def generate_student_data(ndag, size, r=0.8):
order = ndag.getTopologicalOrder()
copulas = []
for k in range(order.getSize()):
d = 1 + ndag.getParents(k).getSize()
R = ot.CorrelationMatrix(d)
for i in range(d):
for j in range(i):
R[i, j] = r
copulas.append(ot.Student(5.0, [0.0]*d, [1.0]*d, R).getCopula())
cbn = otagr.ContinuousBayesianNetwork(ndag, [ot.Uniform(0., 1.)]*ndag.getSize(), copulas)
sample = cbn.getSample(size)
return sample
def generate_student_data(ndag, size, r=0.8):
order = ndag.getTopologicalOrder()
jointDistributions = []
for k in range(order.getSize()):
d = 1 + ndag.getParents(k).getSize()
R = ot.CorrelationMatrix(d)
for i in range(d):
for j in range(i):
R[i, j] = r
jointDistributions.append(
ot.Student(5.0, [0.0] * d, [1.0] * d, R).getCopula())
copula = otagr.ContinuousBayesianNetwork(ndag, jointDistributions)
sample = copula.getSample(size)
return sample
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 an 1-d distribution
dist_1 = ot.Normal()
# Create a 2-d distribution
dist_2 = ot.ComposedDistribution(
[ot.Normal(), ot.Triangular(0.0, 2.0, 3.0)], ot.ClaytonCopula(2.3))
# Create a 3-d distribution
copula_dim3 = ot.Student(5.0, 3).getCopula()
dist_3 = ot.ComposedDistribution(
[ot.Normal(),
ot.Triangular(0.0, 2.0, 3.0),
ot.Exponential(0.2)], copula_dim3)
# %%
# Get the dimension fo the distribution
dist_2.getDimension()
# %%
# Get the 2nd marginal
dist_2.getMarginal(1)
# %%
# Get a 2-d marginal
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
if (ot.ClaytonCopula().__class__.__name__ == 'SklarCopula'):
myStudent = ot.Student(3.0, [1.0] * 2, [3.0] * 2, ot.CorrelationMatrix(2))
copula = ot.SklarCopula(myStudent)
else:
copula = ot.ClaytonCopula()
if copula.getDimension() == 1:
copula = ot.ClaytonCopula(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,
square_axes=True)
View(cdf_graph,
figure=fig,
axes=[cdf_axis],
add_legend=False,
square_axes=True)
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
if (ot.Student().__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.Student().__class__.__name__ == 'CumulativeDistributionNetwork'):
distribution = ot.CumulativeDistributionNetwork(
[ot.Normal(2), ot.Dirichlet([0.5, 1.0, 1.5])],
ot.BipartiteGraph([[0, 1], [0, 1]]))
else:
distribution = ot.Student()
dimension = distribution.getDimension()
if dimension <= 2:
if distribution.getDimension() == 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)
else:
distribution.setDescription(['$x_1$', '$x_2$'])
pdf_graph = distribution.drawPDF()
fig = plt.figure(figsize=(10, 5))
# its PDF.
dim = 2
distribution = ot.Normal(dim)
graph = distribution.drawPDF()
graph.setTitle("Bivariate standard unit gaussian PDF")
view = otv.View(graph)
# %%
# The Student distribution
# ^^^^^^^^^^^^^^^^^^^^^^^^
#
# The :class:`~openturns.Student` distribution is natively multivariate. Here we define a Student distribution in dimension 2 and display its PDF :
dim = 2
R = ot.CorrelationMatrix(dim)
R[1, 0] = -0.2
distribution = ot.Student(4, [0.0, 1.0], [1.0, 1.0], R)
graph = distribution.drawPDF()
graph.setTitle("Bivariate Student PDF")
view = otv.View(graph)
# %%
# The UserDefined distribution
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# We can also define our own distribution with the :class:`~openturns.UserDefined` distribution.
# For instance consider the square :math:`[-1,1] \times [-1, 1]` with some
# random points uniformly drawn. For each point the weight chosen is the square
# of the distance to the origin. The :class:`~openturns.UserDefined` class normalizes the weights.
# %%
# We first generate random points in the square.
size = 20
oneSample = ot.Sample(size, 1)
twoSample = ot.Sample(size, 1)
for i in range(size):
oneSample[i, 0] = 7.0 * sin(-3.5 + (6.5 * i) / (size - 1.0)) + 2.0
twoSample[
i, 0] = -2.0 * oneSample[i, 0] + 3.0 + 0.05 * sin(oneSample[i, 0])
test = lm.LinearModelAlgorithm(oneSample, twoSample)
result = lm.LinearModelResult(test.getResult())
analysis = lm.LinearModelAnalysis(result)
print(analysis)
# Compute confidence level (95%) for coefficients estimate (1-alpha = 0.95)
alpha = 0.05
lower_sample = analysis.getCoefficientsEstimates(
) - analysis.getCoefficientsStandardErrors() * ot.Student(
analysis.getDegreesOfFreedom()).computeQuantile(1 - alpha)
upper_sample = analysis.getCoefficientsEstimates(
) + analysis.getCoefficientsStandardErrors() * ot.Student(
analysis.getDegreesOfFreedom()).computeQuantile(1 - alpha)
# lower and upper bounds as Point
lower_bounds = ot.Point(lower_sample.getSize())
for i in range(lower_bounds.getSize()):
lower_bounds[i] = lower_sample[i][0]
upper_bounds = ot.Point(upper_sample.getSize())
for i in range(upper_bounds.getSize()):
upper_bounds[i] = upper_sample[i][0]
# interval confidence bounds
interval = ot.Interval(lower_bounds, upper_bounds)
print("confidence intervals with level=%1.2f : %s" % (1 - alpha, interval))
print("")
distributionCollection.add(lognormal) continuousDistributionCollection.add(lognormal) logistic = ot.Logistic(1.0, 1.0) distributionCollection.add(logistic) continuousDistributionCollection.add(logistic) normal = ot.Normal(1.0, 2.0) distributionCollection.add(normal) continuousDistributionCollection.add(normal) truncatednormal = ot.TruncatedNormal(1.0, 1.0, 0.0, 3.0) distributionCollection.add(truncatednormal) continuousDistributionCollection.add(truncatednormal) student = ot.Student(10.0, 10.0) distributionCollection.add(student) continuousDistributionCollection.add(student) triangular = ot.Triangular(-1.0, 2.0, 4.0) distributionCollection.add(triangular) continuousDistributionCollection.add(triangular) uniform = ot.Uniform(1.0, 2.0) distributionCollection.add(uniform) continuousDistributionCollection.add(uniform) weibull = ot.WeibullMin(1.0, 1.0, 2.0) distributionCollection.add(weibull) continuousDistributionCollection.add(weibull)
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
if ot.Student().__class__.__name__ == 'Bernoulli':
distribution = ot.Bernoulli(0.7)
elif ot.Student().__class__.__name__ == 'Binomial':
distribution = ot.Binomial(5, 0.2)
elif ot.Student().__class__.__name__ == 'ComposedDistribution':
copula = ot.IndependentCopula(2)
marginals = [ot.Uniform(1.0, 2.0), ot.Normal(2.0, 3.0)]
distribution = ot.ComposedDistribution(marginals, copula)
elif ot.Student().__class__.__name__ == 'CumulativeDistributionNetwork':
coll = [ot.Normal(2), ot.Dirichlet([0.5, 1.0, 1.5])]
distribution = ot.CumulativeDistributionNetwork(
coll, ot.BipartiteGraph([[0, 1], [0, 1]]))
elif ot.Student().__class__.__name__ == 'Histogram':
distribution = ot.Histogram([-1.0, 0.5, 1.0, 2.0], [0.45, 0.4, 0.15])
elif ot.Student().__class__.__name__ == 'KernelMixture':
kernel = ot.Uniform()
sample = ot.Normal().getSample(5)
bandwith = [1.0]
distribution = ot.KernelMixture(kernel, bandwith, sample)
elif ot.Student().__class__.__name__ == 'MaximumDistribution':
coll = [
ot.Uniform(2.5, 3.5),
ot.LogUniform(1.0, 1.2),
ot.Triangular(2.0, 3.0, 4.0)
]
distribution = ot.MaximumDistribution(coll)
elif ot.Student().__class__.__name__ == 'Multinomial':
distribution = ot.Multinomial(5, [0.2])
#! /usr/bin/env python
from __future__ import print_function
import openturns as ot
ot.TESTPREAMBLE()
nu = 4.5
dim = 5
distribution = ot.Student(nu, dim)
transformation = ot.RosenblattEvaluation(distribution)
print("transformation=", repr(transformation))
point = ot.Point(dim, 0.75)
print("transformation(", point, ")=", repr(transformation(point)))
print("transformation parameters gradient=",
repr(transformation.parameterGradient(point)))
print("input dimension=", transformation.getInputDimension())
print("output dimension=", transformation.getOutputDimension())
if False:
marginals = [ot.Uniform(0.0, 1.0) for i in range(order.getSize())]
copulas = list()
for i in range(order.getSize()):
d = 1 + ndag.getParents(i).getSize()
print("i=", i, ", d=", d)
if d == 1:
copulas.append(ot.IndependentCopula(1))
else:
R = ot.CorrelationMatrix(d)
for i in range(d):
for j in range(i):
R[i, j] = 0.5 / d
copulas.append(
ot.Student(5.0, [0.0] * d, [1.0] * d, R).getCopula())
cbn = otagrum.ContinuousBayesianNetwork(ndag, marginals, copulas)
print("cbn=", cbn)
print("cbn pdf=", cbn.computePDF([0.5] * d))
print("cbn realization=", cbn.getRealization())
size = 300
sampleLearn = cbn.getSample(size)
sample = cbn.getSample(size)
sampleLearn.exportToCSVFile("samplelearn.csv", ",")
sample.exportToCSVFile("sample.csv", ",")
print("cbn sample=", sample)
logL = 0.0
pdfSample = cbn.computePDF(sample)
def clean(polynomial):
coefficients = polynomial.getCoefficients()
for i in range(coefficients.getDimension()):
if abs(coefficients[i]) < 1.0e-12:
coefficients[i] = 0.0
return ot.UniVariatePolynomial(coefficients)
iMax = 5
distributionCollection = [
ot.Laplace(1.0, 0.0),
ot.Logistic(0.0, 1.0),
ot.Normal(0.0, 1.0),
ot.Normal(1.0, 1.0),
ot.Rayleigh(1.0),
ot.Student(22.0),
ot.Triangular(-1.0, 0.3, 1.0),
ot.Uniform(-1.0, 1.0),
ot.Uniform(-1.0, 3.0),
ot.Weibull(1.0, 3.0),
ot.Beta(1.0, 3.0, -1.0, 1.0),
ot.Beta(0.5, 1.0, -1.0, 1.0),
ot.Beta(0.5, 1.0, -2.0, 3.0),
ot.Gamma(1.0, 3.0),
ot.Arcsine()
]
for n in range(len(distributionCollection)):
distribution = distributionCollection[n]
name = distribution.getClassName()
polynomialFactory = ot.StandardDistributionPolynomialFactory(
ot.AdaptiveStieltjesAlgorithm(distribution))
# %%
# We draw the fitted distribution
graph = distribution.drawPDF()
graph.setTitle("Fitted Normal distribution")
view = viewer.View(graph)
# %%
# The Student distribution
# ------------------------
#
# The parameters of the Student law are estimated by a mixed method of moments and reduces MLE.
#
# %%
# We generate a sample from a Student distribution with parameters :math:`\nu=5.0`, :math:`\mu = -0.5` and a scale parameter :math:`\sigma=2.0`.
sample = ot.Student(5.0, -0.5, 2.0).getSample(1000)
# %%
# We use the factory to build an estimated distribution :
distribution = ot.StudentFactory().build(sample)
# %%
# We can obtain the estimated parameters with the `getParameter` method :
print(distribution.getParameter())
# %%
# Draw fitted distribution
graph = distribution.drawPDF()
graph.setTitle("Fitted Student distribution")
view = viewer.View(graph)
indices = [1, 2, 3, 5, 6]
print("indices=", indices)
margins = distribution.getMarginal(indices)
print("margins=", margins)
print("margins PDF=%.5f" % margins.computePDF(point))
print("margins CDF=%.5f" % margins.computeCDF(point))
quantile = margins.computeQuantile(0.95)
print("margins quantile=", quantile)
print("margins CDF(quantile)=%.5f" % margins.computeCDF(quantile))
print("margins realization=", margins.getRealization())
# Tests o the isoprobabilistic transformation
# General case with normal standard distribution
print("isoprobabilistic transformation (general normal)=",
distribution.getIsoProbabilisticTransformation())
# General case with non-normal standard distribution
collection[0] = ot.SklarCopula(ot.Student(
3.0, [1.0]*2, [3.0]*2, ot.CorrelationMatrix(2)))
collection.append(ot.Triangular(2.0, 3.0, 4.0))
distribution = ot.BlockIndependentDistribution(collection)
print("isoprobabilistic transformation (general non-normal)=",
distribution.getIsoProbabilisticTransformation())
dim = distribution.getDimension()
x = 0.6
y = [0.2] * (dim - 1)
print("conditional PDF=%.5f" % distribution.computeConditionalPDF(x, y))
print("conditional CDF=%.5f" % distribution.computeConditionalCDF(x, y))
print("conditional quantile=%.5f" %
distribution.computeConditionalQuantile(x, y))
pt = ot.Point(dim)
for i in range(dim):
pt[i] = 0.1 * i + 0.05
print("sequential conditional PDF=",
print("children(", nod, ") : ", ndag.getChildren(nod))
order = ndag.getTopologicalOrder()
marginals = [ot.Uniform(0.0, 1.0) for i in range(order.getSize())]
copulas = list()
for i in range(order.getSize()):
d = 1 + ndag.getParents(i).getSize()
print("i=", i, ", d=", d)
if d == 1:
copulas.append(ot.IndependentCopula(1))
else:
R = ot.CorrelationMatrix(d)
for i in range(d):
for j in range(i):
R[i, j] = 0.5 / d
copulas.append(ot.Student(5.0, [0.0] * d, [1.0] * d, R).getCopula())
cbn = otagrum.ContinuousBayesianNetwork(ndag, marginals, copulas)
size = 300
sample = cbn.getSample(size)
# ContinuousBayesianNetworkFactory
marginalsFactory = ot.UniformFactory()
copulasFactory = ot.BernsteinCopulaFactory()
threshold = 0.1
maxParents = 5
factory = otagrum.ContinuousBayesianNetworkFactory(marginalsFactory,
copulasFactory, ndag,
threshold, maxParents)
cbn = factory.build(sample)
print('cbn=', cbn)
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
if ot.Student().__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.Student().__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.Student().__class__.__name__ == 'Histogram':
distribution = ot.Histogram([-1.0, 0.5, 1.0, 2.0], [0.45, 0.4, 0.15])
else:
distribution = ot.Student()
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()
