Event

class Event(*args)

Event.

Available constructor:

Event()

Event(antecedent, comparisonOperator, threshold)

Event(antecedent, domain)

Event(process, domain)

Parameters:

antecedent : RandomVector of dimension 1

Output variable of interest.

comparisonOperator : ComparisonOperator

Comparison operator used to compare antecedent with threshold.

threshold : float

threshold we want to compare to antecedent.

domain : Domain

Domain failure.

process : Process

Stochastic process.

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. The probability content of the event \cD_f is P_f:

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

Here, the event considered is explicited directly from the limit state function g(\vect{X}\,,\,\vect{d}) : this is the classical structural reliability formulation. However, if the event is a threshold exceedance, it is useful to explicite the variable of interest Z=\tilde{g}(\vect{X}\,,\,\vect{d}), evaluated from the model \tilde{g}(.). In that case, the event considered, associated to the threshold z_s has the formulation:

\cD_f = \{ \vect{X} \in \Rset^n \, / \, Z=\tilde{g}(\vect{X}\,,\,\vect{d}) > z_s \}

and the limit state function is:

g(\vect{X}\,,\,\vect{d}) &= z_s - Z \\
                         &= z_s - \tilde{g}(\vect{X}\,,\,\vect{d})

P_f is the threshold exceedance probability, defined as:

P_f &= P(Z \geq z_s) \\
    &= \int_{g(\vect{X}, \vect{d}) \le 0} \pdf\di{\vect{x}}

Examples

An event 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([50.0, 1.0, 10.0, 5.0], [1.0]*4, ot.IdentityMatrix(4))
>>> vect = ot.RandomVector(myDistribution)
>>> output = ot.RandomVector(myFunction, vect)
>>> myEvent = ot.Event(output, ot.Less(), -3.0)

A composite random vector based event:

>>> X = ot.RandomVector(ot.Normal(2))
>>> model = ot.SymbolicFunction(['x0', 'x1'], ['x0', 'x1'])
>>> Y = ot.RandomVector(model, X)
>>> # The domain: [0, 1]^2
>>> domain = ot.Interval(2)
>>> # The event
>>> event = ot.Event(Y, domain)

A process based event:

>>> # The input process
>>> X = ot.WhiteNoise(ot.Normal(2))
>>> # The domain: [0, 1]^2
>>> domain = ot.Interval(2)
>>> # The event
>>> event = ot.Event(X, domain)

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(*args) 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.
getRealization() Compute one realization of the RandomVector.
getSample(size) Compute realizations of the RandomVector.
getThreshold() Accessor to the threshold of the Event.
isComposite() Accessor to know if the RandomVector is a composite one.
setDescription(description) Accessor to the description of the RandomVector.
setName(name) Accessor to the object’s name.
__init__(*args)

x.__init__(…) initializes x; see help(type(x)) for signature

getAntecedent()

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

Returns:

antecedent : RandomVector

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_name : str

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

getCovariance()

Accessor to the covariance of the RandomVector.

Returns:

covariance : CovarianceMatrix

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:

description : Description

Describes the components of the RandomVector.

getDimension()

Accessor to the dimension of the RandomVector.

Returns:

dimension : positive int

Dimension of the RandomVector.

getDistribution()

Accessor to the distribution of the RandomVector.

Returns:

distribution : Distribution

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:

domain : Domain

Describes the domain of an event.

getFunction()

Accessor to the Function in case of a composite RandomVector.

Returns:

function : Function

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:

id : int

Internal unique identifier.

getImplementation(*args)

Accessor to the underlying implementation.

Returns:

impl : Implementation

The implementation class.

getMarginal(*args)

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

Parameters:

i : int or list of ints, 0\leq i < dim

Indicates the component(s) concerned. dim is the dimension of the RandomVector.

Returns:

vector : RandomVector

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:

mean : Point

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:

name : str

The name of the object.

getOperator()

Accessor to the comparaison operator of the Event.

Returns:

operator : ComparisonOperator

Comparaison operator used to define the Event.

getRealization()

Compute one realization of the RandomVector.

Returns:

aRealization : Point

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).

See also

getSample

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.getRealization())
[0.608202,-1.26617]
>>> print(randomVector.getRealization())
[-0.438266,1.20548]
getSample(size)

Compute realizations of the RandomVector.

Parameters:

n : int, n \geq 0

Number of realizations needed.

Returns:

realizations : Sample

n sequences of values randomly determined from the RandomVector definition. In the case of an event: n realizations of the event (considered as a Bernoulli variable) which are boolean values (1 for the realization of the event and 0 else).

See also

getRealization

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.getSample(3))
    [ X0        X1        ]
0 : [  0.608202 -1.26617  ]
1 : [ -0.438266  1.20548  ]
2 : [ -2.18139   0.350042 ]
getThreshold()

Accessor to the threshold of the Event.

Returns:

threshold : float

Threshold of the Event.

isComposite()

Accessor to know if the RandomVector is a composite one.

Returns:

isComposite : bool

Indicates if the RandomVector is of type Composite or not.

setDescription(description)

Accessor to the description of the RandomVector.

Parameters:

description : str or sequence of str

Describes the components of the RandomVector.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.