FORMResult¶

class FORMResult(*args)¶

Result of a FORM analysis.

Available constructors:: FORMResult(designPoint, limitStateVariable, isInFailureSpace)

Parameters

designPointsequence of float: Design point in the standard space resulting from the optimization algorithm.
limitStateVariableRandomVector: Event of which the probability is calculated.
isInFailureSpacebool: Indicates whether the origin of the standard space is in the failure space.

See also

Analytical, AnalyticalResult, SORM, SORMResult, FORM, StrongMaximumTest

Notes

Structure created by the method run() of the FORM class and obtained thanks to its method getResult().

Methods

`drawEventProbabilitySensitivity`(*args)	Draw the sensitivities of the FORM failure probability.
`drawHasoferReliabilityIndexSensitivity`(*args)	Draw the sensitivity of the Hasofer Reliability Index.
`drawImportanceFactors`(*args)	Draw the importance factors.
`getClassName`()	Accessor to the object's name.
`getEventProbability`()	Accessor to the failure probability $P_f$ .
`getEventProbabilitySensitivity`()	Accessor to the sentivities of the FORM failure probability $P_f$ .
`getGeneralisedReliabilityIndex`()	Accessor to the Generalised Reliability Index.
`getHasoferReliabilityIndex`()	Accessor to the Hasofer Reliability Index.
`getHasoferReliabilityIndexSensitivity`()	Accessor to the sensitivities of the Hasofer Reliability Index.
`getId`()	Accessor to the object's id.
`getImportanceFactors`(*args)	Accessor to the importance factors.
`getIsStandardPointOriginInFailureSpace`()	Accessor to know if the standard point origin is in the failure space.
`getLimitStateVariable`()	Accessor to the event of which the probability is calculated.
`getMeanPointInStandardEventDomain`()	Accessor to the mean point in the standard event domain.
`getName`()	Accessor to the object's name.
`getOptimizationResult`()	Accessor to the result of the optimization problem.
`getPhysicalSpaceDesignPoint`()	Accessor to the design point in the physical space.
`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.
`setIsStandardPointOriginInFailureSpace`(...)	Accessor to specify if the standard point origin is in the failure space.
`setMeanPointInStandardEventDomain`(...)	Accessor to the mean point in the standard event domain.
`setName`(name)	Accessor to the object's name.
`setOptimizationResult`(optimizationResult)	Accessor to the result of the optimization problem.
`setShadowedId`(id)	Accessor to the object's shadowed id.
`setStandardSpaceDesignPoint`(...)	Accessor to the design point in the standard space.
`setVisibility`(visible)	Accessor to the object's visibility state.

__init__(*args)¶

drawEventProbabilitySensitivity(*args)¶

Draw the sensitivities of the FORM failure probability.

Parameters

widthfloat, optional: Value to calculate the shift position of the BarPlot. By default it is 1.0.

Returns

graphCollectionlist of two Graph containing a barplot: The first graph drawing the sensitivities of the FORM failure probability with regards to the parameters of the probabilistic input vector. The second graph drawing the sensitivities of the FORM failure probability with regards to the parameters of the dependence structure of the probabilistic input vector.

drawHasoferReliabilityIndexSensitivity(*args)¶

Draw the sensitivity of the Hasofer Reliability Index.

Parameters

widthfloat, optional: Value to calculate the shift position of the BarPlot. By default it is 1.0.

Returns

graphCollectionsequence of two Graph containing a barplot: The first graph drawing the sensitivity of the Hasofer Reliability Index to the parameters of the marginals of the probabilistic input vector. The second graph drawing the sensitivity of the Hasofer Reliability Index to the parameters of the dependence structure of the probabilistic input vector.

drawImportanceFactors(*args)¶

Draw the importance factors.

Parameters

typeint, optional: See getImportanceFactors()

Returns

graphGraph: Pie of the importance factors of the probabilistic variables.

getClassName()¶

Accessor to the object’s name.

Returns

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

getEventProbability()¶

Accessor to the failure probability $P_f$ .

Returns

probabilitypositive float: The FORM failure probability $P_f$ .

getEventProbabilitySensitivity()¶

Accessor to the sentivities of the FORM failure probability $P_f$ .

Returns

sentivitiesPoint: Sentivities of the FORM failure probability with regards to the parameters of the probabilistic input vector and to parameters of the dependence structure of the probabilistic input vector.

getGeneralisedReliabilityIndex()¶

Accessor to the Generalised Reliability Index.

Returns

indexfloat: Generalised reliability index $\beta_g$ from the FORM failure probability is equal to $\pm$ the Hasofer reliability index $\beta_{HL}$ according to the fact the standard space center fulfills the event or not.

getHasoferReliabilityIndex()¶

Accessor to the Hasofer Reliability Index.

Returns

indexfloat: Hasofer Reliability Index which is the distance of the design point from the origin of the standard space $\beta_{HL}=||\vect{u}^*||$ .

getHasoferReliabilityIndexSensitivity()¶

Accessor to the sensitivities of the Hasofer Reliability Index.

Refer to Sensitivity Factors from FORM method.

Returns

sensitivityPointWithDescription: Sequence containing the sensitivities of the Hasofer Reliability Index to the parameters of the probabilistic input vector (marginals and dependence structure) with a description for each component.

getId()¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

getImportanceFactors(*args)¶

Accessor to the importance factors.

Refer Importance factors from FORM method.

Parameters

typeint, optional

When ot.AnalyticalResult.ELLIPTICAL, the importance factors are evaluated as the square of the co-factors of the design point in the elliptical space of the iso-probabilistic transformation (Y-space).
When ot.AnalyticalResult.CLASSICAL they are evaluated as the square of the co-factors of the design point in the U-space.
When ot.AnalyticalResult.PHYSICAL, the importance factors are evaluated as the square of the physical sensitivities.

By default type = ot.AnalyticalResult.ELLIPTICAL.

Returns

factorsPointWithDescription: Sequence containing the importance factors with a description for each component.

Notes

If the importance factors are evaluated as the square of the co-factors of the design point in the U-space :

$\alpha_i^2 = \frac{(u_i^*)^2} {\beta_{HL}^2}$
If the importance factors are evaluated as the square of the co-factors of the design point in the Y-space :

$\alpha_i^2 = \frac{(y_i^*)^2} {\|\vect{y}^*\|^2}$

where

$Y^* = \left( \begin{array}{c} E^{-1}\circ F_1(X_1^*) \\ E^{-1}\circ F_2(X_2^*) \\ \vdots \\ E^{-1}\circ F_n(X_n^*) \end{array} \right)$

with $\vect{X}^*$ is the design point in the physical space and $E$ the univariate standard CDF of the elliptical space. In the case where the input distribution of $\vect{X}$ has an elliptical copula $C_E$ , then $E$ has the same type as $C_E$ . In the case where the input distribution of $\vect{X}$ has a copula $C$ which is not elliptical, then $E=\Phi$ where $\Phi$ is the CDF of the standard normal.
If the importance factors are evaluated as the square of the physical sensitivities :

$\alpha_i^2 = \displaystyle \frac{s_i^2}{{\|s\|}^2}$

where

$s_i = \displaystyle \frac{\partial \beta}{\partial x_i} (x^*) = \sum_{j=1}^n \frac{\partial \beta}{\partial u_i} \frac{\partial u_j}{\partial x_i} (x^*)$

getIsStandardPointOriginInFailureSpace()¶

Accessor to know if the standard point origin is in the failure space.

Returns

isInFailureSpacebool: Indicates whether the origin of the standard space is in the failure space.

getLimitStateVariable()¶

Accessor to the event of which the probability is calculated.

Returns

limitStateVariableRandomVector: Event of which the probability is calculated.

getMeanPointInStandardEventDomain()¶

Accessor to the mean point in the standard event domain.

Returns

meanPointPoint: Mean point of the standard space distribution restricted to the event domain: $\displaystyle \frac{1}{E_1(-\beta)}\int_{\beta}^{\infty} u_1 p_1(u_1)\di{u_1}$ where $E_1$ is the spheric univariate distribution of the standard space and $\beta$ the reliability index.

getName()¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getOptimizationResult()¶

Accessor to the result of the optimization problem.

Returns

resultOptimizationResult: Contains the design point in the standard space and information concerning the convergence of the optimization algorithm.

getPhysicalSpaceDesignPoint()¶

Accessor to the design point in the physical space.

Returns

designPointPoint: Design point in the physical space resulting from the optimization algorithm.

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns

idint: Internal unique identifier.

getStandardSpaceDesignPoint()¶

Accessor to the design point in the standard space.

Returns

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

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.

setIsStandardPointOriginInFailureSpace(isStandardPointOriginInFailureSpace)¶

Accessor to specify if the standard point origin is in the failure space.

Parameters

isInFailureSpacebool: Indicates whether the origin of the standard space is in the failure space.

setMeanPointInStandardEventDomain(meanPointInStandardEventDomain)¶

Accessor to the mean point in the standard event domain.

Parameters

meanPointsequence of float: Mean point of the standard space distribution restricted to the event domain: $\displaystyle \frac{1}{E_1(-\beta)}\int_{\beta}^{\infty} u_1 p_1(u_1)\di{u_1}$ where $E_1$ is the spheric univariate distribution of the standard space and $\beta$ the reliability index.

setName(name)¶

Accessor to the object’s name.

Parameters

namestr: The name of the object.

setOptimizationResult(optimizationResult)¶

Accessor to the result of the optimization problem.

Parameters

resultOptimizationResult: Contains the design point in the standard space and information concerning the convergence of the optimization algorithm.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters

idint: Internal unique identifier.

setStandardSpaceDesignPoint(standardSpaceDesignPoint)¶

Accessor to the design point in the standard space.

Parameters

designPointsequence of float: Design point in the standard space resulting from the optimization algorithm.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

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