import openturns.viewer as viewer
from matplotlib import pylab as plt
ot.Log.Show(ot.Log.NONE)
# %%
# Copula fitting test using Kendall plot
# --------------------------------------
#
# We first perform a visual goodness-of-fit test for a copula with the Kendall plot test.
# %%
# We create two samples of size 500.
N = 500
# %%
dist1 = ot.ComposedDistribution([ot.Normal()] * 2, ot.GumbelCopula(3.0))
sample1 = dist1.getSample(N)
sample1.setName('sample1')
# %%
dist2 = ot.ComposedDistribution([ot.Normal()] * 2, ot.ClaytonCopula(0.2))
sample2 = dist2.getSample(N)
sample2.setName('sample2')
# %%
# We change the parameter for the evaluation of E(Wi) thanks to the `ResourceMap` :
ot.ResourceMap.SetAsUnsignedInteger('VisualTest-KendallPlot-MonteCarloSize',
25)
# %%
# We can test a specific copula model for a given sample,
from __future__ import print_function
import openturns as ot
from openturns.viewer import View
from time import time
size = 250
copula1 = ot.ClaytonCopula(3.0)
tau = copula1.getKendallTau()[0, 1]
print("tau=", tau)
copula2 = ot.GumbelCopula(1.0 / (1.0 - tau))
sample = copula1.getSample(size)
graph = ot.VisualTest.DrawKendallPlot(sample, copula1)
graph.add(ot.VisualTest.DrawKendallPlot(sample, copula2).getDrawable(1))
graph.setColors(["red", "green", "blue"])
graph.setLegends(
["", "sample vs " + str(copula1), "sample vs " + str(copula2)])
graph.setLegendPosition("bottomright")
graph.setTitle("Copula assessment using Kendall plot")
graph.setXTitle(r"$x$")
graph.setYTitle(r"$y$")
view = View(graph, (800, 600))
view._ax[0].axis("equal")
view.save("../plot_kendall_plot.png")
view.close()
def check_bernstein_copula(est_copula):
"""
Check if an estimated distribution of kind EmpiricalBernstein
is of type copula and all marginals are also.
"""
print("Is estimation a copula ? --> ", est_copula.isCopula())
print("Maginal checking")
dimension = est_copula.getDimension()
for d in range(dimension):
print("Is marginal %d a copula ? --> %s" % (d, est_copula.isCopula()))
try:
coll = [ot.GumbelCopula(3.0), ot.ClaytonCopula(3.0), ot.FrankCopula(3.0)]
size = 100
for i, ref_copula in enumerate(coll):
ref_copula = coll[i]
print("Reference copula", str(ref_copula))
sample = ref_copula.getSample(size)
# Default method: log-likelihood
m = ot.BernsteinCopulaFactory.ComputeLogLikelihoodBinNumber(sample)
print("Log-likelihood m=", m)
est_copula = ot.BernsteinCopulaFactory().build(sample, m)
max_error = compute_max_error(ref_copula, est_copula)
print("Max. error=%.5f" % max_error)
check_bernstein_copula(est_copula)
# AMISE method
m = ot.BernsteinCopulaFactory.ComputeAMISEBinNumber(sample)
print("AMISE m=", m)
# ot.Log.Show(ot.Log.ALL)
coll = []
# case 1: no transformation
coll.append([ot.Normal(), ot.Normal()])
# case 2: same copula
left = ot.ComposedDistribution([ot.Normal(), ot.Gumbel()],
ot.IndependentCopula(2))
right = ot.ComposedDistribution([ot.Triangular()] * 2, ot.IndependentCopula(2))
coll.append([left, right])
# case 3: same standard space
left = ot.ComposedDistribution([ot.Normal(), ot.Gumbel()],
ot.IndependentCopula(2))
right = ot.ComposedDistribution([ot.Triangular()] * 2, ot.GumbelCopula())
coll.append([left, right])
# TODO case 4: different standard space
for left, right in coll:
transformation = ot.DistributionTransformation(left, right)
print('left=', left)
print('right=', right)
print('transformation=', transformation)
inverseTransformation = transformation.inverse()
print('inverseTransformation=', inverseTransformation)
print('-' * 100)
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
if (ot.GumbelCopula().__class__.__name__ == 'SklarCopula'):
myStudent = ot.Student(3.0, [1.0] * 2, [3.0] * 2, ot.CorrelationMatrix(2))
copula = ot.SklarCopula(myStudent)
else:
copula = ot.GumbelCopula()
if copula.getDimension() == 1:
copula = ot.GumbelCopula(2)
copula.setDescription(['$u_1$', '$u_2$'])
pdf_graph = copula.drawPDF()
cdf_graph = copula.drawCDF()
fig = plt.figure(figsize=(10, 4))
plt.suptitle(str(copula))
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)
pdf_axis.set_aspect('equal')
cdf_axis.set_aspect('equal')
expression += "cos(" + str(i + 1) + "*" + inputVar[i] + ")"
formula[0] = expression
model = ot.SymbolicFunction(inputVar, formula)
outputSample = model(inputSample)
cobwebValue = ot.VisualTest.DrawParallelCoordinates(
inputSample, outputSample, 2.5, 3.0, "red", False)
print("cobwebValue = ", cobwebValue)
cobwebQuantile = ot.VisualTest.DrawParallelCoordinates(
inputSample, outputSample, 0.7, 0.9, "red", False)
print("cobwebQuantile = ", cobwebQuantile)
# KendallPlot tests
size = 100
copula1 = ot.FrankCopula(1.5)
copula2 = ot.GumbelCopula(4.5)
sample1 = copula1.getSample(size)
sample1.setName("data 1")
sample2 = copula2.getSample(size)
sample2.setName("data 2")
kendallPlot1 = ot.VisualTest.DrawKendallPlot(sample1, copula2)
print("KendallPlot1 = ", kendallPlot1)
kendallPlot2 = ot.VisualTest.DrawKendallPlot(sample2, sample1)
print("KendallPlot2 = ", kendallPlot2)
# Clouds
sample = ot.Normal(4).getSample(200)
clouds = ot.VisualTest.DrawPairs(sample)
print("Clouds = ", clouds)
distribution = ot.ComposedDistribution(
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
myCop1 = ot.GumbelCopula(2)
myCop2 = ot.NormalCopula(2)
alpha = 0.3
myOrdSumCop = ot.OrdinalSumCopula([myCop1, myCop2], [alpha])
myOrdSumCop.setDescription(['$u_1$', '$u_2$'])
graphPDF = myOrdSumCop.drawPDF()
graphCDF = myOrdSumCop.drawCDF()
fig = plt.figure(figsize=(8, 4))
plt.suptitle("Ordinal Sum of Copulas: Gumbel(2) and Normal(2): 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)
formula_fake_var = 'x1'
formula_y0 = 'cos(0.5*x1) + sin(x2)'
formula_y1 = 'cos(0.5*x1) + sin(x2) + x3'
symbolicModel = persalys.SymbolicPhysicalModel('symbolicModel', [x1, x2, x3], [fake_var, y0, fake_y0, y1], [
formula_fake_var, formula_y0, formula_y0, formula_y1])
myStudy.add(symbolicModel)
# python model ##
code = 'from math import cos, sin, sqrt\n\ndef _exec(x1, x2, x3):\n y0 = cos(0.5*x1) + sin(x2) + sqrt(x3)\n return y0\n'
pythonModel = persalys.PythonPhysicalModel('pythonModel', [x1, x2, x3], [y0], code)
myStudy.add(pythonModel)
filename = 'data.csv'
cDist = ot.ComposedDistribution([ot.Normal(), ot.Gumbel(), ot.Normal(), ot.Uniform()],
ot.ComposedCopula([ot.IndependentCopula(2), ot.GumbelCopula()]))
sample = cDist.getSample(200)
sample.exportToCSVFile(filename, ' ')
# Designs of Experiment ##
# fixed design ##
ot.RandomGenerator.SetSeed(0)
fixedDesign = persalys.FixedDesignOfExperiment('fixedDesign', symbolicModel)
inputSample = ot.LHSExperiment(ot.ComposedDistribution([ot.Uniform(0., 10.), ot.Uniform(0., 10.)]), 10).generate()
inputSample.stack(ot.Sample(10, [0.5]))
fixedDesign.setOriginalInputSample(inputSample)
fixedDesign.run()
myStudy.add(fixedDesign)
# grid ##
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
if ot.GumbelCopula().__class__.__name__ == 'EmpiricalBernsteinCopula':
sample = ot.Dirichlet([1.0, 2.0, 3.0]).getSample(100)
copula = ot.EmpiricalBernsteinCopula(sample, 4)
elif ot.GumbelCopula().__class__.__name__ == 'ExtremeValueCopula':
copula = ot.ExtremeValueCopula(ot.SymbolicFunction("t", "t^3/2-t/2+1"))
elif ot.GumbelCopula(
).__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.GumbelCopula().__class__.__name__ == 'NormalCopula':
R = ot.CorrelationMatrix(2)
R[1, 0] = 0.8
copula = ot.NormalCopula(R)
elif ot.GumbelCopula().__class__.__name__ == 'SklarCopula':
student = ot.Student(3.0, [1.0] * 2, [3.0] * 2, ot.CorrelationMatrix(2))
copula = ot.SklarCopula(student)
else:
copula = ot.GumbelCopula()
if copula.getDimension() == 1:
copula = ot.GumbelCopula(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,
"""
Create the ordinal sum of copulas
=================================
"""
# %%
# In this example we are going to create an ordinal sum of copulas.
# %%
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 copulas
collection = [ot.GumbelCopula(2), ot.NormalCopula(2)]
# %%
# Merge the copulas
bounds = [0.3]
copula = ot.OrdinalSumCopula(collection, bounds)
print(copula)
# %%
# Draw PDF
graph = copula.drawPDF([512]*2)
graph.setXTitle('x')
graph.setYTitle('y')
graph.setLegendPosition('')
view = viewer.View(graph)
plt.show()
ot.Normal(-1.0, 1.0),
ot.Uniform(5.0, 6.0)
]
weights = [0.4, 1.0, 0.2]
# %%
# create the mixture
distribution = ot.Mixture(distributions, weights)
print(distribution)
# %%
# draw PDF
graph = distribution.drawPDF()
view = viewer.View(graph)
# %%
# define a list of copulas and the associated weights
copulas = [ot.GumbelCopula(4.5), ot.ClaytonCopula(2.3)]
weights = [0.2, 0.8]
# %%
# create a mixture of copulas
distribution = ot.Mixture(copulas, weights)
print(distribution)
# %%
# draw PDF
graph = distribution.drawPDF()
view = viewer.View(graph)
plt.show()
formula_fake_var = 'x1+'
formula_y0 = 'cos(0.5*x1) + sin(x2)'
formula_y1 = 'cos(0.5*x1) + sin(x2) + x3'
model1 = persalys.SymbolicPhysicalModel(
'model1', [x1, x2, x3], [fake_var, y0, fake_y0, y1],
[formula_fake_var, formula_y0, formula_y0, formula_y1])
myStudy.add(model1)
# model 3 ##
filename = 'data.csv'
cDist = ot.ComposedDistribution(
[ot.Normal(), ot.Gumbel(),
ot.Normal(), ot.Uniform()],
ot.ComposedCopula([ot.IndependentCopula(2),
ot.GumbelCopula()]))
sample = cDist.getSample(20)
sample.exportToCSVFile(filename, ' ')
model3 = persalys.DataModel('model3', 'data.csv', [0, 2, 3], [1],
['x_0', 'x_2', 'x_3'], ['x_1'])
myStudy.add(model3)
# Design of Experiment ##
probaDesign = persalys.ProbabilisticDesignOfExperiment('probaDesign', model1,
20, "MONTE_CARLO")
probaDesign.run()
myStudy.add(probaDesign)
# 1- meta model1 ##
