gemseo.uncertainty.statistics.empirical_statistics module#

Class for the empirical estimation of statistics from a dataset.

Overview#

The EmpiricalStatistics class inherits from the abstract BaseStatistics class and aims to estimate statistics from a Dataset, based on empirical estimators.

Construction#

A EmpiricalStatistics is built from a Dataset and optionally variables names. In this case, statistics are only computed for these variables. Otherwise, statistics are computed for all the variable available in the dataset. Lastly, the user can give a name to its EmpiricalStatistics object. By default, this name is the concatenation of 'EmpiricalStatistics' and the name of the Dataset.

class EmpiricalStatistics(dataset, variable_names=(), name='')[source]#

Bases: BaseStatistics

A toolbox to compute statistics empirically.

Examples

>>> from gemseo import (
...     create_discipline,
...     create_parameter_space,
...     create_scenario,
... )
>>> from gemseo.uncertainty.statistics.empirical_statistics import (
...     EmpiricalStatistics,
... )
>>>
>>> expressions = {"y1": "x1+2*x2", "y2": "x1-3*x2"}
>>> discipline = create_discipline("AnalyticDiscipline", expressions)
>>>
>>> parameter_space = create_parameter_space()
>>> parameter_space.add_random_variable(
...     "x1", "OTUniformDistribution", minimum=-1, maximum=1
... )
>>> parameter_space.add_random_variable(
...     "x2", "OTUniformDistribution", minimum=-1, maximum=1
... )
>>>
>>> scenario = create_scenario(
...     [discipline],
...     "y1",
...     parameter_space,
...     formulation_name="DisciplinaryOpt",
...     scenario_type="DOE",
... )
>>> scenario.execute(algo_name="OT_MONTE_CARLO", n_samples=100)
>>>
>>> dataset = scenario.to_dataset(opt_naming=False)
>>>
>>> statistics = EmpiricalStatistics(dataset)
>>> mean = statistics.compute_mean()

Parameters:

dataset (Dataset) -- A dataset.
variable_names (Iterable[str]) --
The names of the variables for which to compute statistics. If empty, consider all the variables of the dataset.

By default it is set to ().
name (str) --
A name for the toolbox computing statistics. If empty, concatenate the names of the dataset and the name of the class.

By default it is set to "".

compute_joint_probability(thresh, greater=True)[source]#

Compute the joint probability related to a threshold.

Either \(\mathbb{P}[X \geq x]\) or \(\mathbb{P}[X \leq x]\).

Parameters:

thresh (Mapping[str, float | RealArray]) -- A threshold \(x\) per variable.
greater (bool) --
The type of probability. If True, compute the probability of exceeding the threshold. Otherwise, compute the opposite.

By default it is set to True.

Returns:

The joint probability of the different variables (by definition of the joint probability, this statistics is not computed component-wise).

Return type:

dict[str, float]

compute_maximum()[source]#

Compute the maximum \(\text{Max}[X]\).

Returns:: The component-wise maximum of the different variables.
Return type:: dict[str, RealArray]

compute_mean()[source]#

Compute the mean \(\mathbb{E}[X]\).

Returns:: The component-wise mean of the different variables.
Return type:: dict[str, RealArray]

compute_minimum()[source]#

Compute the \(\text{Min}[X]\).

Returns:: The component-wise minimum of the different variables.
Return type:: dict[str, RealArray]

compute_moment(order)[source]#

Compute the n-th moment \(M[X; n]\).

Parameters:: order (int) -- The order \(n\) of the moment.
Returns:: The component-wise moment of the different variables.
Return type:: dict[str, RealArray]

compute_probability(thresh, greater=True)[source]#

Compute the probability related to a threshold.

Either \(\mathbb{P}[X \geq x]\) or \(\mathbb{P}[X \leq x]\).

Parameters:

thresh (Mapping[str, float | RealArray]) -- A threshold \(x\) per variable.
greater (bool) --
The type of probability. If True, compute the probability of exceeding the threshold. Otherwise, compute the opposite.

By default it is set to True.

Returns:

The component-wise probability of the different variables.

Return type:

dict[str, RealArray]

compute_quantile(prob)[source]#

Compute the quantile \(\mathbb{Q}[X; \alpha]\) related to a probability.

Parameters:: prob (float) -- A probability \(\alpha\) between 0 and 1.
Returns:: The component-wise quantile of the different variables.
Return type:: dict[str, RealArray]

compute_range()[source]#

Compute the range \(R[X]\).

Returns:: The component-wise range of the different variables.
Return type:: dict[str, RealArray]

compute_standard_deviation()[source]#

Compute the standard deviation \(\mathbb{S}[X]\).

Returns:: The component-wise standard deviation of the different variables.
Return type:: dict[str, RealArray]

compute_tolerance_interval(coverage, confidence=0.95, side=ToleranceIntervalSide.BOTH)[source]#

Compute tolerance interval.

Given a confidence level \(1-\alpha\) and a coverage level \(\beta\), the number of samples \(n\) must verify the requirement:

\(1-(1-\beta)^n>=1-\alpha\) for a lower one-sided tolerance interval,
\(1-\beta^n>=1-\alpha\) for a upper one-sided tolerance interval,
\((n-1)\beta^n-n\beta^{n-1}+1>=1-\alpha\)

See [1] and [2] for more information about empirical tolerance intervals.

Statistics,: John Wiley & Sons, 2009.

Parameters:

coverage (float) -- A minimum proportion \(p\in[0,1]\) of belonging to the TI.
confidence (float) --
A level of confidence \(1-\alpha\in[0,1]\).

By default it is set to 0.95.
side (ToleranceIntervalSide) --
The type of the tolerance interval.

By default it is set to "both".

Returns:

The component-wise tolerance intervals of the different variables, expressed as {variable_name: [(lower_bound, upper_bound), ...], ... } where [(lower_bound, upper_bound), ...] are the lower and upper bounds of the tolerance interval of the different components of variable_name.

Raises:

ValueError -- When there are not enough samples.

Return type:

dict[str, list[Bounds]]

gemseo.uncertainty.statistics.empirical_statistics module#

Overview#

Construction#

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