def create_monte_carlo(model, inputRandomVector, coefficient_variation):
"""Create a Monte Carlo algorithm.
Parameters
----------
model : OpenTURNS Function.
inputRandomVector : OpenTURNS RandomVector, vector of random inputs.
coefficient_variation : Float, target for the coefficient of variation of
the estimator.
"""
outputVariableOfInterest = ot.CompositeRandomVector(model, inputRandomVector)
# Create an Event from this RandomVector
threshold = 30
myEvent = ot.ThresholdEvent(outputVariableOfInterest, ot.Greater(), threshold)
myEvent.setName("Deviation > %g cm" % threshold)
# Create a Monte Carlo algorithm
experiment = ot.MonteCarloExperiment()
myAlgoMonteCarlo = ot.ProbabilitySimulationAlgorithm(myEvent, experiment)
myAlgoMonteCarlo.setBlockSize(100)
myAlgoMonteCarlo.setMaximumCoefficientOfVariation(coefficient_variation)
return myAlgoMonteCarlo
def test_UseCaseMonteCarlo(self):
problem = otb.ReliabilityProblem14()
event = problem.getEvent()
# Create a Monte Carlo algorithm
experiment = ot.MonteCarloExperiment()
algo = ot.ProbabilitySimulationAlgorithm(event, experiment)
algo.setMaximumCoefficientOfVariation(0.01)
algo.setBlockSize(int(1.0e3))
algo.setMaximumOuterSampling(int(1e3))
algo.run()
# Retrieve results
result = algo.getResult()
computed_pf = result.getProbabilityEstimate()
exact_pf = problem.getProbability()
print("exact_pf=", exact_pf)
print("computed_pf=", computed_pf)
samplesize = result.getOuterSampling() * result.getBlockSize()
alpha = 0.05
pflen = result.getConfidenceLength(1 - alpha)
print(
"%.2f%% confidence interval = [%f,%f]"
% ((1 - alpha) * 100, computed_pf - pflen / 2, computed_pf + pflen / 2)
)
print("Sample size : ", samplesize)
atol = 1.0e2 / np.sqrt(samplesize)
np.testing.assert_allclose(computed_pf, exact_pf, atol=atol)
def computeCrossingProbability_MonteCarlo(b, t, mu_S, covariance, R, delta_t,
n_block, n_iter, CoV):
X, event = getXEvent(b, t, mu_S, covariance, R, delta_t)
algo = ot.ProbabilitySimulationAlgorithm(event, ot.MonteCarloExperiment())
algo.setBlockSize(n_block)
algo.setMaximumOuterSampling(n_iter)
algo.setMaximumCoefficientOfVariation(CoV)
algo.run()
return algo.getResult().getProbabilityEstimate() / delta_t
def sim_event(ev):
experiment = ot.MonteCarloExperiment()
algo = ot.ProbabilitySimulationAlgorithm(ev, experiment)
algo.setMaximumOuterSampling(2500)
algo.setBlockSize(4)
algo.setMaximumCoefficientOfVariation(-1.0)
algo.run()
result = algo.getResult()
return result.getProbabilityEstimate()
def build_Xy_table(ABM,
size_sample,
size_MC,
SequenceFunction,
SequenceFunctionString,
as_df=True,
to_file=True,
**kwargs):
import os
problem_ABM = ABM.problem()
if SequenceFunction == ot.MonteCarloExperiment:
unscaled_set_X = np.array([
ot.MonteCarloExperiment(ot.Uniform(0, 1), size_sample).generate()
for i in range(problem_ABM['num_vars'])
])
unscaled_set_X = unscaled_set_X.reshape(size_sample,
problem_ABM['num_vars'])
else:
SequenceFunction = SequenceFunction(problem_ABM['num_vars'], **kwargs)
unscaled_set_X = np.array(SequenceFunction.generate(size_sample))
set_X = rescale_sample(unscaled_set_X, problem_ABM['bounds'])
# Evaluate
set_y = [
np.array(Parallel(n_jobs=-1)(delayed(ABM.model)(p) for p in set_X))
for i in range(size_MC)
]
# as DataFrame
df = pd.DataFrame(set_X, columns=problem_ABM['names']).join(
pd.DataFrame(set_y).T.add_prefix('evaluation_'))
if to_file:
# save file
directory = 'ABM_eval_' + problem_ABM['abm_name']
filename = SequenceFunctionString + '_ss' + str(
size_sample) + '_MC' + str(size_MC)
if not os.path.exists(directory):
os.makedirs(directory + '/' + filename)
df.to_csv(directory + '/' + filename, index=False)
print 'Saved file ' + filename
if as_df:
return df
def buildMonteCarlo(self, problem):
"""
Creates a Monte-Carlo algorithm.
We create a MonteCarloExperiment and we create
a ProbabilitySimulationAlgorithm based on the problem event.
Parameters
----------
problem : ot.ReliabilityBenchmarkProblem
The problem.
Returns
----------
algo : ot.ProbabilitySimulationAlgorithm
The Monte-Carlo algorithm for estimating the probability.
"""
myEvent = problem.getEvent()
experiment = ot.MonteCarloExperiment()
algo = ot.ProbabilitySimulationAlgorithm(myEvent, experiment)
return algo
def test_UseCase(self):
problem = otb.ReliabilityProblem60()
event = problem.getEvent()
# Create a Monte Carlo algorithm
experiment = ot.MonteCarloExperiment()
algo = ot.ProbabilitySimulationAlgorithm(event, experiment)
algo.setMaximumCoefficientOfVariation(0.05)
algo.setMaximumOuterSampling(int(1e5))
algo.run()
# Retrieve results
result = algo.getResult()
computed_pf = result.getProbabilityEstimate()
exact_pf = problem.getProbability()
print("exact_pf=", exact_pf)
print("computed_pf=", computed_pf)
samplesize = result.getOuterSampling() * result.getBlockSize()
print("Sample size : ", samplesize)
atol = 1.0 / np.sqrt(samplesize)
np.testing.assert_allclose(computed_pf, exact_pf, atol=atol)
print("asymptotic intervals:")
print("Aggregated first order indices interval = ", interval_fo_asymptotic)
print("Aggregated total order indices interval = ", interval_to_asymptotic)
# special case in dim=2
ot.ResourceMap.SetAsString('SobolIndicesExperiment-SamplingMethod',
'MonteCarlo')
ot.RandomGenerator.SetSeed(0)
distribution = ot.ComposedDistribution([ot.Uniform()] * 2)
size = 1000
model = ot.SymbolicFunction(['X1', 'X2'], ['2*X1 + X2'])
sensitivity_algorithm = ot.SaltelliSensitivityAlgorithm(
distribution, size, model, True)
print(sensitivity_algorithm.getSecondOrderIndices())
ot.RandomGenerator.SetSeed(0)
experiment = ot.MonteCarloExperiment(distribution, size)
sensitivity_algorithm = ot.SaltelliSensitivityAlgorithm(
experiment, model, True)
print(sensitivity_algorithm.getSecondOrderIndices())
ot.RandomGenerator.SetSeed(0)
x = ot.SobolIndicesExperiment(distribution, size, True).generate()
y = model(x)
sensitivity_algorithm = ot.SaltelliSensitivityAlgorithm(x, y, size)
print(sensitivity_algorithm.getSecondOrderIndices())
# null contribution case: X3 not in output formula
model = ot.SymbolicFunction(['X1', 'X2', 'X3'], ['10+3*X1+X2'])
distribution = ot.ComposedDistribution([ot.Uniform(-1.0, 1.0)] *
input_dimension)
size = 10000
for method in methods:
ot.Interval([low] * 2, [up] * 2, [False, True], [True, False]),
ot.Interval([low] * 2, [up] * 2, [False, True], [False, True]),
ot.Interval([low] * 2, [up] * 2, [False, True], [False, False]),
ot.Interval([low] * 2, [up] * 2, [False, False], [True, True]),
ot.Interval([low] * 2, [up] * 2, [False, False], [True, False]),
ot.Interval([low] * 2, [up] * 2, [False, False], [False, True]),
ot.Interval([low] * 2, [up] * 2, [False, False], [False, False])
]
for domain in intervals:
print('#' * 50)
print('domain=\n', domain)
outDim = domain.getDimension()
f = ot.SymbolicFunction(inVars, inVars[0:outDim])
Y = ot.CompositeRandomVector(f, X)
event = ot.ThresholdEvent(Y, domain)
ot.RandomGenerator.SetSeed(0)
# algo = getattr(openturns, algoName)(event)
algo = ot.ProbabilitySimulationAlgorithm(event, ot.MonteCarloExperiment())
algo.run()
res = algo.getResult().getProbabilityEstimate()
print('MC p=%.6g' % res)
ot.RandomGenerator.SetSeed(0)
# algo = getattr(openturns, algoName)(event)
algo = ot.FORM(ot.Cobyla(), event, X.getMean())
algo.run()
res = algo.getResult().getEventProbability()
print('FORM p=%.2f' % res)
import openturns as ot
from openturns.viewer import View
# MonteCarlo
d = ot.MonteCarloExperiment(ot.ComposedDistribution([ot.Uniform()] * 3), 32)
s = d.generate()
s.setDescription(["X1", "X2", "X3"])
g = ot.Graph()
g.setTitle("MonteCarlo experiment")
g.setGridColor("black")
p = ot.Pairs(s)
g.add(p)
View(g)
mostSignificant = 50
significanceFactor = 1.0e-4
truncatureBasisStrategy_2 = ot.CleaningStrategy(
multivariateBasis, maximumConsideredTerms, mostSignificant, significanceFactor, True)
# %%
# STEP 3: Evaluation strategy of the approximation coefficients
# -------------------------------------------------------------
# %%
# The technique illustrated is the Least Squares technique where the points come from an design of experiments. Here : the Monte Carlo sampling technique.
# %%
sampleSize = 100
evaluationCoeffStrategy = ot.LeastSquaresStrategy(
ot.MonteCarloExperiment(sampleSize))
# %%
# You can specify the approximation algorithm. This is the algorithm that generates a sequence of basis using Least Angle Regression.
# %%
basisSequenceFactory = ot.LARS()
# %%
# This algorithm estimates the empirical error on each sub-basis using Leave One Out strategy.
# %%
fittingAlgorithm = ot.CorrectedLeaveOneOut()
# Finally the metamodel selection algorithm embbeded in LeastSquaresStrategy
approximationAlgorithm = ot.LeastSquaresMetaModelSelectionFactory(
basisSequenceFactory, fittingAlgorithm)
mean[3] = 5.0 sigma = [1.0] * dim R = ot.IdentityMatrix(dim) myDistribution = ot.Normal(mean, sigma, R) # We create a 'usual' RandomVector from the Distribution vect = ot.RandomVector(myDistribution) # We create a composite random vector output = ot.CompositeRandomVector(myFunction, vect) # We create an Event from this RandomVector myEvent = ot.Event(output, ot.Less(), -3.0) # Monte Carlo experiments = [ot.MonteCarloExperiment()] # Quasi Monte Carlo experiments.append(ot.LowDiscrepancyExperiment()) # Randomized Quasi Monte Carlo experiment = ot.LowDiscrepancyExperiment() experiment.setRandomize(True) experiments.append(experiment) # Importance sampling mean[0] = 4.99689645939288809018e+01 mean[1] = 1.84194175946153282375e+00 mean[2] = 1.04454036676956398821e+01 mean[3] = 4.66776215562709406726e+00 myImportance = ot.Normal(mean, sigma, R) experiments.append(ot.ImportanceSamplingExperiment(myImportance)) # Randomized LHS experiment = ot.LHSExperiment()
def run_MonteCarlo(
event,
coefVar=0.1,
outerSampling=10000,
blockSize=10,
seed=1234,
verbose=False,
failure_domain=None,
):
"""
Run a Monte Carlo simulation.
Parameters
----------
event : openturns.Event
The failure event or a list of failure event.
coefVar : float
The target coefficient of variation.
outerSampling : int
The maximum number of outer iterations.
Nb of iterations = outerSampling x blockSize.
blockSize : int
The number of samples send to evaluate simultaneously.
seed : int
Seed for the openturns random generator.
logfile : bool
Enable or not to write the log in MonteCarlo.log file.
verbose : bool
Enable or not the display of the result.
activeCache : bool
Enable or not the cache mechanism of the NumericalMathFunction.
activeHistory : bool
Enable or not the history mechanism of the NumericalMathFunction.
failure_domain : string
Type of failure domain form : either 'union' or 'intersection'. Only
needed if the event is a list.
"""
# case with the limit state defined as an intersection or a
# union of the event
if type(event) is list:
n_event = len(event)
antecedent = event[0].getAntecedent()
if failure_domain == "union":
def function_union(X):
sample = ot.NumericalSample(X.getSize(), n_event)
for i in range(n_event):
sample[:, i] = event[i].getFunction()(X)
sample = np.array(sample)
for i in range(n_event):
if (event[i].getOperator().getImplementation(
).getClassName() == "Less" or event[i].getOperator(
).getImplementation().getClassName() == "LessOrEqual"):
sample[:, i] = sample[:, i] < event[i].getThreshold()
if (event[i].getOperator().getImplementation(
).getClassName() == "Greater" or event[i].getOperator(
).getImplementation().getClassName() == "GreaterOrEqual"):
sample[:, i] = sample[:, i] >= event[i].getThreshold()
return np.atleast_2d(sample.sum(axis=1)).T
model = ot.PythonFunction(
event[0].getFunction().getInputDimension(),
event[0].getFunction().getOutputDimension(),
func_sample=function_union,
)
output = ot.RandomVector(model, antecedent)
event = ot.ThresholdEvent(output, ot.Greater(), 0.0)
elif failure_domain == "intersection":
def function_intersection(X):
sample = ot.NumericalSample(X.getSize(), n_event)
for i in range(n_event):
sample[:, i] = event[i].getFunction()(X)
sample = np.array(sample)
for i in range(n_event):
if (event[i].getOperator().getImplementation(
).getClassName() == "Less" or event[i].getOperator(
).getImplementation().getClassName() == "LessOrEqual"):
sample[:, i] = sample[:, i] < event[i].getThreshold()
if (event[i].getOperator().getImplementation(
).getClassName() == "Greater" or event[i].getOperator(
).getImplementation().getClassName() == "GreaterOrEqual"):
sample[:, i] = sample[:, i] >= event[i].getThreshold()
return np.atleast_2d(sample.prod(axis=1)).T
model = ot.PythonFunction(
event[0].getFunction().getInputDimension(),
event[0].getFunction().getOutputDimension(),
func_sample=function_intersection,
)
output = ot.RandomVector(model, antecedent)
new_event = ot.ThresholdEvent(output, ot.Greater(), 0.0)
else:
model = event.getFunction()
new_event = event
# Initialize the random generator
ot.RandomGenerator.SetSeed(seed)
# Run Monte Carlo simulation
experiment = ot.MonteCarloExperiment()
simulation = ot.ProbabilitySimulationAlgorithm(new_event, experiment)
simulation.setMaximumCoefficientOfVariation(coefVar)
simulation.setMaximumOuterSampling(outerSampling)
simulation.setBlockSize(blockSize)
# try:
simulation.run()
# except Exception as e:
# dump_cache(model, 'Cache/physicalModelMathFunction')
# raise e
result = simulation.getResult()
dfResult = pd.DataFrame()
dfResult = dfResult.append(
pd.DataFrame([result.getProbabilityEstimate()],
index=["Probability of failure"]))
dfResult = dfResult.append(
pd.DataFrame(
[result.getCoefficientOfVariation()],
index=["Coefficient of varation"],
))
dfResult = dfResult.append(
pd.DataFrame([result.getConfidenceLength()],
index=["95 % Confidence length"]))
dfResult = dfResult.append(
pd.DataFrame(
[result.getOuterSampling() * result.getBlockSize()],
index=["Number of calls"],
))
dfResult = dfResult.reset_index()
dfResult.columns = ["", "Results - Monte Carlo"]
if verbose:
print(dfResult, "\n")
return simulation
field = ot.Field(mesh, values)
evaluation = ot.P1LagrangeEvaluation(field)
x = [2.3]
y = evaluation(x)
print(y)
ott.assert_almost_equal(y, [0.55])
# Learning sample on meshD
mesher = ot.IntervalMesher([7, 7])
lowerbound = [-1.0, -1.0]
upperBound = [1, 1]
interval = ot.Interval(lowerbound, upperBound)
meshD = mesher.build(interval)
sample = ot.ProcessSample(meshD, 10, 1)
field = ot.Field(meshD, 1)
for k in range(sample.getSize()):
field.setValues(ot.Normal().getSample(64))
sample[k] = field
lagrange = ot.P1LagrangeEvaluation(sample)
# New mesh
mesh = ot.Mesh(
ot.MonteCarloExperiment(ot.ComposedDistribution([ot.Uniform(-1, 1)] * 2),
200).generate())
point = mesh.getVertices()[0]
y = lagrange(point)
print(y)
index = lagrange.getEnclosingSimplexAlgorithm().query(point)
print(index)
assert index == 12, "wrong index"
def myMonteCarloExperiment(distribution, size, model):
experiment = ot.MonteCarloExperiment(distribution, size)
sensitivity_algorithm = ot.SaltelliSensitivityAlgorithm(experiment, model)
return sensitivity_algorithm
S = ot.CorrelationMatrix(2)
S[1, 0] = 0.3
R = ot.NormalCopula.GetCorrelationFromSpearmanCorrelation(S)
copula = ot.NormalCopula(R)
distribution_corr = ot.ComposedDistribution([ot.Normal()] * 2, copula)
# %%
# ANCOVA needs a functional decomposition of the model
enumerateFunction = ot.LinearEnumerateFunction(2)
productBasis = ot.OrthogonalProductPolynomialFactory([ot.HermiteFactory()] * 2,
enumerateFunction)
adaptiveStrategy = ot.FixedStrategy(
productBasis, enumerateFunction.getStrataCumulatedCardinal(4))
samplingSize = 250
projectionStrategy = ot.LeastSquaresStrategy(
ot.MonteCarloExperiment(samplingSize))
algo = ot.FunctionalChaosAlgorithm(model, distribution, adaptiveStrategy,
projectionStrategy)
algo.run()
result = ot.FunctionalChaosResult(algo.getResult())
# %%
# Create the input sample taking account the correlation
size = 2000
sample = distribution_corr.getSample(size)
# %%
# Perform the decomposition
ancova = ot.ANCOVA(result, sample)
# Compute the ANCOVA indices (first order and uncorrelated indices are computed together)
indices = ancova.getIndices()
# %%
# Run FORM
myAlgo = ot.FORM(myCobyla, myEvent, mean)
myAlgo.run()
result = myAlgo.getResult()
print('event probability:', result.getEventProbability())
print('calls number:', myFunction.getCallsNumber())
# %%
# **Stop a simulation algorithm using a time limit**
#
# Here we will create a callback to not exceed a specified simulation time.
# %%
# Create simulation
experiment = ot.MonteCarloExperiment()
myAlgo = ot.ProbabilitySimulationAlgorithm(myEvent, experiment)
myAlgo.setMaximumOuterSampling(1000000)
myAlgo.setMaximumCoefficientOfVariation(-1.0)
# %%
# Define the stopping criterion
timer = ot.TimerCallback(0.01)
myAlgo.setStopCallback(timer)
# %%
# Run algorithm
myAlgo.run()
result = myAlgo.getResult()
print('event probability:', result.getProbabilityEstimate())
print('calls number:', myFunction.getCallsNumber())
# %% # Create the correlated input distribution S = ot.CorrelationMatrix(2) S[1, 0] = 0.3 R = ot.NormalCopula.GetCorrelationFromSpearmanCorrelation(S) copula = ot.NormalCopula(R) distribution_corr = ot.ComposedDistribution([ot.Normal()] * 2, copula) # %% # ANCOVA needs a functional decomposition of the model enumerateFunction = ot.LinearEnumerateFunction(2) productBasis = ot.OrthogonalProductPolynomialFactory([ot.HermiteFactory()]*2, enumerateFunction) adaptiveStrategy = ot.FixedStrategy(productBasis, enumerateFunction.getStrataCumulatedCardinal(4)) samplingSize = 250 projectionStrategy = ot.LeastSquaresStrategy(ot.MonteCarloExperiment(samplingSize)) algo = ot.FunctionalChaosAlgorithm(model, distribution, adaptiveStrategy, projectionStrategy) algo.run() result = ot.FunctionalChaosResult(algo.getResult()) # %% # Create the input sample taking account the correlation size = 2000 sample = distribution_corr.getSample(size) # %% # Perform the decomposition ancova = ot.ANCOVA(result, sample) # Compute the ANCOVA indices (first order and uncorrelated indices are computed together) indices = ancova.getIndices() # Retrieve uncorrelated indices
