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#! /usr/bin/env python
import openturns as ot
import openturns.testing as ott
ot.TESTPREAMBLE()
def cleanScalar(inScalar):
if abs(inScalar) < 1.0e-10:
inScalar = 0.0
return inScalar
def cleanPoint(inPoint):
dim = inPoint.getDimension()
for i in range(dim):
if abs(inPoint[i]) < 1.0e-10:
inPoint[i] = 0.0
return inPoint
ot.PlatformInfo.SetNumericalPrecision(5)
# Instantiate 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()))
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))
# Define a point
point = ot.Point(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(cleanPoint(DDF)))
# PDF value
LPDF = distribution.computeLogPDF(point)
print("log pdf=%.6f" % LPDF)
PDF = distribution.computePDF(point)
print("pdf =%.6f" % PDF)
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)
CCDF = distribution.computeComplementaryCDF(point)
print("ccdf=%.6f" % CCDF)
PDFgr = distribution.computePDFGradient(point)
print("pdf gradient =", repr(cleanPoint(PDFgr)))
# quantile
if dim < 4:
quantile = distribution.computeQuantile(0.95)
print("quantile=", repr(quantile))
print("cdf(quantile)=%.6f" % distribution.computeCDF(quantile))
# Get 95% survival function
inverseSurvival = ot.Point(distribution.computeInverseSurvivalFunction(0.95))
print("InverseSurvival=", repr(inverseSurvival))
print(
"Survival(inverseSurvival)=%.6f"
% distribution.computeSurvivalFunction(inverseSurvival)
)
print("entropy=%.6f" % distribution.computeEntropy())
# Confidence regions
if distribution.getDimension() <= 2:
(
interval,
threshold,
) = distribution.computeMinimumVolumeIntervalWithMarginalProbability(0.95)
print("Minimum volume interval=", interval)
print("threshold=", ot.Point(1, threshold))
levelSet, beta = distribution.computeMinimumVolumeLevelSetWithThreshold(0.95)
print("Minimum volume level set=", levelSet)
print("beta=", ot.Point(1, beta))
(
interval,
beta,
) = distribution.computeBilateralConfidenceIntervalWithMarginalProbability(0.95)
print("Bilateral confidence interval=", interval)
print("beta=", ot.Point(1, beta))
(
interval,
beta,
) = distribution.computeUnilateralConfidenceIntervalWithMarginalProbability(
0.95, False
)
print("Unilateral confidence interval (lower tail)=", interval)
print("beta=", ot.Point(1, beta))
(
interval,
beta,
) = distribution.computeUnilateralConfidenceIntervalWithMarginalProbability(
0.95, True
)
print("Unilateral confidence interval (upper tail)=", interval)
print("beta=", ot.Point(1, beta))
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))
roughness = distribution.getRoughness()
print("roughness=%.6f" % roughness)
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"
% ot.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),
)
x = 0.6
y = [0.2] * (dim - 1)
print("conditional PDF=%.6f" % distribution.computeConditionalPDF(x, y))
print("conditional CDF=%.6f" % distribution.computeConditionalCDF(x, y))
print("conditional quantile=%.6f" % distribution.computeConditionalQuantile(x, y))
pt = ot.Point([i + 1.5 for i in range(dim)])
print(
"sequential conditional PDF=",
distribution.computeSequentialConditionalPDF(point),
)
resCDF = distribution.computeSequentialConditionalCDF(pt)
print("sequential conditional CDF(", pt, ")=", resCDF)
print(
"sequential conditional quantile(",
resCDF,
")=",
distribution.computeSequentialConditionalQuantile(resCDF),
)
# Extract the marginals
for i in range(dim):
margin = distribution.getMarginal(i)
print("margin=", repr(margin))
print("margin PDF=%.6f" % margin.computePDF([0.5]))
print("margin CDF=%.6f" % margin.computeCDF([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 = [1, 0]
print("indices=", repr(indices))
margins = distribution.getMarginal(indices)
print("margins=", repr(margins))
print("margins PDF=%.6f" % margins.computePDF([0.5] * 2))
print("margins CDF=%.6f" % margins.computeCDF([0.5] * 2))
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))
print("invchol=", repr(invChol))
print("chol*t(chol)=", repr((chol * chol.transpose())))
ott.assert_almost_equal((chol * invChol), ot.IdentityMatrix(dim))
ot.Log.Show(ot.Log.TRACE)
validation = ott.DistributionValidation(distribution)
if dim > 2:
validation.skipMoments() # slow
validation.skipCorrelation() # slow
validation.run()
# non-spd cov
dist = ot.Normal(
[0] * 3, ot.CovarianceMatrix([[1.0, 1.0, 0.0], [1.0, 1.0, 0.0], [0.0, 0.0, 1.0]])
)
sample = dist.getSample(10)
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