AnalyticalResult

class AnalyticalResult(*args)

Analytical result.

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.

Methods

drawHasoferReliabilityIndexSensitivity(*args)

Draw the sensitivity of the Hasofer Reliability Index.

drawImportanceFactors(*args)

Draw the importance factors.

getClassName()

Accessor to the object's name.

getHasoferReliabilityIndex()

Accessor to the Hasofer Reliability Index.

getHasoferReliabilityIndexSensitivity()

Accessor to the sensitivities of the Hasofer Reliability Index.

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.

getStandardSpaceDesignPoint()

Accessor to the design point in the standard space.

hasName()

Test if the object is named.

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.

setStandardSpaceDesignPoint(...)

Accessor to the design point in the standard space.

Notes

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

__init__(*args)
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__).

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.

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.

getImportanceFactors(*args)

Accessor to the importance factors.

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 in the standrad event domain.

Notes

This method is used in the context of the FORM approximation of the probability of an event. Let \cD denote the standard event domain.

In the standard space, the transformed random vector \vect{U} follows a spherical distribution. All the univariate marginals are identical. Let E denote the CDF of any univariate marginal and e its PDF.

Let \vect{u}^* denote the design point in the standard space and \beta_{HL} the associated Hasofer-Lind reliability index.

By symmetry in the standard space, the mean point in the standard event domain is computed as follows:

\Expect{\vect{U}|\vect{U} \in \cD} = \Expect{\|\vect{U}\| | \|\vect{U}\| \geq \beta_{HL}} \vect{u}^* = \frac{1}{E(-\beta_{HL})} \left(\int_{\beta}^{+\infty} u e(u)\, \di{u} \right)\vect{u}^*

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.

getStandardSpaceDesignPoint()

Accessor to the design point in the standard space.

Returns:
designPointPoint

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

hasName()

Test if the object is named.

Returns:
hasNamebool

True if the name is not empty.

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

The mean point of the standard space distribution restricted to the event domain.

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.

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.