# Correlation analysis¶

from __future__ import annotations

import pprint

from gemseo import configure_logger
from gemseo.problems.uncertainty.ishigami.ishigami_discipline import IshigamiDiscipline
from gemseo.problems.uncertainty.ishigami.ishigami_space import IshigamiSpace
from gemseo.uncertainty.sensitivity.correlation.analysis import CorrelationAnalysis

configure_logger()
<RootLogger root (INFO)>

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()
WARNING - 08:55:20: No coupling in MDA, switching chain_linearize to True.
INFO - 08:55:20:
INFO - 08:55:20: *** Start CorrelationAnalysisSamplingPhase execution ***
INFO - 08:55:20: CorrelationAnalysisSamplingPhase
INFO - 08:55:20:    Disciplines: IshigamiDiscipline
INFO - 08:55:20:    MDO formulation: MDF
INFO - 08:55:20: Running the algorithm OT_MONTE_CARLO:
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INFO - 08:55:21:    100%|█████████▉| 998/1000 [00:00<00:00, 1068.59 it/sec]
INFO - 08:55:21:    100%|█████████▉| 999/1000 [00:00<00:00, 1068.62 it/sec]
INFO - 08:55:21:    100%|██████████| 1000/1000 [00:00<00:00, 1068.61 it/sec]
INFO - 08:55:21: *** End CorrelationAnalysisSamplingPhase execution (time: 0:00:01.279645) ***

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

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 get the input parameters sorted by decreasing order of influence:

sensitivity_analysis.sort_parameters("y")
['x1', 'x3', 'x2']

We can use the method CorrelationAnalysis.plot() to visualize the different correlation coefficients:

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

Lastly, the sensitivity indices can be exported to a Dataset:

sensitivity_analysis.to_dataset()
GROUP Kendall PCC Pearson PRCC Spearman SRC SRRC SSRC
VARIABLE y y y y y y y y
COMPONENT 0 0 0 0 0 0 0 0
x1 0.314607 0.462821 0.462304 0.472487 0.471779 0.462207 0.471893 0.213635
x2 0.023664 0.030621 0.020591 0.043687 0.032209 0.027121 0.038491 0.000736
x3 0.029798 0.051247 0.049790 0.041961 0.041285 0.045426 0.036966 0.002064

Total running time of the script: (0 minutes 1.522 seconds)

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