ChiSquare distribution¶
(Source code, png, hires.png, pdf)

class
ChiSquare
(*args)¶ distribution.
 Available constructors:
ChiSquare(nu=1.0)
 Parameters
 nufloat,
Degrees of freedom.
Notes
Its probability density function is defined as:
with .
Its first moments are:
Examples
Create a distribution:
>>> import openturns as ot >>> distribution = ot.ChiSquare(2.0)
Draw a sample:
>>> sample = distribution.getSample(5)
Methods
abs
(self)Transform distribution by absolute value function.
acos
(self)Transform distribution by arccosine function.
acosh
(self)Transform distribution by acosh function.
asin
(self)Transform distribution by arcsine function.
asinh
(self)Transform distribution by asinh function.
atan
(self)Transform distribution by arctangent function.
atanh
(self)Transform distribution by atanh function.
cbrt
(self)Transform distribution by cubic root function.
computeBilateralConfidenceInterval
(self, prob)Compute a bilateral confidence interval.
computeBilateralConfidenceIntervalWithMarginalProbability
(…)Compute a bilateral confidence interval.
computeCDF
(self, \*args)Compute the cumulative distribution function.
computeCDFGradient
(self, \*args)Compute the gradient of the cumulative distribution function.
computeCharacteristicFunction
(self, x)Compute the characteristic function.
computeComplementaryCDF
(self, \*args)Compute the complementary cumulative distribution function.
computeConditionalCDF
(self, \*args)Compute the conditional cumulative distribution function.
computeConditionalDDF
(self, x, y)Compute the conditional derivative density function of the last component.
computeConditionalPDF
(self, \*args)Compute the conditional probability density function.
computeConditionalQuantile
(self, \*args)Compute the conditional quantile function of the last component.
computeDDF
(self, \*args)Compute the derivative density function.
computeDensityGenerator
(self, betaSquare)Compute the probability density function of the characteristic generator.
computeDensityGeneratorDerivative
(self, …)Compute the firstorder derivative of the probability density function.
Compute the secondorder derivative of the probability density function.
computeEntropy
(self)Compute the entropy of the distribution.
computeGeneratingFunction
(self, \*args)Compute the probabilitygenerating function.
computeInverseSurvivalFunction
(self, point)Compute the inverse survival function.
computeLogCharacteristicFunction
(self, x)Compute the logarithm of the characteristic function.
computeLogGeneratingFunction
(self, \*args)Compute the logarithm of the probabilitygenerating function.
computeLogPDF
(self, \*args)Compute the logarithm of the probability density function.
computeLogPDFGradient
(self, \*args)Compute the gradient of the log probability density function.
computeMinimumVolumeInterval
(self, prob)Compute the confidence interval with minimum volume.
Compute the confidence interval with minimum volume.
computeMinimumVolumeLevelSet
(self, prob)Compute the confidence domain with minimum volume.
Compute the confidence domain with minimum volume.
computePDF
(self, \*args)Compute the probability density function.
computePDFGradient
(self, \*args)Compute the gradient of the probability density function.
computeProbability
(self, interval)Compute the interval probability.
computeQuantile
(self, \*args)Compute the quantile function.
computeRadialDistributionCDF
(self, radius[, …])Compute the cumulative distribution function of the squared radius.
computeScalarQuantile
(self, prob[, tail])Compute the quantile function for univariate distributions.
computeSequentialConditionalCDF
(self, x)Compute the sequential conditional cumulative distribution functions.
computeSequentialConditionalDDF
(self, x)Compute the sequential conditional derivative density function.
computeSequentialConditionalPDF
(self, x)Compute the sequential conditional probability density function.
computeSequentialConditionalQuantile
(self, q)Compute the conditional quantile function of the last component.
computeSurvivalFunction
(self, \*args)Compute the survival function.
computeUnilateralConfidenceInterval
(self, prob)Compute a unilateral confidence interval.
computeUnilateralConfidenceIntervalWithMarginalProbability
(…)Compute a unilateral confidence interval.
cos
(self)Transform distribution by cosine function.
cosh
(self)Transform distribution by cosh function.
drawCDF
(self, \*args)Draw the cumulative distribution function.
drawLogPDF
(self, \*args)Draw the graph or of isolines of logprobability density function.
drawMarginal1DCDF
(self, marginalIndex, xMin, …)Draw the cumulative distribution function of a margin.
drawMarginal1DLogPDF
(self, marginalIndex, …)Draw the logprobability density function of a margin.
drawMarginal1DPDF
(self, marginalIndex, xMin, …)Draw the probability density function of a margin.
drawMarginal1DSurvivalFunction
(self, …[, …])Draw the cumulative distribution function of a margin.
drawMarginal2DCDF
(self, firstMarginal, …)Draw the cumulative distribution function of a couple of margins.
drawMarginal2DLogPDF
(self, firstMarginal, …)Draw the logprobability density function of a couple of margins.
drawMarginal2DPDF
(self, firstMarginal, …)Draw the probability density function of a couple of margins.
drawMarginal2DSurvivalFunction
(self, …[, …])Draw the cumulative distribution function of a couple of margins.
drawPDF
(self, \*args)Draw the graph or of isolines of probability density function.
drawQuantile
(self, \*args)Draw the quantile function.
drawSurvivalFunction
(self, \*args)Draw the cumulative distribution function.
exp
(self)Transform distribution by exponential function.
getCDFEpsilon
(self)Accessor to the CDF computation precision.
getCenteredMoment
(self, n)Accessor to the componentwise centered moments.
getCholesky
(self)Accessor to the Cholesky factor of the covariance matrix.
getClassName
(self)Accessor to the object’s name.
getCopula
(self)Accessor to the copula of the distribution.
getCorrelation
(self)(ditch me?)
getCovariance
(self)Accessor to the covariance matrix.
getDescription
(self)Accessor to the componentwise description.
getDimension
(self)Accessor to the dimension of the distribution.
getDispersionIndicator
(self)Dispersion indicator accessor.
getId
(self)Accessor to the object’s id.
Accessor to the number of Gauss integration points.
getInverseCholesky
(self)Accessor to the inverse Cholesky factor of the covariance matrix.
Accessor to the inverse isoprobabilistic transformation.
Accessor to the isoprobabilistic transformation.
getKendallTau
(self)Accessor to the Kendall coefficients matrix.
getKurtosis
(self)Accessor to the componentwise kurtosis.
getLinearCorrelation
(self)(ditch me?)
getMarginal
(self, \*args)Accessor to marginal distributions.
getMean
(self)Accessor to the mean.
getMoment
(self, n)Accessor to the componentwise moments.
getName
(self)Accessor to the object’s name.
getNu
(self)Accessor to the degrees of freedom parameter.
getPDFEpsilon
(self)Accessor to the PDF computation precision.
getParameter
(self)Accessor to the parameter of the distribution.
getParameterDescription
(self)Accessor to the parameter description of the distribution.
getParameterDimension
(self)Accessor to the number of parameters in the distribution.
getParametersCollection
(self)Accessor to the parameter of the distribution.
getPearsonCorrelation
(self)Accessor to the Pearson correlation matrix.
getPositionIndicator
(self)Position indicator accessor.
getProbabilities
(self)Accessor to the discrete probability levels.
getRange
(self)Accessor to the range of the distribution.
getRealization
(self)Accessor to a pseudorandom realization from the distribution.
getRoughness
(self)Accessor to roughness of the distribution.
getSample
(self, size)Accessor to a pseudorandom sample from the distribution.
getShadowedId
(self)Accessor to the object’s shadowed id.
getShapeMatrix
(self)Accessor to the shape matrix of the underlying copula if it is elliptical.
getShiftedMoment
(self, n, shift)Accessor to the componentwise shifted moments.
getSingularities
(self)Accessor to the singularities of the PDF function.
getSkewness
(self)Accessor to the componentwise skewness.
getSpearmanCorrelation
(self)Accessor to the Spearman correlation matrix.
getStandardDeviation
(self)Accessor to the componentwise standard deviation.
getStandardDistribution
(self)Accessor to the standard distribution.
getStandardMoment
(self, n)Accessor to the componentwise standard moments.
Accessor to the standard representative distribution in the parametric family.
getSupport
(self, \*args)Accessor to the support of the distribution.
getVisibility
(self)Accessor to the object’s visibility state.
hasEllipticalCopula
(self)Test whether the copula of the distribution is elliptical or not.
hasIndependentCopula
(self)Test whether the copula of the distribution is the independent one.
hasName
(self)Test if the object is named.
hasVisibleName
(self)Test if the object has a distinguishable name.
inverse
(self)Transform distribution by inverse function.
isContinuous
(self)Test whether the distribution is continuous or not.
isCopula
(self)Test whether the distribution is a copula or not.
isDiscrete
(self)Test whether the distribution is discrete or not.
isElliptical
(self)Test whether the distribution is elliptical or not.
isIntegral
(self)Test whether the distribution is integervalued or not.
ln
(self)Transform distribution by natural logarithm function.
log
(self)Transform distribution by natural logarithm function.
setDescription
(self, description)Accessor to the componentwise description.
setIntegrationNodesNumber
(self, …)Accessor to the number of Gauss integration points.
setName
(self, name)Accessor to the object’s name.
setNu
(self, nu)Accessor to the degrees of freedom parameter.
setParameter
(self, parameter)Accessor to the parameter of the distribution.
setParametersCollection
(self, \*args)Accessor to the parameter of the distribution.
setShadowedId
(self, id)Accessor to the object’s shadowed id.
setVisibility
(self, visible)Accessor to the object’s visibility state.
sin
(self)Transform distribution by sine function.
sinh
(self)Transform distribution by sinh function.
sqr
(self)Transform distribution by square function.
sqrt
(self)Transform distribution by square root function.
tan
(self)Transform distribution by tangent function.
tanh
(self)Transform distribution by tanh function.

__init__
(self, \*args)¶ Initialize self. See help(type(self)) for accurate signature.

abs
(self)¶ Transform distribution by absolute value function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

acos
(self)¶ Transform distribution by arccosine function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

acosh
(self)¶ Transform distribution by acosh function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

asin
(self)¶ Transform distribution by arcsine function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

asinh
(self)¶ Transform distribution by asinh function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

atan
(self)¶ Transform distribution by arctangent function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

atanh
(self)¶ Transform distribution by atanh function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

cbrt
(self)¶ Transform distribution by cubic root function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

computeBilateralConfidenceInterval
(self, prob)¶ Compute a bilateral confidence interval.
 Parameters
 alphafloat,
The confidence level.
 Returns
 confInterval
Interval
The confidence interval of level .
 confInterval
Notes
We consider an absolutely continuous measure with density function .
The bilateral confidence interval is the cartesian product where and for all and which verifies .
Examples
Create a sample from a Normal distribution:
>>> import openturns as ot >>> sample = ot.Normal().getSample(10) >>> ot.ResourceMap.SetAsUnsignedInteger('DistributionFactoryDefaultBootstrapSize', 100)
Fit a Normal distribution and extract the asymptotic parameters distribution:
>>> fittedRes = ot.NormalFactory().buildEstimator(sample) >>> paramDist = fittedRes.getParameterDistribution()
Determine the bilateral confidence interval at level 0.9:
>>> confInt = paramDist.computeBilateralConfidenceInterval(0.9)

computeBilateralConfidenceIntervalWithMarginalProbability
(self, prob)¶ Compute a bilateral confidence interval.
Refer to
computeBilateralConfidenceInterval()
 Parameters
 alphafloat,
The confidence level.
 Returns
 confInterval
Interval
The confidence interval of level .
 marginalProbfloat
The value which is the common marginal probability of each marginal interval.
 confInterval
Examples
Create a sample from a Normal distribution:
>>> import openturns as ot >>> sample = ot.Normal().getSample(10) >>> ot.ResourceMap.SetAsUnsignedInteger('DistributionFactoryDefaultBootstrapSize', 100)
Fit a Normal distribution and extract the asymptotic parameters distribution:
>>> fittedRes = ot.NormalFactory().buildEstimator(sample) >>> paramDist = fittedRes.getParameterDistribution()
Determine the bilateral confidence interval at level 0.9 with marginal probability:
>>> confInt, marginalProb = paramDist.computeBilateralConfidenceIntervalWithMarginalProbability(0.9)

computeCDF
(self, \*args)¶ Compute the cumulative distribution function.
 Parameters
 Xsequence of float, 2d sequence of float
CDF input(s).
 Returns
 Ffloat,
Point
CDF value(s) at input(s) .
 Ffloat,
Notes
The cumulative distribution function is defined as:

computeCDFGradient
(self, \*args)¶ Compute the gradient of the cumulative distribution function.
 Parameters
 Xsequence of float
CDF input.
 Returns
 dFdtheta
Point
Partial derivatives of the CDF with respect to the distribution parameters at input .
 dFdtheta

computeCharacteristicFunction
(self, x)¶ Compute the characteristic function.
 Parameters
 tfloat
Characteristic function input.
 Returns
 phicomplex
Characteristic function value at input .
Notes
The characteristic function is defined as:
OpenTURNS features a generic implementation of the characteristic function for all its univariate distributions (both continuous and discrete). This default implementation might be time consuming, especially as the modulus of gets high. Only some univariate distributions benefit from dedicated more efficient implementations.

computeComplementaryCDF
(self, \*args)¶ Compute the complementary cumulative distribution function.
 Parameters
 Xsequence of float, 2d sequence of float
Complementary CDF input(s).
 Returns
 Cfloat,
Point
Complementary CDF value(s) at input(s) .
 Cfloat,
See also
Notes
The complementary cumulative distribution function.
Warning
This is not the survival function (except for 1dimensional distributions).

computeConditionalCDF
(self, \*args)¶ Compute the conditional cumulative distribution function.
 Parameters
 Xnfloat, sequence of float
Conditional CDF input (last component).
 Xcondsequence of float, 2d sequence of float with size
Conditionning values for the other components.
 Returns
 Ffloat, sequence of float
Conditional CDF value(s) at input , .
Notes
The conditional cumulative distribution function of the last component with respect to the other fixed components is defined as follows:

computeConditionalDDF
(self, x, y)¶ Compute the conditional derivative density function of the last component.
With respect to the other fixed components.
 Parameters
 Xnfloat
Conditional DDF input (last component).
 Xcondsequence of float with dimension
Conditionning values for the other components.
 Returns
 dfloat
Conditional DDF value at input , .
See also

computeConditionalPDF
(self, \*args)¶ Compute the conditional probability density function.
Conditional PDF of the last component with respect to the other fixed components.
 Parameters
 Xnfloat, sequence of float
Conditional PDF input (last component).
 Xcondsequence of float, 2d sequence of float with size
Conditionning values for the other components.
 Returns
 Ffloat, sequence of float
Conditional PDF value(s) at input , .
See also

computeConditionalQuantile
(self, \*args)¶ Compute the conditional quantile function of the last component.
Conditional quantile with respect to the other fixed components.
 Parameters
 pfloat, sequence of float,
Conditional quantile function input.
 Xcondsequence of float, 2d sequence of float with size
Conditionning values for the other components.
 Returns
 X1float
Conditional quantile at input , .
See also

computeDDF
(self, \*args)¶ Compute the derivative density function.
 Parameters
 Xsequence of float, 2d sequence of float
PDF input(s).
 Returns
Notes
The derivative density function is the gradient of the probability density function with respect to :

computeDensityGenerator
(self, betaSquare)¶ Compute the probability density function of the characteristic generator.
PDF of the characteristic generator of the elliptical distribution.
 Parameters
 beta2float
Density generator input.
 Returns
 pfloat
Density generator value at input .
See also
Notes
This is the function such that the probability density function rewrites:
This function only exists for elliptical distributions.

computeDensityGeneratorDerivative
(self, betaSquare)¶ Compute the firstorder derivative of the probability density function.
PDF of the characteristic generator of the elliptical distribution.
 Parameters
 beta2float
Density generator input.
 Returns
 pfloat
Density generator firstorder derivative value at input .
See also
Notes
This function only exists for elliptical distributions.

computeDensityGeneratorSecondDerivative
(self, betaSquare)¶ Compute the secondorder derivative of the probability density function.
PDF of the characteristic generator of the elliptical distribution.
 Parameters
 beta2float
Density generator input.
 Returns
 pfloat
Density generator secondorder derivative value at input .
See also
Notes
This function only exists for elliptical distributions.

computeEntropy
(self)¶ Compute the entropy of the distribution.
 Returns
 efloat
Entropy of the distribution.
Notes
The entropy of a distribution is defined by:
Where the random vector follows the probability distribution of interest, and is either the probability density function of if it is continuous or the probability distribution function if it is discrete.

computeGeneratingFunction
(self, \*args)¶ Compute the probabilitygenerating function.
 Parameters
 zfloat or complex
Probabilitygenerating function input.
 Returns
 gfloat
Probabilitygenerating function value at input .
See also
Notes
The probabilitygenerating function is defined as follows:
This function only exists for discrete distributions. OpenTURNS implements this method for univariate distributions only.

computeInverseSurvivalFunction
(self, point)¶ Compute the inverse survival function.
 Parameters
 pfloat,
Level of the survival function.
 Returns
 x
Point
Point such that with isoquantile components.
 x
See also
Notes
The inverse survival function writes: where . OpenTURNS returns the point such that .

computeLogCharacteristicFunction
(self, x)¶ Compute the logarithm of the characteristic function.
 Parameters
 tfloat
Characteristic function input.
 Returns
 phicomplex
Logarithm of the characteristic function value at input .
See also
Notes
OpenTURNS features a generic implementation of the characteristic function for all its univariate distributions (both continuous and discrete). This default implementation might be time consuming, especially as the modulus of gets high. Only some univariate distributions benefit from dedicated more efficient implementations.

computeLogGeneratingFunction
(self, \*args)¶ Compute the logarithm of the probabilitygenerating function.
 Parameters
 zfloat or complex
Probabilitygenerating function input.
 Returns
 lgfloat
Logarithm of the probabilitygenerating function value at input .
See also
Notes
This function only exists for discrete distributions. OpenTURNS implements this method for univariate distributions only.

computeLogPDF
(self, \*args)¶ Compute the logarithm of the probability density function.
 Parameters
 Xsequence of float, 2d sequence of float
PDF input(s).
 Returns
 ffloat,
Point
Logarithm of the PDF value(s) at input(s) .
 ffloat,

computeLogPDFGradient
(self, \*args)¶ Compute the gradient of the log probability density function.
 Parameters
 Xsequence of float
PDF input.
 Returns
 dfdtheta
Point
Partial derivatives of the logPDF with respect to the distribution parameters at input .
 dfdtheta

computeMinimumVolumeInterval
(self, prob)¶ Compute the confidence interval with minimum volume.
 Parameters
 alphafloat,
The confidence level.
 Returns
 confInterval
Interval
The confidence interval of level .
 confInterval
Notes
We consider an absolutely continuous measure with density function .
The minimum volume confidence interval is the cartesian product where and with is the Lebesgue measure on .
This problem resorts to solving univariate non linear equations: for a fixed value , we find each intervals such that:
which consists of finding the bound such that:
To find , we use the Brent algorithm: with a non linear function.
Examples
Create a sample from a Normal distribution:
>>> import openturns as ot >>> sample = ot.Normal().getSample(10) >>> ot.ResourceMap.SetAsUnsignedInteger('DistributionFactoryDefaultBootstrapSize', 100)
Fit a Normal distribution and extract the asymptotic parameters distribution:
>>> fittedRes = ot.NormalFactory().buildEstimator(sample) >>> paramDist = fittedRes.getParameterDistribution()
Determine the confidence interval of the native parameters at level 0.9 with minimum volume:
>>> ot.ResourceMap.SetAsUnsignedInteger('DistributionMinimumVolumeLevelSetSamplingSize', 1000) >>> confInt = paramDist.computeMinimumVolumeInterval(0.9)

computeMinimumVolumeIntervalWithMarginalProbability
(self, prob)¶ Compute the confidence interval with minimum volume.
Refer to
computeMinimumVolumeInterval()
 Parameters
 alphafloat,
The confidence level.
 Returns
 confInterval
Interval
The confidence interval of level .
 marginalProbfloat
The value which is the common marginal probability of each marginal interval.
 confInterval
Examples
Create a sample from a Normal distribution:
>>> import openturns as ot >>> sample = ot.Normal().getSample(10) >>> ot.ResourceMap.SetAsUnsignedInteger('DistributionFactoryDefaultBootstrapSize', 100)
Fit a Normal distribution and extract the asymptotic parameters distribution:
>>> fittedRes = ot.NormalFactory().buildEstimator(sample) >>> paramDist = fittedRes.getParameterDistribution()
Determine the confidence interval of the native parameters at level 0.9 with minimum volume:
>>> ot.ResourceMap.SetAsUnsignedInteger('DistributionMinimumVolumeLevelSetSamplingSize', 1000) >>> confInt, marginalProb = paramDist.computeMinimumVolumeIntervalWithMarginalProbability(0.9)

computeMinimumVolumeLevelSet
(self, prob)¶ Compute the confidence domain with minimum volume.
 Parameters
 alphafloat,
The confidence level.
 Returns
 levelSet
LevelSet
The minimum volume domain of measure .
 levelSet
Notes
We consider an absolutely continuous measure with density function .
The minimum volume confidence domain is the set of minimum volume and which measure is at least . It is defined by:
where is the Lebesgue measure on . Under some general conditions on (for example, no flat regions), the set is unique and realises the minimum: . We show that writes:
for a certain .
If we consider the random variable , with cumulative distribution function , then is defined by:
Thus the minimum volume domain of confidence is the interior of the domain which frontier is the quantile of . It can be determined with simulations of .
Examples
Create a sample from a Normal distribution:
>>> import openturns as ot >>> sample = ot.Normal().getSample(10) >>> ot.ResourceMap.SetAsUnsignedInteger('DistributionFactoryDefaultBootstrapSize', 100)
Fit a Normal distribution and extract the asymptotic parameters distribution:
>>> fittedRes = ot.NormalFactory().buildEstimator(sample) >>> paramDist = fittedRes.getParameterDistribution()
Determine the confidence region of minimum volume of the native parameters at level 0.9:
>>> levelSet = paramDist.computeMinimumVolumeLevelSet(0.9)

computeMinimumVolumeLevelSetWithThreshold
(self, prob)¶ Compute the confidence domain with minimum volume.
Refer to
computeMinimumVolumeLevelSet()
 Parameters
 alphafloat,
The confidence level.
 Returns
 levelSet
LevelSet
The minimum volume domain of measure .
 levelfloat
The value of the density function defining the frontier of the domain.
 levelSet
Examples
Create a sample from a Normal distribution:
>>> import openturns as ot >>> sample = ot.Normal().getSample(10) >>> ot.ResourceMap.SetAsUnsignedInteger('DistributionFactoryDefaultBootstrapSize', 100)
Fit a Normal distribution and extract the asymptotic parameters distribution:
>>> fittedRes = ot.NormalFactory().buildEstimator(sample) >>> paramDist = fittedRes.getParameterDistribution()
Determine the confidence region of minimum volume of the native parameters at level 0.9 with PDF threshold:
>>> levelSet, threshold = paramDist.computeMinimumVolumeLevelSetWithThreshold(0.9)

computePDF
(self, \*args)¶ Compute the probability density function.
 Parameters
 Xsequence of float, 2d sequence of float
PDF input(s).
 Returns
 ffloat,
Point
PDF value(s) at input(s) .
 ffloat,
Notes
The probability density function is defined as follows:

computePDFGradient
(self, \*args)¶ Compute the gradient of the probability density function.
 Parameters
 Xsequence of float
PDF input.
 Returns
 dfdtheta
Point
Partial derivatives of the PDF with respect to the distribution parameters at input .
 dfdtheta

computeProbability
(self, interval)¶ Compute the interval probability.
 Parameters
 interval
Interval
An interval, possibly multivariate.
 interval
 Returns
 Pfloat
Interval probability.
Notes
This computes the probability that the random vector lies in the hyperrectangular region formed by the vectors and :
where the sum runs over the vectors such that with , and is the number of components in such that .

computeQuantile
(self, \*args)¶ Compute the quantile function.
 Parameters
 pfloat (or sequence of float),
Quantile function input (a probability).
 tailbool, optional (default=False)
Whether p should be interpreted as the complementary probability.
 Returns
Notes
The quantile function is also known as the inverse cumulative distribution function:

computeRadialDistributionCDF
(self, radius, tail=False)¶ Compute the cumulative distribution function of the squared radius.
For the underlying standard spherical distribution (for elliptical distributions only).
 Parameters
 r2float,
Squared radius.
 Returns
 Ffloat
CDF value at input .
Notes
This is the CDF of the sum of the squared independent, standard, identically distributed components:

computeScalarQuantile
(self, prob, tail=False)¶ Compute the quantile function for univariate distributions.
 Parameters
 pfloat,
Quantile function input (a probability).
 Returns
 Xfloat
Quantile at probability level .
See also
Notes
The quantile function is also known as the inverse cumulative distribution function:

computeSequentialConditionalCDF
(self, x)¶ Compute the sequential conditional cumulative distribution functions.
 Parameters
 Xsequence of float, with size
Values to be taken sequentially as argument and conditioning part of the CDF.
 Returns
 Fsequence of float
Conditional CDF values at input.
Notes
The sequential conditional cumulative distribution function is defined as follows:
ie its th component is the conditional CDF of at given that . For it reduces to , ie the CDF of the first component at .

computeSequentialConditionalDDF
(self, x)¶ Compute the sequential conditional derivative density function.
 Parameters
 Xsequence of float, with size
Values to be taken sequentially as argument and conditioning part of the DDF.
 Returns
 ddfsequence of float
Conditional DDF values at input.
Notes
The sequential conditional derivative density function is defined as follows:
ie its th component is the conditional DDF of at given that . For it reduces to , ie the DDF of the first component at .

computeSequentialConditionalPDF
(self, x)¶ Compute the sequential conditional probability density function.
 Parameters
 Xsequence of float, with size
Values to be taken sequentially as argument and conditioning part of the PDF.
 Returns
 pdfsequence of float
Conditional PDF values at input.
Notes
The sequential conditional density function is defined as follows:
ie its th component is the conditional PDF of at given that . For it reduces to , ie the PDF of the first component at .

computeSequentialConditionalQuantile
(self, q)¶ Compute the conditional quantile function of the last component.
 Parameters
 qsequence of float in , with size
Values to be taken sequentially as the argument of the conditional quantile.
 Returns
 Qsequence of float
Conditional quantiles values at input.
Notes
The sequential conditional quantile function is defined as follows:
where are defined recursively as and given , : the conditioning part is the set of already computed conditional quantiles.

computeSurvivalFunction
(self, \*args)¶ Compute the survival function.
 Parameters
 xsequence of float, 2d sequence of float
Survival function input(s).
 Returns
 Sfloat,
Point
Survival function value(s) at input(s) x.
 Sfloat,
See also
Notes
The survival function of the random vector is defined as follows:
Warning
This is not the complementary cumulative distribution function (except for 1dimensional distributions).

computeUnilateralConfidenceInterval
(self, prob, tail=False)¶ Compute a unilateral confidence interval.
 Parameters
 alphafloat,
The confidence level.
 tailboolean
True indicates the interval is bounded by an lower value. False indicates the interval is bounded by an upper value. Default value is False.
 Returns
 confInterval
Interval
The unilateral confidence interval of level .
 confInterval
Notes
We consider an absolutely continuous measure .
The left unilateral confidence interval is the cartesian product where for all and which verifies . It means that is the quantile of level of the measure , with isoquantile components.
The right unilateral confidence interval is the cartesian product where for all and which verifies . It means that with isoquantile components, where is the survival function of the measure .
Examples
Create a sample from a Normal distribution:
>>> import openturns as ot >>> sample = ot.Normal().getSample(10) >>> ot.ResourceMap.SetAsUnsignedInteger('DistributionFactoryDefaultBootstrapSize', 100)
Fit a Normal distribution and extract the asymptotic parameters distribution:
>>> fittedRes = ot.NormalFactory().buildEstimator(sample) >>> paramDist = fittedRes.getParameterDistribution()
Determine the right unilateral confidence interval at level 0.9:
>>> confInt = paramDist.computeUnilateralConfidenceInterval(0.9)
Determine the left unilateral confidence interval at level 0.9:
>>> confInt = paramDist.computeUnilateralConfidenceInterval(0.9, True)

computeUnilateralConfidenceIntervalWithMarginalProbability
(self, prob, tail)¶ Compute a unilateral confidence interval.
Refer to
computeUnilateralConfidenceInterval()
 Parameters
 alphafloat,
The confidence level.
 tailboolean
True indicates the interval is bounded by an lower value. False indicates the interval is bounded by an upper value. Default value is False.
 Returns
 confInterval
Interval
The unilateral confidence interval of level .
 marginalProbfloat
The value which is the common marginal probability of each marginal interval.
 confInterval
Examples
Create a sample from a Normal distribution:
>>> import openturns as ot >>> sample = ot.Normal().getSample(10) >>> ot.ResourceMap.SetAsUnsignedInteger('DistributionFactoryDefaultBootstrapSize', 100)
Fit a Normal distribution and extract the asymptotic parameters distribution:
>>> fittedRes = ot.NormalFactory().buildEstimator(sample) >>> paramDist = fittedRes.getParameterDistribution()
Determine the right unilateral confidence interval at level 0.9:
>>> confInt, marginalProb = paramDist.computeUnilateralConfidenceIntervalWithMarginalProbability(0.9, False)
Determine the left unilateral confidence interval at level 0.9:
>>> confInt, marginalProb = paramDist.computeUnilateralConfidenceIntervalWithMarginalProbability(0.9, True)

cos
(self)¶ Transform distribution by cosine function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

cosh
(self)¶ Transform distribution by cosh function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

drawCDF
(self, \*args)¶ Draw the cumulative distribution function.
 Available constructors:
drawCDF(x_min, x_max, pointNumber, logScale)
drawCDF(lowerCorner, upperCorner, pointNbrInd, logScaleX, logScaleY)
drawCDF(lowerCorner, upperCorner)
 Parameters
 x_minfloat, optional
The minvalue of the mesh of the xaxis. Defaults uses the quantile associated to the probability level DistributionQMin from the
ResourceMap
. x_maxfloat, optional,
The maxvalue of the mesh of the yaxis. Defaults uses the quantile associated to the probability level DistributionQMax from the
ResourceMap
. pointNumberint
The number of points that is used for meshing each axis. Defaults uses DistributionImplementationDefaultPointNumber from the
ResourceMap
. logScalebool
Flag to tell if the plot is done on a logarithmic scale. Default is False.
 lowerCornersequence of float, of dimension 2, optional
The lower corner .
 upperCornersequence of float, of dimension 2, optional
The upper corner .
 pointNbrInd
Indices
, of dimension 2 Number of points that is used for meshing each axis.
 logScaleXbool
Flag to tell if the plot is done on a logarithmic scale for X. Default is False.
 logScaleYbool
Flag to tell if the plot is done on a logarithmic scale for Y. Default is False.
 Returns
 graph
Graph
A graphical representation of the CDF.
 graph
See also
Notes
Only valid for univariate and bivariate distributions.
Examples
View the CDF of a univariate distribution:
>>> import openturns as ot >>> dist = ot.Normal() >>> graph = dist.drawCDF() >>> graph.setLegends(['normal cdf'])
View the isolines CDF of a bivariate distribution:
>>> import openturns as ot >>> dist = ot.Normal(2) >>> graph2 = dist.drawCDF() >>> graph2.setLegends(['iso normal cdf']) >>> graph3 = dist.drawCDF([10, 5],[5, 10], [511, 511])

drawLogPDF
(self, \*args)¶ Draw the graph or of isolines of logprobability density function.
 Available constructors:
drawLogPDF(x_min, x_max, pointNumber, logScale)
drawLogPDF(lowerCorner, upperCorner, pointNbrInd, logScaleX, logScaleY)
drawLogPDF(lowerCorner, upperCorner)
 Parameters
 x_minfloat, optional
The minvalue of the mesh of the xaxis. Defaults uses the quantile associated to the probability level DistributionQMin from the
ResourceMap
. x_maxfloat, optional,
The maxvalue of the mesh of the yaxis. Defaults uses the quantile associated to the probability level DistributionQMax from the
ResourceMap
. pointNumberint
The number of points that is used for meshing each axis. Defaults uses DistributionImplementationDefaultPointNumber from the
ResourceMap
. logScalebool
Flag to tell if the plot is done on a logarithmic scale. Default is False.
 lowerCornersequence of float, of dimension 2, optional
The lower corner .
 upperCornersequence of float, of dimension 2, optional
The upper corner .
 pointNbrInd
Indices
, of dimension 2 Number of points that is used for meshing each axis.
 logScaleXbool
Flag to tell if the plot is done on a logarithmic scale for X. Default is False.
 logScaleYbool
Flag to tell if the plot is done on a logarithmic scale for Y. Default is False.
 Returns
 graph
Graph
A graphical representation of the logPDF or its iso_lines.
 graph
See also
Notes
Only valid for univariate and bivariate distributions.
Examples
View the logPDF of a univariate distribution:
>>> import openturns as ot >>> dist = ot.Normal() >>> graph = dist.drawLogPDF() >>> graph.setLegends(['normal logpdf'])
View the isolines logPDF of a bivariate distribution:
>>> import openturns as ot >>> dist = ot.Normal(2) >>> graph2 = dist.drawLogPDF() >>> graph2.setLegends(['iso normal pdf']) >>> graph3 = dist.drawLogPDF([10, 5],[5, 10], [511, 511])

drawMarginal1DCDF
(self, marginalIndex, xMin, xMax, pointNumber, logScale=False)¶ Draw the cumulative distribution function of a margin.
 Parameters
 iint,
The index of the margin of interest.
 x_minfloat
The starting value that is used for meshing the xaxis.
 x_maxfloat,
The ending value that is used for meshing the xaxis.
 n_pointsint
The number of points that is used for meshing the xaxis.
 logScalebool
Flag to tell if the plot is done on a logarithmic scale. Default is False.
 Returns
 graph
Graph
A graphical representation of the CDF of the requested margin.
 graph
See also
Examples
>>> import openturns as ot >>> from openturns.viewer import View >>> distribution = ot.Normal(10) >>> graph = distribution.drawMarginal1DCDF(2, 6.0, 6.0, 100) >>> view = View(graph) >>> view.show()

drawMarginal1DLogPDF
(self, marginalIndex, xMin, xMax, pointNumber, logScale=False)¶ Draw the logprobability density function of a margin.
 Parameters
 iint,
The index of the margin of interest.
 x_minfloat
The starting value that is used for meshing the xaxis.
 x_maxfloat,
The ending value that is used for meshing the xaxis.
 n_pointsint
The number of points that is used for meshing the xaxis.
 logScalebool
Flag to tell if the plot is done on a logarithmic scale. Default is False.
 Returns
 graph
Graph
A graphical representation of the logPDF of the requested margin.
 graph
See also
Examples
>>> import openturns as ot >>> from openturns.viewer import View >>> distribution = ot.Normal(10) >>> graph = distribution.drawMarginal1DLogPDF(2, 6.0, 6.0, 100) >>> view = View(graph) >>> view.show()

drawMarginal1DPDF
(self, marginalIndex, xMin, xMax, pointNumber, logScale=False)¶ Draw the probability density function of a margin.
 Parameters
 iint,
The index of the margin of interest.
 x_minfloat
The starting value that is used for meshing the xaxis.
 x_maxfloat,
The ending value that is used for meshing the xaxis.
 n_pointsint
The number of points that is used for meshing the xaxis.
 logScalebool
Flag to tell if the plot is done on a logarithmic scale. Default is False.
 Returns
 graph
Graph
A graphical representation of the PDF of the requested margin.
 graph
See also
Examples
>>> import openturns as ot >>> from openturns.viewer import View >>> distribution = ot.Normal(10) >>> graph = distribution.drawMarginal1DPDF(2, 6.0, 6.0, 100) >>> view = View(graph) >>> view.show()

drawMarginal1DSurvivalFunction
(self, marginalIndex, xMin, xMax, pointNumber, logScale=False)¶ Draw the cumulative distribution function of a margin.
 Parameters
 iint,
The index of the margin of interest.
 x_minfloat
The starting value that is used for meshing the xaxis.
 x_maxfloat,
The ending value that is used for meshing the xaxis.
 n_pointsint
The number of points that is used for meshing the xaxis.
 logScalebool
Flag to tell if the plot is done on a logarithmic scale. Default is False.
 Returns
 graph
Graph
A graphical representation of the SurvivalFunction of the requested margin.
 graph
See also
Examples
>>> import openturns as ot >>> from openturns.viewer import View >>> distribution = ot.Normal(10) >>> graph = distribution.drawMarginal1DSurvivalFunction(2, 6.0, 6.0, 100) >>> view = View(graph) >>> view.show()

drawMarginal2DCDF
(self, firstMarginal, secondMarginal, xMin, xMax, pointNumber, logScaleX=False, logScaleY=False)¶ Draw the cumulative distribution function of a couple of margins.
 Parameters
 iint,
The index of the first margin of interest.
 jint,
The index of the second margin of interest.
 x_minlist of 2 floats
The starting values that are used for meshing the x and y axes.
 x_maxlist of 2 floats,
The ending values that are used for meshing the x and y axes.
 n_pointslist of 2 ints
The number of points that are used for meshing the x and y axes.
 logScaleXbool
Flag to tell if the plot is done on a logarithmic scale for X. Default is False.
 logScaleYbool
Flag to tell if the plot is done on a logarithmic scale for Y. Default is False.
 Returns
 graph
Graph
A graphical representation of the marginal CDF of the requested couple of margins.
 graph
See also
Examples
>>> import openturns as ot >>> from openturns.viewer import View >>> distribution = ot.Normal(10) >>> graph = distribution.drawMarginal2DCDF(2, 3, [6.0] * 2, [6.0] * 2, [100] * 2) >>> view = View(graph) >>> view.show()

drawMarginal2DLogPDF
(self, firstMarginal, secondMarginal, xMin, xMax, pointNumber, logScaleX=False, logScaleY=False)¶ Draw the logprobability density function of a couple of margins.
 Parameters
 iint,
The index of the first margin of interest.
 jint,
The index of the second margin of interest.
 x_minlist of 2 floats
The starting values that are used for meshing the x and y axes.
 x_maxlist of 2 floats,
The ending values that are used for meshing the x and y axes.
 n_pointslist of 2 ints
The number of points that are used for meshing the x and y axes.
 logScaleXbool
Flag to tell if the plot is done on a logarithmic scale for X. Default is False.
 logScaleYbool
Flag to tell if the plot is done on a logarithmic scale for Y. Default is False.
 Returns
 graph
Graph
A graphical representation of the marginal logPDF of the requested couple of margins.
 graph
See also
Examples
>>> import openturns as ot >>> from openturns.viewer import View >>> distribution = ot.Normal(10) >>> graph = distribution.drawMarginal2DLogPDF(2, 3, [6.0] * 2, [6.0] * 2, [100] * 2) >>> view = View(graph) >>> view.show()

drawMarginal2DPDF
(self, firstMarginal, secondMarginal, xMin, xMax, pointNumber, logScaleX=False, logScaleY=False)¶ Draw the probability density function of a couple of margins.
 Parameters
 iint,
The index of the first margin of interest.
 jint,
The index of the second margin of interest.
 x_minlist of 2 floats
The starting values that are used for meshing the x and y axes.
 x_maxlist of 2 floats,
The ending values that are used for meshing the x and y axes.
 n_pointslist of 2 ints
The number of points that are used for meshing the x and y axes.
 logScaleXbool
Flag to tell if the plot is done on a logarithmic scale for X. Default is False.
 logScaleYbool
Flag to tell if the plot is done on a logarithmic scale for Y. Default is False.
 Returns
 graph
Graph
A graphical representation of the marginal PDF of the requested couple of margins.
 graph
See also
Examples
>>> import openturns as ot >>> from openturns.viewer import View >>> distribution = ot.Normal(10) >>> graph = distribution.drawMarginal2DPDF(2, 3, [6.0] * 2, [6.0] * 2, [100] * 2) >>> view = View(graph) >>> view.show()

drawMarginal2DSurvivalFunction
(self, firstMarginal, secondMarginal, xMin, xMax, pointNumber, logScaleX=False, logScaleY=False)¶ Draw the cumulative distribution function of a couple of margins.
 Parameters
 iint,
The index of the first margin of interest.
 jint,
The index of the second margin of interest.
 x_minlist of 2 floats
The starting values that are used for meshing the x and y axes.
 x_maxlist of 2 floats,
The ending values that are used for meshing the x and y axes.
 n_pointslist of 2 ints
The number of points that are used for meshing the x and y axes.
 logScaleXbool
Flag to tell if the plot is done on a logarithmic scale for X. Default is False.
 logScaleYbool
Flag to tell if the plot is done on a logarithmic scale for Y. Default is False.
 Returns
 graph
Graph
A graphical representation of the marginal SurvivalFunction of the requested couple of margins.
 graph
See also
Examples
>>> import openturns as ot >>> from openturns.viewer import View >>> distribution = ot.Normal(10) >>> graph = distribution.drawMarginal2DSurvivalFunction(2, 3, [6.0] * 2, [6.0] * 2, [100] * 2) >>> view = View(graph) >>> view.show()

drawPDF
(self, \*args)¶ Draw the graph or of isolines of probability density function.
 Available constructors:
drawPDF(x_min, x_max, pointNumber, logScale)
drawPDF(lowerCorner, upperCorner, pointNbrInd, logScaleX, logScaleY)
drawPDF(lowerCorner, upperCorner)
 Parameters
 x_minfloat, optional
The minvalue of the mesh of the xaxis. Defaults uses the quantile associated to the probability level DistributionQMin from the
ResourceMap
. x_maxfloat, optional,
The maxvalue of the mesh of the yaxis. Defaults uses the quantile associated to the probability level DistributionQMax from the
ResourceMap
. pointNumberint
The number of points that is used for meshing each axis. Defaults uses DistributionImplementationDefaultPointNumber from the
ResourceMap
. logScalebool
Flag to tell if the plot is done on a logarithmic scale. Default is False.
 lowerCornersequence of float, of dimension 2, optional
The lower corner .
 upperCornersequence of float, of dimension 2, optional
The upper corner .
 pointNbrInd
Indices
, of dimension 2 Number of points that is used for meshing each axis.
 logScaleXbool
Flag to tell if the plot is done on a logarithmic scale for X. Default is False.
 logScaleYbool
Flag to tell if the plot is done on a logarithmic scale for Y. Default is False.
 Returns
 graph
Graph
A graphical representation of the PDF or its iso_lines.
 graph
See also
Notes
Only valid for univariate and bivariate distributions.
Examples
View the PDF of a univariate distribution:
>>> import openturns as ot >>> dist = ot.Normal() >>> graph = dist.drawPDF() >>> graph.setLegends(['normal pdf'])
View the isolines PDF of a bivariate distribution:
>>> import openturns as ot >>> dist = ot.Normal(2) >>> graph2 = dist.drawPDF() >>> graph2.setLegends(['iso normal pdf']) >>> graph3 = dist.drawPDF([10, 5],[5, 10], [511, 511])

drawQuantile
(self, \*args)¶ Draw the quantile function.
 Parameters
 q_minfloat, in
The min value of the mesh of the xaxis.
 q_maxfloat, in
The max value of the mesh of the xaxis.
 n_pointsint, optional
The number of points that is used for meshing the quantile curve. Defaults uses DistributionImplementationDefaultPointNumber from the
ResourceMap
. logScalebool
Flag to tell if the plot is done on a logarithmic scale. Default is False.
 Returns
 graph
Graph
A graphical representation of the quantile function.
 graph
See also
Notes
This is implemented for univariate and bivariate distributions only. In the case of bivariate distributions, defined by its CDF and its marginals , the quantile of order is the point defined by
Examples
>>> import openturns as ot >>> from openturns.viewer import View >>> distribution = ot.Normal() >>> graph = distribution.drawQuantile() >>> view = View(graph) >>> view.show() >>> distribution = ot.ComposedDistribution([ot.Normal(), ot.Exponential(1.0)], ot.ClaytonCopula(0.5)) >>> graph = distribution.drawQuantile() >>> view = View(graph) >>> view.show()

drawSurvivalFunction
(self, \*args)¶ Draw the cumulative distribution function.
 Available constructors:
drawSurvivalFunction(x_min, x_max, pointNumber, logScale)
drawSurvivalFunction(lowerCorner, upperCorner, pointNbrInd, logScaleX, logScaleY)
drawSurvivalFunction(lowerCorner, upperCorner)
 Parameters
 x_minfloat, optional
The minvalue of the mesh of the xaxis. Defaults uses the quantile associated to the probability level DistributionQMin from the
ResourceMap
. x_maxfloat, optional,
The maxvalue of the mesh of the yaxis. Defaults uses the quantile associated to the probability level DistributionQMax from the
ResourceMap
. pointNumberint
The number of points that is used for meshing each axis. Defaults uses DistributionImplementationDefaultPointNumber from the
ResourceMap
. logScalebool
Flag to tell if the plot is done on a logarithmic scale. Default is False.
 lowerCornersequence of float, of dimension 2, optional
The lower corner .
 upperCornersequence of float, of dimension 2, optional
The upper corner .
 pointNbrInd
Indices
, of dimension 2 Number of points that is used for meshing each axis.
 logScaleXbool
Flag to tell if the plot is done on a logarithmic scale for X. Default is False.
 logScaleYbool
Flag to tell if the plot is done on a logarithmic scale for Y. Default is False.
 Returns
 graph
Graph
A graphical representation of the SurvivalFunction.
 graph
See also
Notes
Only valid for univariate and bivariate distributions.
Examples
View the SurvivalFunction of a univariate distribution:
>>> import openturns as ot >>> dist = ot.Normal() >>> graph = dist.drawSurvivalFunction() >>> graph.setLegends(['normal cdf'])
View the isolines SurvivalFunction of a bivariate distribution:
>>> import openturns as ot >>> dist = ot.Normal(2) >>> graph2 = dist.drawSurvivalFunction() >>> graph2.setLegends(['iso normal cdf']) >>> graph3 = dist.drawSurvivalFunction([10, 5],[5, 10], [511, 511])

exp
(self)¶ Transform distribution by exponential function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

getCDFEpsilon
(self)¶ Accessor to the CDF computation precision.
 Returns
 CDFEpsilonfloat
CDF computation precision.

getCenteredMoment
(self, n)¶ Accessor to the componentwise centered moments.
 Parameters
 kint
The order of the centered moment.
 Returns
 m
Point
Componentwise centered moment of order .
 m
See also
Notes
Centered moments are centered with respect to the firstorder moment:

getCholesky
(self)¶ Accessor to the Cholesky factor of the covariance matrix.
 Returns
 L
SquareMatrix
Cholesky factor of the covariance matrix.
 L
See also

getClassName
(self)¶ Accessor to the object’s name.
 Returns
 class_namestr
The object class name (object.__class__.__name__).

getCopula
(self)¶ Accessor to the copula of the distribution.
 Returns
 C
Distribution
Copula of the distribution.
 C
See also

getCorrelation
(self)¶ (ditch me?)

getCovariance
(self)¶ Accessor to the covariance matrix.
 Returns
 Sigma
CovarianceMatrix
Covariance matrix.
 Sigma
Notes
The covariance is the secondorder centered moment. It is defined as:

getDescription
(self)¶ Accessor to the componentwise description.
 Returns
 description
Description
Description of the components of the distribution.
 description
See also

getDimension
(self)¶ Accessor to the dimension of the distribution.
 Returns
 nint
The number of components in the distribution.

getDispersionIndicator
(self)¶ Dispersion indicator accessor.
Defines a generic metric of the dispersion. When the standard deviation is not defined it falls back to the interquartile. Only available for 1d distributions.
 Returns
 dispersionfloat
Standard deviation or interquartile.

getId
(self)¶ Accessor to the object’s id.
 Returns
 idint
Internal unique identifier.

getIntegrationNodesNumber
(self)¶ Accessor to the number of Gauss integration points.
 Returns
 Nint
Number of integration points.

getInverseCholesky
(self)¶ Accessor to the inverse Cholesky factor of the covariance matrix.
 Returns
 Linv
SquareMatrix
Inverse Cholesky factor of the covariance matrix.
 Linv
See also

getInverseIsoProbabilisticTransformation
(self)¶ Accessor to the inverse isoprobabilistic transformation.
 Returns
 Tinv
Function
Inverse isoprobabilistic transformation.
 Tinv
See also
Notes
The inverse isoprobabilistic transformation is defined as follows:

getIsoProbabilisticTransformation
(self)¶ Accessor to the isoprobabilistic transformation.
Refer to Isoprobabilistic transformations.
 Returns
 T
Function
Isoprobabilistic transformation.
 T
Notes
The isoprobabilistic transformation is defined as follows:
An isoprobabilistic transformation is a diffeomorphism 1 from to that maps realizations of a random vector into realizations of another random vector while preserving probabilities. It is hence defined so that it satisfies:
The present implementation of the isoprobabilistic transformation maps realizations into realizations of a random vector with spherical distribution 2. To be more specific:
if the distribution is elliptical, then the transformed distribution is simply made spherical using the Nataf (linear) transformation.
if the distribution has an elliptical Copula, then the transformed distribution is made spherical using the generalized Nataf transformation.
otherwise, the transformed distribution is the standard multivariate Normal distribution and is obtained by means of the Rosenblatt transformation.
 1
A differentiable map is called a diffeomorphism if it is a bijection and its inverse is differentiable as well. Hence, the isoprobabilistic transformation implements a gradient (and even a Hessian).
 2
A distribution is said to be spherical if it is invariant by rotation. Mathematically, has a spherical distribution if:

getKendallTau
(self)¶ Accessor to the Kendall coefficients matrix.
 Returns
 tau:
SquareMatrix
Kendall coefficients matrix.
 tau:
See also
Notes
The Kendall coefficients matrix is defined as:

getKurtosis
(self)¶ Accessor to the componentwise kurtosis.
 Returns
 k
Point
Componentwise kurtosis.
 k
Notes
The kurtosis is the fourthorder centered moment standardized by the standard deviation:

getLinearCorrelation
(self)¶ (ditch me?)

getMarginal
(self, \*args)¶ Accessor to marginal distributions.
 Parameters
 iint or list of ints,
Component(s) indice(s).
 Returns
 distribution
Distribution
The marginal distribution of the selected component(s).
 distribution

getMoment
(self, n)¶ Accessor to the componentwise moments.
 Parameters
 kint
The order of the moment.
 Returns
 m
Point
Componentwise moment of order .
 m
Notes
The componentwise moment of order is defined as:

getName
(self)¶ Accessor to the object’s name.
 Returns
 namestr
The name of the object.

getNu
(self)¶ Accessor to the degrees of freedom parameter.
 Returns
 nufloat
Degrees of freedom.

getPDFEpsilon
(self)¶ Accessor to the PDF computation precision.
 Returns
 PDFEpsilonfloat
PDF computation precision.

getParameter
(self)¶ Accessor to the parameter of the distribution.
 Returns
 parameter
Point
Parameter values.
 parameter

getParameterDescription
(self)¶ Accessor to the parameter description of the distribution.
 Returns
 description
Description
Parameter names.
 description

getParameterDimension
(self)¶ Accessor to the number of parameters in the distribution.
 Returns
 n_parametersint
Number of parameters in the distribution.
See also

getParametersCollection
(self)¶ Accessor to the parameter of the distribution.
 Returns
 parameters
PointWithDescription
Dictionarylike object with parameters names and values.
 parameters

getPearsonCorrelation
(self)¶ Accessor to the Pearson correlation matrix.
 Returns
 R
CorrelationMatrix
Pearson’s correlation matrix.
 R
See also
Notes
Pearson’s correlation is defined as the normalized covariance matrix:

getPositionIndicator
(self)¶ Position indicator accessor.
Defines a generic metric of the position. When the mean is not defined it falls back to the median. Available only for 1d distributions.
 Returns
 positionfloat
Mean or median of the distribution.

getProbabilities
(self)¶ Accessor to the discrete probability levels.
 Returns
 probabilities
Point
The probability levels of a discrete distribution.
 probabilities

getRange
(self)¶ Accessor to the range of the distribution.
 Returns
 range
Interval
Range of the distribution.
 range
See also
Notes
The mathematical range is the smallest closed interval outside of which the PDF is zero. The numerical range is the interval outside of which the PDF is rounded to zero in double precision.

getRealization
(self)¶ Accessor to a pseudorandom realization from the distribution.
Refer to Distribution realizations.
 Returns
 point
Point
A pseudorandom realization of the distribution.
 point
See also

getRoughness
(self)¶ Accessor to roughness of the distribution.
 Returns
 rfloat
Roughness of the distribution.
See also
Notes
The roughness of the distribution is defined as the norm of its PDF:

getSample
(self, size)¶ Accessor to a pseudorandom sample from the distribution.
 Parameters
 sizeint
Sample size.
 Returns
 sample
Sample
A pseudorandom sample of the distribution.
 sample
See also

getShadowedId
(self)¶ Accessor to the object’s shadowed id.
 Returns
 idint
Internal unique identifier.

getShapeMatrix
(self)¶ Accessor to the shape matrix of the underlying copula if it is elliptical.
 Returns
 shape
CorrelationMatrix
Shape matrix of the elliptical copula of a distribution.
 shape
See also
Notes
This is not the Pearson correlation matrix.

getShiftedMoment
(self, n, shift)¶ Accessor to the componentwise shifted moments.
 Parameters
 kint
The order of the shifted moment.
 shiftsequence of float
The shift of the moment.
 Returns
 m
Point
Componentwise centered moment of order .
 m
See also
Notes
The moments are centered with respect to the given shift :math:vect{s}:

getSingularities
(self)¶ Accessor to the singularities of the PDF function.
It is defined for univariate distributions only, and gives all the singularities (ie discontinuities of any order) strictly inside of the range of the distribution.
 Returns
 singularities
Point
The singularities of the PDF of an univariate distribution.
 singularities

getSkewness
(self)¶ Accessor to the componentwise skewness.
 Returns
 d
Point
Componentwise skewness.
 d
Notes
The skewness is the thirdorder centered moment standardized by the standard deviation:

getSpearmanCorrelation
(self)¶ Accessor to the Spearman correlation matrix.
 Returns
 R
CorrelationMatrix
Spearman’s correlation matrix.
 R
See also
Notes
Spearman’s (rank) correlation is defined as the normalized covariance matrix of the copula (ie that of the uniform margins):

getStandardDeviation
(self)¶ Accessor to the componentwise standard deviation.
The standard deviation is the square root of the variance.
 Returns
 sigma
Point
Componentwise standard deviation.
 sigma
See also

getStandardDistribution
(self)¶ Accessor to the standard distribution.
 Returns
 standard_distribution
Distribution
Standard distribution.
 standard_distribution
See also
Notes
The standard distribution is determined according to the distribution properties. This is the target distribution achieved by the isoprobabilistic transformation.

getStandardMoment
(self, n)¶ Accessor to the componentwise standard moments.
 Parameters
 kint
The order of the standard moment.
 Returns
 m
Point
Componentwise standard moment of order .
 m
See also
Notes
Standard moments are the raw moments of the standard representative of the parametric family of distributions.

getStandardRepresentative
(self)¶ Accessor to the standard representative distribution in the parametric family.
 Returns
 std_repr_dist
Distribution
Standard representative distribution.
 std_repr_dist
Notes
The standard representative distribution is defined on a distribution by distribution basis, most of the time by scaling the distribution with bounded support to or by standardizing (ie zero mean, unit variance) the distributions with unbounded support. It is the member of the family for which orthonormal polynomials will be built using generic algorithms of orthonormalization.

getSupport
(self, \*args)¶ Accessor to the support of the distribution.
 Parameters
 interval
Interval
An interval to intersect with the support of the discrete part of the distribution.
 interval
 Returns
 support
Interval
The intersection of the support of the discrete part of the distribution with the given interval.
 support
See also
Notes
The mathematical support of the discrete part of a distribution is the collection of points with nonzero probability.
This is yet implemented for discrete distributions only.

getVisibility
(self)¶ Accessor to the object’s visibility state.
 Returns
 visiblebool
Visibility flag.

hasEllipticalCopula
(self)¶ Test whether the copula of the distribution is elliptical or not.
 Returns
 testbool
Answer.
See also

hasIndependentCopula
(self)¶ Test whether the copula of the distribution is the independent one.
 Returns
 testbool
Answer.

hasName
(self)¶ Test if the object is named.
 Returns
 hasNamebool
True if the name is not empty.

hasVisibleName
(self)¶ Test if the object has a distinguishable name.
 Returns
 hasVisibleNamebool
True if the name is not empty and not the default one.

inverse
(self)¶ Transform distribution by inverse function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

isContinuous
(self)¶ Test whether the distribution is continuous or not.
 Returns
 testbool
Answer.

isCopula
(self)¶ Test whether the distribution is a copula or not.
 Returns
 testbool
Answer.
Notes
A copula is a distribution with uniform margins on [0; 1].

isDiscrete
(self)¶ Test whether the distribution is discrete or not.
 Returns
 testbool
Answer.

isElliptical
(self)¶ Test whether the distribution is elliptical or not.
 Returns
 testbool
Answer.
Notes
A multivariate distribution is said to be elliptical if its characteristic function is of the form:
for specified vector and positivedefinite matrix . The function is known as the characteristic generator of the elliptical distribution.

isIntegral
(self)¶ Test whether the distribution is integervalued or not.
 Returns
 testbool
Answer.

ln
(self)¶ Transform distribution by natural logarithm function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

log
(self)¶ Transform distribution by natural logarithm function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

setDescription
(self, description)¶ Accessor to the componentwise description.
 Parameters
 descriptionsequence of str
Description of the components of the distribution.

setIntegrationNodesNumber
(self, integrationNodesNumber)¶ Accessor to the number of Gauss integration points.
 Parameters
 Nint
Number of integration points.

setName
(self, name)¶ Accessor to the object’s name.
 Parameters
 namestr
The name of the object.

setNu
(self, nu)¶ Accessor to the degrees of freedom parameter.
 Parameters
 nufloat,
Degrees of freedom.

setParameter
(self, parameter)¶ Accessor to the parameter of the distribution.
 Parameters
 parametersequence of float
Parameter values.

setParametersCollection
(self, \*args)¶ Accessor to the parameter of the distribution.
 Parameters
 parameters
PointWithDescription
Dictionarylike object with parameters names and values.
 parameters

setShadowedId
(self, id)¶ Accessor to the object’s shadowed id.
 Parameters
 idint
Internal unique identifier.

setVisibility
(self, visible)¶ Accessor to the object’s visibility state.
 Parameters
 visiblebool
Visibility flag.

sin
(self)¶ Transform distribution by sine function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

sinh
(self)¶ Transform distribution by sinh function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

sqr
(self)¶ Transform distribution by square function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

sqrt
(self)¶ Transform distribution by square root function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

tan
(self)¶ Transform distribution by tangent function.
 Returns
 dist
Distribution
The transformed distribution.
 dist

tanh
(self)¶ Transform distribution by tanh function.
 Returns
 dist
Distribution
The transformed distribution.
 dist