Correlation analysis

from __future__ import annotations

import pprint

from gemseo.uncertainty.sensitivity.correlation.analysis import CorrelationAnalysis
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 CorrelationAnalysis:

sensitivity_analysis = CorrelationAnalysis([discipline], uncertain_space, 1000)
sensitivity_analysis.compute_indices()

{'KENDALL': {'y': [{'x1': array([0.31460661]), 'x2': array([0.02366366]), 'x3': array([0.0297978])}]}, 'PCC': {'y': [{'x1': array([0.46282123]), 'x2': array([0.0306214]), 'x3': array([0.0512473])}]}, 'PEARSON': {'y': [{'x1': array([0.46230446]), 'x2': array([0.02059071]), 'x3': array([0.04979026])}]}, 'PRCC': {'y': [{'x1': array([0.47248718]), 'x2': array([0.04368652]), 'x3': array([0.0419606])}]}, 'SPEARMAN': {'y': [{'x1': array([0.47177886]), 'x2': array([0.03220921]), 'x3': array([0.04128518])}]}, 'SRC': {'y': [{'x1': array([0.46220697]), 'x2': array([0.027121]), 'x3': array([0.04542585])}]}, 'SRRC': {'y': [{'x1': array([0.4718925]), 'x2': array([0.03849088]), 'x3': array([0.03696642])}]}, 'SSRC': {'y': [{'x1': array([0.21363529]), 'x2': array([0.00073555]), 'x3': array([0.00206351])}]}}

The resulting indices are

• the Pearson correlation coefficients,

• the Spearman correlation coefficients,

• the Partial Correlation Coefficients (PCC),

• the Partial Rank Correlation Coefficients (PRCC),

• the Standard Regression Coefficients (SRC),

• the Standard Rank Regression Coefficient (SRRC),

• the Signed Standard Rank Regression Coefficient (SSRRC):

pprint.pprint(sensitivity_analysis.indices)

{'KENDALL': {'y': [{'x1': array([0.31460661]),
                    'x2': array([0.02366366]),
                    'x3': array([0.0297978])}]},
 'PCC': {'y': [{'x1': array([0.46282123]),
                'x2': array([0.0306214]),
                'x3': array([0.0512473])}]},
 'PEARSON': {'y': [{'x1': array([0.46230446]),
                    'x2': array([0.02059071]),
                    'x3': array([0.04979026])}]},
 'PRCC': {'y': [{'x1': array([0.47248718]),
                 'x2': array([0.04368652]),
                 'x3': array([0.0419606])}]},
 'SPEARMAN': {'y': [{'x1': array([0.47177886]),
                     'x2': array([0.03220921]),
                     'x3': array([0.04128518])}]},
 'SRC': {'y': [{'x1': array([0.46220697]),
                'x2': array([0.027121]),
                'x3': array([0.04542585])}]},
 'SRRC': {'y': [{'x1': array([0.4718925]),
                 'x2': array([0.03849088]),
                 'x3': array([0.03696642])}]},
 'SSRC': {'y': [{'x1': array([0.21363529]),
                 'x2': array([0.00073555]),
                 'x3': array([0.00206351])}]}}

The main indices corresponds to the Spearman correlation indices (this main method can be changed with CorrelationAnalysis.main_method):

pprint.pprint(sensitivity_analysis.main_indices)

{'y': [{'x1': array([0.47177886]),
        'x2': array([0.03220921]),
        'x3': array([0.04128518])}]}

We can also sort the input parameters by decreasing order of influence:

print(sensitivity_analysis.sort_parameters("y"))

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

Lastly, we can use the method CorrelationAnalysis.plot() to visualize the different correlation coefficients:

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