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"""
Estimate Sobol' indices for the beam by simulation algorithm
============================================================
"""
# %%
#
# In this example, we estimate the Sobol' indices for the
# :ref:`cantilever beam <use-case-cantilever-beam>` by simulation algorithm.
#
# %%
# Introduction
# ------------
#
# In this example we are going to quantify the correlation between the input
# variables and the output variable of a model thanks to Sobol indices.
#
# Sobol indices are designed to evaluate the importance of a single variable
# or a specific set of variables.
# Here the Sobol indices are estimated by sampling from the distributions of
# the input variables and propagating uncertainty through a function.
#
# In theory, Sobol indices range from 0 to 1; the closer an index value is
# to 1, the better the associated input variable explains the function output.
#
# Let us denote by :math:`d` the input dimension of the model.
#
# Sobol' indices can be computed at different orders.
#
# * First order indices evaluate the importance of one input variable
# at a time.
#
# * Total indices give the relative importance of one input variable
# and all its interactions with other variables.
# Alternatively, they can be viewed as measuring how much wriggle room
# remains to the output when all but one input variables are fixed.
#
# * In general, we are only interested in first order and total Sobol' indices.
# There are situations, however, where we want to estimate interactions.
# Second order indices evaluate the importance of every pair of input variables.
# The number of second order indices is:
#
# .. math::
# \binom{d}{2} = \frac{d \times \left( d-1\right)}{2}.
#
# In practice, when the number of input variables is not small (say,
# when :math:`d>5`), then the number of second order indices is too large
# to be easily analyzed.
# In these situations, we limit the analysis to the first order and total
# Sobol' indices.
# %%
# Define the model
# ----------------
# %%
from openturns.usecases import cantilever_beam
import openturns as ot
import openturns.viewer as otv
# %%
# We load the distribution and model from the example:
beam = cantilever_beam.CantileverBeam()
distribution = beam.independentDistribution
model = beam.model
# %%
# Estimate the Sobol' indices
# ---------------------------
# %%
# We first create the algorithm from `SobolSimulationAlgorithm` with the Saltelli estimator
# it will allow one to control the number of evaluations by convergence instead of using a fixed-size experiment
estimator = ot.SaltelliSensitivityAlgorithm()
estimator.setUseAsymptoticDistribution(True)
algo = ot.SobolSimulationAlgorithm(distribution, model, estimator)
algo.setMaximumOuterSampling(50) # number of iterations
algo.setExperimentSize(1000) # size of Sobol experiment at each iteration
algo.setBlockSize(4) # number of points evaluated simultaneously
# algo.setIndexQuantileLevel(0.05) # alpha
# algo.setIndexQuantileEpsilon(1e-2) # epsilon
algo.run()
# %%
# Extract the results
# -------------------
result = algo.getResult()
fo = result.getFirstOrderIndicesEstimate()
to = result.getTotalOrderIndicesEstimate()
foDist = result.getFirstOrderIndicesDistribution()
graph = result.draw()
_ = otv.View(graph)
# %%
# Using a different estimator
# ---------------------------
#
# We have used the `SaltelliSensitivityAlgorithm` class to estimate the indices.
# Others are available in the library:
#
# * `SaltelliSensitivityAlgorithm`
# * `MartinezSensitivityAlgorithm`
# * `JansenSensitivityAlgorithm`
# * `MauntzKucherenkoSensitivityAlgorithm`
#
# %%
otv.View.ShowAll()
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