gemseo / uncertainty / sensitivity / morris

# analysis module¶

Class for the estimation of Morris indices.

## OAT technique¶

The purpose of the One-At-a-Time (OAT) methodology is to quantify the elementary effect

$df_i = f(X_1+dX_1,\ldots,X_{i-1}+dX_{i-1},X_i+dX_i,\ldots,X_d) - f(X_1+dX_1,\ldots,X_{i-1}+dX_{i-1},X_i,\ldots,X_d)$

associated with a small variation $$dX_i$$ of $$X_i$$ with

$df_1 = f(X_1+dX_1,\ldots,X_d)-f(X_1,\ldots,X_d)$

The elementary effects $$df_1,\ldots,df_d$$ are computed sequentially from an initial point

$X=(X_1,\ldots,X_d)$

From these elementary effects, we can compare their absolute values $$|df_1|,\ldots,|df_d|$$ and sort $$X_1,\ldots,X_d$$ accordingly.

## Morris technique¶

Then, the purpose of the Morris’ methodology is to repeat the OAT method from different initial points $$X^{(1)},\ldots,X^{(r)}$$ and compare the parameters in terms of mean

$\mu_i^* = \frac{1}{r}\sum_{j=1}^r|df_i^{(j)}|$

and standard deviation

$\sigma_i = \frac{1}{r}\sum_{j=1}^r\left(|df_i^{(j)}|-\mu_i\right)^2$

where $$\mu_i = \frac{1}{r}\sum_{j=1}^rdf_i^{(j)}$$.

This methodology relies on the MorrisAnalysis class.

Classes:

 MorrisAnalysis(discipline, parameter_space, ...) Sensitivity analysis based on the Morris' indices.
class gemseo.uncertainty.sensitivity.morris.analysis.MorrisAnalysis(discipline, parameter_space, n_samples, algo=None, algo_options=None, n_replicates=5, step=0.05)[source]

Sensitivity analysis based on the Morris’ indices.

MorrisAnalysis.indices contains both $$\mu^*$$, $$\mu$$ and $$\sigma$$ while MorrisAnalysis.main_indices represents $$\mu^*$$. Lastly, the MorrisAnalysis.plot() method represents the parameters as a scatter plot where $$X_i$$ has as coordinates $$(\mu_i^*,\sigma_i)$$. The bigger $$\mu_i^*$$ is, the more significant $$X_i$$ is. Concerning $$\sigma_i$$, it highlights non-linear effects along $$X_i$$ or cross-effects between $$X_i$$ and other parameter(s).

The user can specify the DOE algorithm name to select the initial points, as well as the number of replicates and the relative step for the input variations.

default_output

The default outputs of interest.

Type

list(str)

dataset

The dataset containing the discipline evaluations.

Type

Dataset

mu_

The mean effects with the following structure:

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

dict

mu_star

The mean absolute effects with the following structure:

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

dict

sigma

The variability of the effects with the following structure:

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

dict

relative_sigma

The relative variability of the effects with the following structure:

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

dict

min

The minimum effect with the following structure:

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

dict

max

The maximum effect with the following structure:

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

dict

Examples

>>> from numpy import pi
>>> from gemseo.api import create_discipline, create_parameter_space
>>> from gemseo.uncertainty.sensitivity.morris.analysis import MorrisAnalysis
>>>
>>> expressions = {"y": "sin(x1)+7*sin(x2)**2+0.1*x3**4*sin(x1)"}
>>> discipline = create_discipline(
...     "AnalyticDiscipline", expressions_dict=expressions
... )
>>>
>>> parameter_space = create_parameter_space()
...     "x1", "OTUniformDistribution", minimum=-pi, maximum=pi
... )
...     "x2", "OTUniformDistribution", minimum=-pi, maximum=pi
... )
...     "x3", "OTUniformDistribution", minimum=-pi, maximum=pi
... )
>>>
>>> analysis = MorrisAnalysis(
...     discipline, parameter_space, n_samples=None, n_replicates=5
... )
>>> indices = analysis.compute_indices()

Initialize self. See help(type(self)) for accurate signature.

Parameters
• discipline (MDODiscipline) – A discipline.

• parameter_space (DesignSpace) – A parameter space.

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

• algo (Optional[str]) –

The name of the DOE algorithm. If None, use the DEFAULT_DRIVER.

By default it is set to None.

• algo_options (Optional[Mapping[str,DOELibraryOptionType]]) –

The options of the DOE algorithm.

By default it is set to None.

• n_replicates (int) –

The number of times the OAT method is repeated. Used only if n_samples is None. Otherwise, this number is the greater integer $$r$$ such that $$r(d+1)\leq$$ n_samples and $$r(d+1)$$ is the number of samples actually carried out.

By default it is set to 5.

• step (float) –

The finite difference step of the OAT method.

By default it is set to 0.05.

Return type

None

Attributes:

 DEFAULT_DRIVER indices The sensitivity indices. inputs_names The names of the inputs. main_indices The main sensitivity indices. main_method The name of the main method. n_replicates The number of OAT replicates. outputs_bounds The empirical bounds of the outputs.

Methods:

 compute_indices([outputs, normalize]) Compute the sensitivity indices. Convert SensitivityAnalysis.indices into a Dataset. plot(output[, inputs, title, save, show, ...]) Plot the Morris indices for each input variable. plot_bar(outputs[, inputs, standardize, ...]) Plot the sensitivity indices on a bar chart. plot_comparison(indices, output[, inputs, ...]) Plot a comparison between the current sensitivity indices and other ones. plot_field(output[, mesh, inputs, ...]) Plot the sensitivity indices related to a 1D or 2D functional output. plot_radar(outputs[, inputs, standardize, ...]) Plot the sensitivity indices on a radar chart. sort_parameters(output) Return the parameters sorted in descending order. standardize_indices(indices) Standardize the sensitivity indices for each output component.
DEFAULT_DRIVER = 'lhs'
compute_indices(outputs=None, normalize=False)[source]

Compute the sensitivity indices.

Parameters
• outputs (Optional[Sequence[str]]) –

The outputs for which to display sensitivity indices. If None, use the default outputs, that are all the discipline outputs.

By default it is set to None.

• normalize (bool) –

Whether to normalize the indices with the empirical bounds of the outputs.

By default it is set to False.

Returns

The sensitivity indices.

With the following structure:

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

Return type

Dict[str, Dict[str, List[Dict[str, numpy.ndarray]]]]

export_to_dataset()

Convert SensitivityAnalysis.indices into a Dataset.

Returns

The sensitivity indices.

Return type

Dataset

property indices

The sensitivity indices.

With the following structure:

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

The names of the inputs.

property main_indices

The main sensitivity indices.

With the following structure:

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

The name of the main method.

property n_replicates

The number of OAT replicates.

property outputs_bounds

The empirical bounds of the outputs.

plot(output, inputs=None, title=None, save=True, show=False, file_path=None, directory_path=None, file_name=None, file_format=None, offset=1, lower_mu=None, lower_sigma=None)[source]

Plot the Morris indices for each input variable.

For $$i\in\{1,\ldots,d\}$$, plot $$\mu_i^*$$ in function of $$\sigma_i$$.

Parameters
• output (Union[str, Tuple[str, int]]) – 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 (Optional[Iterable[str]]) –

The inputs to display. If None, display all.

By default it is set to None.

• title (Optional[str]) –

The title of the plot. If None, no title.

By default it is set to None.

• 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 (Optional[Union[str, pathlib.Path]]) –

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.

By default it is set to None.

• file_format (Optional[str]) –

A file format, e.g. ‘png’, ‘pdf’, ‘svg’, … Used when file_path does not have any extension. If None, use a default file extension.

By default it is set to None.

• offset (float) –

The offset to display the inputs names, expressed as a percentage applied to both x-range and y-range.

By default it is set to 1.

• lower_mu (Optional[float]) –

The lower bound for $$\mu$$. If None, use a default value.

By default it is set to None.

• lower_sigma (Optional[float]) –

The lower bound for $$\sigma$$. If None, use a default value.

By default it is set to None.

• directory_path (Optional[Union[str, pathlib.Path]]) –

By default it is set to None.

• file_name (Optional[str]) –

By default it is set to None.

Return type

None

plot_bar(outputs, inputs=None, standardize=False, title=None, save=True, show=False, file_path=None, directory_path=None, file_name=None, file_format=None, **options)

Plot the sensitivity indices on a bar chart.

This method may consider one or more outputs, as well as all inputs (default behavior) or a subset.

Parameters
• outputs (Union[str, Tuple[str, int], Sequence[Union[str, Tuple[str, int]]]]) – The outputs for which to display sensitivity indices, either a name, a list of names, a (name, component) tuple, a list of such tuples or a list mixing such tuples and names. When a name is specified, all its components are considered. If None, use the default outputs.

• inputs (Optional[Iterable[str]]) –

The inputs to display. If None, display all.

By default it is set to None.

• standardize (bool) –

If True, standardize the indices between 0 and 1 for each output.

By default it is set to False.

• title (Optional[str]) –

The title of the plot. If None, no title.

By default it is set to None.

• 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 (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 from directory_path, file_name and file_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 post-processing.

By default it is set to None.

• file_format (Optional[str]) –

A file extension, e.g. ‘png’, ‘pdf’, ‘svg’, … If None, use a default file extension.

By default it is set to None.

• **options (int) –

Returns

A bar chart representing the sensitivity indices.

Return type

gemseo.post.dataset.bars.BarPlot

plot_comparison(indices, output, inputs=None, title=None, use_bar_plot=True, save=True, show=False, file_path=None, directory_path=None, file_name=None, file_format=None, **options)

Plot a comparison between the current sensitivity indices and other ones.

This method allows to use either a bar chart (default option) or a radar one.

Parameters
• indices (List[gemseo.uncertainty.sensitivity.analysis.SensitivityAnalysis]) – The sensitivity indices.

• output (Union[str, Tuple[str, int]]) – 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 (Optional[Iterable[str]]) –

The inputs to display. If None, display all.

By default it is set to None.

• title (Optional[str]) –

The title of the plot. If None, no title.

By default it is set to None.

• use_bar_plot (bool) –

The type of graph. If True, use a bar plot. Otherwise, use a radar chart.

By default it is set to True.

• 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 (Optional[Union[str, pathlib.Path]]) –

The path of the file to save the figures. If None, create a file path from directory_path, file_name and file_format.

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 post-processing.

By default it is set to None.

• file_format (Optional[str]) –

A file format, e.g. ‘png’, ‘pdf’, ‘svg’, … If None, use a default file extension.

By default it is set to None.

• **options (bool) – The options passed to the underlying DatasetPlot.

Returns

A graph comparing sensitivity indices.

Return type
plot_field(output, mesh=None, inputs=None, standardize=False, title=None, save=True, show=False, file_path=None, directory_path=None, file_name=None, file_format=None, properties=None)

Plot the sensitivity indices related to a 1D or 2D functional output.

The output is considered as a 1D or 2D functional variable, according to the shape of the mesh on which it is represented.

Parameters
• output (Union[str, Tuple[str, int]]) – 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.

• mesh (Optional[numpy.ndarray]) –

The mesh on which the p-length output is represented. Either a (1, p) array for a 1D functional output or a (2, p) array for a 2D one. If None, assume a 1D functional output.

By default it is set to None.

• inputs (Optional[Iterable[str]]) –

The inputs to display. If None, display all inputs.

By default it is set to None.

• standardize (bool) –

If True, standardize the indices between 0 and 1 for each output.

By default it is set to False.

• title (Optional[str]) –

The title of the plot. If None, no title is displayed.

By default it is set to None.

• 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 (Optional[Union[str, pathlib.Path]]) –

The path of the file to save the figures. If None, create a file path from directory_path, file_name and file_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 post-processing.

By default it is set to None.

• file_format (Optional[str]) –

A file extension, e.g. ‘png’, ‘pdf’, ‘svg’, … If None, use a default file extension.

By default it is set to None.

• properties (Optional[Mapping[str, Union[str, int, float, Sequence[Union[str, int, float]]]]]) –

The general properties of a DatasetPlot.

By default it is set to None.

Returns

A bar plot representing the sensitivity indices.

Raises

NotImplementedError – If the dimension of the mesh is greater than 2.

Return type

Plot the sensitivity indices on a radar chart.

This method may consider one or more outputs, as well as all inputs (default behavior) or a subset.

For visualization purposes, it is also possible to change the minimum and maximum radius values.

Parameters
• outputs (Union[str, Tuple[str, int], Sequence[Union[str, Tuple[str, int]]]]) – The outputs for which to display sensitivity indices, either a name, a list of names, a (name, component) tuple, a list of such tuples or a list mixing such tuples and names. When a name is specified, all its components are considered. If None, use the default outputs.

• inputs (Optional[Iterable[str]]) –

The inputs to display. If None, display all.

By default it is set to None.

• standardize (bool) –

If True, standardize the indices between 0 and 1 for each output.

By default it is set to False.

• title (Optional[str]) –

The title of the plot. If None, no title.

By default it is set to None.

• 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 (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 from directory_path, file_name and file_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 post-processing.

By default it is set to None.

• file_format (Optional[str]) –

A file extension, e.g. ‘png’, ‘pdf’, ‘svg’, … If None, use a default file extension.

By default it is set to None.

The minimal radial value. If None, from data.

By default it is set to None.

The maximal radial value. If None, from data.

By default it is set to None.

• **options (bool) –

Returns

A radar chart representing the sensitivity indices.

Return type

sort_parameters(output)

Return the parameters sorted in descending order.

Parameters

output (Union[str, Tuple[str, int]]) – An output of the form (name, component), where name is the output name and component is the output component. If a string is passed, the tuple (name, 0) will be considered corresponding to the first component of the output name.

Returns

The input parameters sorted in descending order.

Return type

List[str]

static standardize_indices(indices)

Standardize the sensitivity indices for each output component.

Each index is replaced by its absolute value divided by the largest index. Thus, the standardized indices belong to the interval $$[0,1]$$.

Parameters

indices (Dict[str, List[Dict[str, numpy.ndarray]]]) – The indices to be standardized.

Returns

The standardized indices.

Return type

Dict[str, List[Dict[str, numpy.ndarray]]]