composed module¶
Joint probability distribution.
Overview¶
ComposedDistribution
is an abstract class
implementing the concept of joint probability distribution.
The joint probability distribution of a set of random variables is the probability distribution of the random vector consisting of these random variables.
It takes into account both the marginal probability distributions of these random variables and their dependency structure.
A ComposedDistribution
is defined
from a list of Distribution
instances
defining the marginals of the random variables
and a copula defining the dependency 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.
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 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.
- class gemseo.uncertainty.distributions.composed.ComposedDistribution(distributions, copula=None, variable='')[source]
Bases:
Distribution
Joint probability distribution.
- Parameters:
distributions (Sequence[Distribution]) – The marginal distributions.
copula (Any) – A copula distribution defining the dependency structure between random variables; if
None
, consider an independent copula.variable (str) –
The name of the variable, if any; otherwise, concatenate the names of the random variables defined by
distributions
.By default it is set to “”.
- compute_samples(n_samples=1)[source]
Sample the random variable.
- dimension: int
The number of dimensions of the random variable.
- distribution: type
The probability distribution of the random variable.
- distribution_name: str
The name of the probability distribution.
- math_lower_bound: ndarray
The mathematical lower bound of the random variable.
- math_upper_bound: ndarray
The mathematical upper bound of the random variable.
- property mean: ndarray
The analytical mean 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.
- property standard_deviation: ndarray
The analytical standard deviation of the random variable.
- standard_parameters: dict[str, str] | None
The standard representation of the parameters of the distribution, used for its string representation.
- transformation: str
The transformation applied to the random variable, e.g. ‘sin(x)’.
- variable_name: str
The name of the random variable.