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

-
class
Binomial
(*args)¶ Binomial distribution.
- Available constructors:
- Binomial(n=1, p=0.5)
Parameters: - n : int,
The number of Bernoulli trials.
- p : float,
The success probability of the Bernoulli trial.
See also
Notes
Its probability density function is defined as:
with
and
.
Its first moments are:
Examples
Create a distribution:
>>> import openturns as ot >>> distribution = ot.Binomial(10, 0.5)
Draw a sample:
>>> sample = distribution.getSample(5)
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 first-order derivative of the probability density function. computeDensityGeneratorSecondDerivative
(…)Compute the second-order derivative of the probability density function. computeEntropy
()Compute the entropy of the distribution. computeGeneratingFunction
(z)Compute the probability-generating function. computeInverseSurvivalFunction
(point)Compute the inverse survival function. computeLogCharacteristicFunction
(x)Compute the logarithm of the characteristic function. computeLogGeneratingFunction
(z)Compute the logarithm of the probability-generating 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 iso-lines of log-probability density function. drawMarginal1DCDF
(marginalIndex, xMin, xMax, …)Draw the cumulative distribution function of a margin. drawMarginal1DLogPDF
(marginalIndex, xMin, …)Draw the log-probability 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 log-probability 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 iso-lines 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. getIntegrationNodesNumber
()Accessor to the number of Gauss integration points. getInverseCholesky
()Accessor to the inverse Cholesky factor of the covariance matrix. getInverseIsoProbabilisticTransformation
()Accessor to the inverse iso-probabilistic transformation. getIsoProbabilisticTransformation
()Accessor to the iso-probabilistic 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. getN
()Accessor to the number of trials. getName
()Accessor to the object’s name. getP
()Accessor to the success probability parameter. 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 pseudo-random realization from the distribution. getRoughness
()Accessor to roughness of the distribution. getSample
(size)Accessor to a pseudo-random sample from the distribution. getShadowedId
()Accessor to the object’s shadowed id. 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. getVisibility
()Accessor to the object’s visibility state. hasEllipticalCopula
()Test whether the copula of the distribution is elliptical or not. hasIndependentCopula
()Test whether the copula of the distribution is the independent one. hasName
()Test if the object is named. hasVisibleName
()Test if the object has a distinguishable name. 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 integer-valued or not. ln
()Transform distribution by natural logarithm function. log
()Transform distribution by natural logarithm function. setDescription
(description)Accessor to the componentwise description. setIntegrationNodesNumber
(integrationNodesNumber)Accessor to the number of Gauss integration points. setN
(n)Accessor to the number of trials. setName
(name)Accessor to the object’s name. setP
(p)Accessor to the success probability parameter. setParameter
(parameter)Accessor to the parameter of the distribution. setParametersCollection
(*args)Accessor to the parameter of the distribution. setShadowedId
(id)Accessor to the object’s shadowed id. setVisibility
(visible)Accessor to the object’s visibility state. 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. getSupportEpsilon setSupportEpsilon -
__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('DistributionFactory-DefaultBootstrapSize', 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)
- alpha : float,
-
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('DistributionFactory-DefaultBootstrapSize', 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)
- alpha : float,
-
computeCDF
(*args)¶ Compute the cumulative distribution function.
Parameters: - X : sequence of float, 2-d 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, 2-d 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 1-dimensional distributions).
-
computeConditionalCDF
(*args)¶ Compute the conditional cumulative distribution function.
Parameters: - Xn : float, sequence of float
Conditional CDF input (last component).
- Xcond : sequence of float, 2-d 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, 2-d 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, 2-d sequence of float with size
Conditionning values for the other components.
Returns: - X1 : float
Conditional quantile at input p, Xcond.
See also
- p : float, sequence of float,
-
computeDDF
(*args)¶ Compute the derivative density function.
Parameters: - X : sequence of float, 2-d 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 first-order 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 first-order derivative value at input X.
See also
Notes
This function only exists for elliptical distributions.
-
computeDensityGeneratorSecondDerivative
(betaSquare)¶ Compute the second-order 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 second-order 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
(z)¶ Compute the probability-generating function.
Parameters: - z : float or complex
Probability-generating function input.
Returns: - g : float
Probability-generating function value at input X.
See also
Notes
The probability-generating function is defined as follows:
This function only exists for discrete distributions. OpenTURNS implements this method for univariate distributions only.
-
computeInverseSurvivalFunction
(point)¶ Compute the inverse survival function.
Parameters: - p : float,
Level of the survival function.
Returns: - x :
Point
Point
such that
with iso-quantile components.
See also
Notes
The inverse survival function writes:
where
. OpenTURNS returns the point
such that
.
- p : float,
-
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
(z)¶ Compute the logarithm of the probability-generating function.
Parameters: - z : float or complex
Probability-generating function input.
Returns: - lg : float
Logarithm of the probability-generating 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, 2-d 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('DistributionFactory-DefaultBootstrapSize', 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('Distribution-MinimumVolumeLevelSetSamplingSize', 1000) >>> confInt = paramDist.computeMinimumVolumeInterval(0.9)
- alpha : float,
-
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('DistributionFactory-DefaultBootstrapSize', 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('Distribution-MinimumVolumeLevelSetSamplingSize', 1000) >>> confInt, marginalProb = paramDist.computeMinimumVolumeIntervalWithMarginalProbability(0.9)
- alpha : float,
-
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('DistributionFactory-DefaultBootstrapSize', 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)
- alpha : float,
-
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('DistributionFactory-DefaultBootstrapSize', 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)
- alpha : float,
-
computePDF
(*args)¶ Compute the probability density function.
Parameters: - X : sequence of float, 2-d 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 hyper-rectangular 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:
- p : float,
-
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:
- r2 : float,
-
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:
- p : float,
-
computeSurvivalFunction
(*args)¶ Compute the survival function.
Parameters: - x : sequence of float, 2-d 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 1-dimensional 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 iso-quantile components.
The right unilateral confidence interval
is the cartesian product
where
for all i and which verifies
. It means that
with iso-quantile 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('DistributionFactory-DefaultBootstrapSize', 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)
- alpha : float,
-
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('DistributionFactory-DefaultBootstrapSize', 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)
- alpha : float,
-
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 min-value of the mesh of the x-axis. Defaults uses the quantile associated to the probability level Distribution-QMin from the
ResourceMap
.- x_max : float, optional,
The max-value of the mesh of the y-axis. Defaults uses the quantile associated to the probability level Distribution-QMax from the
ResourceMap
.- pointNumber : int
The number of points that is used for meshing each axis. Defaults uses DistributionImplementation-DefaultPointNumber 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 iso-lines 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 iso-lines of log-probability density function.
- Available constructors:
drawLogPDF(x_min, x_max, pointNumber)
drawLogPDF(lowerCorner, upperCorner, pointNbrInd)
drawLogPDF(lowerCorner, upperCorner)
Parameters: - x_min : float, optional
The min-value of the mesh of the x-axis. Defaults uses the quantile associated to the probability level Distribution-QMin from the
ResourceMap
.- x_max : float, optional,
The max-value of the mesh of the y-axis. Defaults uses the quantile associated to the probability level Distribution-QMax from the
ResourceMap
.- pointNumber : int
The number of points that is used for meshing each axis. Defaults uses DistributionImplementation-DefaultPointNumber 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 log-PDF or its iso_lines.
See also
Notes
Only valid for univariate and bivariate distributions.
Examples
View the log-PDF of a univariate distribution:
>>> import openturns as ot >>> dist = ot.Normal() >>> graph = dist.drawLogPDF() >>> graph.setLegends(['normal log-pdf'])
View the iso-lines log-PDF 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 x-axis.
- x_max : float,
The ending value that is used for meshing the x-axis.
- n_points : int
The number of points that is used for meshing the x-axis.
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()
- i : int,
-
drawMarginal1DLogPDF
(marginalIndex, xMin, xMax, pointNumber)¶ Draw the log-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 x-axis.
- x_max : float,
The ending value that is used for meshing the x-axis.
- n_points : int
The number of points that is used for meshing the x-axis.
Returns: - graph :
Graph
A graphical representation of the log-PDF 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()
- i : int,
-
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 x-axis.
- x_max : float,
The ending value that is used for meshing the x-axis.
- n_points : int
The number of points that is used for meshing the x-axis.
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()
- i : int,
-
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()
- i : int,
-
drawMarginal2DLogPDF
(firstMarginal, secondMarginal, xMin, xMax, pointNumber)¶ Draw the log-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 log-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.drawMarginal2DLogPDF(2, 3, [-6.0] * 2, [6.0] * 2, [100] * 2) >>> view = View(graph) >>> view.show()
- i : int,
-
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()
- i : int,
-
drawPDF
(*args)¶ Draw the graph or of iso-lines 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 min-value of the mesh of the x-axis. Defaults uses the quantile associated to the probability level Distribution-QMin from the
ResourceMap
.- x_max : float, optional,
The max-value of the mesh of the y-axis. Defaults uses the quantile associated to the probability level Distribution-QMax from the
ResourceMap
.- pointNumber : int
The number of points that is used for meshing each axis. Defaults uses DistributionImplementation-DefaultPointNumber 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 iso-lines 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 x-axis.
- q_max : float, in
The max value of the mesh of the x-axis.
- n_points : int, optional
The number of points that is used for meshing the quantile curve. Defaults uses DistributionImplementation-DefaultPointNumber 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()
- q_min : float, in
-
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 first-order 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 second-order 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 1-d distributions.
Returns: - dispersion : float
Standard deviation or interquartile.
-
getId
()¶ Accessor to the object’s id.
Returns: - id : int
Internal unique identifier.
-
getIntegrationNodesNumber
()¶ Accessor to the number of Gauss integration points.
Returns: - N : int
Number of integration points.
-
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 iso-probabilistic transformation.
Returns: - Tinv :
Function
Inverse iso-probabilistic transformation.
See also
Notes
The inverse iso-probabilistic transformation is defined as follows:
- Tinv :
-
getIsoProbabilisticTransformation
()¶ Accessor to the iso-probabilistic transformation.
Refer to Isoprobabilistic transformations.
Returns: - T :
Function
Iso-probabilistic transformation.
Notes
The iso-probabilistic transformation is defined as follows:
An iso-probabilistic 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 iso-probabilistic 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 iso-probabilistic 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 fourth-order 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).
- i : int or list of ints,
-
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:
-
getN
()¶ Accessor to the number of trials.
Returns: - n : int
The number of Bernoulli trials.
-
getName
()¶ Accessor to the object’s name.
Returns: - name : str
The name of the object.
-
getP
()¶ Accessor to the success probability parameter.
Returns: - p : float
The success probability of the Bernoulli trial.
-
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
Dictionary-like 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 1-d 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 pseudo-random realization from the distribution.
Refer to Distribution realizations.
Returns: - point :
Point
A pseudo-random 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 pseudo-random sample from the distribution.
Parameters: - size : int
Sample size.
Returns: - sample :
Sample
A pseudo-random sample of the distribution.
See also
-
getShadowedId
()¶ Accessor to the object’s shadowed id.
Returns: - id : int
Internal unique identifier.
-
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 third-order 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 iso-probabilistic 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 :
-
getVisibility
()¶ Accessor to the object’s visibility state.
Returns: - visible : bool
Visibility flag.
-
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.
-
hasName
()¶ Test if the object is named.
Returns: - hasName : bool
True if the name is not empty.
-
hasVisibleName
()¶ Test if the object has a distinguishable name.
Returns: - hasVisibleName : bool
True if the name is not empty and not the default one.
-
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 positive-definite matrix
. The function
is known as the characteristic generator of the elliptical distribution.
-
isIntegral
()¶ Test whether the distribution is integer-valued 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.
-
setIntegrationNodesNumber
(integrationNodesNumber)¶ Accessor to the number of Gauss integration points.
Parameters: - N : int
Number of integration points.
-
setN
(n)¶ Accessor to the number of trials.
Parameters: - n : int,
The number of Bernoulli trials.
- n : int,
-
setName
(name)¶ Accessor to the object’s name.
Parameters: - name : str
The name of the object.
-
setP
(p)¶ Accessor to the success probability parameter.
Parameters: - p : float,
The success probability of the Bernoulli trial.
- p : float,
-
setParameter
(parameter)¶ 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
Dictionary-like object with parameters names and values.
- parameters :
-
setShadowedId
(id)¶ Accessor to the object’s shadowed id.
Parameters: - id : int
Internal unique identifier.
-
setVisibility
(visible)¶ Accessor to the object’s visibility state.
Parameters: - visible : bool
Visibility flag.
-
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 :