StandardEvent¶

class StandardEvent(*args)¶

Event defined in the standard space.

Available constructor:

StandardEvent(antecedent, comparisonOperator, threshold)

StandardEvent(event)

Parameters

antecedentRandomVector of dimension 1: Output variable of interest.
comparisonOperatorComparisonOperator: Comparison operator used to compare antecedent with threshold.
thresholdfloat: threshold we want to compare to antecedent.
eventRandomVector: Physical event associated with the standard event to be created.

See also

Analytical, SORM, FORM, SORMResult, FORMResult, StrongMaximumTest, ImportanceSamplingExperiment

Notes

An event is defined as follows:

$\cD_f = \{\vect{X} \in \Rset^n \, / \, g(\vect{X},\vect{d}) \le 0\}$

where $\vect{X}$ denotes a random input vector, representing the sources of uncertainties, $\vect{d}$ is a determinist vector, representing the fixed variables and $g(\vect{X},\vect{d})$ is the limit state function of the model.

One way to evaluate the probability content $P_f$ of the event $\cD_f$ :

$P_f = \int_{g(\vect{X},\vect{d})\le 0}f_\vect{X}(\vect{x})\di{\vect{x}}$

is to use an isoprobabilistic transformation to move from the physical space to a standard normal space (U-space) where distributions are spherical (invariant by rotation by definition), with zero mean, unit variance and unit correlation matrix. The usual isoprobabilistic transformations are the Generalized Nataf transformation and Rosenblatt transformation.

In that new U-space, the event can be expressed in terms of the transformed limit state function of the model $g$

$\cD_f = \{\vect{U} \in \Rset^n \, | \, g(\vect{U}\,,\,\vect{d}) \le 0\}$

Examples

A StandardEvent created from a limit state function :

>>> import openturns as ot
>>> myFunction = ot.SymbolicFunction(['E', 'F', 'L', 'I'], ['-F*L^3/(3*E*I)'])
>>> myDistribution = ot.Normal(4)
>>> vect = ot.RandomVector(myDistribution)
>>> output = ot.CompositeRandomVector(myFunction, vect)
>>> myStandardEvent = ot.StandardEvent(output, ot.Less(), 1.0)

A StandardEvent based on an event :

>>> myEvent = ot.ThresholdEvent(output, ot.Less(), 1.0)
>>> myStandardEvent = ot.StandardEvent(myEvent)

Methods

`getAntecedent`()	Accessor to the antecedent RandomVector in case of a composite RandomVector.
`getClassName`()	Accessor to the object's name.
`getCovariance`()	Accessor to the covariance of the RandomVector.
`getDescription`()	Accessor to the description of the RandomVector.
`getDimension`()	Accessor to the dimension of the RandomVector.
`getDistribution`()	Accessor to the distribution of the RandomVector.
`getDomain`()	Accessor to the domain of the Event.
`getFunction`()	Accessor to the Function in case of a composite RandomVector.
`getId`()	Accessor to the object's id.
`getImplementation`()	Accessor to the underlying implementation.
`getMarginal`(*args)	Get the random vector corresponding to the $i^{th}$ marginal component(s).
`getMean`()	Accessor to the mean of the RandomVector.
`getName`()	Accessor to the object's name.
`getOperator`()	Accessor to the comparaison operator of the Event.
`getParameter`()	Accessor to the parameter of the distribution.
`getParameterDescription`()	Accessor to the parameter description of the distribution.
`getRealization`()	Compute one realization of the RandomVector.
`getSample`(size)	Compute realizations of the RandomVector.
`getThreshold`()	Accessor to the threshold of the Event.
`intersect`(other)	Intersection of two events.
`isComposite`()	Accessor to know if the RandomVector is a composite one.
`isEvent`()	Whether the random vector is an event.
`join`(other)	Union of two events.
`setDescription`(description)	Accessor to the description of the RandomVector.
`setName`(name)	Accessor to the object's name.
`setParameter`(parameters)	Accessor to the parameter of the distribution.

__init__(*args)¶

getAntecedent()¶

Accessor to the antecedent RandomVector in case of a composite RandomVector.

Returns

antecedentRandomVector: Antecedent RandomVector $\vect{X}$ in case of a CompositeRandomVector such as: $\vect{Y}=f(\vect{X})$ .

getClassName()¶

Accessor to the object’s name.

Returns

class_namestr: The object class name (object.__class__.__name__).

getCovariance()¶

Accessor to the covariance of the RandomVector.

Returns

covarianceCovarianceMatrix: Covariance of the considered UsualRandomVector.

Examples

>>> import openturns as ot
>>> distribution = ot.Normal([0.0, 0.5], [1.0, 1.5], ot.CorrelationMatrix(2))
>>> randomVector = ot.RandomVector(distribution)
>>> ot.RandomGenerator.SetSeed(0)
>>> print(randomVector.getCovariance())
[[ 1    0    ]
 [ 0    2.25 ]]

getDescription()¶

Accessor to the description of the RandomVector.

Returns

descriptionDescription: Describes the components of the RandomVector.

getDimension()¶

Accessor to the dimension of the RandomVector.

Returns

dimensionpositive int: Dimension of the RandomVector.

getDistribution()¶

Accessor to the distribution of the RandomVector.

Returns

distributionDistribution: Distribution of the considered UsualRandomVector.

Examples

>>> import openturns as ot
>>> distribution = ot.Normal([0.0, 0.0], [1.0, 1.0], ot.CorrelationMatrix(2))
>>> randomVector = ot.RandomVector(distribution)
>>> ot.RandomGenerator.SetSeed(0)
>>> print(randomVector.getDistribution())
Normal(mu = [0,0], sigma = [1,1], R = [[ 1 0 ]
 [ 0 1 ]])

getDomain()¶

Accessor to the domain of the Event.

Returns

domainDomain: Describes the domain of an event.

getFunction()¶

Accessor to the Function in case of a composite RandomVector.

Returns

functionFunction: Function used to define a CompositeRandomVector as the image through this function of the antecedent $\vect{X}$ : $\vect{Y}=f(\vect{X})$ .

getId()¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

getImplementation()¶

Accessor to the underlying implementation.

Returns

implImplementation: The implementation class.

getMarginal(*args)¶

Get the random vector corresponding to the $i^{th}$ marginal component(s).

Parameters

iint or list of ints, $0\leq i < dim$: Indicates the component(s) concerned. $dim$ is the dimension of the RandomVector.

Returns

vectorRandomVector: RandomVector restricted to the concerned components.

Notes

Let’s note $\vect{Y}=\Tr{(Y_1,\dots,Y_n)}$ a random vector and $I \in [1,n]$ a set of indices. If $\vect{Y}$ is a UsualRandomVector, the subvector is defined by $\tilde{\vect{Y}}=\Tr{(Y_i)}_{i \in I}$ . If $\vect{Y}$ is a CompositeRandomVector, defined by $\vect{Y}=f(\vect{X})$ with $f=(f_1,\dots,f_n)$ , $f_i$ some scalar functions, the subvector is $\tilde{\vect{Y}}=(f_i(\vect{X}))_{i \in I}$ .

Examples

>>> import openturns as ot
>>> distribution = ot.Normal([0.0, 0.0], [1.0, 1.0], ot.CorrelationMatrix(2))
>>> randomVector = ot.RandomVector(distribution)
>>> ot.RandomGenerator.SetSeed(0)
>>> print(randomVector.getMarginal(1).getRealization())
[0.608202]
>>> print(randomVector.getMarginal(1).getDistribution())
Normal(mu = 0, sigma = 1)

getMean()¶

Accessor to the mean of the RandomVector.

Returns

meanPoint: Mean of the considered UsualRandomVector.

Examples

>>> import openturns as ot
>>> distribution = ot.Normal([0.0, 0.5], [1.0, 1.5], ot.CorrelationMatrix(2))
>>> randomVector = ot.RandomVector(distribution)
>>> ot.RandomGenerator.SetSeed(0)
>>> print(randomVector.getMean())
[0,0.5]

getName()¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getOperator()¶

Accessor to the comparaison operator of the Event.

Returns

operatorComparisonOperator: Comparaison operator used to define the RandomVector.

getParameter()¶

Accessor to the parameter of the distribution.

Returns

parameterPoint: Parameter values.

getParameterDescription()¶

Accessor to the parameter description of the distribution.

Returns

descriptionDescription: Parameter names.

getRealization()¶

Compute one realization of the RandomVector.

Returns

aRealizationPoint: Sequence of values randomly determined from the RandomVector definition. In the case of an event: one realization of the event (considered as a Bernoulli variable) which is a boolean value (1 for the realization of the event and 0 else).

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