"""
Estimate a probability with Monte-Carlo on axial stressed beam: a quick start guide to reliability
==================================================================================================
"""
# %%
#
# The goal of this example is to show a simple practical example of
# probability estimation in a reliability study with the `ProbabilitySimulationAlgorithm` class.
# The `ThresholdEvent` is used to define the event. We use the Monte-Carlo
# method thanks to the `MonteCarloExperiment` class to estimate this probability and its confidence interval.
# We use the :ref:`axial stressed beam ` model as an illustrative example.
# %%
# Definition of the model
# -----------------------
# %%
from openturns.usecases import stressed_beam
import openturns as ot
import numpy as np
import openturns.viewer as viewer
ot.Log.Show(ot.Log.NONE)
# %%
# We load the model from the usecases module :
sm = stressed_beam.AxialStressedBeam()
# %%
# The limit state function is defined as a symbolic function in the `model` parameter of the `AxialStressedBeam` data class:
limitStateFunction = sm.model
# %%
# Before using the function within an algorithm, we check that the limit state function is correctly evaluated.
# %%
x = [3.0e6, 750.0]
print("x=", x)
print("G(x)=", limitStateFunction(x))
# %%
# Probabilistic model
# -------------------
# %%
# We load the first marginal, a univariate `LogNormal` distribution, parameterized by its mean and standard deviation:
R = sm.distribution_R
# %%
# We draw the PDF of the first marginal.
graph = R.drawPDF()
view = viewer.View(graph)
# %%
# Our second marginal is a `Normal` univariate distribution:
F = sm.distribution_F
# %%
# We draw the PDF of the second marginal.
graph = F.drawPDF()
view = viewer.View(graph)
# %%
# In order to create the input distribution, we use the `ComposedDistribution` class which associates
# the distribution marginals and a copula.
# If no copula is supplied to the constructor, it selects the independent copula as default.
# That is implemented in the data class:
# %%
myDistribution = sm.distribution
# %%
# We create a `RandomVector` from the `Distribution`, which will then be fed to the limit state function.
# %%
inputRandomVector = ot.RandomVector(myDistribution)
# %%
# Finally we create a `CompositeRandomVector` by associating the limit state function with the input random vector.
# %%
outputRandomVector = ot.CompositeRandomVector(limitStateFunction, inputRandomVector)
# %%
# Exact computation
# -----------------
# %%
# The simplest method is to perform an exact computation based on the arithmetic of distributions.
# %%
D = 0.02
# %%
G = R - F / (D ** 2 / 4 * np.pi)
# %%
G.computeCDF(0.0)
# %%
# This is the exact result from the description of this example.
# %%
# Distribution of the output
# --------------------------
# %%
# Plot the distribution of the output.
# %%
sampleSize = 500
sampleG = outputRandomVector.getSample(sampleSize)
graph = ot.HistogramFactory().build(sampleG).drawPDF()
view = viewer.View(graph)
# %%
# Estimate the probability with Monte-Carlo
# -----------------------------------------
# %%
# We first create a `ThresholdEvent` based on the output `RandomVector`, the operator and the threshold.
# %%
myEvent = ot.ThresholdEvent(outputRandomVector, ot.Less(), 0.0)
# %%
# The `ProbabilitySimulationAlgorithm` is the main tool to estimate a probability.
# It is based on a specific design of experiments: in this example, we use the simplest of all, the `MonteCarloExperiment`.
# %%
maximumCoV = 0.05 # Coefficient of variation
maximumNumberOfBlocks = 100000
experiment = ot.MonteCarloExperiment()
algoMC = ot.ProbabilitySimulationAlgorithm(myEvent, experiment)
algoMC.setMaximumOuterSampling(maximumNumberOfBlocks)
algoMC.setBlockSize(1)
algoMC.setMaximumCoefficientOfVariation(maximumCoV)
# %%
# In order to gather statistics about the algorithm, we get the initial number of function calls
# (this is not mandatory, but will prove to be convenient later in the study).
# %%
initialNumberOfCall = limitStateFunction.getEvaluationCallsNumber()
# %%
# Now comes the costly part: the `run` method performs the required simulations.
# The algorithm stops when the coefficient of variation of the probability estimate becomes lower than 0.5.
# %%
algoMC.run()
# %%
# We can then get the results of the algorithm.
# %%
result = algoMC.getResult()
probability = result.getProbabilityEstimate()
numberOfFunctionEvaluations = (
limitStateFunction.getEvaluationCallsNumber() - initialNumberOfCall
)
print("Number of calls to the limit state =", numberOfFunctionEvaluations)
print("Pf = ", probability)
print("CV =", result.getCoefficientOfVariation())
# %%
# The `drawProbabilityConvergence` method plots the probability estimate depending on the number of function evaluations.
# The order of convergence is :math:`O \left( 1/N^2 \right)` with :math:`N` being the number of function evaluations.
# This is why we use a logarithmic scale for the X axis of the graphics.
# %%
graph = algoMC.drawProbabilityConvergence()
graph.setLogScale(ot.GraphImplementation.LOGX)
view = viewer.View(graph)
# %%
# We see that the 95% confidence interval becomes smaller and smaller and stabilizes at the end of the simulation.
#
# In order to compute the confidence interval, we use the `getConfidenceLength` method, which returns the length of the interval.
# In order to compute the bounds of the interval, we divide this length by 2.
# %%
alpha = 0.05
# %%
pflen = result.getConfidenceLength(1 - alpha)
print(
"%.2f%% confidence interval = [%f,%f]"
% ((1 - alpha) * 100, probability - pflen / 2, probability + pflen / 2)
)
# %%
# This interval is consistent with the exact probability :math:`P_f=0.02920`.
# %%
# Appendix: derivation of the failure probability
# -----------------------------------------------
#
# The failure probability is:
#
# .. math::
# P_f = \text{Prob}(R-S \leq 0) = \int_{r-s \leq 0} f_{R, S}(r, s)drds
#
#
# where :math:`f_{R, S}` is the probability distribution function of the random vector :math:`(R,S)`.
# If R and S are independent, then:
#
# .. math::
# f_{R, S}(r, s) = f_R(r) f_S(s)
#
#
# for any :math:`r,s\in\mathbb{R}`,
# where :math:`f_S` is the probability distribution function of the random
# variable :math:`S` and :math:`f_R` is the probability distribution function of the random variable :math:`R`.
# Therefore,
#
# .. math::
# P_f = \int_{r-s \leq 0} f_R(r) f_S(s) drds.
#
#
# This implies:
#
# .. math::
# P_f = \int_{-\infty}^{+\infty} \left(\int_{r \leq s} f_R(r) dr \right) f_S(s) ds.
#
# Therefore,
#
# .. math::
# P_f = \int_{-\infty}^{+\infty}f_S(s)F_R(s)ds
#
#
# where :math:`F_R` is the cumulative distribution function of the random variable :math:`R`.
#