# Exploitation of simulation algorithm results¶

In this example we are going to retrieve all the results proposed by probability simulation algorithms:

• the probability estimate

• the estimator variance

• the confidence interval

• the convergence graph of the estimator

• the stored input and output numerical samples

• importance factors

:

from __future__ import print_function
import openturns as ot


Create the joint distribution of the parameters.

:

distribution_R = ot.LogNormalMuSigma(300.0, 30.0, 0.0).getDistribution()
distribution_F = ot.Normal(75e3, 5e3)
marginals = [distribution_R, distribution_F]
distribution = ot.ComposedDistribution(marginals)


Create the model.

:

model = ot.SymbolicFunction(['R', 'F'], ['R-F/(pi_*100.0)'])

:

modelCallNumberBefore = model.getEvaluationCallsNumber()
modelHessianCallNumberBefore = model.getHessianCallsNumber()


To have access to the input and output samples after the simulation, activate the History mechanism.

:

model = ot.MemoizeFunction(model)


Remove all the values stored in the history mechanism. Care : it is done regardless the status of the History mechanism.

:

model.clearHistory()


Create the event whose probability we want to estimate.

:

vect = ot.RandomVector(distribution)
G = ot.CompositeRandomVector(model, vect)
event = ot.ThresholdEvent(G, ot.Less(), 0.0)


Create a Monte Carlo algorithm.

:

experiment = ot.MonteCarloExperiment()
algo = ot.ProbabilitySimulationAlgorithm(event, experiment)
algo.setMaximumCoefficientOfVariation(0.1)
algo.setMaximumStandardDeviation(0.001)
algo.setMaximumOuterSampling(int(1e4))


Define the HistoryStrategy to store the values of and used ot draw the convergence graph. Compact strategy : N points

:

N = 1000
algo.setConvergenceStrategy(ot.Compact(N))
algo.run()


Retrieve result structure.

:

result = algo.getResult()


Display the simulation event probability.

:

result.getProbabilityEstimate()

:

0.030580075662042867

:

#  Criteria 3 : Display the Standard Deviation of the estimator
result.getStandardDeviation()

:

0.003057093060511293


Display the variance of the simulation probability estimator.

:

result.getVarianceEstimate()

:

9.345817980626304e-06

:

#  Criteria 2 : Display the number of iterations of the simulation
result.getOuterSampling()

:

3172

:

# Display the total number of evaluations of the model
result.getOuterSampling() * result.getBlockSize()

:

3172


Save the number of calls to the model, its gradient and hessian done so far.

:

modelCallNumberAfter = model.getEvaluationCallsNumber()
modelHessianCallNumberAfter = model.getHessianCallsNumber()


Display the number of iterations executed and the number of evaluations of the model.

:

modelCallNumberAfter - modelCallNumberBefore

:

3172


Get the mean point in event domain care : only for Monte Carlo and LHS sampling methods.

:

result.getMeanPointInEventDomain()

:


[245.843,80222]

Get the associated importance factors care : only for Monte Carlo and LHS sampling methods.

:

result.getImportanceFactors()

:


[X0 : 0.776373, X1 : 0.223627]

:

result.drawImportanceFactors()

: Display the confidence interval length centered around the MonteCarlo probability. The confidence interval is with level 0.95, where is the estimated probability and is the confidence interval length.

:

probability = result.getProbabilityEstimate()
length95 = result.getConfidenceLength(0.95)
print("0.95 Confidence Interval length = ", length95)
print("IC at 0.95 = [", probability - 0.5*length95, "; ", probability + 0.5*length95, "]")

0.95 Confidence Interval length =  0.011983584591978923
IC at 0.95 = [ 0.024588283366053405 ;  0.03657186795803233 ]


Draw the convergence graph and the confidence interval of level alpha. By default, alpha = 0.95.

:

alpha = 0.90
algo.drawProbabilityConvergence(alpha)

: Get the numerical samples of the input and output random vectors stored according to the History Strategy specified and used to evaluate the probability estimator and its variance.

:

inputSampleStored = model.getInputHistory()
outputSampleStored = model.getOutputHistory()
inputSampleStored

:

 v0 v1 0 317.18215872560563 68669.13448891672 1 285.74211953628173 81027.39100414288 2 240.13738518981802 76750.21043265145 ... 3169 266.95634238937606 74830.12233857176 3170 282.28504401568824 78480.19105287212 3171 247.17242375744675 81708.23104995265

Get the values of the estimator and its variance stored according to the History Strategy specified and used to draw the convergence graph.

:

estimator_probability_sample = algo.getConvergenceStrategy().getSample()
estimator_variance_sample = algo.getConvergenceStrategy().getSample()
print(estimator_probability_sample, estimator_variance_sample)

[0,-1] [0.25,0.046875]