IndependentCopula¶

(Source code, png)

class IndependentCopula(*args)¶

Independent copula.

Parameters:

nint: Dimension of the copula, $n \geq 1$ . Default is $n=2$ .

See also

Distribution

Notes

The Independent copula is defined by :

$C(u_1, \cdots, u_n) = \prod_{i=1}^n u_i$

for $u_i \in [0, 1]$

Examples

Create a distribution:

>>> import openturns as ot
>>> copula = ot.IndependentCopula(3)

Draw a sample:

>>> sample = copula.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`(point)	Compute the gradient of the cumulative distribution function.
`computeCharacteristicFunction`(*args)	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`(*args)	Compute the probability-generating function.
`computeInverseSurvivalFunction`(point)	Compute the inverse survival function.
`computeLogCharacteristicFunction`(*args)	Compute the logarithm of the characteristic function.
`computeLogGeneratingFunction`(*args)	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.
`computeLowerExtremalDependenceMatrix`()	Compute the lower extremal dependence coefficients.
`computeLowerTailDependenceMatrix`()	Compute the lower tail dependence coefficients.
`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`(point)	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.
`computeSequentialConditionalCDF`(x)	Compute the sequential conditional cumulative distribution functions.
`computeSequentialConditionalDDF`(x)	Compute the sequential conditional derivative density function.
`computeSequentialConditionalPDF`(x)	Compute the sequential conditional probability density function.
`computeSequentialConditionalQuantile`(q)	Compute the conditional quantile function of the last component.
`computeSurvivalFunction`(*args)	Compute the survival function.
`computeUnilateralConfidenceInterval`(prob[, tail])	Compute a unilateral confidence interval.
`computeUnilateralConfidenceIntervalWithMarginalProbability`(...)	Compute a unilateral confidence interval.
`computeUpperExtremalDependenceMatrix`()	Compute the upper extremal dependence coefficients.
`computeUpperTailDependenceMatrix`()	Compute the upper tail dependence coefficients.
`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.
`drawLowerExtremalDependenceFunction`()	Draw the lower extremal dependence function.
`drawLowerTailDependenceFunction`()	Draw the lower tail dependence 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.
`drawMarginal1DSurvivalFunction`(...[, logScale])	Draw the cumulative distribution 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.
`drawMarginal2DSurvivalFunction`(...[, ...])	Draw the cumulative distribution 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.
`drawSurvivalFunction`(*args)	Draw the cumulative distribution function.
`drawUpperExtremalDependenceFunction`()	Draw the upper extremal dependence function.
`drawUpperTailDependenceFunction`()	Draw the upper tail dependence function.
`exp`()	Transform distribution by exponential function.
`getCDFEpsilon`()	Accessor to the CDF computation precision.
`getCentralMoment`(n)	Accessor to the componentwise central 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.
`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 pseudo-random realization from the distribution.
`getRoughness`()	Accessor to roughness of the distribution.
`getSample`(size)	Accessor to a pseudo-random sample from the distribution.
`getSampleByInversion`(size)	Accessor to a pseudo-random sample from the distribution.
`getSampleByQMC`(size)	Accessor to a low discrepancy 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.
`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.
`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.
`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.

__init__(*args)¶

abs()¶

Transform distribution by absolute value function.

Returns:

distDistribution: The transformed distribution.

acos()¶

Transform distribution by arccosine function.

Returns:

distDistribution: The transformed distribution.

acosh()¶

Transform distribution by acosh function.

Returns:

distDistribution: The transformed distribution.

asin()¶

Transform distribution by arcsine function.

Returns:

distDistribution: The transformed distribution.

asinh()¶

Transform distribution by asinh function.

Returns:

distDistribution: The transformed distribution.

atan()¶

Transform distribution by arctangent function.

Returns:

distDistribution: The transformed distribution.

atanh()¶

Transform distribution by atanh function.

Returns:

distDistribution: The transformed distribution.

cbrt()¶

Transform distribution by cubic root function.

Returns:

distDistribution: The transformed distribution.

computeBilateralConfidenceInterval(prob)¶

Compute a bilateral confidence interval.

Parameters:

alphafloat, $\alpha \in [0,1]$: The confidence level.

Returns:

confIntervalInterval: The confidence interval of level $\alpha$ .

Notes

We consider an absolutely continuous measure $\mu$ with density function $p$ .

The bilateral confidence interval $I^*_{\alpha}$ is the cartesian product $I^*_{\alpha} = [a_1, b_1] \times \dots \times [a_d, b_d]$ where $a_i = F_i^{-1}((1-\beta)/2)$ and $b_i = F_i^{-1}((1+\beta)/2)$ for all $i$ and which verifies $\mu(I^*_{\alpha}) = \alpha$ .

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)

computeBilateralConfidenceIntervalWithMarginalProbability(prob)¶

Compute a bilateral confidence interval.

Refer to computeBilateralConfidenceInterval()

Parameters:

alphafloat, $\alpha \in [0,1]$: The confidence level.

Returns:

confIntervalInterval: The confidence interval of level $\alpha$ .
marginalProbfloat: The value $\beta$ 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)

computeCDF(*args)¶

Compute the cumulative distribution function.

Parameters:

Xsequence of float, 2-d sequence of float: CDF input(s).

Returns:

Ffloat, Point: CDF value(s) at input(s) $X$ .

Notes

The cumulative distribution function is defined as:

$F_{\vect{X}}(\vect{x}) = \Prob{\bigcap_{i=1}^n X_i \leq x_i}, \quad \vect{x} \in \supp{\vect{X}}$

computeCDFGradient(point)¶

Compute the gradient of the cumulative distribution function.

Parameters:

Xsequence of float: CDF input.

Returns:

dFdthetaPoint: Partial derivatives of the CDF with respect to the distribution parameters at input $X$ .

computeCharacteristicFunction(*args)¶

Compute the characteristic function.

Parameters:

tfloat: Characteristic function input.

Returns:

phicomplex: Characteristic function value at input $t$ .

Notes

The characteristic function is defined as:

$\phi_X(t) = \mathbb{E}\left[\exp(- i t X)\right], \quad t \in \Rset$

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:

Xsequence of float, 2-d sequence of float: Complementary CDF input(s).

Returns:

Cfloat, Point: Complementary CDF value(s) at input(s) $X$ .

See also

computeSurvivalFunction

Notes

The complementary cumulative distribution function.

$1 - F_{\vect{X}}(\vect{x}) = 1 - \Prob{\bigcap_{i=1}^n X_i \leq x_i}, \quad \vect{x} \in \supp{\vect{X}}$

Warning

This is not the survival function (except for 1-dimensional distributions).

computeConditionalCDF(*args)¶

Compute the conditional cumulative distribution function.

Parameters:

Xnfloat, sequence of float: Conditional CDF input (last component).
Xcondsequence of float, 2-d sequence of float with size $n-1$: Conditionning values for the other components.

Returns:

Ffloat, sequence of float: Conditional CDF value(s) at input $X_n$ , $X_{cond}$ .

Notes

The conditional cumulative distribution function of the last component with respect to the other fixed components is defined as follows:

$F_{X_n \mid X_1, \ldots, X_{n - 1}}(x_n) = \Prob{X_n \leq x_n \mid X_1=x_1, \ldots, X_{n-1}=x_{n-1}}, \quad x_n \in \supp{X_n}$

computeConditionalDDF(x, y)¶

Compute the conditional derivative density function of the last component.

With respect to the other fixed components.

Parameters:

Xnfloat: Conditional DDF input (last component).
Xcondsequence of float with dimension $n-1$: Conditionning values for the other components.

Returns:

dfloat: Conditional DDF value at input $X_n$ , $X_{cond}$ .

See also

computeDDF, computeConditionalCDF

computeConditionalPDF(*args)¶

Compute the conditional probability density function.

Conditional PDF of the last component with respect to the other fixed components.

Parameters:

Xnfloat, sequence of float: Conditional PDF input (last component).
Xcondsequence of float, 2-d sequence of float with size $n-1$: Conditionning values for the other components.

Returns:

Ffloat, sequence of float: Conditional PDF value(s) at input $X_n$ , $X_{cond}$ .

See also

computePDF, computeConditionalCDF

computeConditionalQuantile(*args)¶

Compute the conditional quantile function of the last component.

Conditional quantile with respect to the other fixed components.

Parameters:

pfloat, sequence of float, $0 < p < 1$: Conditional quantile function input.
Xcondsequence of float, 2-d sequence of float with size $n-1$: Conditionning values for the other components.

Returns:

X1float: Conditional quantile at input $p$ , $X_{cond}$ .

See also

computeQuantile, computeConditionalCDF

computeDDF(*args)¶

Compute the derivative density function.

Parameters:

Xsequence of float, 2-d sequence of float: PDF input(s).

Returns:

dPoint, Sample: DDF value(s) at input(s) $X$ .

Notes

The derivative density function is the gradient of the probability density function with respect to $\vect{x}$ :

$\vect{\nabla}_{\vect{x}} f_{\vect{X}}(\vect{x}) = \Tr{\left(\frac{\partial f_{\vect{X}}(\vect{x})}{\partial x_i}, \quad i = 1, \ldots, n\right)}, \quad \vect{x} \in \supp{\vect{X}}$

computeDensityGenerator(betaSquare)¶

Compute the probability density function of the characteristic generator.

PDF of the characteristic generator of the elliptical distribution.

Parameters:

beta2float: Density generator input.

Returns:

pfloat: Density generator value at input $X$ .

See also

isElliptical, computePDF

Notes

This is the function $\phi$ such that the probability density function rewrites:

$f_{\vect{X}}(\vect{x}) = \phi\left(\Tr{\left(\vect{x} - \vect{\mu}\right)} \mat{\Sigma}^{-1} \left(\vect{x} - \vect{\mu}\right) \right), \quad \vect{x} \in \supp{\vect{X}}$

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:

beta2float: Density generator input.

Returns:

pfloat: Density generator first-order derivative value at input $X$ .

See also

isElliptical, computeDensityGenerator

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:

beta2float: Density generator input.

Returns:

pfloat: Density generator second-order derivative value at input $X$ .

See also

isElliptical, computeDensityGenerator

Notes

This function only exists for elliptical distributions.

computeEntropy()¶

Compute the entropy of the distribution.

Returns:

efloat: Entropy of the distribution.

Notes

The entropy of a distribution is defined by:

$\cE_X = \Expect{-\log(p_X(\vect{X}))}$

Where the random vector $\vect{X}$ follows the probability distribution of interest, and $p_X$ is either the probability density function of $\vect{X}$ if it is continuous or the probability distribution function if it is discrete.

computeGeneratingFunction(*args)¶

Compute the probability-generating function.

Parameters:

zfloat or complex: Probability-generating function input.

Returns:

gfloat: Probability-generating function value at input $X$ .

See also

isDiscrete

Notes

The probability-generating function is defined as follows:

$G_X(z) = \Expect{z^X}, \quad z \in \Cset$

This function only exists for discrete distributions. OpenTURNS implements this method for univariate distributions only.

computeInverseSurvivalFunction(point)¶

Compute the inverse survival function.

Parameters:

pfloat, $p \in [0; 1]$: Level of the survival function.

Returns:

xPoint: Point $\vect{x}$ such that $S_{\vect{X}}(\vect{x}) = p$ with iso-quantile components.

See also

computeQuantile, computeSurvivalFunction

Notes

The inverse survival function writes: $S^{-1}(p) = \vect{x}^p$ where $S( \vect{x}^p) = \Prob{\bigcap_{i=1}^d X_i > x_i^p}$ . OpenTURNS returns the point $\vect{x}^p$ such that $\Prob{ X_1 > x_1^p} = \dots = \Prob{ X_d > x_d^p}$ .

computeLogCharacteristicFunction(*args)¶

Compute the logarithm of the characteristic function.

Parameters:

tfloat: Characteristic function input.

Returns:

phicomplex: Logarithm of the characteristic function value at input $t$ .

See also

computeCharacteristicFunction

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 probability-generating function.

Parameters:

zfloat or complex: Probability-generating function input.

Returns:

lgfloat: Logarithm of the probability-generating function value at input $X$ .

See also

isDiscrete, computeGeneratingFunction

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:

Xsequence of float, 2-d sequence of float: PDF input(s).

Returns:

ffloat, Point: Logarithm of the PDF value(s) at input(s) $X$ .

computeLogPDFGradient(*args)¶

Compute the gradient of the log probability density function.

Parameters:

Xsequence of float: PDF input.

Returns:

dfdthetaPoint: Partial derivatives of the logPDF with respect to the distribution parameters at input $X$ .

computeLowerExtremalDependenceMatrix()¶

Compute the lower extremal dependence coefficients.

We assume that the distribution is $d$ -dimensional, with $d \geq 2$ and that its copula is denoted by $C$ . The lower extremal dependence matrix is $(\bar{\chi}_L^{ij})_{1 \leq i, j \leq d}$ where $\\bar{\chi}_L^{ij}$ is the lower extremal dependence coefficient of the bivariate distribution $(X_i, X_j)$ . It is defined by:

$\bar{\chi}_L^{ij} = \lim_{u \to 0} \bar{\chi}_L^{ij}(u)$

where $u \mapsto \bar{\chi}_L^{ij}(u)$ is the lower extremal dependence function of the bivariate distribution $(X_i, X_j)$ defined by:

$\bar{\chi}_L(u) = \frac{2 \log u}{\log C(u,u)} - 1, \forall u \in [0,1]$

Refer to Tail dependence coefficients to get more details.

Returns:

depCorrelationMatrix: The dependence matrix.

Examples

>>> import openturns as ot
>>> copula = ot.FrankCopula()
>>> chiLb = copula.computeLowerExtremalDependenceMatrix()[1, 0]

computeLowerTailDependenceMatrix()¶

Compute the lower tail dependence coefficients.

We assume that the distribution is $d$ -dimensional, with $d \geq 2$ and that its copula is denoted by $C$ . The lower tail dependence matrix is $(\chi_L^{ij})_{1 \leq i, j \leq d}$ where $\chi_L^{ij}$ is the lower tail dependence coefficient of the bivariate distribution $(X_i, X_j)$ . It is defined by:

$\chi_L^{ij} = \lim_{u \to 0} \chi_L^{ij}(u)$

where $u \mapsto \chi_L^{ij}(u)$ is the lower tail dependence function of the bivariate distribution $(X_i, X_j)$ defined by:

$\chi_L(u) = \frac{\log (1 - C(u,u))}{\log (1-u)}, \forall u \in [0,1]$

Refer to Tail dependence coefficients to get more details.

Returns:

depCorrelationMatrix: The dependence matrix.

Examples

>>> import openturns as ot
>>> copula = ot.FrankCopula()
>>> chiL = copula.computeLowerTailDependenceMatrix()[1, 0]

computeMinimumVolumeInterval(prob)¶

Compute the confidence interval with minimum volume.

Parameters:

alphafloat, $\alpha \in [0,1]$: The confidence level.

Returns:

confIntervalInterval: The confidence interval of level $\alpha$ .

Notes

We consider an absolutely continuous measure $\mu$ with density function $p$ .

The minimum volume confidence interval $I^*_{\alpha}$ is the cartesian product $I^*_{\alpha} = [a_1, b_1] \times \dots \times [a_d, b_d]$ where $[a_i, b_i] = \argmin_{I \in \Rset \, | \, \mu_i(I) = \beta} \lambda_i(I)$ and $\mu(I^*_{\alpha}) = \alpha$ with $\lambda$ is the Lebesgue measure on $\Rset^d$ .

This problem resorts to solving $d$ univariate non linear equations: for a fixed value $\beta$ , we find each intervals $[a_i, b_i]$ such that:

$\begin{eqnarray*} F_i(b_i) - F_i(a_i) & = & \beta \\ p_i(b_i) & = & p_i(a_i) \end{eqnarray*}$

which consists of finding the bound $a_i$ such that:

$p_i(a_i) = p_i(F_i^{-1}(\beta + F_i(a_i)))$

To find $\beta$ , we use the Brent algorithm: $\mu([\vect{a}(\beta); \vect{b}(\beta)] = g(\beta) = \alpha$ 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)
>>> ot.ResourceMap.Reload()

computeMinimumVolumeIntervalWithMarginalProbability(prob)¶

Compute the confidence interval with minimum volume.

Refer to computeMinimumVolumeInterval()

Parameters:

alphafloat, $\alpha \in [0,1]$: The confidence level.

Returns:

confIntervalInterval: The confidence interval of level

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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IndependentCopula¶