def _PODgaussModelCl(self, defects, intercept, slope, stderr, detection):
class buildPODModel():
def __init__(self, intercept, slope, sigmaEpsilon, detection):
self.intercept = intercept
self.slope = slope
self.sigmaEpsilon = sigmaEpsilon
self.detection = detection
def PODmodel(self, x):
t = (self.detection -
(self.intercept + self.slope * x)) / self.sigmaEpsilon
return ot.DistFunc.pNormal(t, True)
N = defects.getSize()
X = ot.Sample(N, [1, 0])
X[:, 1] = defects
X = ot.Matrix(X)
covMatrix = X.computeGram(True).solveLinearSystem(ot.IdentityMatrix(2))
sampleNormal = ot.Normal([0, 0],
ot.CovarianceMatrix(
covMatrix.getImplementation())).getSample(
self._simulationSize)
sampleSigmaEpsilon = (ot.Chi(N - 2).inverse() * np.sqrt(N - 2) *
stderr).getSample(self._simulationSize)
PODcoll = []
for i in range(self._simulationSize):
sigmaEpsilon = sampleSigmaEpsilon[i][0]
interceptSimu = sampleNormal[i][0] * sigmaEpsilon + intercept
slopeSimu = sampleNormal[i][1] * sigmaEpsilon + slope
PODcoll.append(
buildPODModel(interceptSimu, slopeSimu, sigmaEpsilon,
detection).PODmodel)
return PODcoll
# %% from __future__ import print_function import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt import math as m import time ot.Log.Show(ot.Log.NONE) # %% # Define an event to compute a probability myFunction = ot.SymbolicFunction(['E', 'F', 'L', 'I'], ['-F*L^3/(3.0*E*I)']) dim = myFunction.getInputDimension() mean = [50.0, 1.0, 10.0, 5.0] sigma = [1.0] * dim R = ot.IdentityMatrix(dim) myDistribution = ot.Normal(mean, sigma, R) vect = ot.RandomVector(myDistribution) output = ot.CompositeRandomVector(myFunction, vect) myEvent = ot.ThresholdEvent(output, ot.Less(), -3.0) # %% # **Stop a FORM algorithm using a calls number limit** # # A FORM algorithm termination can be controlled by the maximum number of iterations # # of its underlying optimization solver, but not directly by a maximum number of evaluations. # %% # Create the optimization algorithm myCobyla = ot.Cobyla()
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
f = ot.SymbolicFunction(['x'], ['17-exp(0.1*(x-1.0))'])
graph = f.draw(0.0, 12.0)
dist = ot.Normal([5.0, 15.0], [1.0, 0.25], ot.IdentityMatrix(2))
N = 1000
sample = dist.getSample(N)
sample1 = ot.Sample(0, 2)
sample2 = ot.Sample(0, 2)
for X in sample:
x, y = X
if f([x])[0] > y:
sample1.add(X)
else:
sample2.add(X)
cloud = ot.Cloud(sample1)
cloud.setColor('green')
cloud.setPointStyle('square')
graph.add(cloud)
cloud = ot.Cloud(sample2)
cloud.setColor('red')
cloud.setPointStyle('square')
graph.add(cloud)
graph.setTitle('Monte Carlo simulation (Pf=0.048, N=1000)')
graph.setLegends(['domain Df', 'simulations'])
import sys
ot.TESTPREAMBLE()
ot.PlatformInfo.SetNumericalPrecision(3)
m = 10
x = [[0.5 + i] for i in range(m)]
inVars = ["a", "b", "c", "x"]
formulas = ["a + b * exp(c * x)", "(a * x^2 + b) / (c + x^2)"]
model = ot.SymbolicFunction(inVars, formulas)
p_ref = [2.8, 1.2, 0.5]
params = [0, 1, 2]
modelX = ot.ParametricFunction(model, params, p_ref)
y = modelX(x)
y += ot.Normal([0.0] * 2, [0.05] * 2, ot.IdentityMatrix(2)).getSample(m)
candidate = [1.0] * 3
priorCovariance = ot.CovarianceMatrix(3)
for i in range(3):
priorCovariance[i, i] = 3.0 + (1.0 + i) * (1.0 + i)
for j in range(i):
priorCovariance[i, j] = 1.0 / (1.0 + i + j)
errorCovariance = ot.CovarianceMatrix(2)
for i in range(2):
errorCovariance[i, i] = 2.0 + (1.0 + i) * (1.0 + i)
for j in range(i):
errorCovariance[i, j] = 1.0 / (1.0 + i + j)
globalErrorCovariance = ot.CovarianceMatrix(2 * m)
for i in range(2 * m):
globalErrorCovariance[i, i] = 2.0 + (1.0 + i) * (1.0 + i)
for j in range(i):
graph.add(cloud)
return graph, s
# %%
# **Definition of some IFS**
# %%
# Spiral
rho1 = 0.9
theta1 = 137.5 * m.pi / 180.0
f1 = [[0.0]*2, ot.SquareMatrix(2, [rho1 * m.cos(theta1), -rho1 * m.sin(theta1), \
rho1 * m.sin(theta1), rho1 * m.cos(theta1)])]
rho2 = 0.15
f2 = [[1.0, 0.0], rho2 * ot.IdentityMatrix(2)]
f_i = [f1, f2]
graph, s = drawIFS(f_i, skip = 100, iterations = 100000, batch_size = 1, name="Spiral", color="blue")
print("Box counting dimension=%.3f" % s)
view = viewer.View(graph)
# %%
# Fern
f1 = [[0.0]*2, ot.SquareMatrix(2, [0.0, 0.0, 0.0, 0.16])]
f2 = [[0.0, 1.6], ot.SquareMatrix(2, [0.85, 0.04, -0.04, 0.85])]
f3 = [[0.0, 1.6], ot.SquareMatrix(2, [0.2, -0.26, 0.23, 0.22])]
f4 = [[0.0, 0.44], ot.SquareMatrix(2, [-0.15, 0.28, 0.26, 0.24])]
f_i = [f1, f2, f3, f4]
graph, s = drawIFS(f_i, skip = 100, iterations = 100000, batch_size = 1, name="Fern", color="green")
print("Box counting dimension=%.3f" % s)
view = viewer.View(graph)
from __future__ import print_function
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
import math as m
Id = ot.IdentityMatrix(2)
atoms = [
ot.Normal([1.0, 2.0], [0.5, 0.8], Id),
ot.Normal([1.0, -2.0], [0.9, 0.8], Id),
ot.Normal([-1.0, 0.0], [0.5, 0.6], Id)
]
weights = [0.3, 0.3, 0.4]
mixture = ot.Mixture(atoms, weights)
data = mixture.getSample(1000)
classifier = ot.MixtureClassifier(mixture)
graph = mixture.drawPDF(data.getMin(), data.getMax())
graph.setLegendPosition("")
graph.setTitle("MixtureClassifier example")
classes = classifier.classify(data)
palette = ot.Drawable.BuildDefaultPalette(len(atoms))
symbols = ot.Drawable.GetValidPointStyles()
for i in range(classes.getSize()):
index = classes[i]
graph.add(
ot.Cloud([data[i]], palette[index % len(palette)],
symbols[index % len(symbols)]))
fig = plt.figure(figsize=(4, 4))
axis = fig.add_subplot(111)
axis.set_xlim(auto=True)
print('Distribution Parameters ', distParam)
non_native = distParam.getValues()
desc = distParam.getDescription()
print('non-native=', non_native, desc)
native = distParam.evaluate()
print('native=', native)
non_native = distParam.inverse(native)
print('non-native=', non_native)
print('built dist=', distParam.getDistribution())
# derivative of the native parameters with regards the parameters of the
# distribution
print(distParam.gradient())
# by the finite difference technique
eps = 1e-5
dim = len(non_native)
nativeParamGrad = ot.SquareMatrix(ot.IdentityMatrix(dim))
for i in range(dim):
for j in range(dim):
xp = list(non_native)
xp[i] += eps
xm = list(non_native)
xm[i] -= eps
nativeParamGrad[i, j] = 0.5 * \
(distParam(xp)[j] - distParam(xm)[j]) / eps
print(nativeParamGrad)
# %%
# First, import the python modules:
# %%
import openturns as ot
from openturns.viewer import View
# %%
# Create the probabilistic model :math:`Y = g(X)`
# -----------------------------------------------
# %%
# Create the input random vector :math:`X`:
# %%
X = ot.RandomVector(ot.Normal([0.25] * 2, [1] * 2, ot.IdentityMatrix(2)))
# %%
# Create the function :math:`g`:
# %%
g = ot.SymbolicFunction(['x1', 'x2'], ['20-(x1-x2)^2-8*(x1+x2-4)^3'])
print('function g: ', g)
# %%
# In order to be able to get the subset samples used in the algorithm, it is necessary to transform the *SymbolicFunction* into a *MemoizeFunction*:
# %%
g = ot.MemoizeFunction(g)
# %%
Theta1 = ot.Dirac(trueParameter[0])
Theta2 = ot.Dirac(trueParameter[1])
Theta3 = ot.Dirac(trueParameter[2])
inputRandomVector = ot.ComposedDistribution([Theta1, Theta2, Theta3])
candidate = ot.Point([8.0, 9.0, -6.0])
calibratedIndices = [0, 1, 2]
model = ot.ParametricFunction(g, calibratedIndices, candidate)
outputObservationNoiseSigma = 0.01
meanNoise = ot.Point(outputDimension)
covarianceNoise = ot.Point(outputDimension, outputObservationNoiseSigma)
R = ot.IdentityMatrix(outputDimension)
observationOutputNoise = ot.Normal(meanNoise, covarianceNoise, R)
size = 100
inputObservations = ot.Sample(size, 0)
# Generate exact outputs
inputSample = inputRandomVector.getSample(size)
outputStress = g(inputSample)
# Add noise
sampleNoiseH = observationOutputNoise.getSample(size)
outputObservations = outputStress + sampleNoiseH
priorCovariance = ot.CovarianceMatrix(inputDimension)
for i in range(inputDimension):
priorCovariance[i, i] = 3.0 + (1.0 + i) * (1.0 + i)
for measure in measures:
print(measure, '(continuous)', measure(x))
N = 10000
experiment = ot.LHSExperiment(N)
factory = otrobopt.MeasureFactory(experiment)
discretizedMeasure = factory.build(measure)
print(discretizedMeasure, '(discretized LHS)', discretizedMeasure(x))
N = 4
experiment = ot.GaussProductExperiment([N])
factory = otrobopt.MeasureFactory(experiment)
discretizedMeasure = factory.build(measure)
print(discretizedMeasure, '(discretized Gauss)', discretizedMeasure(x))
# Second test: theta of dimension 2
thetaDist = ot.Normal([2.0] * 2, [0.1] * 2, ot.IdentityMatrix(2))
f_base = ot.SymbolicFunction(['x', 'theta0', 'theta1'], ['x*theta0+theta1'])
f = ot.ParametricFunction(f_base, [1, 2], thetaDist.getMean())
x = [1.0]
measures = [otrobopt.MeanMeasure(f, thetaDist),
otrobopt.VarianceMeasure(f, thetaDist),
otrobopt.WorstCaseMeasure(
f, ot.ComposedDistribution([ot.Uniform(-1.0, 4.0)] * 2)),
otrobopt.WorstCaseMeasure(
f, ot.ComposedDistribution(
[ot.Uniform(-1.0, 4.0)] * 2), False),
otrobopt.JointChanceMeasure(
f, thetaDist, ot.GreaterOrEqual(), 0.95),
otrobopt.IndividualChanceMeasure(
#! /usr/bin/env python
import openturns as ot
ot.TESTPREAMBLE()
# Instantiate one distribution object
dimension = 3
meanPoint = ot.Point([0.5, -0.5, 1])
sigma = [2, 3, 1]
sample = ot.Sample(0, dimension)
# Create a collection of distribution
aCollection = ot.DistributionCollection()
aCollection.add(ot.Normal(meanPoint, sigma, ot.IdentityMatrix(dimension)))
sample.add(meanPoint)
meanPoint += [1.0] * dimension
aCollection.add(ot.Normal(meanPoint, sigma, ot.IdentityMatrix(dimension)))
sample.add(meanPoint)
meanPoint += [1.0] * dimension
aCollection.add(ot.Normal(meanPoint, sigma, ot.IdentityMatrix(dimension)))
sample.add(meanPoint)
# Instantiate one distribution object
distribution = ot.KernelMixture(ot.Normal(), sigma, sample)
print("Distribution ", repr(distribution))
print("Distribution ", distribution)
distributionRef = ot.Mixture(aCollection)
# Is this distribution elliptical ?
def run(self):
"""
Build the POD models.
Notes
-----
This method build the polynomial chaos model. First the censored data
are filtered if needed. The Box Cox transformation is performed if it is
enabled. Then it builds the POD models, the Monte Carlo simulation is
performed for each given defect sizes. The confidence interval is
computed by simulating new coefficients of the polynomial chaos, then
Monte Carlo simulations are performed.
"""
# run the chaos algorithm and get result if not given
if not self._userChaos:
if self._verbose:
print('Start build polynomial chaos model...')
self._algoChaos = self._buildChaosAlgo(self._input, self._signals)
self._algoChaos.run()
if self._verbose:
print('Polynomial chaos model completed')
self._chaosResult = self._algoChaos.getResult()
# get the metamodel
self._chaosPred = self._chaosResult.getMetaModel()
# get the basis, coef and transformation, needed for the confidence interval
self._chaosCoefs = self._chaosResult.getCoefficients()
self._reducedBasis = self._chaosResult.getReducedBasis()
self._transformation = self._chaosResult.getTransformation()
self._basisFunction = ot.NumericalMathFunction(
ot.NumericalMathFunction(self._reducedBasis), self._transformation)
# compute the residuals and stderr
inputSize = self._input.getSize()
basisSize = self._reducedBasis.getSize()
self._residuals = self._signals - self._chaosPred(
self._input) # residuals
self._stderr = np.sqrt(
np.sum(np.array(self._residuals)**2) / (inputSize - basisSize - 1))
# Check the quality of the chaos model
R2 = self.getR2()
Q2 = self.getQ2()
if self._verbose:
print('Polynomial chaos validation R2 (>0.8) : {:0.4f}'.format(R2))
print('Polynomial chaos validation Q2 (>0.8) : {:0.4f}'.format(Q2))
# Compute the POD values for each defect sizes
self.POD = self._computePOD(self._defectSizes, self._chaosCoefs)
# create the interpolate function
interpModel = interp1d(self._defectSizes, self.POD, kind='linear')
self._PODmodel = ot.PythonFunction(1, 1, interpModel)
####################### confidence interval ############################
dof = inputSize - basisSize - 1
varEpsilon = (ot.ChiSquare(dof).inverse() * dof *
self._stderr**2).getRealization()[0]
gramBasis = ot.Matrix(self._basisFunction(self._input)).computeGram()
covMatrix = gramBasis.solveLinearSystem(
ot.IdentityMatrix(basisSize)) * varEpsilon
self._coefsDist = ot.Normal(
np.hstack(self._chaosCoefs),
ot.CovarianceMatrix(covMatrix.getImplementation()))
coefsRandom = self._coefsDist.getSample(self._simulationSize)
self._PODPerDefect = ot.NumericalSample(self._simulationSize,
self._defectNumber)
for i, coefs in enumerate(coefsRandom):
self._PODPerDefect[i, :] = self._computePOD(
self._defectSizes, coefs)
if self._verbose:
updateProgress(i, self._simulationSize,
'Computing POD per defect')
# This is capital J: J(x,xi) = calJ(x+xi), the parametric objective function
JFull = ot.ComposedFunction(calJ, noise)
J = ot.ParametricFunction(JFull, [2, 3], [0.0] * 2)
# This is g, the parametric constraints
gFull = ot.ComposedFunction(calG, noise)
g = ot.ParametricFunction(gFull, [2, 3], [0.0] * 2)
bounds = ot.Interval([-3.0] * 2, [3.0] * 2)
solver = ot.NLopt('LD_SLSQP')
solver.setMaximumIterationNumber(100)
for sigma_xi in [0.1, 0.2, 0.3, 0.4, 0.5]:
thetaDist = ot.Normal([0.0] * 2, [sigma_xi] * 2, ot.IdentityMatrix(2))
robustnessMeasure = otrobopt.MeanMeasure(J, thetaDist)
reliabilityMeasure = otrobopt.JointChanceMeasure(g, thetaDist, ot.Less(),
0.9)
problem = otrobopt.RobustOptimizationProblem(robustnessMeasure,
reliabilityMeasure)
problem.setBounds(bounds)
algo = otrobopt.SequentialMonteCarloRobustAlgorithm(problem, solver)
algo.setMaximumIterationNumber(11)
algo.setMaximumAbsoluteError(1e-6)
algo.setInitialSamplingSize(2) # size of initial xsi discretization
algo.setSamplingSizeIncrement(
ot.PythonFunction(1, 1, lambda x: [1.0 * x[0]]))
algo.setInitialSearch(
1000) # number of multi-start tries, uniform law using bounds
def run_ImportanceSampling(
event,
pstar,
sd=1.0,
coefVar=0.05,
outerSampling=1000,
blockSize=10,
seed=1234,
verbose=False,
failure_domain=None,
):
"""
Run an importance sampling simulation.
Parameters
----------
event : openturns.Event
The failure event.
pstar : list of points
Design points in the standard space where to centered the instrumental
distribution.
sd : positive float
The standard deviation of the instrumental distribution.
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 ImportanceSampling.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)
dim = model.getInputDimension()
pstar = np.atleast_2d(pstar)
nPoint = pstar.shape[0]
stdev = [sd] * dim
corr = ot.IdentityMatrix(dim)
if nPoint > 1:
distribution_list = list()
for point in pstar:
distribution_list.append(ot.Normal(point, stdev, corr))
instrumental_distribution = ot.Mixture(distribution_list)
elif nPoint == 1:
instrumental_distribution = ot.Normal(pstar[0], stdev, corr)
# Run importance sampling simulation
experiment = ot.ImportanceSamplingExperiment(instrumental_distribution)
simulation = ot.ProbabilitySimulationAlgorithm(ot.StandardEvent(new_event),
experiment)
simulation.setMaximumOuterSampling(outerSampling)
simulation.setBlockSize(blockSize)
simulation.setMaximumCoefficientOfVariation(coefVar)
# 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 - Importance Sampling"]
if verbose:
print(dfResult, "\n")
return simulation
description = ndag.getDescription()
print("description=", description)
for node in order:
print(" parents(", description[node], ") : ",
[description[i] for i in ndag.getParents(node)])
print("children(", description[node], ") : ",
[description[i] for i in ndag.getChildren(node)])
# Conditional linear coefficients, b matrix:
# X3, X5, X6, X1, X2, X4
b = [[], [0.5], [0.0, 0.75], [0.0, 0.00, 0.0], [0.5, 0.00, 0.0, 0.75],
[0.0, 1.00, 0.0, 0.00, 1.0]]
T = buildT(b)
I = ot.IdentityMatrix(T.getDimension())
sigma = ot.CovarianceMatrix(np.array(T.solveLinearSystem(I)))
sigma_ordered = ot.CovarianceMatrix(len(b))
for i in range(len(b)):
for j in range(i + 1):
sigma_ordered[order[i], order[j]] = sigma[i, j]
print("Sigma matrix: ", sigma_ordered)
# Marginal mean, mu vector:
mu = [0.0] * 6
distribution = ot.Normal(mu, sigma_ordered)
distribution.setDescription(description)
size = 100000
sample = distribution.getSample(size)
data = ot.Sample([[53, 1], [57, 1], [58, 1], [63, 1], [66, 0], [67,
0], [67, 0],
[67, 0], [68, 0], [69, 0], [70, 0], [70, 0], [70,
1], [70, 1],
[72, 0], [73, 0], [75, 0], [75, 1], [76, 0], [76, 0],
[78, 0], [79, 0], [81, 0]])
data.setDescription(['Temp. (°F)', 'Failure'])
print(data)
fun = ot.SymbolicFunction(
["alpha", "beta", "x"],
["exp(alpha + beta * x) / (1 + exp(alpha + beta * x))"])
linkFunction = ot.ParametricFunction(fun, [2], [0.0])
instrumental = ot.Normal([0.0] * 2, [0.5, 0.05], ot.IdentityMatrix(2))
target = ot.ComposedDistribution([ot.Uniform(-100.0, 100.0)] * 2)
rwmh = ot.RandomWalkMetropolisHastings(target, [0.0] * 2, instrumental)
conditional = ot.Bernoulli()
observations = data[:, 1]
covariates = data[:, 0]
rwmh.setLikelihood(conditional, observations, linkFunction, covariates)
# try to generate a sample
sample = rwmh.getSample(10000)
mu = sample.computeMean()
sigma = sample.computeStandardDeviation()
print('mu=', mu, 'sigma=', sigma)
ott.assert_almost_equal(mu, [14.8747, -0.230384])
ott.assert_almost_equal(sigma, [7.3662, 0.108103])
