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

# -*- coding: utf-8 -*-
# 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 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+dX_i,\ldots,X_d)-f(X_1,\ldots,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_i,\ldots,X_d)-f(X_1,\ldots,X_i,\ldots,X_d)

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

.. math::


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 division, unicode_literals

import logging
from typing import Dict, Iterable, Mapping, Optional, Sequence, Tuple, Union

import matplotlib.pyplot as plt
from numpy import abs as np_abs
from numpy import array

from gemseo.algos.design_space import DesignSpace
from gemseo.algos.doe.lib_pydoe import PyDOE
from gemseo.core.discipline import MDODiscipline
from gemseo.uncertainty.sensitivity.analysis import IndicesType, SensitivityAnalysis
from gemseo.uncertainty.sensitivity.morris.oat import OATSensitivity
from gemseo.utils.py23_compat import Path

LOGGER = logging.getLogger(__name__)

[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. Attributes: mu_ (dict): The mean effects with the following structure: .. code-block:: python { "output_name": [ { "input_name": data_array, } ] } mu_star (dict): The mean absolute effects with the following structure: .. code-block:: python { "output_name": [ { "input_name": data_array, } ] } sigma (dict): The variability of the effects with the following structure: .. code-block:: python { "output_name": [ { "input_name": data_array, } ] } Examples: >>> from numpy import pi >>> from gemseo.api 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_dict=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_replicates=5) >>> indices = analysis.compute_indices() """ DEFAULT_DRIVER = PyDOE.PYDOE_LHS def __init__( self, discipline, # type: MDODiscipline parameter_space, # type:DesignSpace n_samples, # type: int algo=None, # type: Optional[str] algo_options=None, # type: Optional[Mapping] n_replicates=5, # type: int step=0.05, # type: float ): # type: (...) -> None # noqa: D205,D212,D415 r""" Args: n_replicates (int, optional): The number of times the OAT method is repeated. Used only if :attr:`n_samples` is None. Otherwise, this number is the greater integer :math:`r` such that :math:`r(d+1)\leq` :attr:`n_samples` and :math:`r(d+1)` is the number of samples actually carried out. step (float, optional): The finite difference step of the OAT method. """ self.mu_ = None self.mu_star = None self.sigma = None self.__step = step if n_samples is None: self.__n_replicates = n_replicates else: self.__n_replicates = n_samples // (parameter_space.dimension + 1) self.__outputs = discipline.get_output_data_names() if parameter_space.dimension != len(parameter_space.variables_names): raise ValueError("Each input dimension must be equal to 1.") self.__diff_discipline = OATSensitivity(discipline, parameter_space, step) super(MorrisAnalysis, self).__init__( self.__diff_discipline, parameter_space, n_replicates, algo, algo_options ) self._main_method = "Morris(mu*)" self.default_output = list(discipline.get_output_data_names()) @property def n_replicates(self): # type: (...) -> int """The number of OAT replicates.""" return self.__n_replicates
[docs] def compute_indices( self, outputs=None # type: Optional[Sequence[str]] ): # type: (...) -> Dict[str,IndicesType] # noqa: D102 fd_data = self.dataset.get_data_by_group(self.dataset.OUTPUT_GROUP, True) output_names = outputs or self.default_output if not isinstance(output_names, list): 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} for fd_name, value in fd_data.items(): output_name, input_name = self.__diff_discipline.get_io_names(fd_name) if output_name in output_names: 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) for output_name in output_names: length = len(next(iter(self.sigma[output_name].values()))) self.mu_[output_name] = [ {name: array([val[idx]]) for name, val in self.mu_[output_name].items()} for idx in range(length) ] self.mu_star[output_name] = [ { name: array([val[idx]]) for name, val in self.mu_star[output_name].items() } for idx in range(length) ] self.sigma[output_name] = [ { name: array([val[idx]]) for name, val in self.sigma[output_name].items() } for idx in range(length) ] return self.indices
@property def indices( self, ): # type: (...) -> IndicesType # noqa: D102 return {"mu": self.mu_, "mu_star": self.mu_star, "sigma": self.sigma} @property def main_indices( self, ): # type: (...) -> IndicesType # noqa: D102 return self.mu_star
[docs] def plot( self, output, # type: Union[str,Tuple[str,int]] inputs=None, # type: Optional[Iterable[str]] title=None, # type: Optional[str] save=True, # type: bool show=False, # type: bool file_path=None, # type: Optional[Union[str,Path]] directory_path=None, # type: Optional[Union[str,Path]] file_name=None, # type: Optional[str] file_format=None, # type: Optional[str] offset=1, # type: float lower_mu=None, # type: Optional[float] lower_sigma=None, # type: Optional[float] ): # type: (...) -> None # noqa: D417 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. """ if not isinstance(output, tuple): output = (output, 0) names = self.dataset.get_names(self.dataset.INPUT_GROUP) names = self._filter_names(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"$\sigma$") output = "{}({})".format(output[0], output[1]) default_title = "Sampling: {}(size={}) - Relative step: {} - Output: {}" default_title = default_title.format( self._algo_name, self.__n_replicates, self.__step, output ) ax.set_xlim(left=lower_mu) ax.set_ylim(bottom=lower_sigma) ax.set_title(title or default_title) 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, )