gemseo / uncertainty / sensitivity / sobol

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analysis module

Class for the estimation of Sobol’ indices.

Let us consider the model \(Y=f(X_1,\ldots,X_d)\) where:

  • \(X_1,\ldots,X_d\) are independent random variables,

  • \(E\left[f(X_1,\ldots,X_d)^2\right]<\infty\).

Then, the following decomposition is unique:

\[Y=f_0 + \sum_{i=1}^df_i(X_i) + \sum_{i,j=1\atop i\neq j}^d f_{i,j}(X_i,X_j) + \sum_{i,j,k=1\atop i\neq j\neq k}^d f_{i,j,k}(X_i,X_j,X_k) + \ldots + f_{1,\ldots,d}(X_1,\ldots,X_d)\]

where:

  • \(f_0=E[Y]\),

  • \(f_i(X_i)=E[Y|X_i]-f_0\),

  • \(f_{i,j}(X_i,X_j)=E[Y|X_i,X_j]-f_i(X_i)-f_j(X_j)-f_0\)

  • and so on.

Then, the shift to variance leads to:

\[V[Y]=\sum_{i=1}^dV\left[f_i(X_i)\right] + \sum_{i,j=1\atop j\neq i}^d V\left[f_{i,j}(X_i,X_j)\right] + \ldots + V\left[f_{1,\ldots,d}(X_1,\ldots,X_d)\right]\]

and the Sobol’ indices are obtained by dividing by the variance and sum up to 1:

\[1=\sum_{i=1}^dS_i + \sum_{i,j=1\atop j\neq i}^d S_{i,j} + \sum_{i,j,k=1\atop i\neq j\neq k}^d S_{i,j,k} + \ldots + S_{1,\ldots,d}\]

A Sobol’ index represents the share of output variance explained by a parameter or a group of parameters. For the parameter \(X_i\),

  • \(S_i\) is the first-order Sobol’ index measuring the individual effect of \(X_i\),

  • \(S_{i,j}\) is the second-order Sobol’ index measuring the joint effect between \(X_i\) and \(X_j\),

  • \(S_{i,j,k}\) is the third-order Sobol’ index measuring the joint effect between \(X_i\), \(X_j\) and \(X_k\),

  • and so on.

In practice, we only consider the first-order Sobol’ index:

\[S_i=\frac{V[E[Y|X_i]]}{V[Y]}\]

and the total-order Sobol’ index:

\[S_i^T=\sum_{u\subset\{1,\ldots,d\}\atop u \ni i}S_u\]

The latter represents the sum of the individual effect of \(X_i\) and the joint effects between \(X_i\) and any parameter or group of parameters.

This methodology relies on the SobolAnalysis class. Precisely, SobolAnalysis.indices contains both SobolAnalysis.first_order_indices and SobolAnalysis.total_order_indices while SobolAnalysis.main_indices represents total-order Sobol’ indices. Lastly, the SobolAnalysis.plot() method represents the estimations of both first-order and total-order Sobol’ indices along with their confidence intervals whose default level is 95%.

The user can select the algorithm to estimate the Sobol’ indices. The computation relies on OpenTURNS capabilities.

class gemseo.uncertainty.sensitivity.sobol.analysis.SobolAnalysis(disciplines, parameter_space, n_samples, output_names=None, algo=None, algo_options=None, formulation='MDF', compute_second_order=True, use_asymptotic_distributions=True, **formulation_options)[source]

Bases: SensitivityAnalysis

Sensitivity analysis based on the Sobol’ indices.

Examples

>>> from numpy import pi
>>> from gemseo import create_discipline, create_parameter_space
>>> from gemseo.uncertainty.sensitivity.sobol.analysis import SobolAnalysis
>>>
>>> expressions = {"y": "sin(x1)+7*sin(x2)**2+0.1*x3**4*sin(x1)"}
>>> discipline = create_discipline(
...     "AnalyticDiscipline", expressions=expressions
... )
>>>
>>> parameter_space = create_parameter_space()
>>> parameter_space.add_random_variable(
...     "x1", "OTUniformDistribution", minimum=-pi, maximum=pi
... )
>>> parameter_space.add_random_variable(
...     "x2", "OTUniformDistribution", minimum=-pi, maximum=pi
... )
>>> parameter_space.add_random_variable(
...     "x3", "OTUniformDistribution", minimum=-pi, maximum=pi
... )
>>>
>>> analysis = SobolAnalysis([discipline], parameter_space, n_samples=10000)
>>> indices = analysis.compute_indices()
Parameters:

  • disciplines (Collection[MDODiscipline]) – The discipline or disciplines to use for the analysis.

  • parameter_space (ParameterSpace) – A parameter space.

  • n_samples (int) – A number of samples. If None, the number of samples is computed by the algorithm.

  • output_names (Iterable[str] | None) – The disciplines’ outputs to be considered for the analysis. If None, use all the outputs.

  • algo (str | None) – The name of the DOE algorithm. If None, use the SensitivityAnalysis.DEFAULT_DRIVER.

  • algo_options (Mapping[str, DOELibraryOptionType] | None) – The options of the DOE algorithm.

  • formulation (str) –

    The name of the MDOFormulation to sample the disciplines.

    By default it is set to “MDF”.

  • compute_second_order (bool) –

    Whether to compute the second-order indices.

    By default it is set to True.

  • use_asymptotic_distributions (bool) –

    Whether to estimate the confidence intervals of the first- and total-order Sobol’ indices with the asymptotic distributions; otherwise, use bootstrap.

    By default it is set to True.

  • **formulation_options (Any) – The options of the MDOFormulation.

Notes

The estimators of Sobol’ indices rely on the same DOE algorithm. This algorithm starts with two independent input datasets composed of \(N\) independent samples and this number \(N\) is the usual sampling size for Sobol’ analysis. When compute_second_order=False or when the input dimension \(d\) is equal to 2, \(N=\frac{n_\text{samples}}{2+d}\). Otherwise, \(N=\frac{n_\text{samples}}{2+2d}\). The larger \(N\), the more accurate the estimators of Sobol’ indices are. Therefore, for a small budget n_samples, the user can choose to set compute_second_order to False to ensure a better estimation of the first- and second-order indices.

class Algorithm(value)[source]

Bases: PascalCaseStrEnum

The algorithms to estimate the Sobol’ indices.

JANSEN = 'Jansen'
MARTINEZ = 'Martinez'
MAUNTZ_KUCHERENKO = 'MauntzKucherenko'
SALTELLI = 'Saltelli'
class Method(value)[source]

Bases: StrEnum

The names of the sensitivity methods.

FIRST = 'first'

The first-order Sobol’ index.

TOTAL = 'total'

The total-order Sobol’ index.

compute_indices(outputs=None, algo=Algorithm.SALTELLI, confidence_level=0.95)[source]

Compute the sensitivity indices.

Parameters:

  • outputs (str | Sequence[str] | None) – The output(s) for which to display the sensitivity indices. If None, use the default outputs set at instantiation.

  • algo (Algorithm) –

    The name of the algorithm to estimate the Sobol’ indices.

    By default it is set to “Saltelli”.

  • confidence_level (float) –

    The level of the confidence intervals.

    By default it is set to 0.95.

Returns:

The sensitivity indices.

With the following structure:

{
    "method_name": {
        "output_name": [
            {
                "input_name": data_array,
            }
        ]
    }
}

Return type:

dict[str, FirstOrderIndicesType]

get_intervals(first_order=True)[source]

Get the confidence intervals for the Sobol’ indices.

Warning

You must first call compute_indices().

Parameters:

first_order (bool) –

If True, compute the intervals for the first-order indices. Otherwise, for the total-order indices.

By default it is set to True.

Returns:

The confidence intervals for the Sobol’ indices.

With the following structure:

{
    "output_name": [
        {
            "input_name": data_array,
        }
    ]
}

Return type:

Dict[str, List[Dict[str, ndarray]]]

plot(output, inputs=None, title=None, save=True, show=False, file_path=None, directory_path=None, file_name=None, file_format=None, sort=True, sort_by_total=True)[source]

Plot the first- and total-order Sobol’ indices.

For \(i\in\{1,\ldots,d\}\), plot \(S_i^{1}\) and \(S_T^{1}\) with their confidence intervals.

Parameters:

  • output (VariableType) – The output for which to display sensitivity indices, either a name or a tuple of the form (name, component). If name, its first component is considered.

  • inputs (Iterable[str] | None) – The inputs to display. If None, display all.

  • title (str | None) – The title of the plot. If None, no title.

  • save (bool) –

    If True, save the figure.

    By default it is set to True.

  • show (bool) –

    If True, show the figure.

    By default it is set to False.

  • file_path (str | Path | None) – A file path. Either a complete file path, a directory name or a file name. If None, use a default file name and a default directory. The file extension is inferred from filepath extension, if any.

  • directory_path (str | Path | None) – The description is missing.

  • file_name (str | None) – The description is missing.

  • file_format (str | None) – A file format, e.g. ‘png’, ‘pdf’, ‘svg’, … Used when file_path does not have any extension. If None, use a default file extension.

  • sort (bool) –

    The sorting option. If True, sort variables before display.

    By default it is set to True.

  • sort_by_total (bool) –

    The type of sorting. If True, sort variables according to total-order Sobol’ indices. Otherwise, use first-order Sobol’ indices.

    By default it is set to True.

Return type:

None

unscale_indices(indices, use_variance=True)[source]

Unscale the Sobol’ indices.

Parameters:

  • indices (FirstOrderIndicesType | SecondOrderIndicesType) – The Sobol’ indices.

  • use_variance (bool) –

    Whether to express an unscaled Sobol’ index as a share of output variance; otherwise, express it as the square root of this part and therefore with the same unit as the output.

    By default it is set to True.

Returns:

The unscaled Sobol’ indices.

Return type:

FirstOrderIndicesType | SecondOrderIndicesType

DEFAULT_DRIVER: ClassVar[str] = 'OT_SOBOL_INDICES'
dataset: IODataset

The dataset containing the discipline evaluations.

default_output: list[str]

The default outputs of interest.

property first_order_indices: Dict[str, List[Dict[str, ndarray]]]

The first-order Sobol’ indices.

With the following structure:

{
    "output_name": [
        {
            "input_name": data_array,
        }
    ]
}
property indices: dict[str, FirstOrderIndicesType | SecondOrderIndicesType]

The sensitivity indices.

With the following structure:

{
    "method_name": {
        "output_name": [
            {
                "input_name": data_array,
            }
        ]
    }
}
output_standard_deviations: dict[str, NDArray[float]]

The standard deviations of the output variables.

output_variances: dict[str, NDArray[float]]

The variances of the output variables.

property second_order_indices: Dict[str, List[Dict[str, Dict[str, ndarray]]]]

The second-order Sobol’ indices.

With the following structure:

{
    "output_name": [
        {
            {"input_name": {"other_input_name": data_array},
        }
    ]
}
property total_order_indices: Dict[str, List[Dict[str, ndarray]]]

The total-order Sobol’ indices.

With the following structure:

{
    "output_name": [
        {
            "input_name": data_array,
        }
    ]
}

Examples using SobolAnalysis

Sobol’ analysis

Sobol' analysis