composed module¶
Abstract classes defining the concept of joint probability distribution.
Overview¶
The abstract ComposedDistribution
class
implements the concept of joint probability distribution,
which is a mathematical function giving the probabilities of occurrence
of different possible outcomes of several random variables for an experiment.
In the style of OpenTURNS, a ComposedDistribution
is defined
from a list of Distribution
instances
defining the marginals of the random variables
and a copula defining the dependence structure between them.
Note
A copula is a mathematical function used to define the dependence between random variables from their cumulative density functions. See more.
By definition, a joint probability distribution is a probability distribution
Therefore, ComposedDistribution
inherits
from the abstract class Distribution
.
Construction¶
The ComposedDistribution
of a list of given uncertain variables is built
from a list of Distribution
objects
implementing the probability distributions of these variables
and from a copula name.
Capabilities¶
Because ComposedDistribution
inherits from Distribution
,
we can easily get statistics, such as ComposedDistribution.mean
,
ComposedDistribution.standard_deviation
.
We can also get the numerical ComposedDistribution.range
and
mathematical ComposedDistribution.support
.
Note
We call mathematical support the set of values that the random variable can take in theory, e.g. \(]\infty,+\infty[\) for a Gaussian variable, and numerical range the set of values that it can 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
(ComposedDistribution.compute_cdf()
)
for the different marginals of the random variable,
as well as the inverse cumulative density function
(ComposedDistribution.compute_inverse_cdf()
). We can plot them,
either for a given marginal (ComposedDistribution.plot()
)
or for all marginals (ComposedDistribution.plot_all()
).
Lastly, we can compute realizations of the random variable
by means of the ComposedDistribution.compute_samples()
method.
Classes:

Composed distribution. 
 class gemseo.uncertainty.distributions.composed.ComposedDistribution(distributions, copula='independent_copula')[source]¶
Bases:
gemseo.uncertainty.distributions.distribution.Distribution
Composed distribution.
 Parameters
distributions (Sequence[Distribution]) – The distributions.
copula (str) –
A name of copula.
By default it is set to independent_copula.
 Return type
None
Attributes:
The analytical mean of the random variable.
The numerical range.
The analytical standard deviation of the random variable.
The mathematical support.
Methods:
compute_cdf
(vector)Evaluate the cumulative density function (CDF).
compute_inverse_cdf
(vector)Evaluate the inverse of the cumulative density function (ICDF).
compute_samples
([n_samples])Sample the random variable.
plot
([index, show, save, file_path, ...])Plot both probability and cumulative density functions for a given component.
plot_all
([show, save, file_path, ...])Plot both probability and cumulative density functions for all components.
 AVAILABLE_COPULA_MODELS = ['independent_copula']¶
 compute_cdf(vector)¶
Evaluate the cumulative density function (CDF).
Evaluate the CDF of the components of the random variable for a given realization of this random variable.
 Parameters
vector (Iterable[float]) – A realization of the random variable.
 Returns
The CDF values of the components of the random variable.
 Return type
numpy.ndarray
 compute_inverse_cdf(vector)¶
Evaluate the inverse of the cumulative density function (ICDF).
 Parameters
vector (Iterable[float]) – 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.
 Return type
numpy.ndarray
 compute_samples(n_samples=1)[source]¶
Sample the random variable.
 Parameters
n_samples (int) –
The number of samples.
By default it is set to 1.
 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.
 Return type
numpy.ndarray
 property mean¶
The analytical mean of the random variable.
 plot(index=0, show=True, save=False, file_path=None, directory_path=None, file_name=None, file_extension=None)¶
Plot both probability and cumulative density functions for a given component.
 Parameters
index (int) –
The index of a component of the random variable.
By default it is set to 0.
save (bool) –
If True, save the figure.
By default it is set to False.
show (bool) –
If True, display the figure.
By default it is set to True.
file_path (Optional[Union[str, pathlib.Path]]) –
The path of the file to save the figures. If the extension is missing, use
file_extension
. If None, create a file path fromdirectory_path
,file_name
andfile_extension
.By default it is set to None.
directory_path (Optional[Union[str, pathlib.Path]]) –
The path of the directory to save the figures. If None, use the current working directory.
By default it is set to None.
file_name (Optional[str]) –
The name of the file to save the figures. If None, use a default one generated by the postprocessing.
By default it is set to None.
file_extension (Optional[str]) –
A file extension, e.g. ‘png’, ‘pdf’, ‘svg’, … If None, use a default file extension.
By default it is set to None.
 Returns
The figure.
 Return type
matplotlib.figure.Figure
 plot_all(show=True, save=False, file_path=None, directory_path=None, file_name=None, file_extension=None)¶
Plot both probability and cumulative density functions for all components.
 Parameters
save (bool) –
If True, save the figure.
By default it is set to False.
show (bool) –
If True, display the figure.
By default it is set to True.
file_path (Optional[Union[str, pathlib.Path]]) –
The path of the file to save the figures. If the extension is missing, use
file_extension
. If None, create a file path fromdirectory_path
,file_name
andfile_extension
.By default it is set to None.
directory_path (Optional[Union[str, pathlib.Path]]) –
The path of the directory to save the figures. If None, use the current working directory.
By default it is set to None.
file_name (Optional[str]) –
The name of the file to save the figures. If None, use a default one generated by the postprocessing.
By default it is set to None.
file_extension (Optional[str]) –
A file extension, e.g. ‘png’, ‘pdf’, ‘svg’, … If None, use a default file extension.
By default it is set to None.
 Returns
The figures.
 Return type
List[matplotlib.figure.Figure]
 property range¶
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.
 property standard_deviation¶
The analytical standard deviation of the random variable.
 property support¶
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.