Sobol’ analysis

from __future__ import annotations

import pprint

from gemseo.uncertainty.sensitivity.sobol.analysis import SobolAnalysis
from gemseo.uncertainty.use_cases.ishigami.ishigami_discipline import IshigamiDiscipline
from gemseo.uncertainty.use_cases.ishigami.ishigami_space import IshigamiSpace

In this example, we consider the Ishigami function [IH90]

\[f(x_1,x_2,x_3)=\sin(x_1)+7\sin(x_2)^2+0.1x_3^4\sin(x_1)\]

implemented as an MDODiscipline by the IshigamiDiscipline. It is commonly used with the independent random variables \(X_1\), \(X_2\) and \(X_3\) uniformly distributed between \(-\pi\) and \(\pi\) and defined in the IshigamiSpace.

discipline = IshigamiDiscipline()
uncertain_space = IshigamiSpace()

Then, we run sensitivity analysis of type SobolAnalysis:

sensitivity_analysis = SobolAnalysis([discipline], uncertain_space, 10000)
sensitivity_analysis.main_method = "total"
sensitivity_analysis.compute_indices()

{<Method.FIRST: 'first'>: {'y': [{'x1': array([0.33668934]), 'x2': array([0.48974171]), 'x3': array([0.01387372])}]}, 'second': {'y': [{'x1': {'x1': array([[0.]]), 'x2': array([[-0.06556589]]), 'x3': array([[0.17784496]])}, 'x2': {'x1': array([[-0.06556589]]), 'x2': array([[0.]]), 'x3': array([[-0.05604046]])}, 'x3': {'x1': array([[0.17784496]]), 'x2': array([[-0.05604046]]), 'x3': array([[0.]])}}]}, <Method.TOTAL: 'total'>: {'y': [{'x1': array([0.49769038]), 'x2': array([0.46903493]), 'x3': array([0.23786094])}]}}

The resulting indices are the first-order and total Sobol’ indices:

pprint.pprint(sensitivity_analysis.indices)

{<Method.FIRST: 'first'>: {'y': [{'x1': array([0.33668934]),
                                  'x2': array([0.48974171]),
                                  'x3': array([0.01387372])}]},
 'second': {'y': [{'x1': {'x1': array([[0.]]),
                          'x2': array([[-0.06556589]]),
                          'x3': array([[0.17784496]])},
                   'x2': {'x1': array([[-0.06556589]]),
                          'x2': array([[0.]]),
                          'x3': array([[-0.05604046]])},
                   'x3': {'x1': array([[0.17784496]]),
                          'x2': array([[-0.05604046]]),
                          'x3': array([[0.]])}}]},
 <Method.TOTAL: 'total'>: {'y': [{'x1': array([0.49769038]),
                                  'x2': array([0.46903493]),
                                  'x3': array([0.23786094])}]}}

They can also be accessed separately:

pprint.pprint(sensitivity_analysis.first_order_indices)
pprint.pprint(sensitivity_analysis.total_order_indices)

{'y': [{'x1': array([0.33668934]),
        'x2': array([0.48974171]),
        'x3': array([0.01387372])}]}
{'y': [{'x1': array([0.49769038]),
        'x2': array([0.46903493]),
        'x3': array([0.23786094])}]}

One can also obtain their confidence intervals:

pprint.pprint(sensitivity_analysis.get_intervals())
pprint.pprint(sensitivity_analysis.get_intervals(first_order=False))

{'y': [{'x1': (array([0.27106596]), array([0.40231271])),
        'x2': (array([0.42038752]), array([0.55909591])),
        'x3': (array([-0.06311772]), array([0.09086516]))}]}
{'y': [{'x1': (array([0.42529255]), array([0.57008822])),
        'x2': (array([0.42022722]), array([0.51784263])),
        'x3': (array([0.19308657]), array([0.2826353]))}]}

The main indices are the total Sobol’ indices (SobolAnalysis.main_method can also be set to "first" to use the first-order indices as main indices):

pprint.pprint(sensitivity_analysis.main_indices)

{'y': [{'x1': array([0.49769038]),
        'x2': array([0.46903493]),
        'x3': array([0.23786094])}]}

These main indices are used to sort the input parameters by decreasing order of influence:

print(sensitivity_analysis.sort_parameters("y"))

['x1', 'x2', 'x3']

Lastly, we can use the method SobolAnalysis.plot() to visualize both first-order and total Sobol’ indices:

sensitivity_analysis.plot("y", save=False, show=True)