Morris analysis

from __future__ import annotations

import pprint

from gemseo import configure_logger
from gemseo.uncertainty.sensitivity.morris.analysis import MorrisAnalysis
from gemseo.uncertainty.use_cases.ishigami.ishigami_discipline import IshigamiDiscipline
from gemseo.uncertainty.use_cases.ishigami.ishigami_space import IshigamiSpace

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 MorrisAnalysis:

sensitivity_analysis = MorrisAnalysis([discipline], uncertain_space, n_samples=None)
sensitivity_analysis.compute_indices()

 WARNING - 13:07:22: No coupling in MDA, switching chain_linearize to True.
 WARNING - 13:07:22: No coupling in MDA, switching chain_linearize to True.
    INFO - 13:07:22:
    INFO - 13:07:22: *** Start MorrisAnalysisSamplingPhase execution ***
    INFO - 13:07:22: MorrisAnalysisSamplingPhase
    INFO - 13:07:22:    Disciplines: _OATSensitivity
    INFO - 13:07:22:    MDO formulation: MDF
    INFO - 13:07:22: Running the algorithm lhs:
    INFO - 13:07:22:     20%|██        | 1/5 [00:00<00:00, 42.38 it/sec]
    INFO - 13:07:22:     40%|████      | 2/5 [00:00<00:00, 68.17 it/sec]
    INFO - 13:07:22:     60%|██████    | 3/5 [00:00<00:00, 86.38 it/sec]
    INFO - 13:07:22:     80%|████████  | 4/5 [00:00<00:00, 100.64 it/sec]
    INFO - 13:07:22:    100%|██████████| 5/5 [00:00<00:00, 110.65 it/sec]
    INFO - 13:07:22: *** End MorrisAnalysisSamplingPhase execution (time: 0:00:00.061520) ***

{'MU': {'y': [{'x1': array([-0.36000398]), 'x2': array([0.77781853]), 'x3': array([-0.70990541])}]}, 'MU_STAR': {'y': [{'x1': array([0.67947346]), 'x2': array([0.88906579]), 'x3': array([0.72694219])}]}, 'SIGMA': {'y': [{'x1': array([0.98724949]), 'x2': array([0.79064599]), 'x3': array([0.8074493])}]}, 'RELATIVE_SIGMA': {'y': [{'x1': array([1.45296254]), 'x2': array([0.88929976]), 'x3': array([1.11074761])}]}, 'MIN': {'y': [{'x1': array([0.0338188]), 'x2': array([0.11821721]), 'x3': array([8.72820113e-05])}]}, 'MAX': {'y': [{'x1': array([2.2360336]), 'x2': array([1.83987522]), 'x3': array([2.12052546])}]}}

The resulting indices are the empirical means and the standard deviations of the absolute output variations due to input changes.

pprint.pprint(sensitivity_analysis.indices)

{'MAX': {'y': [{'x1': array([2.2360336]),
                'x2': array([1.83987522]),
                'x3': array([2.12052546])}]},
 'MIN': {'y': [{'x1': array([0.0338188]),
                'x2': array([0.11821721]),
                'x3': array([8.72820113e-05])}]},
 'MU': {'y': [{'x1': array([-0.36000398]),
               'x2': array([0.77781853]),
               'x3': array([-0.70990541])}]},
 'MU_STAR': {'y': [{'x1': array([0.67947346]),
                    'x2': array([0.88906579]),
                    'x3': array([0.72694219])}]},
 'RELATIVE_SIGMA': {'y': [{'x1': array([1.45296254]),
                           'x2': array([0.88929976]),
                           'x3': array([1.11074761])}]},
 'SIGMA': {'y': [{'x1': array([0.98724949]),
                  'x2': array([0.79064599]),
                  'x3': array([0.8074493])}]}}

The main indices corresponds to these empirical means (this main method can be changed with MorrisAnalysis.main_method):

pprint.pprint(sensitivity_analysis.main_indices)

{'y': [{'x1': array([0.67947346]),
        'x2': array([0.88906579]),
        'x3': array([0.72694219])}]}

and can be interpreted with respect to the empirical bounds of the outputs:

pprint.pprint(sensitivity_analysis.outputs_bounds)

{'y': [array([-1.00881748]), array([14.89344259])]}

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

sensitivity_analysis.sort_parameters("y")

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

We can use the method MorrisAnalysis.plot() to visualize the different series of indices:

sensitivity_analysis.plot("y", save=False, show=True, lower_mu=0, lower_sigma=0)

<Figure size 640x480 with 1 Axes>

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

sensitivity_analysis.to_dataset()

GROUP	MU	MU_STAR	SIGMA	RELATIVE_SIGMA	MIN	MAX
VARIABLE	y	y	y	y	y	y
COMPONENT	0	0	0	0	0	0
x1	-0.360004	0.679473	0.987249	1.452963	0.033819	2.236034
x2	0.777819	0.889066	0.790646	0.889300	0.118217	1.839875
x3	-0.709905	0.726942	0.807449	1.110748	0.000087	2.120525