Morris analysis¶

from __future__ import annotations

import pprint

from gemseo.algos.parameter_space import ParameterSpace
from gemseo.api import create_discipline
from gemseo.uncertainty.sensitivity.morris.analysis import MorrisAnalysis
from matplotlib import pyplot as plt
from numpy import pi


In this example, we consider a function from $$[-\pi,\pi]^3$$ to $$\mathbb{R}^3$$:

$(y_1,y_2)=\left(f(x_1,x_2,x_3),f(x_2,x_1,x_3)\right)$

where $$f(a,b,c)=\sin(a)+7\sin(b)^2+0.1*c^4\sin(a)$$ is the Ishigami function:

expressions = {
"y1": "sin(x1)+7*sin(x2)**2+0.1*x3**4*sin(x1)",
"y2": "sin(x2)+7*sin(x1)**2+0.1*x3**4*sin(x2)",
}
discipline = create_discipline(
"AnalyticDiscipline", expressions=expressions, name="Ishigami2"
)


Then, we consider the case where the deterministic variables $$x_1$$, $$x_2$$ and $$x_3$$ are replaced with the uncertain variables $$X_1$$, $$X_2$$ and $$X_3$$. The latter are independent and identically distributed according to a uniform distribution between $$-\pi$$ and $$\pi$$:

space = ParameterSpace()
for variable in ["x1", "x2", "x3"]:
variable, "OTUniformDistribution", minimum=-pi, maximum=pi
)


From that, we would like to carry out a sensitivity analysis with the random outputs $$Y_1=f(X_1,X_2,X_3)$$ and $$Y_2=f(X_2,X_1,X_3)$$. For that, we can compute the correlation coefficients from a MorrisAnalysis:

morris = MorrisAnalysis([discipline], space, 10)
morris.compute_indices()

{'mu': {'y2': [{'x1': array([-0.29766709]), 'x2': array([0.26848457]), 'x3': array([-0.7755748])}], 'y1': [{'x1': array([-0.36000398]), 'x2': array([0.77781853]), 'x3': array([-0.70990541])}]}, 'mu_star': {'y2': [{'x1': array([1.33011973]), 'x2': array([0.38907897]), 'x3': array([1.00221431])}], 'y1': [{'x1': array([0.67947346]), 'x2': array([0.88906579]), 'x3': array([0.72694219])}]}, 'sigma': {'y2': [{'x1': array([1.46392293]), 'x2': array([0.39387241]), 'x3': array([1.38465263])}], 'y1': [{'x1': array([0.98724949]), 'x2': array([0.79064599]), 'x3': array([0.8074493])}]}, 'relative_sigma': {'y2': [{'x1': array([1.10059485]), 'x2': array([1.01231995]), 'x3': array([1.38159335])}], 'y1': [{'x1': array([1.45296254]), 'x2': array([0.88929976]), 'x3': array([1.11074761])}]}, 'min': {'y2': [{'x1': array([0.46488117]), 'x2': array([0.02015985]), 'x3': array([3.93670669e-05])}], 'y1': [{'x1': array([0.0338188]), 'x2': array([0.11821721]), 'x3': array([8.72820113e-05])}]}, 'max': {'y2': [{'x1': array([2.14896136]), 'x2': array([0.85930239]), 'x3': array([3.33216248])}], 'y1': [{'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(morris.indices)

{'max': {'y1': [{'x1': array([2.2360336]),
'x2': array([1.83987522]),
'x3': array([2.12052546])}],
'y2': [{'x1': array([2.14896136]),
'x2': array([0.85930239]),
'x3': array([3.33216248])}]},
'min': {'y1': [{'x1': array([0.0338188]),
'x2': array([0.11821721]),
'x3': array([8.72820113e-05])}],
'y2': [{'x1': array([0.46488117]),
'x2': array([0.02015985]),
'x3': array([3.93670669e-05])}]},
'mu': {'y1': [{'x1': array([-0.36000398]),
'x2': array([0.77781853]),
'x3': array([-0.70990541])}],
'y2': [{'x1': array([-0.29766709]),
'x2': array([0.26848457]),
'x3': array([-0.7755748])}]},
'mu_star': {'y1': [{'x1': array([0.67947346]),
'x2': array([0.88906579]),
'x3': array([0.72694219])}],
'y2': [{'x1': array([1.33011973]),
'x2': array([0.38907897]),
'x3': array([1.00221431])}]},
'relative_sigma': {'y1': [{'x1': array([1.45296254]),
'x2': array([0.88929976]),
'x3': array([1.11074761])}],
'y2': [{'x1': array([1.10059485]),
'x2': array([1.01231995]),
'x3': array([1.38159335])}]},
'sigma': {'y1': [{'x1': array([0.98724949]),
'x2': array([0.79064599]),
'x3': array([0.8074493])}],
'y2': [{'x1': array([1.46392293]),
'x2': array([0.39387241]),
'x3': array([1.38465263])}]}}


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

pprint.pprint(morris.main_indices)

{'y1': [{'x1': array([0.67947346]),
'x2': array([0.88906579]),
'x3': array([0.72694219])}],
'y2': [{'x1': array([1.33011973]),
'x2': array([0.38907897]),
'x3': array([1.00221431])}]}


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

pprint.pprint(morris.outputs_bounds)

{'y1': [array([-1.42959705]), array([14.89344259])],
'y2': [array([-1.81332358]), array([14.77920445])]}


We can also sort the input parameters by decreasing order of influence and observe that this ranking is not the same for both outputs:

print(morris.sort_parameters("y1"))
print(morris.sort_parameters("y2"))

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


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

morris.plot("y1", save=False, show=False, lower_mu=0, lower_sigma=0)
morris.plot("y2", save=False, show=False, lower_mu=0, lower_sigma=0)
# Workaround for HTML rendering, instead of show=True
plt.show()


Total running time of the script: ( 0 minutes 0.572 seconds)

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