Uniform distribution¶

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class Uniform(*args)¶

Uniform distribution.

Parameters:

afloat

Lower bound.

Default value is -1.0.

bfloat, $b > a$

Upper bound.

Default value is 1.0.

Notes

Its probability density function is defined as:

$f_X(x) = \frac{1}{b-a}, \quad x \in [a; b]$

with $a < b$ .

Its first moments are defined as:

$\begin{eqnarray*} \Expect{X} & = & \frac{a+b}{2} \\ \Var{X} & = & \frac{(b-a)^2}{12} \end{eqnarray*}$

Examples

Create a distribution:

>>> import openturns as ot
>>> distribution = ot.Uniform(2.0, 3.0)

Create it from the alternative parametrization $(\mu, \sigma)$ :

>>> distribution2 = ot.UniformMuSigma(3.0, 1.15).getDistribution()

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.
`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`(*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.
`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.
`getA`()	Accessor to the distribution's lower bound.
`getB`()	Accessor to the distribution's upper bound.
`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`()	Accessor to the correlation matrix.
`getCovariance`()	Accessor to the covariance matrix.
`getDescription`()	Accessor to the componentwise description.
`getDimension`()	Accessor to the dimension of the distribution.
`getDispersionIndicator`()	Dispersion indicator accessor.
`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.
`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.
`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.
`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.
`setA`(a)	Accessor to the distribution's lower bound.
`setB`(b)	Accessor to the distribution's upper bound.
`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`(parameter)	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)¶

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 $I^*_{\alpha} = [a_1, b_1] \times \dots \times [a_d, b_d]$ such that there exists $\beta \in [0,1]$ which satisfies the equations $a_i = F_i^{-1}((1 - \beta) / 2)$ and $b_i = F_i^{-1}((1 + \beta) / 2)$ for all $i$ and $\mu(I^*_{\alpha}) = \alpha$ .

Examples

We consider a Normal(2) distribution with zero mean, unit standard deviation and independent components. We note $\Phi_2$ its cdf. Due to symetries of the distribution, the bilateral confidence interval is $I^*_{\alpha} = [-a, a] \times \times [-a, a]$ where $a = \Phi^{-1}((1+\beta)/2)$ where $\Phi$ is the marginal cdf of each component. Then $\beta$ is such that $\Phi_2(I^*_{\alpha}) = \alpha$ . As $\Phi_2(I^*_{\alpha}) = (2\Phi(a) - 1)^2 = _beta^2$ , then, $\beta$ is equal to $\beta = \sqrt{\alpha} \simeq 0.9486$ and $a \simeq -1.9488$ .

>>> import openturns as ot
>>> dist = ot.Normal(2)
>>> confInt = dist.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.
betafloat: The probability $\beta$ .

Examples

We consider a Normal(2) distribution with zero mean, unit standard deviation and independent components. We note $\Phi_2$ its cdf. Due to symetries of the distribution, the bilateral confidence interval is $I^*_{\alpha} = [-a, a] \times [-a, a]$ where $a = \Phi^{-1}((1 + \beta) / 2)$ where $\Phi$ is the marginal cdf of each 1D marginal standard Gaussian component. Then $\beta$ is such that $\Phi_2(I^*_{\alpha}) = \alpha$ . As $\Phi_2(I^*_{\alpha}) = (2\Phi(a) - 1)^2 = \beta^2$ , then, $\beta$ is equal to $\beta = \sqrt{\alpha} = 0.9486$ and $a = -1.9488$ with 4 significant digits.

>>> import openturns as ot
>>> dist = ot.Normal(2)
>>> confInt, beta = dist.computeBilateralConfidenceIntervalWithMarginalProbability(0.9)

computeCDF(*args)¶

Compute the cumulative distribution function.

Parameters:

xsequence of float, 2-d sequence of float: Point in $\Rset^d$ .

Returns:

Ffloat, Point: CDF value at x.

Notes

The cumulative distribution function is defined as:

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

computeCDFGradient(*args)¶

Compute the gradient of the cumulative distribution function.

Parameters:

xsequence of float: Point in $\Rset^d$ .

Returns:

dFdthetaPoint: Partial derivatives of the CDF with respect to the distribution parameters at x.

computeCharacteristicFunction(x)¶

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: Point in $\Rset^d$ .

Returns:

Cfloat, Point: Complementary CDF value at x.

See also

computeSurvivalFunction

Notes

The complementary cumulative distribution function.

$1 - F_{\vect{X}}(\vect{x}) = 1 - \Prob{\bigcap_{i = 1}^d \left\{X_i \leq x_i \right\}}, \qquad \vect{x} \in \Rset^d$

Warning

The complementary CDF is different from the survival function (except for scalar distributions).

computeConditionalCDF(*args)¶

Compute the conditional cumulative distribution function.

Parameters:

xjfloat, sequence of float: Conditional CDF input.
xcondsequence of float, 2-d sequence of float with size $j-1$ , $j \leq d$: Conditioning values for the components $(X_{1}, \dots, X_{j-1})$ .

Returns:

pfloat, sequence of float: Conditional CDF value at xj given xcond.

Notes

Let $\vect{X}$ be a random vector of dimension $d$ . The conditional cumulative distribution function of the component $X_j$ given that the components of indices $k \leq j-1$ are fixed to $(x_1, \dots, x_{j-1})$ is defined by:

$F_{X_j \mid X_1, \ldots, X_{j - 1}}(x_j; x_1, \dots, x_{j-1}) = \Prob{X_j \leq x_j \mid X_1=x_1, \ldots, X_{j-1}=x_{j-1}}.$

For $j=1$ , it reduces to $F_{X_1}(x_1)$ .

computeConditionalDDF(x, y)¶

Compute the conditional derivative density function of the last component.

With respect to the other fixed components.

Parameters:

xjfloat, sequence of float: Conditional CDF input.
xcondsequence of float, 2-d sequence of float with size $j-1$ , $j \leq d$: Conditioning values for the components $(X_{1}, \dots, X_{j-1})$ .

Returns:

ddffloat,: Conditional DDF value at xj given xcond.

See also

computeDDF, computeConditionalCDF

Notes

Let $\vect{X}$ be a random vector of dimension $d$ . The conditional derivative density function of the component $X_j$ given that the components of indices $k \leq j-1$ are fixed to $(x_1, \dots, x_{j-1})$ is defined by:

$\dfrac{d^2}{d\,x_j^2}F_{X_j \mid X_1, \ldots, X_{j - 1}}(x_j; x_1, \dots, x_{j-1})$

where for $2 \leq j \leq d$ :

$F_{X_j \mid X_1, \ldots, X_{j - 1}}(x_j; x_1, \dots, x_{j-1}) = \Prob{X_j \leq x_j \mid X_1=x_1, \ldots, X_{j-1}=x_{j-1}}.$

For $j=1$ , it reduces to $\dfrac{d^2}{d\,x_1^2}F_{X_1}(x_1)$ , ie the DDF of the first component at $x_1$ .

computeConditionalPDF(*args)¶

Compute the conditional probability density function.

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

Parameters:

xjfloat, sequence of float,: Conditional CDF input.
xcondsequence of float, 2-d sequence of float with size $j-1$ , $j \leq d$: Conditioning values for the components $(X_{1}, \dots, X_{j-1})$ .

Returns:

cpdffloat, sequence of float: Conditional PDF at xd, given xcond.

See also

computePDF, computeConditionalCDF

Notes

Let $\vect{X}$ be a random vector of dimension $d$ . The conditional probability density function of the component $X_j$ given that the components of indices $k \leq j-1$ are fixed to $(x_1, \dots, x_{j-1})$ is defined by:

$\dfrac{d}{d\,x_j}F_{X_j \mid X_1, \ldots, X_{j - 1}}(x_j; x_1, \dots, x_{j-1})$

where for $2 \leq j \leq d$ :

$F_{X_j \mid X_1, \ldots, X_{j - 1}}(x_j; x_1, \dots, x_{j-1}) = \Prob{X_j \leq x_j \mid X_1=x_1, \ldots, X_{j-1}=x_{j-1}}.$

For $j=1$ , it reduces to $\dfrac{d}{d\,x_1}F_{X_1}(x_1)$ .

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, $p \in [0, 1]$: Conditional quantile.
xcondsequence of float, 2-d sequence of float with size $j-1$ , $j \leq d$: Conditioning values for the components $(X_{1}, \dots, X_{j-1})$ .

Returns:

xjfloat: Conditional quantile of order p of the component $X_j$ given that the components of indices $k \leq j-1$ are fixed and equal to xcond.

See also

computeQuantile, computeConditionalCDF

Notes

Let $\vect{X}$ be a random vector of dimension $d$ . The conditional quantile of order $p$ of the component $X_j$ given that the components of indices $k \leq j-1$ are fixed to $(x_1, \dots, x_{j-1})$ is defined by:

$F^{-1}_{X_j \mid X_1, \ldots, X_{j - 1}}(p; x_1, \dots, x_{j-1})$

where $F^{-1}$ is the quantile function. For $j=1$ , it reduces to $F^{-1}_{X_1}(p)$ .

computeDDF(*args)¶

Compute the derivative density function.

Parameters:

xsequence of float, 2-d sequence of float: The input value where the conditional derivative density function must be evaluated.

Returns:

ddfPoint, Sample: DDF value at x.

Notes

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

$\vect{\nabla}_{\vect{x}} f_{\vect{X}}(\vect{x}) = \Tr{\left(\frac{\partial f_{\vect{X}}(\vect{x})}{\partial x_1}, \dots, \frac{\partial f_{\vect{X}}(\vect{x})}{\partial x_d}\right)}, \quad \vect{x} \in \Rset^d$

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 z.

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, $p \in [0; 1]$: Value 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

Among the points $\vect{x}$ that satisfy

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