gemseo / uncertainty / statistics

# empirical module¶

Class for the empirical estimation of statistics from a dataset.

## Overview¶

The EmpiricalStatistics class inherits from the abstract Statistics 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 gemseo.uncertainty.statistics.empirical.EmpiricalStatistics(dataset, variable_names=(), name='')[source]

Bases: Statistics

A toolbox to compute statistics empirically.

Unless otherwise stated, the statistics are computed variable-wise and component-wise, i.e. variable-by-variable and component-by-component. So, for the sake of readability, the methods named as compute_statistic() return dict[str, ndarray] objects whose values are the names of the variables and the values are the statistic estimated for the different component.

Examples

>>> from gemseo import (
...     create_discipline,
...     create_parameter_space,
...     create_scenario)
>>> from gemseo.uncertainty.statistics.empirical import EmpiricalStatistics
>>>
>>> expressions = {"y1": "x1+2*x2", "y2": "x1-3*x2"}
>>> discipline = create_discipline(
...     "AnalyticDiscipline", expressions=expressions
... )
>>>
>>> parameter_space = create_parameter_space()
...     "x1", "OTUniformDistribution", minimum=-1, maximum=1
... )
...     "x2", "OTUniformDistribution", minimum=-1, maximum=1
... )
>>>
>>> scenario = create_scenario(
...     [discipline],
...     "DisciplinaryOpt",
...     "y1",
...     parameter_space,
...     scenario_type="DOE"
... )
>>> scenario.execute({'algo': '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_a_value()

Compute the A-value $$\text{Aval}[X]$$.

The A-value is the lower bound of the left-sided tolerance interval associated with a coverage level equal to 99% and a confidence level equal to 95%.

Returns:

The component-wise A-value of the different variables.

Return type:
compute_b_value()

Compute the B-value $$\text{Bval}[X]$$.

The B-value is the lower bound of the left-sided tolerance interval associated with a coverage level equal to 90% and a confidence level equal to 95%.

Returns:

The component-wise B-value of the different variables.

Return type:
classmethod compute_expression(variable_name, statistic_name, show_name=False, **options)

Return the expression of a statistical function applied to a variable.

E.g. “P[X >= 1.0]” for the probability that X exceeds 1.0.

Parameters:
• variable_name (str) – The name of the variable, e.g. "X".

• statistic_name (str) – The name of the statistic, e.g. "probability".

• show_name (bool) –

If True, show option names. Otherwise, only show option values.

By default it is set to False.

• **options (bool | float | int) – The options passed to the statistical function, e.g. {"greater": True, "thresh": 1.0}.

Returns:

The expression of the statistical function applied to the variable.

Return type:

str

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 | ndarray]) – 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_margin(std_factor)

Compute a margin $$\text{Margin}[X]=\mathbb{E}[X]+\kappa\mathbb{S}[X]$$.

Parameters:

std_factor (float) – The weight $$\kappa$$ of the standard deviation.

Returns:

The component-wise margin for the different variables.

Return type:
compute_maximum()[source]

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

Returns:

The component-wise maximum of the different variables.

Return type:
compute_mean()[source]

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

Returns:

The component-wise mean of the different variables.

Return type:
compute_mean_std(std_factor)

Compute a margin $$\text{Margin}[X]=\mathbb{E}[X]+\kappa\mathbb{S}[X]$$.

Parameters:

std_factor (float) – The weight $$\kappa$$ of the standard deviation.

Returns:

The component-wise margin for the different variables.

Return type:
compute_median()

Compute the median $$\text{Med}[X]$$.

Returns:

The component-wise median of the different variables.

Return type:
compute_minimum()[source]

Compute the $$\text{Min}[X]$$.

Returns:

The component-wise minimum of the different variables.

Return type:
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:
compute_percentile(order)

Compute the n-th percentile $$\text{p}[X; n]$$.

Parameters:

order (int) – The order $$n\in\{0,1,2,...100\}$$ of the percentile.

Returns:

The component-wise percentile of the different variables.

Raises:

ValueError – When $$n\notin\{0,1,2,...100\}$$.

Return type:
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 | ndarray]) – 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, ndarray]

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:
compute_quartile(order)

Compute the n-th quartile $$q[X; n]$$.

Parameters:

order (int) – The order $$n\in\{1,2,3\}$$ of the quartile.

Returns:

The component-wise quartile of the different variables.

Raises:

ValueError – When $$n\notin\{1,2,3\}$$.

Return type:
compute_range()[source]

Compute the range $$R[X]$$.

Returns:

The component-wise range of the different variables.

Return type:
compute_standard_deviation()[source]

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

Returns:

The component-wise standard deviation of the different variables.

Return type:
compute_tolerance_interval(coverage, confidence=0.95, side=ToleranceIntervalSide.BOTH)

Compute a $$(p,1-\alpha)$$ tolerance interval $$\text{TI}[X]$$.

The tolerance interval $$\text{TI}[X]$$ is defined to contain at least a proportion $$p$$ of the values of $$X$$ with a level of confidence $$1-\alpha$$. $$p$$ is also called the coverage level of the TI.

Typically, $$\alpha=0.05$$ or equivalently $$1-\alpha=0.95$$.

The tolerance interval can be either

• lower-sided (side="LOWER": $$[L, +\infty[$$),

• upper-sided (side="UPPER": $$]-\infty, U]$$) or

• both-sided (side="BOTH": $$[L, U]$$).

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.

Return type:
compute_variance()[source]

Compute the variance $$\mathbb{V}[X]$$.

Returns:

The component-wise variance of the different variables.

Return type:
compute_variation_coefficient()

Compute the coefficient of variation $$CoV[X]$$.

This is the standard deviation normalized by the expectation: $$CoV[X]=\mathbb{E}[S]/\mathbb{E}[X]$$.

Returns:

The component-wise coefficient of variation of the different variables.

Return type:
SYMBOLS = {'a_value': 'Aval', 'b_value': 'Bval', 'margin': 'Margin', 'maximum': 'Max', 'mean': 'E', 'mean_std': 'E_StD', 'median': 'Med', 'minimum': 'Min', 'moment': 'M', 'percentile': 'p', 'probability': 'P', 'quantile': 'Q', 'quartile': 'q', 'range': 'R', 'standard_deviation': 'StD', 'tolerance_interval': 'TI', 'variance': 'V', 'variation_coefficient': 'CoV'}
dataset: Dataset

The dataset.

n_samples: int

The number of samples.

n_variables: int

The number of variables.

name: str

The name of the object.

## Examples using EmpiricalStatistics¶

Empirical estimation of statistics

Empirical estimation of statistics