def test_multivariate_aggregation():
"""Check different aggregation functions for multivariate aggregation, as well as a copula."""
# For sum
marginal_pdfs = [
[1 / 3, 1 / 3, 1 / 3],
[1 / 4, 1 / 4, 2 / 4],
[1 / 6, 1 / 6, 2 / 3],
]
marginal_cdfs = np.cumsum(marginal_pdfs, axis=1)
cdf_p, cdf_v = multivariate_marginal_to_univariate_joint_cdf(marginal_cdfs,
a=10,
b=100)
assert all(np.diff(cdf_v) >= 0) and all(
np.diff(cdf_p) >= 0) # Check for non-decreasing cdf
assert cdf_v[
0] == 30 and cdf_v[-1] == 300 # Check range of aggregated outcomes
assert (cdf_p[0] == 1 / 3 * 1 / 4 * 1 / 6
and cdf_p[-1] == 1) # Check range of cumulative probabilities
# For mean
cdf_p, cdf_v = multivariate_marginal_to_univariate_joint_cdf(
marginal_cdfs, agg_function=np.mean, a=10, b=100)
assert all(np.diff(cdf_v) >= 0) and all(
np.diff(cdf_p) >= 0) # Check for non-decreasing cdf
assert cdf_v[
0] == 10 and cdf_v[-1] == 100 # Check range of aggregated outcomes
assert (cdf_p[0] == 1 / 3 * 1 / 4 * 1 / 6
and cdf_p[-1] == 1) # Check range of cumulative probabilities
# For a normal copula with a correlation matrix with positive correlation between 1st and 2nd variable
R = ot.CorrelationMatrix(len(marginal_cdfs))
R[0, 1] = 0.25
cdf_p, cdf_v = multivariate_marginal_to_univariate_joint_cdf(
marginal_cdfs, copula=ot.NormalCopula(R), a=10, b=100)
assert all(np.diff(cdf_v) >= 0) and all(
np.diff(cdf_p) >= 0) # Check for non-decreasing cdf
assert cdf_v[
0] == 30 and cdf_v[-1] == 300 # Check range of aggregated outcomes
assert (cdf_p[0] > 1 / 3 * 1 / 4 * 1 / 6
and cdf_p[-1] == 1) # Check range of cumulative probabilities
# For a normal copula with a correlation matrix with negative correlation between 1st and 2nd variable
R = ot.CorrelationMatrix(len(marginal_cdfs))
R[0, 1] = -0.25
cdf_p, cdf_v = multivariate_marginal_to_univariate_joint_cdf(
marginal_cdfs, copula=ot.NormalCopula(R), a=10, b=100)
assert all(np.diff(cdf_v) >= 0) and all(
np.diff(cdf_p) >= 0) # Check for non-decreasing cdf
assert cdf_v[
0] == 30 and cdf_v[-1] == 300 # Check range of aggregated outcomes
assert (cdf_p[0] < 1 / 3 * 1 / 4 * 1 / 6
and cdf_p[-1] == 1) # Check range of cumulative probabilities
def generateSampleWithConditionalIndependance3(size=1000):
dim = 5
# 0T4 | 1,2,3 <=> r34 = (-r01*r13*r23*r24+r01*r14*r23^2+r02*r13^2*r24-r02*r13*r14*r23-r03*r12*r13*r24-r03*r12*r14
# *r23+2*r04*r12*r13*r23+r01*r12*r24+r02*r12*r14+r03*r13*r14+r03*r23*r24-r04*r12^2-r04*r13^2-r04*r23^2-r01*r14-r02
# *r24+r04)/(r01*r12*r23+r02*r12*r13-r03*r12^2-r01*r13-r02*r23+r03)
R = ot.CorrelationMatrix(dim)
R[0, 1] = 0.9
R[0, 2] = 0.9
R[0, 3] = 0.9
R[0, 4] = 0.9
R[1, 2] = 0.9
R[1, 3] = 0.9
R[1, 4] = 0.9
R[2, 3] = 0.9
R[2, 4] = 0.95
R[3,
4] = (-R[0, 1] * R[1, 3] * R[2, 3] * R[2, 4] +
R[0, 1] * R[1, 4] * R[2, 3]**2 + R[0, 2] * R[1, 3]**2 * R[2, 4] -
R[0, 2] * R[1, 3] * R[1, 4] * R[2, 3] - R[0, 3] * R[1, 2] *
R[1, 3] * R[2, 4] - R[0, 3] * R[1, 2] * R[1, 4] * R[2, 3] +
2 * R[0, 4] * R[1, 2] * R[1, 3] * R[2, 3] +
R[0, 1] * R[1, 2] * R[2, 4] + R[0, 2] * R[1, 2] * R[1, 4] +
R[0, 3] * R[1, 3] * R[1, 4] + R[0, 3] * R[2, 3] * R[2, 4] -
R[0, 4] * R[1, 2]**2 - R[0, 4] * R[1, 3]**2 -
R[0, 4] * R[2, 3]**2 - R[0, 1] * R[1, 4] - R[0, 2] * R[2, 4] +
R[0, 4]) / (R[0, 1] * R[1, 2] * R[2, 3] +
R[0, 2] * R[1, 2] * R[1, 3] - R[0, 3] * R[1, 2]**2 -
R[0, 1] * R[1, 3] - R[0, 2] * R[2, 3] + R[0, 3])
copula = ot.NormalCopula(R)
copula.setDescription(["X" + str(i) for i in range(dim)])
return copula.getSample(
size) # .exportToCSVFile("conditional_independence_04_123.csv")
def test_multivariate_aggregation_with_unmatched_bins_and_dependence(
multivariate_test_cdfs):
"""Check multivariate aggregation where the outcomes of each variable are completely different,
and the variables are correlated."""
marginal_cdf_p, marginal_cdf_v = multivariate_test_cdfs
dim = len(marginal_cdf_p)
# Make a correlation matrix with positive correlation between each pair of adjacent variables (needs at least 2D)
R = ot.CorrelationMatrix(dim)
for d in range(1, dim):
R[d - 1, d] = 0.25
cdf_p, cdf_v = multivariate_marginal_to_univariate_joint_cdf(
marginal_cdf_p,
marginal_cdfs_v=marginal_cdf_v,
copula=ot.NormalCopula(R))
assert all(np.diff(cdf_v) >= 0) and all(
np.diff(cdf_p) >= 0) # Check for non-decreasing cdf
assert (cdf_v[0] >= 10 * dim
and cdf_v[-1] <= 100 * dim) # Check range of aggregated outcomes
assert cdf_p[0] >= 0 and (cdf_p[-1] < 1 or cdf_p[-1] == approx(1)
) # Check range of cumulative probabilities
cdf_p_2, cdf_v_2 = multivariate_marginal_to_univariate_joint_cdf(
marginal_cdf_p,
marginal_cdfs_v=marginal_cdf_v,
copula=ot.NormalCopula(R),
n_draws=1000,
)
assert len(cdf_p_2) == 1000
def generateDataForSpecificInstance(size): R = ot.CorrelationMatrix(3) R[0, 1] = 0.5 R[0, 2] = 0.45 collection = [ot.FrankCopula(3.0), ot.NormalCopula(R), ot.ClaytonCopula(2.0)] copula = ot.ComposedCopula(collection) return copula.getSample(size)
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 test_marginal_distributions_with_residual_probability():
"""Aggregate three time slots with CDFs that are not fully specified.
That means each has a residual probability that their outcome is higher than the highest value given.
"""
# Make sure incomplete cdf functions can still be transformed (a higher outcome with cp=1 can be assumed to exist)
marginal_cdfs = [
[1 / 3, 2 / 4, 3 / 4],
[1 / 4, 4 / 6, 5 / 6],
[1 / 6, 5 / 9, 8 / 9],
]
cdf_p, _ = multivariate_marginal_to_univariate_joint_cdf(marginal_cdfs)
assert all(np.diff(cdf_p) >= 0) # Check for non-decreasing cdf
assert (
cdf_p[-1] < 1
) # Check that the assumed outcome with cp=1 is not actually returned
a = (
cdf_p[-1] - cdf_p[-2]
) # The probability of the highest outcome for each of the three variables
b = 1 / 4 * 1 / 6 * 3 / 9 # The probability for independent random variables
assert a == approx(b) # Check the expected outcome
# Make a correlation matrix with negative correlation between the first and second variable
R = ot.CorrelationMatrix(len(marginal_cdfs))
R[0, 1] = -0.25
cdf_p, _ = multivariate_marginal_to_univariate_joint_cdf(
marginal_cdfs, copula=ot.NormalCopula(R))
assert all(np.diff(cdf_p) >= 0) # Check for non-decreasing cdf
a = (
cdf_p[-1] - cdf_p[-2]
) # The probability of the highest outcome for each of the three variables
assert (
a < b
) # Check the expected outcome is now lower (if x1 is high, than x2 is less likely to be high)
# Make a correlation matrix with positive correlation between the first and second variable
R = ot.CorrelationMatrix(len(marginal_cdfs))
R[0, 1] = 0.25
cdf_p, _ = multivariate_marginal_to_univariate_joint_cdf(
marginal_cdfs, copula=ot.NormalCopula(R))
assert all(np.diff(cdf_p) >= 0) # Check for non-decreasing cdf
a = (
cdf_p[-1] - cdf_p[-2]
) # The probability of the highest outcome for each of the three variables
assert (
a > b
) # Check the expected outcome is now higher (if x1 is high, than x2 is likely to be high, too)
def generate_gaussian_copulas(ndag, r=0.8):
lcc = []
for k in range(ndag.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
lcc.append(ot.Normal([0.0]*d, [1.0]*d, R).getCopula())
return lcc
def generateSampleWithConditionalIndependance1(size=1000):
dim = 3
# 0T2 | 1 <=> r12 = r02/r01
R = ot.CorrelationMatrix(dim)
R[0, 1] = 0.95
R[0, 2] = 0.9
R[1, 2] = R[0, 2] / R[0, 1]
copula = ot.NormalCopula(R)
copula.setDescription(["X" + str(i) for i in range(dim)])
return copula.getSample(
size) # ".exportToCSVFile("conditional_independence_02_1.csv")
def generate_gaussian_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.NormalCopula(R))
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
def _getRho(self):
if self._alpha is None:
self.getAlpha()
dim = len(self._alpha)
self._rho = ot.CorrelationMatrix(dim)
for i in range(dim):
for j in range(dim):
if i < j:
self._rho[i, j] = np.dot(self._alpha[i], self._alpha[j])
return self._rho
def generateSampleWithConditionalIndependance2(size=1000):
dim = 4
# 0T3 | 1,2 <=> r23 = (r03*(r12^2-1)+r13*(r01-r02*r12)) / (r01*r12-r02)
R = ot.CorrelationMatrix(dim)
R[0, 1] = 0.9
R[0, 2] = 0.9
R[0, 3] = 0.9
R[1, 2] = 0.9
R[1, 3] = 0.95
R[2, 3] = (R[0, 3] * (R[1, 2]**2 - 1.0) + R[1, 3] *
(R[0, 1] - R[0, 2] * R[1, 2])) / (R[0, 1] * R[1, 2] - R[0, 2])
copula = ot.NormalCopula(R)
copula.setDescription(["X" + str(i) for i in range(dim)])
return copula.getSample(
size) # .exportToCSVFile("conditional_independence_03_12.csv")
def test_CrossCutDistribution2(self):
# Create a Funky distribution
corr = ot.CorrelationMatrix(2)
corr[0, 1] = 0.2
copula = ot.NormalCopula(corr)
x1 = ot.Normal(-1.0, 1.0)
x2 = ot.Normal(2.0, 1.0)
x_funk = ot.ComposedDistribution([x1, x2], copula)
# Create a Punk distribution
x1 = ot.Normal(1.0, 1.0)
x2 = ot.Normal(-2.0, 1.0)
x_punk = ot.ComposedDistribution([x1, x2], copula)
distribution = ot.Mixture([x_funk, x_punk], [0.5, 1.0])
referencePoint = distribution.getMean()
crossCut = otbenchmark.CrossCutDistribution(distribution)
# Avoid failing on CircleCi
# _tkinter.TclError: no display name and no $DISPLAY environment variable
try:
_ = crossCut.drawMarginalPDF()
_ = crossCut.drawConditionalPDF(referencePoint)
except Exception as e:
print(e)
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])
def hill_climbing(D, max_parents=4, restart=1):
N = D.getDimension()
# Compute the estimate of the gaussian copula
kendall_tau = D.computeKendallTau()
#print(kendall_tau)
pearson_r = ot.CorrelationMatrix(np.sin((np.pi / 2) * kendall_tau))
# Create the gaussian copula with parameters pearson_r
# if pearson_r isn't PSD, a regularization is done
eps = 1e-6
done = False
while not done:
try:
gaussian_copula = ot.NormalCopula(pearson_r)
done = True
except:
print("Regularization")
for i in range(pearson_r.getDimension()):
for j in range(i):
pearson_r[i, j] /= 1 + eps
# Initialization
G = du.create_empty_dag(N)
score = sc.bic_score(D, gaussian_copula, G)
best_graph = G
best_score = score
for r in range(restart):
if r != 0:
G = du.create_random_dag(N, max_parents)
G, score = one_hill_climbing(D, gaussian_copula, G, max_parents)
if score > best_score:
best_graph = G
best_score = score
return gaussian_copula, best_graph, best_score
"""
# %%
# In this example we are going to draw clouds of points from a data 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 2-d samples to visualize
N = 500
R = ot.CorrelationMatrix(2)
R[0, 1] = -0.7
# 2d N(1,1) with correlation
sample1 = ot.Normal([1.0] * 2, [1.0] * 2, R).getSample(N)
sample2 = ot.Normal(2).getSample(N) # 2d N(0,1) independent
# %%
# Create cloud drawables
cloud1 = ot.Cloud(sample1, 'blue', 'fsquare', 'First Cloud')
cloud2 = ot.Cloud(sample2, 'red', 'fsquare', 'Second Cloud')
# Then, assemble it into a graph
myGraph2d = ot.Graph('2d clouds', 'x1', 'x2', True, 'topright')
myGraph2d.add(cloud1)
myGraph2d.add(cloud2)
view = viewer.View(myGraph2d)
# %%
# First, we define a regular 2-d mesh
discretization = [10, 5]
mesher = ot.IntervalMesher(discretization)
lowerBound = [0.0, 0.0]
upperBound = [2.0, 1.0]
interval = ot.Interval(lowerBound, upperBound)
mesh = mesher.build(interval)
graph = mesh.draw()
graph.setTitle('Regular 2-d mesh')
view = viewer.View(graph)
# %%
# We now create a field from a mesh and some values
values = ot.Normal([0.0] * 2, [1.0] * 2,
ot.CorrelationMatrix(2)).getSample(len(mesh.getVertices()))
for i in range(len(values)):
x = values[i]
values[i] = 0.05 * x / x.norm()
field = ot.Field(mesh, values)
# %%
# We can export the `field` to a VTK files. It can be
# read later with an external program such as Paraview.
field.exportToVTKFile('field.vtk')
# %%
# Display figures
plt.show()
for i in range(obsSize):
for j in range(chainDim):
p[i, j] = (-2 + 5. * i / 9.)**j
print('p=', p)
fullModel = ot.SymbolicFunction(['p1', 'p2', 'p3', 'x1', 'x2', 'x3'],
['p1*x1+p2*x2+p3*x3', '1.0'])
linkFunction = ot.ParametricFunction(fullModel, range(chainDim),
[0.0] * chainDim)
# instrumental distribution
instrumental = ot.Uniform(-1., 1.)
# prior distribution
sigma0 = [10.0] * chainDim
Q0 = ot.CorrelationMatrix(chainDim) # precision matrix
Q0_inv = ot.CorrelationMatrix(chainDim) # variance matrix
for i in range(chainDim):
Q0_inv[i, i] = sigma0[i] * sigma0[i]
Q0[i, i] = 1.0 / Q0_inv[i, i]
print('Q0=', Q0)
mu0 = [0.0] * chainDim
prior = ot.Normal(mu0, Q0_inv) # x0 ~ N(mu0, sigma0)
print('x~', prior)
# start from te mean x0=(0.,0.,0.)
print('x0=', mu0)
# conditional distribution y~N(z, 1.0)
conditional = ot.Normal()
#marginals piping
marginals_p =[distribution_D, distribution_D_cover,
distribution_d70, distribution_h_exit,
distribution_k,
distribution_L, distribution_m_p]
distribution_p = ot.ComposedDistribution(marginals_p)
#create marginal distribution
marginals = [distribution_D, distribution_D_cover,
distribution_d70, distribution_h_exit,
distribution_i_ch, distribution_k,
distribution_L, distribution_m_p,
distribution_m_u]
#create copula uplift
RS_u = ot.CorrelationMatrix(len(marginals_u))
R_u = ot.NormalCopula.GetCorrelationFromSpearmanCorrelation(RS_u)
copula_u = ot.NormalCopula(R_u)
#create copula heave
RS_h = ot.CorrelationMatrix(len(marginals_h))
R_h = ot.NormalCopula.GetCorrelationFromSpearmanCorrelation(RS_h)
copula_h = ot.NormalCopula(R_h)
#create copula piping
RS_p = ot.CorrelationMatrix(len(marginals_p))
R_p = ot.NormalCopula.GetCorrelationFromSpearmanCorrelation(RS_p)
copula_p = ot.NormalCopula(R_p)
#create copula
RS = ot.CorrelationMatrix(len(marginals))
#! /usr/bin/env python
import openturns as ot
ot.TESTPREAMBLE()
# Instantiate one distribution object
dimension = 3
meanPoint = ot.Point(dimension, 1.0)
meanPoint[0] = 0.5
meanPoint[1] = -0.5
sigma = ot.Point(dimension, 1.0)
sigma[0] = 2.0
sigma[1] = 3.0
R = ot.CorrelationMatrix(dimension)
for i in range(1, dimension):
R[i, i - 1] = 0.5
# Create a collection of distribution
aCollection = ot.DistributionCollection()
aCollection.add(ot.Normal(meanPoint, sigma, R))
meanPoint += ot.Point(meanPoint.getDimension(), 1.0)
aCollection.add(ot.Normal(meanPoint, sigma, R))
meanPoint += ot.Point(meanPoint.getDimension(), 1.0)
aCollection.add(ot.Normal(meanPoint, sigma, R))
# Instantiate one distribution object
distribution = ot.Mixture(aCollection, ot.Point(aCollection.getSize(), 2.0))
print("Distribution ", repr(distribution))
print("Weights = ", repr(distribution.getWeights()))
# **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}
# 1 & 0 & 0 & 0 \\
# 0 & 1 & 0 & 0 \\
# 0 & 0 & 1 & -0.2 \\
# 0 & 0 & -0.2 & 1 \\
# \end{pmatrix}
# %%
# We implement this correlation:
# Create the Spearman correlation matrix of the input random vector
RS = ot.CorrelationMatrix(4)
RS[2, 3] = -0.2
# Evaluate the correlation matrix of the Normal copula from RS
R = ot.NormalCopula.GetCorrelationFromSpearmanCorrelation(RS)
# Create the Normal copula parametrized by R
mycopula = ot.NormalCopula(R)
# %%
# And we endly create the composed input probability distribution.
inputDistribution = ot.ComposedDistribution([E, F, L, I], mycopula)
inputDistribution.setDescription(("E", "F", "L", "I"))
# %%
# Create the event whose probability we want to estimate:
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)
test_model(myModel)
myDefautModel = ot.FractionalBrownianMotionModel(2.0, 3.0, 0.25)
print('myDefautModel = ', myDefautModel)
test_model(myDefautModel)
myModel = ot.SphericalModel([2.0] * inputDimension, [3.0], 4.5)
test_model(myModel)
myDefautModel = ot.DiracCovarianceModel()
print('myDefautModel = ', myDefautModel)
test_model(myDefautModel)
amplitude = [1.5 + 2.0 * k for k in range(2)]
dimension = 2
spatialCorrelation = ot.CorrelationMatrix(dimension)
for j in range(dimension):
for i in range(j + 1, dimension):
spatialCorrelation[i,
j] = (i + 1.0) / dimension - (j + 1.0) / dimension
myModel = ot.DiracCovarianceModel(inputDimension, amplitude,
spatialCorrelation)
test_model(myModel, x1=[0.5, 0.0], x2=[0.5, 0.0])
myDefautModel = ot.ProductCovarianceModel()
print('myDefautModel = ', myDefautModel)
test_model(myDefautModel)
cov1 = ot.AbsoluteExponential([2.0], [3.0])
cov2 = ot.SquaredExponential([2.0], [3.0])
myModel = ot.ProductCovarianceModel([cov1, cov2])
#! /usr/bin/env python
import openturns as ot
ot.TESTPREAMBLE()
marginals = [ot.Normal(), ot.Uniform()]
R = ot.CorrelationMatrix(2, [1.0, 0.5, 0.5, 1.0])
copula = ot.NormalCopula(R)
factories = [
ot.SoizeGhanemFactory(ot.ComposedDistribution(marginals)),
ot.SoizeGhanemFactory(ot.ComposedDistribution(marginals, copula), False),
ot.SoizeGhanemFactory(ot.ComposedDistribution(marginals, copula), True)
]
x = [0.5] * 2
kMax = 5
ot.ResourceMap.SetAsUnsignedInteger("IteratedQuadrature-MaximumSubIntervals",
2048)
ot.ResourceMap.SetAsScalar("IteratedQuadrature-MaximumError", 1.0e-6)
for soize in factories:
distribution = soize.getMeasure()
print('SoizeGhanem=', soize)
functions = list()
for k in range(kMax):
functions.append(soize.build(k))
print('SoizeGhanem(', k, ')=', functions[k].getEvaluation())
print('SoizeGhanem(', k, ')(', x, '=', functions[k](x))
M = ot.SymmetricMatrix(kMax)
for m in range(kMax):
for n in range(m + 1):
# %% 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 data to visualize # Create the model Y = x1^2 + x2 model = ot.SymbolicFunction(["x1", "x2"], ["x1^2+x2"]) # Create the input distribution and random vector X myCorMat = ot.CorrelationMatrix(2) myCorMat[0, 1] = -0.6 inputDist = ot.Normal([0., 0.], myCorMat) inputDist.setDescription(['X1', 'X2']) inputVector = ot.RandomVector(inputDist) # Create the output random vector Y=model(X) output = ot.CompositeRandomVector(model, inputVector) # Generate the input sample N = 500 X = inputVector.getSample(N) # Evaluate the associated output sample Y = model(X)
def cleanPoint(inPoint):
dim = inPoint.getDimension()
for i in range(dim):
if (fabs(inPoint[i]) < 1.e-10):
inPoint[i] = 0.0
return inPoint
ot.PlatformInfo.SetNumericalPrecision(5)
# Instanciate one distribution object
for dim in range(1, 5):
meanPoint = [0.0] * dim
sigma = [1.0 + i for i in range(dim)]
R = ot.CorrelationMatrix(dim)
for i in range(1, dim):
R[i, i - 1] = 0.5
distribution = ot.Normal(meanPoint, sigma, R)
distribution.setName("A normal distribution")
description = ["Marginal " + str(1 + i) for i in range(dim)]
distribution.setDescription(description)
print("Parameters collection=",
repr(distribution.getParametersCollection()))
for i in range(6):
print("standard moment n=", i, " value=",
distribution.getStandardMoment(i))
print("Standard representative=", distribution.getStandardRepresentative())
import sys
try:
from openturns.viewer import View
import openturns as ot
# Curve
graph = ot.Normal().drawCDF()
# graph.draw('curve1.png')
view = View(graph, pixelsize=(800, 600), plot_kw={'color': 'blue'})
# view.save('curve1.png')
view.show()
# Contour
graph = ot.Normal([1, 2], [3, 5], ot.CorrelationMatrix(2)).drawPDF()
# graph.draw('curve2.png')
view = View(graph)
# view.save('curve2.png')
view.show()
# Histogram tests
normal = ot.Normal(1)
size = 100
sample = normal.getSample(size)
graph = ot.HistogramFactory().build(sample, 10).drawPDF()
# graph.draw('curve3.png')
view = View(graph)
# view.save('curve3.png')
view.show()
from __future__ import print_function
import openturns as ot
import sys
import openturns.testing as ott
import math as m
ot.TESTPREAMBLE()
ot.ResourceMap.SetAsBool('Distribution-MinimumVolumeLevelSetBySampling', True)
ot.ResourceMap.SetAsUnsignedInteger(
'Distribution-MinimumVolumeLevelSetSamplingSize', 500)
# 2-d test
dists = [
ot.Normal([-1.0, 2.0], [1.0] * 2, ot.CorrelationMatrix(2)),
ot.Normal([1.0, -2.0], [1.5] * 2, ot.CorrelationMatrix(2))
]
mixture = ot.Mixture(dists)
# 3-d test
R1 = ot.CovarianceMatrix(3)
R1[2, 1] = -0.25
R2 = ot.CovarianceMatrix(3)
R2[1, 0] = 0.5
R2[2, 1] = -0.3
R2[0, 0] = 1.3
print(R2)
dists = [ot.Normal([1.0, -2.0, 3.0], R1), ot.Normal([-1.0, 2.0, -2.0], R2)]
mixture = ot.Mixture(dists, [2.0 / 3.0, 1.0 / 3.0])
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))
# Using some reference values
# See https://en.wikipedia.org/wiki/Kernel_(statistics)#Kernel_functions_in_common_use
# First Normal dist with default ctor
distribution = ot.Normal()
ott.assert_almost_equal(distribution.getRoughness(),
0.5 /m.sqrt(m.pi))
# Dimension 2 (Fix https://github.com/openturns/openturns/issues/1485)
# Indep copula : product of integrales
distribution = ot.Normal(2)
ott.assert_almost_equal(distribution.getRoughness(),
compute_roughness_sampling(distribution))
# 2D Normal with scale & correlation
# This allows checking that Normal::getRoughness is well implemented
corr = ot.CorrelationMatrix(2)
corr[1, 0] = 0.3
distribution = ot.Normal([0, 0], [1, 2], corr)
ott.assert_almost_equal(distribution.getRoughness(),
compute_roughness_sampling(distribution))
distribution = ot.Epanechnikov()
ott.assert_almost_equal(distribution.getRoughness(), 3/5)
distribution = ot.Triangular()
ott.assert_almost_equal(distribution.getRoughness(), 2/3)
distribution = ot.Distribution(Quartic())
ott.assert_almost_equal(distribution.getRoughness(), 5/7)
# Testing Histogram ==> getSingularities
