# 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
# License version 3 as published by the Free Software Foundation.
#
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# 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.
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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"""Abstract class defining the concept of probability distribution.
Overview
--------
The abstract :class:`.Distribution` class implements the concept of
`probability distribution <https://en.wikipedia.org/wiki/Probability_distribution>`_,
which is a mathematical function giving the probabilities of occurrence
of different possible outcomes of a random variable for an experiment.
The `normal distribution <https://en.wikipedia.org/wiki/Normal_distribution>`_
with its famous *bell curve* is a well-known example of probability distribution.
.. seealso::
This abstract class is enriched by concrete ones,
such as :class:`.OTDistribution` interfacing the OpenTURNS probability distributions
and :class:`.SPDistribution` interfacing the SciPy probability distributions.
Construction
------------
The :class:`.Distribution` of a given uncertain variable is built
from a recognized distribution name (e.g. 'Normal' for OpenTURNS or 'norm' for SciPy),
a variable dimension, a set of parameters
and optionally a standard representation of these parameters.
Capabilities
------------
From a :class:`.Distribution`, we can easily get statistics,
such as :attr:`.Distribution.mean`,
:attr:`.Distribution.standard_deviation`. We can also get the
numerical :attr:`.Distribution.range` and
mathematical :attr:`.Distribution.support`.
.. note::
We call mathematical *support* the set of values that the random variable
can take in theory, e.g. :math:`]-\infty,+\infty[` for a Gaussian variable,
and numerical *range* the set of values that it can take in practice,
taking into account the values rounded to zero double precision.
Both support and range are described in terms of lower and upper bounds
We can also evaluate the cumulative density function
(:meth:`.Distribution.compute_cdf`)
for the different marginals of the random variable,
as well as the inverse cumulative density function
(:meth:`.Distribution.compute_inverse_cdf`). We can plot them,
either for a given marginal (:meth:`.Distribution.plot`)
or for all marginals (:meth:`.Distribution.plot_all`).
Lastly, we can compute realizations of the random variable
by means of the :meth:`.Distribution.compute_samples` method.
"""
from __future__ import annotations
from abc import abstractmethod
from collections.abc import Iterable
from collections.abc import Mapping
from typing import TYPE_CHECKING
from typing import Any
from typing import Callable
from typing import ClassVar
from typing import Final
from typing import Union
import matplotlib.pyplot as plt
from numpy import arange
from numpy import array
from numpy import ndarray
from gemseo.utils.file_path_manager import FilePathManager
from gemseo.utils.matplotlib_figure import save_show_figure
from gemseo.utils.metaclasses import ABCGoogleDocstringInheritanceMeta
from gemseo.utils.string_tools import pretty_str
from gemseo.utils.string_tools import repr_variable
if TYPE_CHECKING:
from pathlib import Path
from matplotlib.figure import Figure
from gemseo.uncertainty.distributions.composed import ComposedDistribution
StandardParametersType = Mapping[str, Union[str, int, float]]
ParametersType = Union[tuple[str, int, float], StandardParametersType]
[docs]
class Distribution(metaclass=ABCGoogleDocstringInheritanceMeta):
"""Probability distribution related to a random variable."""
math_lower_bound: ndarray
"""The mathematical lower bound of the random variable."""
math_upper_bound: ndarray
"""The mathematical upper bound of the random variable."""
num_lower_bound: ndarray
"""The numerical lower bound of the random variable."""
num_upper_bound: ndarray
"""The numerical upper bound of the random variable."""
distribution: type
"""The probability distribution of the random variable."""
marginals: list[type]
"""The marginal distributions of the components of the random variable."""
dimension: int
"""The number of dimensions of the random variable."""
variable_name: str
"""The name of the random variable."""
distribution_name: str
"""The name of the probability distribution."""
transformation: str
"""The transformation applied to the random variable, e.g. 'sin(x)'."""
parameters: tuple[Any] | dict[str, Any]
"""The parameters of the probability distribution."""
standard_parameters: dict[str, str] | None
"""The standard representation of the parameters of the distribution, used for its
string representation."""
_LOC: Final[str] = "loc"
_LOWER: Final[str] = "lower"
_MODE: Final[str] = "mode"
_MU: Final[str] = "mu"
_RATE: Final[str] = "rate"
_SCALE: Final[str] = "scale"
_SHAPE: Final[str] = "shape"
_LOCATION: Final[str] = "location"
_SIGMA: Final[str] = "sigma"
_UPPER: Final[str] = "upper"
DEFAULT_VARIABLE_NAME: Final[str] = "x"
"""The default name of the variable."""
COMPOSED_DISTRIBUTION_CLASS: ClassVar[type[ComposedDistribution] | None] = None
"""The class of the joint distribution associated with this distribution, if any."""
# TODO: remove the argument dimension / use ComposedDistribution for random vectors
def __init__(
self,
variable: str,
interfaced_distribution: str,
parameters: ParametersType,
dimension: int = 1,
standard_parameters: StandardParametersType | None = None,
) -> None:
"""
Args:
variable: The name of the random variable.
interfaced_distribution: The name of the probability distribution,
typically the name of a class wrapped from an external library,
such as ``"Normal"`` for OpenTURNS or ``"norm"`` for SciPy.
parameters: The parameters of the probability distribution.
dimension: The dimension of the random variable.
If greater than 1,
the probability distribution is applied
to all components of the random variable under the hypothesis
that these components are stochastically independent.
To be removed in a future version;
use a :class:`.ComposedDistribution` instead.
standard_parameters: The parameters of the probability distribution
used for string representation only
(use ``parameters`` for computation).
If ``None``, use ``parameters`` instead.
For instance,
let us consider an interfaced distribution
named ``"Dirac"``
with positional parameters
(this is the case of :class:`.OTDistribution`).
Then,
the string representation of
``Distribution("x", "Dirac", (1,), 1, {"loc": 1})``
is ``"Dirac(loc=1)"``
while the string representation of
``Distribution("x", "Dirac", (1,))``
is ``"Dirac(1)"``.
The same mechanism works for keyword parameters
(this is the case of :class:`.SPDistribution`).
""" # noqa: D205,D212,D415
self.math_lower_bound = None
self.math_upper_bound = None
self.num_lower_bound = None
self.num_upper_bound = None
self.distribution = None
self.marginals = None
self.dimension = dimension
self.variable_name = variable
self.distribution_name = interfaced_distribution
self.transformation = variable
self.parameters = parameters or self._get_empty_parameter_set()
if standard_parameters is None:
self.standard_parameters = self.parameters
else:
self.standard_parameters = standard_parameters
self.__file_path_manager = FilePathManager(
FilePathManager.FileType.FIGURE,
default_name=f"distribution_{self.variable_name}",
)
def _get_empty_parameter_set(self) -> dict:
"""Return an empty parameter set."""
return {}
def __repr__(self) -> str:
prefix = "" if self.dimension == 1 else f"[{self.dimension}]"
return (
f"{self.distribution_name}{prefix}("
f"{pretty_str(self.standard_parameters, sort=False)}"
f")"
)
[docs]
@abstractmethod
def compute_samples(
self,
n_samples: int = 1,
) -> ndarray:
"""Sample the random variable.
Args:
n_samples: The number of samples.
Returns:
The samples of the random variable,
The number of columns is equal to the dimension of the variable
and the number of lines is equal to the number of samples.
"""
[docs]
@abstractmethod
def compute_cdf(
self,
vector: Iterable[float],
) -> ndarray:
"""Evaluate the cumulative density function (CDF).
Evaluate the CDF of the components of the random variable
for a given realization of this random variable.
Args:
vector: A realization of the random variable.
Returns:
The CDF values of the components of the random variable.
"""
[docs]
@abstractmethod
def compute_inverse_cdf(
self,
vector: Iterable[float],
) -> ndarray:
"""Evaluate the inverse of the cumulative density function (ICDF).
Args:
vector: A vector of values comprised between 0 and 1
whose length is equal to the dimension of the random variable.
Returns:
The ICDF values of the components of the random variable.
"""
@property
@abstractmethod
def mean(self) -> ndarray:
"""The analytical mean of the random variable."""
@property
@abstractmethod
def standard_deviation(self) -> ndarray:
"""The analytical standard deviation of the random variable."""
@property
def range(self) -> list[ndarray]: # noqa: A003
"""The numerical range.
The numerical range is the interval defined by the lower and upper bounds
numerically reachable by the random variable.
Here, the numerical range of the random variable is defined by one array for
each component of the random variable, whose first element is the lower bound of
this component while the second one is its upper bound.
"""
return [
array([l_b, u_b])
for l_b, u_b in zip(self.num_lower_bound, self.num_upper_bound)
]
@property
def support(self) -> list[ndarray]:
"""The mathematical support.
The mathematical support is the interval defined by the theoretical lower and
upper bounds of the random variable.
Here, the mathematical range of the random variable is defined by one array for
each component of the random variable, whose first element is the lower bound of
this component while the second one is its upper bound.
"""
return [
array([l_b, u_b])
for l_b, u_b in zip(self.math_lower_bound, self.math_upper_bound)
]
[docs]
def plot_all(
self,
show: bool = True,
save: bool = False,
file_path: str | Path = "",
directory_path: str | Path = "",
file_name: str = "",
file_extension: str = "",
) -> list[Figure]:
"""Plot both probability and cumulative density functions for all components.
Args:
save: If ``True``, save the figure.
show: If ``True``, display the figure.
file_path: The path of the file to save the figures.
If the extension is missing, use ``file_extension``.
If empty,
create a file path
from ``directory_path``, ``file_name`` and ``file_extension``.
directory_path: The path of the directory to save the figures.
If empty, use the current working directory.
file_name: The name of the file to save the figures.
If empty, use a default one generated by the post-processing.
file_extension: A file extension, e.g. ``'png'``, ``'pdf'``, ``'svg'``, ...
If empty, use a default file extension.
Returns:
The figures.
"""
return [
self.plot(
index=index,
show=show,
save=save,
file_path=file_path,
file_name=file_name,
file_extension=file_extension,
directory_path=directory_path,
)
for index in range(self.dimension)
]
[docs]
def plot(
self,
index: int = 0,
show: bool = True,
save: bool = False,
file_path: str | Path = "",
directory_path: str | Path = "",
file_name: str = "",
file_extension: str = "",
) -> Figure:
"""Plot both probability and cumulative density functions for a given component.
Args:
index: The index of a component of the random variable.
save: If ``True``, save the figure.
show: If ``True``, display the figure.
file_path: The path of the file to save the figures.
If the extension is missing, use ``file_extension``.
If empty,
create a file path
from ``directory_path``, ``file_name`` and ``file_extension``.
directory_path: The path of the directory to save the figures.
If empty, use the current working directory.
file_name: The name of the file to save the figures.
If empty, use a default one generated by the post-processing.
file_extension: A file extension, e.g. ``'png'``, ``'pdf'``, ``'svg'``, ...
If empty, use a default file extension.
Returns:
The figure.
"""
variable_name = repr_variable(self.variable_name, index, self.dimension)
l_b = self.num_lower_bound[index]
u_b = self.num_upper_bound[index]
x_values = arange(l_b, u_b, (u_b - l_b) / 100)
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(6.4, 3.2))
fig.suptitle(f"Probability distribution of {variable_name}")
ax1.plot(x_values, [self._pdf(index)(x_value) for x_value in x_values])
ax1.grid()
ax1.set_xlabel(variable_name)
ax1.set_ylabel("Probability density function")
ax1.set_box_aspect(1)
ax2.plot(x_values, [self._cdf(index)(x_value) for x_value in x_values])
ax2.grid()
ax2.set_xlabel(variable_name)
ax2.set_ylabel("Cumulative distribution function")
ax2.yaxis.tick_right()
ax2.set_box_aspect(1)
if save:
file_path = self.__file_path_manager.create_file_path(
file_path=file_path,
file_name=file_name,
directory_path=directory_path,
file_extension=file_extension,
)
if self.dimension > 1:
file_path = self.__file_path_manager.add_suffix(file_path, str(index))
else:
file_path = ""
save_show_figure(fig, show, file_path)
return fig
def _pdf(
self,
index: int,
) -> Callable:
"""Get the probability density function of a marginal.
Args:
index: The index of a component of the random variable.
Return:
The probability density function
of the given component of the random variable.
"""
def pdf(
point: float,
) -> float:
"""Probability Density Function (PDF).
Args:
point: An evaluation point.
Returns:
The PDF value at the evaluation point.
"""
raise NotImplementedError
return pdf
def _cdf(
self,
index: int,
) -> Callable:
"""Get the cumulative density function of a marginal.
Args:
index: The index of a component of the random variable.
Return:
The cumulative density function
of the given component of the random variable.
"""
def cdf(
level: float,
) -> float:
"""Cumulative Density Function (CDF).
Args:
level: A probability level.
Returns:
The CDF value for the probability level.
"""
raise NotImplementedError
return cdf
