def define_distribution():
"""
Define the distribution of the training example (beam).
Return a ComposedDistribution object from openTURNS
"""
sample_E = ot.Sample.ImportFromCSVFile("sample_E.csv")
kernel_smoothing = ot.KernelSmoothing(ot.Normal())
bandwidth = kernel_smoothing.computeSilvermanBandwidth(sample_E)
E = kernel_smoothing.build(sample_E, bandwidth)
E.setDescription(['Young modulus'])
F = ot.LogNormal()
F.setParameter(ot.LogNormalMuSigma()([30000, 9000, 15000]))
F.setDescription(['Load'])
L = ot.Uniform(250, 260)
L.setDescription(['Length'])
I = ot.Beta(2.5, 4, 310, 450)
I.setDescription(['Inertia'])
marginal_distributions = [F, E, L, I]
SR_cor = ot.CorrelationMatrix(len(marginal_distributions))
SR_cor[2, 3] = -0.2
copula = ot.NormalCopula(ot.NormalCopula.GetCorrelationFromSpearmanCorrelation(SR_cor))
return(ot.ComposedDistribution(marginal_distributions, copula))
def __init__(self):
self.dim = 4 # number of inputs
# Young's modulus E
self.E = ot.Beta(0.9, 3.5, 65.0e9, 75.0e9) # in N/m^2
self.E.setDescription("E")
self.E.setName("Young modulus")
# Load F
self.F = ot.LogNormal() # in N
self.F.setParameter(ot.LogNormalMuSigma()([300.0, 30.0, 0.0]))
self.F.setDescription("F")
self.F.setName("Load")
# Length L
self.L = ot.Uniform(2.5, 2.6) # in m
self.L.setDescription("L")
self.L.setName("Length")
# Moment of inertia I
self.I = ot.Beta(2.5, 4.0, 1.3e-7, 1.7e-7) # in m^4
self.I.setDescription("I")
self.I.setName("Inertia")
# physical model
self.model = ot.SymbolicFunction(['E', 'F', 'L', 'I'],
['F*L^3/(3*E*I)'])
# correlation matrix
self.R = ot.CorrelationMatrix(self.dim)
self.R[2, 3] = -0.2
self.copula = ot.NormalCopula(
ot.NormalCopula.GetCorrelationFromSpearmanCorrelation(self.R))
self.distribution = ot.ComposedDistribution(
[self.E, self.F, self.L, self.I], self.copula)
# special case of an independent copula
self.independentDistribution = ot.ComposedDistribution(
[self.E, self.F, self.L, self.I])
# -*- coding: utf-8 -*- # Copyright 2016 EDF. This software was developed with the collaboration of # Phimeca Engineering (Sylvain Girard, [email protected]). """Run multiple FMU simulations in parallel.""" #§ Identifying the platform import platform key_platform = (platform.system(), platform.architecture()[0]) # Call to either 'platform.system' or 'platform.architecture' *after* # importing pyfmi causes a segfault. dict_platform = {("Linux", "64bit"): "linux64", ("Windows", "64bit"): "win64"} #§ Define the input distribution import openturns as ot E = ot.Beta(0.93, 3.2, 28000000.0, 48000000.0) F = ot.LogNormalMuSigma(3.0e4, 9000.0, 15000.0).getDistribution() L = ot.Uniform(250.0, 260.0) I = ot.Beta(2.5, 4.0, 310.0, 450.0) # Create the input probability distribution of dimension 4 inputDistribution = ot.ComposedDistribution([E, F, L, I]) # Give a description of each component of the input distribution inputDistribution.setDescription(("E", "F", "L", "I")) # Create the input random vector inputRandomVector = ot.RandomVector(inputDistribution) #§ FMU model import otfmi from pyfmi.fmi import FMUException import sys
from matplotlib import pylab as plt
ot.Log.Show(ot.Log.NONE)
# %%
# Create a function R^n --> R^p
# For example R^4 --> R
myModel = ot.SymbolicFunction(['x1', 'x2', 'x3', 'x4'],
['1+x1*x2 + 2*x3^2+x4^4'])
# Create a distribution of dimension n
# for example n=3 with indpendent components
distribution = ot.ComposedDistribution([
ot.Normal(),
ot.Uniform(),
ot.Gamma(2.75, 1.0),
ot.Beta(2.5, 1.0, -1.0, 2.0)
])
# %%
# Prepare the input/output samples
sampleSize = 250
X = distribution.getSample(sampleSize)
Y = myModel(X)
dimension = X.getDimension()
# %%
# build the orthogonal basis
coll = [
ot.StandardDistributionPolynomialFactory(distribution.getMarginal(i))
for i in range(dimension)
]
# The following parameter value leads to fast simulations. # %% maxDegree = 4 # %% # For real tests, we suggest using the following parameter value: # %% # maxDegree = 7 # %% # Let us define the parameters of the cantilever beam problem. # %% dist_E = ot.Beta(0.9, 2.2, 2.8e7, 4.8e7) dist_E.setDescription(["E"]) F_para = ot.LogNormalMuSigma(3.0e4, 9.0e3, 15.0e3) # in N dist_F = ot.ParametrizedDistribution(F_para) dist_F.setDescription(["F"]) dist_L = ot.Uniform(250.0, 260.0) # in cm dist_L.setDescription(["L"]) dist_I = ot.Beta(2.5, 1.5, 310.0, 450.0) # in cm^4 dist_I.setDescription(["I"]) myDistribution = ot.ComposedDistribution([dist_E, dist_F, dist_L, dist_I]) dim_input = 4 # dimension of the input dim_output = 1 # dimension of the output
print('parameters =', repr(parameters))
print('parameters (ref)=',
repr(referenceDistribution[testCase].getParametersCollection()))
print('parameter =', repr(distribution[testCase].getParameter()))
print('parameter desc =',
repr(distribution[testCase].getParameterDescription()))
print('marginal 0 =', repr(distribution[testCase].getMarginal(0)))
print('Standard representative=',
referenceDistribution[testCase].getStandardRepresentative())
# Check simplification
candidates = [
ot.Normal(1.0, 2.0),
ot.Uniform(1.0, 2.0),
ot.Exponential(1.0, 2.0),
ot.TruncatedDistribution(ot.WeibullMin(), 1.5, 7.8),
ot.Beta(1.5, 6.3, -1.0, 2.0)
]
intervals = [
ot.Interval(-1.0, 4.0),
ot.Interval(0.2, 2.4),
ot.Interval(2.5, 65.0),
ot.Interval(2.5, 6.0),
ot.Interval(-2.5, 6.0)
]
for i in range(len(candidates)):
d = ot.TruncatedDistribution(candidates[i], intervals[i])
print("d=", d, "simplified=", d.getSimplifiedVersion())
# Check that we can set the bounds independently
truncated = ot.TruncatedDistribution()
truncated.setDistribution(ot.Normal(20.0, 7.5))
ot.Log.Show(ot.Log.NONE) # %% # We first create the data : distribution = ot.Normal(2.0, 0.5) sample1 = distribution.getSample(100) # %% # We draw the Henry line plot and expect a good fitting : graph = ot.VisualTest.DrawHenryLine(sample1) view = viewer.View(graph) # %% # For comparison sake e draw the Henry line plot for a Beta distribution. The result is expected to be bad. sample2 = ot.Beta(0.7, 0.9, 0.0, 2.0).getSample(100) graph = ot.VisualTest.DrawHenryLine(sample2) view = viewer.View(graph) # %% # Normality tests # --------------- # # We use two tests to check whether a sample follows a normal distribution : # # - the Anderson-Darling test # - the Cramer-Von Mises test # # %% # We first generate two samples, one from a standard unit gaussian and another from a Gumbel
simulationResult = ot.ProbabilitySimulationResult(ot.ThresholdEvent(), 0.5,
0.01, 150, 4)
myStudy.add('simulationResult', simulationResult)
cNameList = [
'LHS', 'DirectionalSampling', 'SimulationSensitivityAnalysis',
'ProbabilitySimulationAlgorithm'
]
for cName in cNameList:
otClass = getattr(ot, cName)
instance = otClass()
print('--', cName, instance)
myStudy.add(cName, instance)
# Create a Beta distribution
beta = ot.Beta(3.0, 2.0, -1.0, 4.0)
myStudy.add('beta', beta)
# Create an analytical Function
input = ot.Description(3)
input[0] = 'a'
input[1] = 'b'
input[2] = 'c'
formulas = ot.Description(3)
formulas[0] = 'a+b+c'
formulas[1] = 'a-b*c'
formulas[2] = '(a+2*b^2+3*c^3)/6'
analytical = ot.SymbolicFunction(input, formulas)
analytical.setName('analytical')
analytical.setOutputDescription(['z1', 'z2', 'z3'])
myStudy.add('analytical', analytical)
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
if ot.FrankCopula().__class__.__name__ == 'EmpiricalBernsteinCopula':
sample = ot.Dirichlet([1.0, 2.0, 3.0]).getSample(100)
copula = ot.EmpiricalBernsteinCopula(sample, 4)
elif ot.FrankCopula().__class__.__name__ == 'ExtremeValueCopula':
copula = ot.ExtremeValueCopula(ot.SymbolicFunction("t", "t^3/2-t/2+1"))
elif ot.FrankCopula(
).__class__.__name__ == 'MaximumEntropyOrderStatisticsCopula':
marginals = [ot.Beta(1.5, 3.2, 0.0, 1.0), ot.Beta(2.0, 4.3, 0.5, 1.2)]
copula = ot.MaximumEntropyOrderStatisticsCopula(marginals)
elif ot.FrankCopula().__class__.__name__ == 'NormalCopula':
R = ot.CorrelationMatrix(2)
R[1, 0] = 0.8
copula = ot.NormalCopula(R)
elif ot.FrankCopula().__class__.__name__ == 'SklarCopula':
student = ot.Student(3.0, [1.0] * 2, [3.0] * 2, ot.CorrelationMatrix(2))
copula = ot.SklarCopula(student)
else:
copula = ot.FrankCopula()
if copula.getDimension() == 1:
copula = ot.FrankCopula(2)
copula.setDescription(['$u_1$', '$u_2$'])
pdf_graph = copula.drawPDF()
cdf_graph = copula.drawCDF()
fig = plt.figure(figsize=(10, 4))
pdf_axis = fig.add_subplot(121)
cdf_axis = fig.add_subplot(122)
View(pdf_graph,
figure=fig,
# %%
from __future__ import print_function
import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt
ot.Log.Show(ot.Log.NONE)
# %%
# Create a function R^n --> R^p
# For example R^4 --> R
myModel = ot.SymbolicFunction(['x1', 'x2', 'x3', 'x4'], ['1+x1*x2 + 2*x3^2+x4^4'])
# Create a distribution of dimension n
# for example n=3 with indpendent components
distribution = ot.ComposedDistribution(
[ot.Normal(), ot.Uniform(), ot.Gamma(2.75, 1.0), ot.Beta(2.5, 1.0, -1.0, 2.0)])
# %%
# Prepare the input/output samples
sampleSize = 250
X = distribution.getSample(sampleSize)
Y = myModel(X)
dimension = X.getDimension()
# %%
# build the orthogonal basis
coll = [ot.StandardDistributionPolynomialFactory(distribution.getMarginal(i)) for i in range(dimension)]
enumerateFunction = ot.LinearEnumerateFunction(dimension)
productBasis = ot.OrthogonalProductPolynomialFactory(coll, enumerateFunction)
# %%
def Beta(r=2.0, t=4.0, a=-1.0, b=1.0):
"""
InverseGamma compatibility shim.
"""
return ot.Beta(r, t - r, a, b)
#! /usr/bin/env python from __future__ import print_function import openturns as ot from math import fabs import openturns.testing as ott ot.TESTPREAMBLE() ot.RandomGenerator.SetSeed(0) continuousDistributionCollection = ot.DistributionCollection() discreteDistributionCollection = ot.DistributionCollection() distributionCollection = ot.DistributionCollection() beta = ot.Beta(2.0, 1.0, 0.0, 1.0) distributionCollection.add(beta) continuousDistributionCollection.add(beta) gamma = ot.Gamma(1.0, 2.0, 3.0) distributionCollection.add(gamma) continuousDistributionCollection.add(gamma) gumbel = ot.Gumbel(1.0, 2.0) distributionCollection.add(gumbel) continuousDistributionCollection.add(gumbel) lognormal = ot.LogNormal(1.0, 1.0, 2.0) distributionCollection.add(lognormal) continuousDistributionCollection.add(lognormal) logistic = ot.Logistic(1.0, 1.0)
E.setDescription(['Young modulus']) # In[3]: F = ot.LogNormal() F.setParameter(ot.LogNormalMuSigma()([30000, 9000, 15000])) F.setDescription(['Load']) # In[4]: L = ot.Uniform(250, 260) L.setDescription(['Length']) # In[5]: I = ot.Beta(2.5, 4, 310, 450) I.setDescription(['Inertia']) # We now fix the order of the marginal distributions in the joint distribution. Order must match in the implementation of the physical model (to come). # In[6]: marginal_distributions = [F, E, L, I] # Let then define the dependence structure as a Normal copula with a single non-zero Spearman correlation between components 2 and 3 of the final random vector, that is $L$ and $I$. # In[7]: SR_cor = ot.CorrelationMatrix(len(marginal_distributions)) SR_cor[2, 3] = -0.2 copula = ot.NormalCopula(
import openturns as ot
import numpy as np
from matplotlib import pyplot as plt
import sys
from openturns import viewer
from scipy.stats import beta, betaprime
Beta = ot.Beta(5.957 + 1, 20.151 + 2 + 5.957, -7.223, 52.559)
Beta_PDF = Beta.drawPDF()
Beta_draw = Beta_PDF.getDrawable(0)
Beta_draw.setColor('blue')
viewer.View(Beta_draw)
# Beta2 = ot.Beta(2.61339,7.55733,-2.50357,27.9764)
# Beta2_PDF = Beta2.drawPDF()
# Beta2_PDF.add(Beta_draw)
# viewer.View(Beta2_PDF)
x = np.linspace(-8, 60, 1002)
plt.figure(1)
plt.plot(x,
beta.pdf(x, 5.957 + 1, 20.151 + 1, loc=-7.223, scale=59.782),
c='black')
plt.show()
estimatedDistribution = factory.build(sample)
print("distribution=", repr(distribution))
print("Estimated distribution=", repr(estimatedDistribution))
distribution = ot.Chi(1.0)
sample = distribution.getSample(size)
estimatedDistribution = factory.build(sample)
print("distribution=", repr(distribution))
print("Estimated distribution=", repr(estimatedDistribution))
distribution = ot.Chi(2.5)
sample = distribution.getSample(size)
estimatedDistribution = factory.build(sample)
print("distribution=", repr(distribution))
print("Estimated distribution=", repr(estimatedDistribution))
estimatedDistribution = factory.build()
print("Default distribution=", estimatedDistribution)
estimatedDistribution = factory.build(distribution.getParameter())
print("Distribution from parameters=", estimatedDistribution)
estimatedChi = factory.buildAsChi(sample)
print("Chi =", distribution)
print("Estimated chi=", estimatedChi)
estimatedChi = factory.buildAsChi()
print("Default chi=", estimatedChi)
estimatedChi = factory.buildAsChi(distribution.getParameter())
print("Chi from parameters=", estimatedChi)
# overflow in boost::math::gamma_q_inv
true_dist = ot.Beta(0.9, 2., 3e7, 5e7)
ot.RandomGenerator.SetSeed(0)
data = true_dist.getSample(229)
ot.ChiFactory().build(data)
# %% # # In this example we are going to evaluate the min and max values of the output variable of interest from a sample and to evaluate the gradient of the limit state function defining the output variable of interest at a particular point. # %% from __future__ import print_function import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt import math as m ot.Log.Show(ot.Log.NONE) # %% # Create the marginal distributions of the parameters dist_E = ot.Beta(0.93, 2.27, 2.8e7, 4.8e7) dist_F = ot.LogNormalMuSigma(30000, 9000, 15000).getDistribution() dist_L = ot.Uniform(250, 260) dist_I = ot.Beta(2.5, 1.5, 3.1e2, 4.5e2) marginals = [dist_E, dist_F, dist_L, dist_I] distribution = ot.ComposedDistribution(marginals) # %% # Sample inputs sampleX = distribution.getSample(100) # %% # Create the model model = ot.SymbolicFunction(['E', 'F', 'L', 'I'], ['F*L^3/(3*E*I)']) # %%
#! /usr/bin/env python
from __future__ import print_function
import openturns as ot
ot.PlatformInfo.SetNumericalPrecision(3)
distribution = ot.Beta(2.3, 2.2, -1.0, 1.0)
print('distribution=', distribution)
sample = distribution.getSample(1000)
factory = ot.MaximumLikelihoodFactory(ot.Beta())
inf_distribution = factory.build(sample)
print('estimated distribution=', inf_distribution)
# set (a,b) out of (r, t, a, b)
factory.setKnownParameter([-1.0, 1.0], [2, 3])
inf_distribution = factory.build(sample)
print('estimated distribution with bounds=', inf_distribution)
factory = ot.MaximumLikelihoodFactory(ot.Exponential())
factory.setKnownParameter([0.1], [1])
print(factory.build())
print(factory.build([3, 0]))
import openturns as ot
ot.TESTPREAMBLE()
ot.PlatformInfo.SetNumericalPrecision(3)
size = 10000
distribution = ot.Gumbel(1.5, -0.5)
print('distribution=', distribution)
sample = distribution.getSample(size)
factory = ot.MethodOfMomentsFactory(ot.Gumbel())
inf_distribution = factory.build(sample)
print('estimated distribution=', inf_distribution)
# set (a,b) out of (r, t, a, b)
distribution = ot.Beta(2.3, 2.2, -1.0, 1.0)
print('distribution=', distribution)
sample = distribution.getSample(size)
factory = ot.MethodOfMomentsFactory(ot.Beta())
factory.setKnownParameter([-1.0, 1.0], [2, 3])
inf_distribution = factory.build(sample)
print('estimated distribution=', inf_distribution)
# with bounds
data = [
0.6852, 0.9349, 0.5884, 1.727, 1.581, 0.3193, -0.5701, 1.623, 2.210,
-0.3440, -0.1646
]
sample = ot.Sample([[x] for x in data])
size = sample.getSize()
xMin = sample.getMin()[0]
model_fmu = otfmi.FMUFunction(path_fmu,
inputs_fmu=["E", "F", "L", "I"],
outputs_fmu="y")
# %%
# We test the function wrapping the deviation model on a point:
import openturns as ot
point = ot.Point([3e7, 2e4, 255, 350])
model_evaluation = model_fmu(point)
print("Running the FMU: deviation = {}".format(model_evaluation))
# %%
# We define probability laws on the 4 uncertain inputs:
E = ot.Beta(0.93, 3.2, 2.8e7, 4.8e7)
F = ot.LogNormal()
F.setParameter(ot.LogNormalMuSigma()([30.e3, 9e3, 15.e3]))
L = ot.Uniform(250.0, 260.0)
I = ot.Beta(2.5, 4.0, 310.0, 450.0)
# %%
# According to the laws of mechanics, when the length L increases, the moment
# of inertia I decreases.
# The variables L and I are thus negatively correlated.
#
# **We assume that the random variables E, F, L and I are dependent and
# associated with a gaussian copula which correlation matrix:**
#
# .. math::
# \begin{pmatrix}
# f(x) = \frac{(x-a)^{\alpha - 1} (b - x)^{\beta - 1}}{(b - a)^{\alpha + \beta - 1} B(\alpha, \beta)}
#
#
# for any :math:`x\in[a,b]`, where :math:`B` is Euler's beta function.
#
# For any :math:`y,z>0`, the beta function is:
#
# .. math::
# B(y,z) = \int_0^1 t^{y-1} (1-t)^{z-1} dt.
#
# %%
# The `Beta` class uses the native parametrization.
# %%
distribution = ot.Beta(2.5, 2.5, -1, 2)
graph = distribution.drawPDF()
view = viewer.View(graph)
# %%
# The `BetaMuSigma` class provides another parametrization, based on the expectation :math:`\mu` and the standard deviation :math:`\sigma` of the random variable:
#
# .. math::
# \mu = a + \frac{(b-a)\alpha}{\alpha+\beta}
#
#
# and
#
# .. math::
# \sigma^2 = \left(\frac{b-a}{\alpha+\beta}\right)^2 \frac{\alpha\beta}{\alpha+\beta+1}.
#
import openturns.viewer as viewer
from matplotlib import pylab as plt
ot.Log.Show(ot.Log.NONE)
# %%
# We define the symbolic function which evaluates the output Y depending on the inputs E, F, L and I.
# %%
model = ot.SymbolicFunction(["E", "F", "L", "I"], ["F*L^3/(3*E*I)"])
# %%
# Then we define the distribution of the input random vector.
# %%
# Young's modulus E
E = ot.Beta(0.9, 3.5, 2.5e7, 5.0e7) # in N/m^2
E.setDescription("E")
# Load F
F = ot.LogNormal() # in N
F.setParameter(ot.LogNormalMuSigma()([30.e3, 9e3, 15.e3]))
F.setDescription("F")
# Length L
L = ot.Uniform(250., 260.) # in cm
L.setDescription("L")
# Moment of inertia I
I = ot.Beta(2.5, 4, 310, 450) # in cm^4
I.setDescription("I")
# %%
# Finally, we define the dependency using a `NormalCopula`.
sigma = [2.0, 3.0]
R = ot.CorrelationMatrix(dim)
for i in range(1, dim):
R[i, i - 1] = 0.5
distribution = ot.Normal(meanPoint, sigma, R)
discretization = 100
kernel = ot.KernelSmoothing()
sample = distribution.getSample(discretization)
kernels = ot.DistributionCollection(0)
kernels.add(ot.Normal())
kernels.add(ot.Epanechnikov())
kernels.add(ot.Uniform())
kernels.add(ot.Triangular())
kernels.add(ot.Logistic())
kernels.add(ot.Beta(2.0, 2.0, -1.0, 1.0))
kernels.add(ot.Beta(3.0, 3.0, -1.0, 1.0))
meanExact = distribution.getMean()
covarianceExact = distribution.getCovariance()
for i in range(kernels.getSize()):
kernel = kernels[i]
print("kernel=", kernel.getName())
smoother = ot.KernelSmoothing(kernel)
smoothed = smoother.build(sample)
bw = smoother.getBandwidth()
print("kernel bandwidth=[ %.6g" % bw[0], ", %.6g" % bw[1], "]")
meanSmoothed = smoothed.getMean()
print("mean(smoothed)=[ %.6g" % meanSmoothed[0],
", %.6g" % meanSmoothed[1], "] mean(exact)=[", meanExact[0], ", ",
meanExact[1], "]")
covarianceSmoothed = smoothed.getCovariance()
#! /usr/bin/env python
from __future__ import print_function
import openturns as ot
ot.TESTPREAMBLE()
ot.RandomGenerator.SetSeed(0)
# Instanciate one distribution object
distribution = ot.MaximumEntropyOrderStatisticsDistribution([
ot.Trapezoidal(-2.0, -1.1, -1.0, 1.0),
ot.LogUniform(1.0, 1.2),
ot.Triangular(3.0, 4.5, 5.0),
ot.Beta(2.5, 3.5, 4.7, 5.2)
])
dim = distribution.getDimension()
print("Distribution ", distribution)
# Is this distribution elliptical ?
print("Elliptical = ", distribution.isElliptical())
# Test for realization of distribution
oneRealization = distribution.getRealization()
print("oneRealization=", repr(oneRealization))
# Test for sampling
size = 10000
oneSample = distribution.getSample(size)
print("oneSample first=", repr(oneSample[0]), " last=",
repr(oneSample[size - 1]))
import openturns.viewer as viewer
from matplotlib import pylab as plt
ot.Log.Show(ot.Log.NONE)
# %%
# We define the symbolic function which evaluates the output Y depending on the inputs E, F, L and I.
# %%
model = ot.SymbolicFunction(["E", "F", "L", "I"], ["F*L^3/(3*E*I)"])
# %%
# Then we define the distribution of the input random vector.
# %%
# Young's modulus E
E = ot.Beta(0.9, 2.27, 2.5e7, 5.0e7) # in N/m^2
E.setDescription("E")
# Load F
F = ot.LogNormal() # in N
F.setParameter(ot.LogNormalMuSigma()([30.e3, 9e3, 15.e3]))
F.setDescription("F")
# Length L
L = ot.Uniform(250., 260.) # in cm
L.setDescription("L")
# Moment of inertia I
I = ot.Beta(2.5, 1.5, 310, 450) # in cm^4
I.setDescription("I")
# %%
# Finally, we define the dependency using a `NormalCopula`.
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
myDist = [ot.Beta(1.5, 3.2, 0.0, 1.0), ot.Beta(2.0, 4.3, 0.5, 1.2)]
myOrderStatCop = ot.MaximumEntropyOrderStatisticsCopula(myDist)
myOrderStatCop.setDescription(['$u_1$', '$u_2$'])
graphPDF = myOrderStatCop.drawPDF()
graphCDF = myOrderStatCop.drawCDF()
fig = plt.figure(figsize=(8, 4))
plt.suptitle("Max Entropy Order Statistics Copula: pdf and cdf")
pdf_axis = fig.add_subplot(121)
cdf_axis = fig.add_subplot(122)
pdf_axis.set_xlim(auto=True)
cdf_axis.set_xlim(auto=True)
View(graphPDF, figure=fig, axes=[pdf_axis], add_legend=True)
View(graphCDF, figure=fig, axes=[cdf_axis], add_legend=True)
print('covariance (ref)=', repr(
cleanCovariance(referenceDistribution[testCase].getCovariance())))
parameters = distribution[testCase].getParametersCollection()
print('parameters =', repr(parameters))
print('parameters (ref)=', repr(
referenceDistribution[testCase].getParametersCollection()))
print('parameter =', repr(distribution[testCase].getParameter()))
print('parameter desc =',
repr(distribution[testCase].getParameterDescription()))
print('marginal 0 =', repr(distribution[testCase].getMarginal(0)))
for i in range(6):
print('standard moment n=', i, ' value=',
referenceDistribution[testCase].getStandardMoment(i))
print('Standard representative=', referenceDistribution[
testCase].getStandardRepresentative())
# Check simplification
candidates = [ot.Normal(1.0, 2.0), ot.Uniform(1.0, 2.0), ot.Exponential(1.0, 2.0),
ot.TruncatedDistribution(ot.Weibull(), 1.5, 7.8),
ot.Beta(1.5, 7.8, -1.0, 2.0)]
intervals = [ot.Interval(-1.0, 4.0), ot.Interval(0.2, 2.4), ot.Interval(2.5, 65.0),
ot.Interval(2.5, 6.0), ot.Interval(-2.5, 6.0)]
for i in range(len(candidates)):
d = ot.TruncatedDistribution(candidates[i], intervals[i])
print("d=", d, "simplified=", d.getSimplifiedVersion())
# Check that we can set the bounds independently
truncated = ot.TruncatedDistribution()
truncated.setDistribution(ot.Normal(20.0, 7.5))
truncated.setBounds(ot.Interval([0], [80], [True], [False]))
print('after setbounds [email protected]=', truncated.computeQuantile(0.9))
# %% # In this example we are going to build maximum entropy statistics distribution, which yields ordered realizations: # # .. math:: # X_1 \leq \dots \leq X_n # # %% import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) # %% # create a collection of distribution coll = [ot.Beta(1.5, 1.7, 0.0, 1.0), ot.Beta(2.0, 2.3, 0.5, 1.2)] # %% # create the distribution distribution = ot.MaximumEntropyOrderStatisticsDistribution(coll) print(distribution) # %% # draw a sample (ordered!) distribution.getSample(10) # %% # draw PDF graph = distribution.drawPDF() view = viewer.View(graph) plt.show()
cloud = ot.Cloud(sample, "blue", "fsquare", "") # Create the cloud
graph.add(cloud) # Then, add it to the graph
view = viewer.View(graph)
# %%
# We see that the sample is quite different from the previous sample with independent copula.
# %%
# Draw several distributions in the same plot
# -------------------------------------------
# %%
# It is sometimes convenient to create a plot presenting the PDF and CDF on the same graphics. This is possible thanks to Matplotlib.
# %%
beta = ot.Beta(5, 7, 9, 10)
pdfbeta = beta.drawPDF()
cdfbeta = beta.drawCDF()
exponential = ot.Exponential(3)
pdfexp = exponential.drawPDF()
cdfexp = exponential.drawCDF()
# %%
import openturns.viewer as otv
# %% slideshow={"slide_type": "subslide"}
import pylab as plt
fig = plt.figure(figsize=(12, 4))
ax = fig.add_subplot(2, 2, 1)
_ = otv.View(pdfbeta, figure=fig, axes=[ax])
ax = fig.add_subplot(2, 2, 2)
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
if ot.Beta().__class__.__name__ == 'ComposedDistribution':
correlation = ot.CorrelationMatrix(2)
correlation[1, 0] = 0.25
aCopula = ot.NormalCopula(correlation)
marginals = [ot.Normal(1.0, 2.0), ot.Normal(2.0, 3.0)]
distribution = ot.ComposedDistribution(marginals, aCopula)
elif ot.Beta().__class__.__name__ == 'CumulativeDistributionNetwork':
distribution = ot.CumulativeDistributionNetwork([ot.Normal(2),ot.Dirichlet([0.5, 1.0, 1.5])], ot.BipartiteGraph([[0,1], [0,1]]))
elif ot.Beta().__class__.__name__ == 'Histogram':
distribution = ot.Histogram([-1.0, 0.5, 1.0, 2.0], [0.45, 0.4, 0.15])
else:
distribution = ot.Beta()
dimension = distribution.getDimension()
if dimension == 1:
distribution.setDescription(['$x$'])
pdf_graph = distribution.drawPDF()
cdf_graph = distribution.drawCDF()
fig = plt.figure(figsize=(10, 4))
plt.suptitle(str(distribution))
pdf_axis = fig.add_subplot(121)
cdf_axis = fig.add_subplot(122)
View(pdf_graph, figure=fig, axes=[pdf_axis], add_legend=False)
View(cdf_graph, figure=fig, axes=[cdf_axis], add_legend=False)
elif dimension == 2:
distribution.setDescription(['$x_1$', '$x_2$'])
pdf_graph = distribution.drawPDF()
fig = plt.figure(figsize=(10, 5))
plt.suptitle(str(distribution))
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
myDist = [ot.Beta(1.5, 1.7, 0.0, 1.0), ot.Beta(2.0, 2.3, 0.5, 1.2)]
myOrderStatDist = ot.MaximumEntropyOrderStatisticsDistribution(myDist)
myOrderStatDist.setDescription(['$x_1$', '$x_2$'])
graphPDF = myOrderStatDist.drawPDF()
graphCDF = myOrderStatDist.drawCDF()
fig = plt.figure(figsize=(8, 4))
pdf_axis = fig.add_subplot(121)
cdf_axis = fig.add_subplot(122)
pdf_axis.set_xlim(auto=True)
cdf_axis.set_xlim(auto=True)
View(graphPDF, figure=fig, axes=[pdf_axis], add_legend=True)
View(graphCDF, figure=fig, axes=[cdf_axis], add_legend=True)
fig.suptitle("Max Entropy Order Statistics Distribution: pdf and cdf")