx]]) for name, val in func[output_name].items()} for idx in range(length) ] self._indices = { "MU": self.mu_, "MU_STAR": self.mu_star, "SIGMA": self.sigma, "RELATIVE_SIGMA": self.relative_sigma, "MIN": self.min, "MAX": self.max, } return self._indices
[docs] def plot( self, output: VariableType, inputs: Iterable[str] = (), title: str = "", save: bool = True, show: bool = False, file_path: str | Path = "", directory_path: str | Path = "", file_name: str = "", file_format: str = "", offset: float = 1, lower_mu: float | None = None, lower_sigma: float | None = None, ) -> None: r"""Plot the Morris indices for each input variable. For :math:`i\in\{1,\ldots,d\}`, plot :math:`\mu_i^*` in function of :math:`\sigma_i`. Args: offset: The offset to display the inputs names, expressed as a percentage applied to both x-range and y-range. lower_mu: The lower bound for :math:`\mu`. If ``None``, use a default value. lower_sigma: The lower bound for :math:`\sigma`. If ``None``, use a default value. """ # noqa: D415 D417 if not isinstance(output, tuple): output = (output, 0) names = self._filter_names(self._input_names, inputs) x_val = [self.mu_star[output[0]][output[1]][name] for name in names] y_val = [self.sigma[output[0]][output[1]][name] for name in names] fig, ax = plt.subplots() ax.scatter(x_val, y_val) ax.set_xlabel(r"$\mu^*$") ax.set_ylabel(r"$\sigma$") default_title = "Sampling: {}(size={}) - Relative step: {} - Output: {}".format( self._algo_name, self.__n_replicates, self.__step, repr_variable(*output, size=len(self.sigma[output[0]])), ) ax.set_xlim(left=lower_mu) ax.set_ylim(bottom=lower_sigma) ax.set_title(title or default_title) ax.set_axisbelow(True) ax.grid() x_offset = offset * (max(x_val) - min(x_val)) / 100.0 y_offset = offset * (max(y_val) - min(y_val)) / 100.0 for index, txt in enumerate(names): ax.annotate(txt, (x_val[index] + x_offset, y_val[index] + y_offset)) 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, )