StrongMaximumTest¶

class StrongMaximumTest(*args)¶

Strong Maximum Test.

Refer to Strong Maximum Test.

Available constructors:

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

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

Parameters

eventStandardEvent: Failure event $\cD_f$ defining in the standard space (U-space).
designPointsequence of float: Design point in the standard space resulting from the optimization algorithm.
importanceLevelfloat, $0 < \varepsilon < 1$: Importance level.
accuracyLevelfloat, $\tau > 0$: Accuracy level. It is recommanded to take $\tau \leq 4$ .
confidenceLevelpositive float, $0 < (1-q) < 1$: Confidence level.
pointNumberint, $N > 0$: Number of points used to perform the Strong Maximum Test, evaluated by the limit state function.

See also

Analytical, SORM, FORM, SORMResult, FORMResult

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.CompositeRandomVector(myFunction, vect)
>>> myEvent = ot.ThresholdEvent(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)¶

getAccuracyLevel()¶

Accessor to the accuracy level.

Returns

accuracypositive float: Accuracy level $\tau$ .

getClassName()¶

Accessor to the object’s name.

Returns

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

getConfidenceLevel()¶

Accessor to the confidence level.

Returns

levelpositive float: Confidence level $(1-q)$ .

getDeltaEpsilon()¶

Accessor to the parameter $\delta_{\varepsilon}$ .

Returns

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

getDesignPointVicinity()¶

Accessor to the design point vinicity.

Returns

vicinityfloat, $v > 0$: Design point vinicity $v = \frac{1} {1 + \tau \delta_{\varepsilon}}$ .

getEvent()¶

Accessor to the event in the standard space.

Returns

eventStandardEvent: 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

pointsSample: 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

valuesSample: 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

pointsSample: 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

valuesSample: 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

idint: Internal unique identifier.

getImportanceLevel()¶

Accessor to the importance level.

Returns

levelfloat: Importance level $\varepsilon$ .

getName()¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getNearDesignPointVerifyingEventPoints()¶

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

Returns

pointsSample: 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

valuesSample: 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

pointSample: 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

valuesSample: 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

numberint, $N > 0$: 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

idint: Internal unique identifier.

getStandardSpaceDesignPoint()¶

Accessor to the design point in the standard space.

Returns

pointPoint: Design point in the standard space.

getVisibility()¶

Accessor to the object’s visibility state.

Returns

visiblebool: Visibility flag.

hasName()¶

Test if the object is named.

Returns

hasNamebool: True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns

hasVisibleNamebool: 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

namestr: The name of the object.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters

idint: Internal unique identifier.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

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

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