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 recommended 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.

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.
`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.
`getStandardSpaceDesignPoint`()	Accessor to the design point in the standard space.
`hasName`()	Test if the object is named.
`run`()	Perform the Strong Maximum Test.
`setName`(name)	Accessor to the object's name.

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()

__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.

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.

getStandardSpaceDesignPoint()¶

Accessor to the design point in the standard space.

Returns:

pointPoint: Design point in the standard space.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

run()¶: Perform the Strong Maximum Test.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

Examples using the class¶

Test the design point with the Strong Maximum Test

OpenTURNS

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

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StrongMaximumTest¶

Examples using the class¶