Copula¶

class
Copula
(*args)¶ Base class for copulas.
See also
ArchimedeanCopula
,NormalCopula
,ComposedCopula
,SklarCopula
,IndependentCopula
,MinCopula
,OrdinalSumCopula
,FarlieGumbelMorgensternCopula
Notes
To define the joined probability density function of the random input vector by composition, one needs:
 the specification of the copula of interest with its parameters,
 the specification of the marginal laws of interest of the input variables .
The joined cumulative density function is therefore defined by :
Copulas allow to represent the part of the joined cumulative density function which is not described by the marginal laws. It enables to represent the dependence structure of the input variables. A copula is a special cumulative density function defined on whose marginal distributions are uniform on . The choice of the dependence structure is disconnected from the choice of the marginal distributions.
A copula, restricted to is a dimensional cumulative density function with uniform marginals such as:
 For all box
,
we have , where :
 , the summation being made over the vertices of .
 if for an even number of ’s, otherwise.
Methods
abs
()Transform distribution by absolute value function. acos
()Transform distribution by arccosine function. acosh
()Transform distribution by acosh function. asin
()Transform distribution by arcsine function. asinh
()Transform distribution by asinh function. atan
()Transform distribution by arctangent function. atanh
()Transform distribution by atanh function. cbrt
()Transform distribution by cubic root function. computeBilateralConfidenceInterval
(prob)Compute a bilateral confidence interval. computeBilateralConfidenceIntervalWithMarginalProbability
(prob)Compute a bilateral confidence interval. computeCDF
(*args)Compute the cumulative distribution function. computeCDFGradient
(*args)Compute the gradient of the cumulative distribution function. computeCharacteristicFunction
(x)Compute the characteristic function. computeComplementaryCDF
(*args)Compute the complementary cumulative distribution function. computeConditionalCDF
(*args)Compute the conditional cumulative distribution function. computeConditionalDDF
(x, y)Compute the conditional derivative density function of the last component. computeConditionalPDF
(*args)Compute the conditional probability density function. computeConditionalQuantile
(*args)Compute the conditional quantile function of the last component. computeDDF
(*args)Compute the derivative density function. computeDensityGenerator
(betaSquare)Compute the probability density function of the characteristic generator. computeDensityGeneratorDerivative
(betaSquare)Compute the firstorder derivative of the probability density function. computeDensityGeneratorSecondDerivative
(…)Compute the secondorder derivative of the probability density function. computeEntropy
()Compute the entropy of the distribution. computeGeneratingFunction
(*args)Compute the probabilitygenerating function. computeInverseSurvivalFunction
(prob)Compute the inverse survival function. computeLogCharacteristicFunction
(x)Compute the logarithm of the characteristic function. computeLogGeneratingFunction
(*args)Compute the logarithm of the probabilitygenerating function. computeLogPDF
(*args)Compute the logarithm of the probability density function. computeLogPDFGradient
(*args)Compute the gradient of the log probability density function. computeMinimumVolumeInterval
(prob)Compute the confidence interval with minimum volume. computeMinimumVolumeIntervalWithMarginalProbability
(prob)Compute the confidence interval with minimum volume. computeMinimumVolumeLevelSet
(prob)Compute the confidence domain with minimum volume. computeMinimumVolumeLevelSetWithThreshold
(prob)Compute the confidence domain with minimum volume. computePDF
(*args)Compute the probability density function. computePDFGradient
(*args)Compute the gradient of the probability density function. computeProbability
(interval)Compute the interval probability. computeQuantile
(*args)Compute the quantile function. computeRadialDistributionCDF
(radius[, tail])Compute the cumulative distribution function of the squared radius. computeScalarQuantile
(prob[, tail])Compute the quantile function for univariate distributions. computeSurvivalFunction
(*args)Compute the survival function. computeUnilateralConfidenceInterval
(prob[, tail])Compute a unilateral confidence interval. computeUnilateralConfidenceIntervalWithMarginalProbability
(…)Compute a unilateral confidence interval. cos
()Transform distribution by cosine function. cosh
()Transform distribution by cosh function. drawCDF
(*args)Draw the cumulative distribution function. drawLogPDF
(*args)Draw the graph or of isolines of logprobability density function. drawMarginal1DCDF
(marginalIndex, xMin, xMax, …)Draw the cumulative distribution function of a margin. drawMarginal1DLogPDF
(marginalIndex, xMin, …)Draw the logprobability density function of a margin. drawMarginal1DPDF
(marginalIndex, xMin, xMax, …)Draw the probability density function of a margin. drawMarginal2DCDF
(firstMarginal, …)Draw the cumulative distribution function of a couple of margins. drawMarginal2DLogPDF
(firstMarginal, …)Draw the logprobability density function of a couple of margins. drawMarginal2DPDF
(firstMarginal, …)Draw the probability density function of a couple of margins. drawPDF
(*args)Draw the graph or of isolines of probability density function. drawQuantile
(*args)Draw the quantile function. exp
()Transform distribution by exponential function. getCDFEpsilon
()Accessor to the CDF computation precision. getCenteredMoment
(n)Accessor to the componentwise centered moments. getCholesky
()Accessor to the Cholesky factor of the covariance matrix. getClassName
()Accessor to the object’s name. getCopula
()Accessor to the copula of the distribution. getCorrelation
()(ditch me?) getCovariance
()Accessor to the covariance matrix. getDescription
()Accessor to the componentwise description. getDimension
()Accessor to the dimension of the distribution. getDispersionIndicator
()Dispersion indicator accessor. getId
()Accessor to the object’s id. getImplementation
()Accessor to the underlying implementation. getInverseCholesky
()Accessor to the inverse Cholesky factor of the covariance matrix. getInverseIsoProbabilisticTransformation
()Accessor to the inverse isoprobabilistic transformation. getIsoProbabilisticTransformation
()Accessor to the isoprobabilistic transformation. getKendallTau
()Accessor to the Kendall coefficients matrix. getKurtosis
()Accessor to the componentwise kurtosis. getLinearCorrelation
()(ditch me?) getMarginal
(*args)Accessor to marginal distributions. getMean
()Accessor to the mean. getMoment
(n)Accessor to the componentwise moments. getName
()Accessor to the object’s name. getPDFEpsilon
()Accessor to the PDF computation precision. getParameter
()Accessor to the parameter of the distribution. getParameterDescription
()Accessor to the parameter description of the distribution. getParameterDimension
()Accessor to the number of parameters in the distribution. getParametersCollection
()Accessor to the parameter of the distribution. getPearsonCorrelation
()Accessor to the Pearson correlation matrix. getPositionIndicator
()Position indicator accessor. getProbabilities
()Accessor to the discrete probability levels. getRange
()Accessor to the range of the distribution. getRealization
()Accessor to a pseudorandom realization from the distribution. getRoughness
()Accessor to roughness of the distribution. getSample
(size)Accessor to a pseudorandom sample from the distribution. getShapeMatrix
()Accessor to the shape matrix of the underlying copula if it is elliptical. getShiftedMoment
(n, shift)Accessor to the componentwise shifted moments. getSingularities
()Accessor to the singularities of the PDF function. getSkewness
()Accessor to the componentwise skewness. getSpearmanCorrelation
()Accessor to the Spearman correlation matrix. getStandardDeviation
()Accessor to the componentwise standard deviation. getStandardDistribution
()Accessor to the standard distribution. getStandardMoment
(n)Accessor to the componentwise standard moments. getStandardRepresentative
()Accessor to the standard representative distribution in the parametric family. getSupport
(*args)Accessor to the support of the distribution. hasEllipticalCopula
()Test whether the copula of the distribution is elliptical or not. hasIndependentCopula
()Test whether the copula of the distribution is the independent one. inverse
()Transform distribution by inverse function. isContinuous
()Test whether the distribution is continuous or not. isCopula
()Test whether the distribution is a copula or not. isDiscrete
()Test whether the distribution is discrete or not. isElliptical
()Test whether the distribution is elliptical or not. isIntegral
()Test whether the distribution is integervalued or not. ln
()Transform distribution by natural logarithm function. log
()Transform distribution by natural logarithm function. setDescription
(description)Accessor to the componentwise description. setName
(name)Accessor to the object’s name. setParameter
(parameters)Accessor to the parameter of the distribution. setParametersCollection
(*args)Accessor to the parameter of the distribution. sin
()Transform distribution by sine function. sinh
()Transform distribution by sinh function. sqr
()Transform distribution by square function. sqrt
()Transform distribution by square root function. tan
()Transform distribution by tangent function. tanh
()Transform distribution by tanh function. 
__init__
(*args)¶ Initialize self. See help(type(self)) for accurate signature.

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

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

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

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

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

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

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

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

computeBilateralConfidenceInterval
(prob)¶ Compute a bilateral confidence interval.
Parameters:  alpha : float,
The confidence level.
Returns:  confInterval :
Interval
The confidence interval of level .
Notes
We consider an absolutely continuous measure with density function p.
The bilateral confidence interval is the cartesian product where and for all i 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
(prob)¶ Compute a bilateral confidence interval.
Refer to
computeBilateralConfidenceInterval()
Parameters:  alpha : float,
The confidence level.
Returns:  confInterval :
Interval
The confidence interval of level .
 marginalProb : float
The value which is the common marginal probability of each marginal interval.
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
(*args)¶ Compute the cumulative distribution function.
Parameters:  X : sequence of float, 2d sequence of float
CDF input(s).
Returns:  F : float,
Point
CDF value(s) at input(s) X.
Notes
The cumulative distribution function is defined as:

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

computeCharacteristicFunction
(x)¶ Compute the characteristic function.
Parameters:  t : float
Characteristic function input.
Returns:  phi : complex
Characteristic function value at input t.
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 t gets high. Only some univariate distributions benefit from dedicated more efficient implementations.

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

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

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

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

computeConditionalQuantile
(*args)¶ Compute the conditional quantile function of the last component.
Conditional quantile with respect to the other fixed components.
Parameters:  p : float, sequence of float,
Conditional quantile function input.
 Xcond : sequence of float, 2d sequence of float with size
Conditionning values for the other components.
Returns:  X1 : float
Conditional quantile at input p, Xcond.
See also

computeDDF
(*args)¶ Compute the derivative density function.
Parameters:  X : sequence 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
(betaSquare)¶ Compute the probability density function of the characteristic generator.
PDF of the characteristic generator of the elliptical distribution.
Parameters:  beta2 : float
Density generator input.
Returns:  p : float
Density generator value at input X.
See also
Notes
This is the function such that the probability density function rewrites:
This function only exists for elliptical distributions.

computeDensityGeneratorDerivative
(betaSquare)¶ Compute the firstorder derivative of the probability density function.
PDF of the characteristic generator of the elliptical distribution.
Parameters:  beta2 : float
Density generator input.
Returns:  p : float
Density generator firstorder derivative value at input X.
See also
Notes
This function only exists for elliptical distributions.

computeDensityGeneratorSecondDerivative
(betaSquare)¶ Compute the secondorder derivative of the probability density function.
PDF of the characteristic generator of the elliptical distribution.
Parameters:  beta2 : float
Density generator input.
Returns:  p : float
Density generator secondorder derivative value at input X.
See also
Notes
This function only exists for elliptical distributions.

computeEntropy
()¶ Compute the entropy of the distribution.
Returns:  e : float
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
(*args)¶ Compute the probabilitygenerating function.
Parameters:  z : float or complex
Probabilitygenerating function input.
Returns:  g : float
Probabilitygenerating function value at input X.
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
(prob)¶ Compute the inverse survival function.
Parameters:  p : float,
Level of the survival function.
Returns:  x :
Point
Point such that with isoquantile components.
See also
Notes
The inverse survival function writes: where . OpenTURNS returns the point such that .

computeLogCharacteristicFunction
(x)¶ Compute the logarithm of the characteristic function.
Parameters:  t : float
Characteristic function input.
Returns:  phi : complex
Logarithm of the characteristic function value at input t.
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 t gets high. Only some univariate distributions benefit from dedicated more efficient implementations.

computeLogGeneratingFunction
(*args)¶ Compute the logarithm of the probabilitygenerating function.
Parameters:  z : float or complex
Probabilitygenerating function input.
Returns:  lg : float
Logarithm of the probabilitygenerating function value at input X.
See also
Notes
This function only exists for discrete distributions. OpenTURNS implements this method for univariate distributions only.

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

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

computeMinimumVolumeInterval
(prob)¶ Compute the confidence interval with minimum volume.
Parameters:  alpha : float,
The confidence level.
Returns:  confInterval :
Interval
The confidence interval of level .
Notes
We consider an absolutely continuous measure with density function p.
The minimum volume confidence interval is the cartesian product where and with is the Lebesgue measure on .
This problem resorts to solving d 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 g 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
(prob)¶ Compute the confidence interval with minimum volume.
Refer to
computeMinimumVolumeInterval()
Parameters:  alpha : float,
The confidence level.
Returns:  confInterval :
Interval
The confidence interval of level .
 marginalProb : float
The value which is the common marginal probability of each marginal interval.
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
(prob)¶ Compute the confidence domain with minimum volume.
Parameters:  alpha : float,
The confidence level.
Returns:  levelSet :
LevelSet
The minimum volume domain of measure .
Notes
We consider an absolutely continuous measure with density function p.
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
(prob)¶ Compute the confidence domain with minimum volume.
Refer to
computeMinimumVolumeLevelSet()
Parameters:  alpha : float,
The confidence level.
Returns:  levelSet :
LevelSet
The minimum volume domain of measure .
 level : float
The value of the density function defining the frontier of the domain.
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
(*args)¶ Compute the probability density function.
Parameters:  X : sequence of float, 2d sequence of float
PDF input(s).
Returns:  f : float,
Point
PDF value(s) at input(s) X.
Notes
The probability density function is defined as follows:

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

computeProbability
(interval)¶ Compute the interval probability.
Parameters:  interval :
Interval
An interval, possibly multivariate.
Returns:  P : float
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 .
 interval :

computeQuantile
(*args)¶ Compute the quantile function.
Parameters:  p : float,
Quantile function input (a probability).
Returns:  X :
Point
Quantile at probability level p.
Notes
The quantile function is also known as the inverse cumulative distribution function:

computeRadialDistributionCDF
(radius, tail=False)¶ Compute the cumulative distribution function of the squared radius.
For the underlying standard spherical distribution (for elliptical distributions only).
Parameters:  r2 : float,
Squared radius.
Returns:  F : float
CDF value at input r2.
Notes
This is the CDF of the sum of the squared independent, standard, identically distributed components:

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

computeSurvivalFunction
(*args)¶ Compute the survival function.
Parameters:  x : sequence of float, 2d sequence of float
Survival function input(s).
Returns:  S : float,
Point
Survival function value(s) at input(s) x.
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
(prob, tail=False)¶ Compute a unilateral confidence interval.
Parameters:  alpha : float,
The confidence level.
 tail : boolean
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 .
Notes
We consider an absolutely continuous measure .
The left unilateral confidence interval is the cartesian product where for all i 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 i 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
(prob, tail)¶ Compute a unilateral confidence interval.
Refer to
computeUnilateralConfidenceInterval()
Parameters:  alpha : float,
The confidence level.
 tail : boolean
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 .
 marginalProb : float
The value which is the common marginal probability of each marginal interval.
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
()¶ Transform distribution by cosine function.
Returns:  dist :
Distribution
The transformed distribution.
 dist :

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

drawCDF
(*args)¶ Draw the cumulative distribution function.
 Available constructors:
drawCDF(x_min, x_max, pointNumber)
drawCDF(lowerCorner, upperCorner, pointNbrInd)
drawCDF(lowerCorner, upperCorner)
Parameters:  x_min : float, optional
The minvalue of the mesh of the xaxis. Defaults uses the quantile associated to the probability level DistributionQMin from the
ResourceMap
. x_max : float, optional,
The maxvalue of the mesh of the yaxis. Defaults uses the quantile associated to the probability level DistributionQMax from the
ResourceMap
. pointNumber : int
The number of points that is used for meshing each axis. Defaults uses DistributionImplementationDefaultPointNumber from the
ResourceMap
. lowerCorner : sequence of float, of dimension 2, optional
The lower corner .
 upperCorner : sequence of float, of dimension 2, optional
The upper corner .
 pointNbrInd :
Indices
, of dimension 2 Number of points that is used for meshing each axis.
Returns:  graph :
Graph
A graphical representation of the CDF.
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
(*args)¶ Draw the graph or of isolines of logprobability density function.
 Available constructors:
drawLogPDF(x_min, x_max, pointNumber)
drawLogPDF(lowerCorner, upperCorner, pointNbrInd)
drawLogPDF(lowerCorner, upperCorner)
Parameters:  x_min : float, optional
The minvalue of the mesh of the xaxis. Defaults uses the quantile associated to the probability level DistributionQMin from the
ResourceMap
. x_max : float, optional,
The maxvalue of the mesh of the yaxis. Defaults uses the quantile associated to the probability level DistributionQMax from the
ResourceMap
. pointNumber : int
The number of points that is used for meshing each axis. Defaults uses DistributionImplementationDefaultPointNumber from the
ResourceMap
. lowerCorner : sequence of float, of dimension 2, optional
The lower corner .
 upperCorner : sequence of float, of dimension 2, optional
The upper corner .
 pointNbrInd :
Indices
, of dimension 2 Number of points that is used for meshing each axis.
Returns:  graph :
Graph
A graphical representation of the logPDF or its iso_lines.
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
(marginalIndex, xMin, xMax, pointNumber)¶ Draw the cumulative distribution function of a margin.
Parameters:  i : int,
The index of the margin of interest.
 x_min : float
The starting value that is used for meshing the xaxis.
 x_max : float,
The ending value that is used for meshing the xaxis.
 n_points : int
The number of points that is used for meshing the xaxis.
Returns:  graph :
Graph
A graphical representation of the CDF of the requested margin.
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
(marginalIndex, xMin, xMax, pointNumber)¶ Draw the logprobability density function of a margin.
Parameters:  i : int,
The index of the margin of interest.
 x_min : float
The starting value that is used for meshing the xaxis.
 x_max : float,
The ending value that is used for meshing the xaxis.
 n_points : int
The number of points that is used for meshing the xaxis.
Returns:  graph :
Graph
A graphical representation of the logPDF of the requested margin.
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
(marginalIndex, xMin, xMax, pointNumber)¶ Draw the probability density function of a margin.
Parameters:  i : int,
The index of the margin of interest.
 x_min : float
The starting value that is used for meshing the xaxis.
 x_max : float,
The ending value that is used for meshing the xaxis.
 n_points : int
The number of points that is used for meshing the xaxis.
Returns:  graph :
Graph
A graphical representation of the PDF of the requested margin.
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()

drawMarginal2DCDF
(firstMarginal, secondMarginal, xMin, xMax, pointNumber)¶ Draw the cumulative distribution function of a couple of margins.
Parameters:  i : int,
The index of the first margin of interest.
 j : int,
The index of the second margin of interest.
 x_min : list of 2 floats
The starting values that are used for meshing the x and y axes.
 x_max : list of 2 floats,
The ending values that are used for meshing the x and y axes.
 n_points : list of 2 ints
The number of points that are used for meshing the x and y axes.
Returns:  graph :
Graph
A graphical representation of the marginal CDF of the requested couple of margins.
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
(firstMarginal, secondMarginal, xMin, xMax, pointNumber)¶ Draw the logprobability density function of a couple of margins.
Parameters:  i : int,
The index of the first margin of interest.
 j : int,
The index of the second margin of interest.
 x_min : list of 2 floats
The starting values that are used for meshing the x and y axes.
 x_max : list of 2 floats,
The ending values that are used for meshing the x and y axes.
 n_points : list of 2 ints
The number of points that are used for meshing the x and y axes.
Returns:  graph :
Graph
A graphical representation of the marginal logPDF of the requested couple of margins.
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
(firstMarginal, secondMarginal, xMin, xMax, pointNumber)¶ Draw the probability density function of a couple of margins.
Parameters:  i : int,
The index of the first margin of interest.
 j : int,
The index of the second margin of interest.
 x_min : list of 2 floats
The starting values that are used for meshing the x and y axes.
 x_max : list of 2 floats,
The ending values that are used for meshing the x and y axes.
 n_points : list of 2 ints
The number of points that are used for meshing the x and y axes.
Returns:  graph :
Graph
A graphical representation of the marginal PDF of the requested couple of margins.
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()

drawPDF
(*args)¶ Draw the graph or of isolines of probability density function.
 Available constructors:
drawPDF(x_min, x_max, pointNumber)
drawPDF(lowerCorner, upperCorner, pointNbrInd)
drawPDF(lowerCorner, upperCorner)
Parameters:  x_min : float, optional
The minvalue of the mesh of the xaxis. Defaults uses the quantile associated to the probability level DistributionQMin from the
ResourceMap
. x_max : float, optional,
The maxvalue of the mesh of the yaxis. Defaults uses the quantile associated to the probability level DistributionQMax from the
ResourceMap
. pointNumber : int
The number of points that is used for meshing each axis. Defaults uses DistributionImplementationDefaultPointNumber from the
ResourceMap
. lowerCorner : sequence of float, of dimension 2, optional
The lower corner .
 upperCorner : sequence of float, of dimension 2, optional
The upper corner .
 pointNbrInd :
Indices
, of dimension 2 Number of points that is used for meshing each axis.
Returns:  graph :
Graph
A graphical representation of the PDF or its iso_lines.
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
(*args)¶ Draw the quantile function.
Parameters:  q_min : float, in
The min value of the mesh of the xaxis.
 q_max : float, in
The max value of the mesh of the xaxis.
 n_points : int, optional
The number of points that is used for meshing the quantile curve. Defaults uses DistributionImplementationDefaultPointNumber from the
ResourceMap
.
Returns:  graph :
Graph
A graphical representation of the quantile function.
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()

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

getCDFEpsilon
()¶ Accessor to the CDF computation precision.
Returns:  CDFEpsilon : float
CDF computation precision.

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

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

getClassName
()¶ Accessor to the object’s name.
Returns:  class_name : str
The object class name (object.__class__.__name__).

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

getCorrelation
()¶ (ditch me?)

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

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

getDimension
()¶ Accessor to the dimension of the distribution.
Returns:  n : int
The number of components in the distribution.

getDispersionIndicator
()¶ 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:  dispersion : float
Standard deviation or interquartile.

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

getImplementation
()¶ Accessor to the underlying implementation.
Returns:  impl : Implementation
The implementation class.

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

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

getIsoProbabilisticTransformation
()¶ Accessor to the isoprobabilistic transformation.
Returns:  T :
Function
Isoprobabilistic transformation.
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 [Nataf1962], [Lebrun2009a].
 if the distribution has an elliptical Copula, then the transformed distribution is made spherical using the generalized Nataf transformation [Lebrun2009b].
 otherwise, the transformed distribution is the standard multivariate Normal distribution and is obtained by means of the Rosenblatt transformation [Rosenblatt1952], [Lebrun2009c].
[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 is invariant by rotation. Mathematically, has a spherical distribution if:
 T :

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

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

getLinearCorrelation
()¶ (ditch me?)

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

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

getName
()¶ Accessor to the object’s name.
Returns:  name : str
The name of the object.

getPDFEpsilon
()¶ Accessor to the PDF computation precision.
Returns:  PDFEpsilon : float
PDF computation precision.

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

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

getParameterDimension
()¶ Accessor to the number of parameters in the distribution.
Returns:  n_parameters : int
Number of parameters in the distribution.
See also

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

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

getPositionIndicator
()¶ 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:  position : float
Mean or median of the distribution.

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

getRange
()¶ Accessor to the range of the distribution.
Returns:  range :
Interval
Range of the distribution.
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.
 range :

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

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

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

getShapeMatrix
()¶ 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.
See also
Notes
This is not the Pearson correlation matrix.
 shape :

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

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

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

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

getStandardDistribution
()¶ Accessor to the standard distribution.
Returns:  standard_distribution :
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.
 standard_distribution :

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

getStandardRepresentative
()¶ Accessor to the standard representative distribution in the parametric family.
Returns:  std_repr_dist :
Distribution
Standard representative distribution.
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.
 std_repr_dist :

getSupport
(*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.
Returns:  support :
Interval
The intersection of the support of the discrete part of the distribution with the given interval.
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.
 interval :

hasEllipticalCopula
()¶ Test whether the copula of the distribution is elliptical or not.
Returns:  test : bool
Answer.
See also

hasIndependentCopula
()¶ Test whether the copula of the distribution is the independent one.
Returns:  test : bool
Answer.

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

isContinuous
()¶ Test whether the distribution is continuous or not.
Returns:  test : bool
Answer.

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

isDiscrete
()¶ Test whether the distribution is discrete or not.
Returns:  test : bool
Answer.

isElliptical
()¶ Test whether the distribution is elliptical or not.
Returns:  test : bool
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
()¶ Test whether the distribution is integervalued or not.
Returns:  test : bool
Answer.

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

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

setDescription
(description)¶ Accessor to the componentwise description.
Parameters:  description : sequence of str
Description of the components of the distribution.

setName
(name)¶ Accessor to the object’s name.
Parameters:  name : str
The name of the object.

setParameter
(parameters)¶ Accessor to the parameter of the distribution.
Parameters:  parameter : sequence of float
Parameter values.

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

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

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

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

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

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

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