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#! /usr/bin/env python
from __future__ import print_function
from openturns import *
from math import *
TESTPREAMBLE()
RandomGenerator.SetSeed(0)
def cleanScalar(inScalar):
if (fabs(inScalar) < 1.e-10):
inScalar = 0.0
return inScalar
def cleanNumericalPoint(inNumericalPoint):
dim = inNumericalPoint.getDimension()
for i in range(dim):
if (fabs(inNumericalPoint[i]) < 1.e-10):
inNumericalPoint[i] = 0.0
return inNumericalPoint
try:
PlatformInfo.SetNumericalPrecision(5)
# Instanciate one distribution object
dim = 3
meanPoint = NumericalPoint(dim, 1.0)
meanPoint[0] = 0.5
meanPoint[1] = -0.5
sigma = NumericalPoint(dim, 1.0)
sigma[0] = 2.0
sigma[1] = 3.0
R = CorrelationMatrix(dim)
for i in range(1, dim):
R[i, i - 1] = 0.5
distribution = Normal(meanPoint, sigma, R)
distribution.setName("A normal distribution")
description = Description(dim)
description[0] = "Marginal 1"
description[1] = "Marginal 2"
description[2] = "Marginal 3"
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())
print("Distribution ", repr(distribution))
print("Distribution ", distribution)
print("Covariance ", repr(distribution.getCovariance()))
# Is this distribution elliptical ?
print("Elliptical = ", distribution.isElliptical())
# Is this distribution continuous ?
print("Continuous = ", distribution.isContinuous())
# 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]))
print("mean=", repr(oneSample.computeMean()))
print("covariance=", repr(oneSample.computeCovariance()))
# Define a point
point = NumericalPoint(distribution.getDimension(), 0.5)
print("Point= ", repr(point))
# Show PDF and CDF of point
eps = 1e-5
# derivative of PDF with respect to its arguments
DDF = distribution.computeDDF(point)
print("ddf =", repr(cleanNumericalPoint(DDF)))
# by the finite difference technique
ddfFD = NumericalPoint(dim)
for i in range(dim):
pointEps = point
pointEps[i] += eps
ddfFD[i] = distribution.computePDF(pointEps)
pointEps[i] -= 2.0 * eps
ddfFD[i] -= distribution.computePDF(pointEps)
ddfFD[i] /= 2.0 * eps
print("ddf (FD)=", repr(cleanNumericalPoint(ddfFD)))
# PDF value
LPDF = distribution.computeLogPDF(point)
print("log pdf=%.6f" % LPDF)
PDF = distribution.computePDF(point)
print("pdf =%.6f" % PDF)
# by the finite difference technique from CDF
if dim == 1:
print("pdf (FD)=%.6f" % cleanScalar((distribution.computeCDF(
point + NumericalPoint(1, eps)) - distribution.computeCDF(point + NumericalPoint(1, -eps))) / (2.0 * eps)))
CF = distribution.computeCharacteristicFunction(point)
print("characteristic function=%.6f+%.6fi" % (CF.real, CF.imag))
LCF = distribution.computeLogCharacteristicFunction(point)
print("log characteristic function=%.6f+%.6fi" % (LCF.real, LCF.imag))
CDF = distribution.computeCDF(point)
print("cdf=%.6f" % CDF)
PDFgr = distribution.computePDFGradient(point)
print("pdf gradient =", repr(PDFgr))
# by the finite difference technique
PDFgrFD = NumericalPoint(2 * dim)
for i in range(dim):
meanPoint[i] += eps
distributionLeft = Normal(meanPoint, sigma, R)
meanPoint[i] -= 2.0 * eps
distributionRight = Normal(meanPoint, sigma, R)
PDFgrFD[i] = (distributionLeft.computePDF(point)
- distributionRight.computePDF(point)) / (2.0 * eps)
meanPoint[i] += eps
for i in range(dim):
sigma[i] += eps
distributionLeft = Normal(meanPoint, sigma, R)
sigma[i] -= 2.0 * eps
distributionRight = Normal(meanPoint, sigma, R)
PDFgrFD[dim + i] = (distributionLeft.computePDF(
point) - distributionRight.computePDF(point)) / (2.0 * eps)
sigma[i] += eps
print("pdf gradient (FD)=", repr(cleanNumericalPoint(PDFgrFD)))
# derivative of the PDF with regards the parameters of the distribution
# CDFgr = distribution.computeCDFGradient( point )
# print "cdf gradient =" , CDFgr
# quantile
quantile = distribution.computeQuantile(0.95)
print("quantile=", repr(quantile))
print("cdf(quantile)=%.6f" % distribution.computeCDF(quantile))
mean = distribution.getMean()
print("mean=", repr(mean))
standardDeviation = distribution.getStandardDeviation()
print("standard deviation=", repr(standardDeviation))
skewness = distribution.getSkewness()
print("skewness=", repr(skewness))
kurtosis = distribution.getKurtosis()
print("kurtosis=", repr(kurtosis))
covariance = distribution.getCovariance()
print("covariance=", repr(covariance))
parameters = distribution.getParametersCollection()
print("parameters=", repr(parameters))
# Specific to this distribution
beta = point.normSquare()
densityGenerator = distribution.computeDensityGenerator(beta)
print("density generator=%.6f" % densityGenerator)
print("pdf via density generator=%.6f" %
EllipticalDistribution.computePDF(distribution, point))
densityGeneratorDerivative = distribution.computeDensityGeneratorDerivative(
beta)
print("density generator derivative =%.6f" %
densityGeneratorDerivative)
print("density generator derivative (FD)=%.6f" % cleanScalar((distribution.computeDensityGenerator(
beta + eps) - distribution.computeDensityGenerator(beta - eps)) / (2.0 * eps)))
densityGeneratorSecondDerivative = distribution.computeDensityGeneratorSecondDerivative(
beta)
print("density generator second derivative =%.6f" %
densityGeneratorSecondDerivative)
print("density generator second derivative (FD)=%.6f" % cleanScalar((distribution.computeDensityGeneratorDerivative(
beta + eps) - distribution.computeDensityGeneratorDerivative(beta - eps)) / (2.0 * eps)))
# Compute the radial CDF
radius = 2.0
print("Radial CDF(%.6f" % radius, ")=%.6f" %
distribution.computeRadialDistributionCDF(radius))
# Extract the marginals
for i in range(dim):
margin = distribution.getMarginal(i)
print("margin=", repr(margin))
print("margin PDF=%.6f" % margin.computePDF(NumericalPoint(1, 0.5)))
print("margin CDF=%.6f" % margin.computeCDF(NumericalPoint(1, 0.5)))
print("margin quantile=", repr(margin.computeQuantile(0.95)))
print("margin realization=", repr(margin.getRealization()))
if (dim >= 2):
# Extract a 2-D marginal
indices = Indices(2, 0)
indices[0] = 1
indices[1] = 0
print("indices=", repr(indices))
margins = distribution.getMarginal(indices)
print("margins=", repr(margins))
print("margins PDF=%.6f" % margins.computePDF(NumericalPoint(2, 0.5)))
print("margins CDF=%.6f" % margins.computeCDF(NumericalPoint(2, 0.5)))
quantile = margins.computeQuantile(0.95)
print("margins quantile=", repr(quantile))
print("margins CDF(qantile)=%.6f" % margins.computeCDF(quantile))
print("margins realization=", repr(margins.getRealization()))
chol = distribution.getCholesky()
invChol = distribution.getInverseCholesky()
print("chol=", repr(chol.clean(1e-6)))
print("invchol=", repr(invChol.clean(1e-6)))
print("chol*t(chol)=", repr((chol * chol.transpose()).clean(1e-6)))
print("chol*invchol=", repr((chol * invChol).clean(1e-6)))
except:
import sys
print("t_Normal_std.py", sys.exc_info()[0], sys.exc_info()[1])
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