"""
Analyse the central tendency of a cantilever beam
=================================================
"""

# %%
# In this example we perform a central tendency analysis of a random variable Y using the various methods available.
# We consider the :ref:`cantilever beam <use-case-cantilever-beam>` example and show
# how to use the `TaylorExpansionMoments` and `ExpectationSimulationAlgorithm` classes.

# %%
from openturns.usecases import cantilever_beam
import openturns as ot
import openturns.viewer as otv


# %%
# We first load the data class from the usecases module :
cb = cantilever_beam.CantileverBeam()

# %%
# We want to create the random variable of interest :math:`Y=g(X)` where :math:`g(.)` is the physical model and :math:`X` is the input vectors.
#
# We create the input parameters distribution and make a random vector.
# For the sake of this example, we consider an independent copula.

distribution = ot.JointDistribution([cb.E, cb.F, cb.L, cb.II])
X = ot.RandomVector(distribution)
X.setDescription(["E", "F", "L", "I"])

# %%
# `f` is the cantilever beam model :
f = cb.model

# %%
# The random variable of interest Y is then
Y = ot.CompositeRandomVector(f, X)
Y.setDescription("Y")

# %%
# Taylor expansion
# ----------------

# %%
# Perform Taylor approximation to get the expected value of Y and the importance factors.

# %%
taylor = ot.TaylorExpansionMoments(Y)
taylor_mean_fo = taylor.getMeanFirstOrder()
taylor_mean_so = taylor.getMeanSecondOrder()
taylor_cov = taylor.getCovariance()
taylor_if = taylor.getImportanceFactors()
print("model evaluation calls number=", f.getGradientCallsNumber())
print("model gradient calls number=", f.getGradientCallsNumber())
print("model hessian calls number=", f.getHessianCallsNumber())
print("taylor mean first order=", taylor_mean_fo)
print("taylor variance=", taylor_cov)
print("taylor importance factors=", taylor_if)

# %%
graph = taylor.drawImportanceFactors()
view = otv.View(graph)

# %%
# We see that, at first order, the variable :math:`F` explains about 70% of the variance of the output :math:`Y`.
# On the other hand, the variable :math:`E` is the least significant in the variance of the output: :math:`E` only explains about 5% of the output variance.

# %%
# Monte-Carlo simulation
# ----------------------

# %%
# Perform a Monte Carlo simulation of Y to estimate its mean.

# %%
algo = ot.ExpectationSimulationAlgorithm(Y)
algo.setMaximumOuterSampling(1000)
algo.setCoefficientOfVariationCriterionType("NONE")
algo.run()
print("model evaluation calls number=", f.getEvaluationCallsNumber())
expectation_result = algo.getResult()
expectation_mean = expectation_result.getExpectationEstimate()
print(
    "monte carlo mean=",
    expectation_mean,
    "var=",
    expectation_result.getVarianceEstimate(),
)

# %%
# Central dispersion analysis based on a sample
# ---------------------------------------------

# %%
# Directly compute statistical moments based on a sample of Y. Sometimes the probabilistic model is not available and the study needs to start from the data.

# %%
Y_s = Y.getSample(1000)
y_mean = Y_s.computeMean()
y_stddev = Y_s.computeStandardDeviation()
y_quantile_95p = Y_s.computeQuantilePerComponent(0.95)
print("mean=", y_mean, "stddev=", y_stddev, "quantile@95%", y_quantile_95p)

# %%
graph = ot.KernelSmoothing().build(Y_s).drawPDF()
graph.setTitle("Kernel smoothing approximation of the output distribution")
view = otv.View(graph)

# %%
otv.View.ShowAll()