StrongMaximumTest

class StrongMaximumTest(*args)

Strong Maximum Test.

Available constructors:

StrongMaximumTest(event, designPoint, importanceLevel, accuracyLevel, confidenceLevel)

StrongMaximumTest(event, designPoint, importanceLevel, accuracyLevel, pointNumber)

Parameters:

event : StandardEvent

Failure event \cD_f defining in the standard space (U-space).

designPoint : sequence of float

Design point in the standard space resulting from the optimization algorithm.

importanceLevel : float, 0 < \varepsilon < 1

Importance level.

accuracyLevel : float, \tau > 0

Accuracy level. It is recommanded to take \tau \leq 4.

confidenceLevel : positive float, 0 < (1-q) < 1

Confidence level.

pointNumber : int, N > 0

Number of points used to perform the Strong Maximum Test, evaluated by the limit state function.

Notes

The Strong Maximum Test helps to evaluate the quality of the design point P^* resulting from the optimization algorithm launched for example by the Analytical class or its derived classes FORM and SORM. It checks whether the design point computed is :

  • the true design point, which means a global maximum point,
  • a strong design point, which means that there is no other local maximum located on the event boundary and which likelihood is slightly inferior to the design point one.

The Strong Maximum Test will sample the sphere centered on the origin of the standard space and of radius : R = \beta_{HL} (1 + \tau \delta_{\varepsilon}) with \tau > 0, \delta_{\varepsilon}=\sqrt{1 - 2 \frac{ln(\varepsilon)}{\beta_{HL}^2}} - 1 and \beta_{HL} the Hasofer-Lind reliability index.

The number N of the simulations sampling the sphere of radius R is determined to ensure that the test detects with a probability greater than (1-q) any point of \cD_f outside the design point vicinity which contribution to P_f is not negligeable (i.e. which density value in the U-space is greater than \varepsilon times the density value at the design point).

Examples

>>> 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)
>>> # FORM analyse to get the design point
>>> myAlgo = ot.FORM(ot.AbdoRackwitz(), myEvent, [50.0, 1.0, 10.0, 5.0])
>>> myAlgo.run()
>>> FORMresult = myAlgo.getResult()
>>> designPoint = FORMresult.getStandardSpaceDesignPoint()
>>> # Strong Max Test
>>> myStandardEvent = ot.StandardEvent(myEvent)
>>> myTest = ot.StrongMaximumTest(myStandardEvent, designPoint, 0.15, 3, 0.99)
>>> myTest.run()

Methods

getAccuracyLevel() Accessor to the accuracy level.
getClassName() Accessor to the object’s name.
getConfidenceLevel() Accessor to the confidence level.
getDeltaEpsilon() Accessor to the parameter \delta_{\varepsilon}.
getDesignPointVicinity() Accessor to the design point vinicity.
getEvent() Accessor to the event in the standard space.
getFarDesignPointVerifyingEventPoints() Accessor to the points verifying the event and far of the design point.
getFarDesignPointVerifyingEventValues() Accessor to values of the limit state function.
getFarDesignPointViolatingEventPoints() Accessor to the points not verifying the event and far of the design point.
getFarDesignPointViolatingEventValues() Accessor to values of the limit state function.
getId() Accessor to the object’s id.
getImportanceLevel() Accessor to the importance level.
getName() Accessor to the object’s name.
getNearDesignPointVerifyingEventPoints() Accessor to the points verifying the event and near of the design point.
getNearDesignPointVerifyingEventValues() Accessor to values of the limit state function.
getNearDesignPointViolatingEventPoints() Accessor to the points not verifying the event and near of the design point.
getNearDesignPointViolatingEventValues() Accessor to values of the limit state function.
getPointNumber() Accessor to the number of points.
getShadowedId() Accessor to the object’s shadowed id.
getStandardSpaceDesignPoint() Accessor to the design point in the standard space.
getVisibility() Accessor to the object’s visibility state.
hasName() Test if the object is named.
hasVisibleName() Test if the object has a distinguishable name.
run() Perform the Strong Maximum Test.
setName(name) Accessor to the object’s name.
setShadowedId(id) Accessor to the object’s shadowed id.
setVisibility(visible) Accessor to the object’s visibility state.
__init__(*args)

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

getAccuracyLevel()

Accessor to the accuracy level.

Returns:

accuracy : positive float

Accuracy level \tau.

getClassName()

Accessor to the object’s name.

Returns:

class_name : str

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

getConfidenceLevel()

Accessor to the confidence level.

Returns:

level : positive float

Confidence level (1-q).

getDeltaEpsilon()

Accessor to the parameter \delta_{\varepsilon}.

Returns:

delta_epsilon : float

\delta_{\varepsilon} = \sqrt{1 - 2 \frac{ln(\varepsilon)}{\beta^2}} - 1.

getDesignPointVicinity()

Accessor to the design point vinicity.

Returns:

vicinity : float, v > 0

Design point vinicity v = \frac{1} {1 + \tau \delta_{\varepsilon}}.

getEvent()

Accessor to the event in the standard space.

Returns:

event : StandardEvent

Failure event \cD_f in the standard space on which is based the Strong Maximum Test.

getFarDesignPointVerifyingEventPoints()

Accessor to the points verifying the event and far of the design point.

Returns:

points : Sample

The points of the discretized sphere which are out of the vicinity of the standard design point and which verify the event.

getFarDesignPointVerifyingEventValues()

Accessor to values of the limit state function.

Returns:

values : Sample

The values of the limit state function on the points of the discretized sphere which are out of the vicinity of the standard design point and which verify the event.

getFarDesignPointViolatingEventPoints()

Accessor to the points not verifying the event and far of the design point.

Returns:

points : Sample

The points of the discretized sphere which are out of the vicinity of the standard design point and which don’t verify the event.

getFarDesignPointViolatingEventValues()

Accessor to values of the limit state function.

Returns:

values : Sample

The values of the limit state function on the points of the discretized sphere which are out of the vicinity of the standard design point and which don’t verify the event.

getId()

Accessor to the object’s id.

Returns:

id : int

Internal unique identifier.

getImportanceLevel()

Accessor to the importance level.

Returns:

level : float

Importance level \varepsilon.

getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getNearDesignPointVerifyingEventPoints()

Accessor to the points verifying the event and near of the design point.

Returns:

points : Sample

The points of the discretized sphere which are inside the vicinity of the standard design point and which verify the event.

getNearDesignPointVerifyingEventValues()

Accessor to values of the limit state function.

Returns:

values : Sample

The values of the limit state function on the points of the discretized sphere which are inside the vicinity of the standard design point and which verify the event.

getNearDesignPointViolatingEventPoints()

Accessor to the points not verifying the event and near of the design point.

Returns:

point : Sample

The points of the discretized sphere which are out of the vicinity of the standard design point and which don’t verify the event.

getNearDesignPointViolatingEventValues()

Accessor to values of the limit state function.

Returns:

values : Sample

The values of the limit state function on the points of the discretized sphere which are inside the vicinity of the standard design point and which don’t verify the event.

getPointNumber()

Accessor to the number of points.

Returns:

number : int, N > 0

Number of points used to perform the Strong Maximum Test, evaluated by the limit state function.

getShadowedId()

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.

getStandardSpaceDesignPoint()

Accessor to the design point in the standard space.

Returns:

point : Point

Design point in the standard space.

getVisibility()

Accessor to the object’s visibility state.

Returns:

visible : bool

Visibility flag.

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.

run()

Perform the Strong Maximum Test.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

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.