%feature("docstring") OT::AnalyticalResult
"Analytical result.

Available constructors:
    AnalyticalResult(*designPoint, limitStateVariable, isInFailureSpace*)

Notes
-----
Structure created by the method run() of the :class:`~openturns.Analytical`
class and obtained thanks to its method getAnalyticalResult().

Parameters
----------
designPoint : sequence of float
    Design point in the standard space resulting from the optimization
    algorithm.
limitStateVariable : :class:`~openturns.Event`
    Event of which the probability is calculated.
isInFailureSpace : bool
    Indicates whether the origin of the standard space is in the failure space."

// ---------------------------------------------------------------------

%feature("docstring") OT::AnalyticalResult::drawHasoferReliabilityIndexSensitivity
"Draw the sensitivity of the Hasofer Reliability Index.

Parameters
----------
width : float, optional
    Value to calculate the shift position of the :class:`~openturns.BarPlot`.
    By default it is 1.0.

Returns
-------
graphCollection : sequence of two :class:`~openturns.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."

// ---------------------------------------------------------------------

%feature("docstring") OT::AnalyticalResult::drawImportanceFactors
"Draw the importance factors.

Parameters
----------
type : int, optional
    See :meth:`getImportanceFactors`

Returns
-------
graph : :class:`~openturns.Graph`
    Pie of the importance factors of the probabilistic variables."

// ---------------------------------------------------------------------

%feature("docstring") OT::AnalyticalResult::getImportanceFactors
"Accessor to the importance factors.

Parameters
----------
type : int, 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*.

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

  .. math::

    \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 :

  .. math::

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

  where

  .. math::

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

- If the importance factors are evaluated as the square of the physical sensitivities :

  .. math::
      \alpha_i^2 = \displaystyle \frac{s_i^2}{{\|s\|}^2}

  where

  .. math::
      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^*)

Returns
-------
factors : :class:`~openturns.PointWithDescription`
    Sequence containing the importance factors with a description for each
    component."

// ---------------------------------------------------------------------

%feature("docstring") OT::AnalyticalResult::getOptimizationResult
"Accessor to the result of the optimization problem.

Returns
-------
result : :class:`~openturns.OptimizationResult`
    Contains the design point in the standard space and information concerning
    the convergence of the optimization algorithm."

// ---------------------------------------------------------------------

%feature("docstring") OT::AnalyticalResult::setOptimizationResult
"Accessor to the result of the optimization problem.

Parameters
----------
result : :class:`~openturns.OptimizationResult`
    Contains the design point in the standard space and information concerning
    the convergence of the optimization algorithm."

// ---------------------------------------------------------------------

%feature("docstring") OT::AnalyticalResult::getHasoferReliabilityIndex
"Accessor to the Hasofer Reliability Index.

Returns
-------
index : float
    Hasofer Reliability Index which is the distance of the design point from
    the origin of the standard space :math:`\beta_{HL}=||\vect{u}^*||`."

// ---------------------------------------------------------------------

%feature("docstring") OT::AnalyticalResult::getHasoferReliabilityIndexSensitivity
"Accessor to the sensitivities of the Hasofer Reliability Index.

Returns
-------
sensitivity : :class:`~openturns.PointWithDescription`
    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."

// ---------------------------------------------------------------------

%feature("docstring") OT::AnalyticalResult::getLimitStateVariable
"Accessor to the event of which the probability is calculated.

Returns
-------
limitStateVariable : :class:`~openturns.Event`
    Event of which the probability is calculated."

// ---------------------------------------------------------------------

%feature("docstring") OT::AnalyticalResult::getMeanPointInStandardEventDomain
"Accessor to the mean point in the standard event domain.

Returns
-------
meanPoint : :class:`~openturns.Point`
    Mean point of the standard space distribution restricted to the event
    domain:
    :math:`\displaystyle \frac{1}{E_1(-\beta)}\int_{\beta}^{\infty} u_1 p_1(u_1)\di{u_1}`
    where :math:`E_1` is the spheric univariate distribution of the standard
    space and :math:`\beta` the reliability index."

// ---------------------------------------------------------------------

%feature("docstring") OT::AnalyticalResult::setMeanPointInStandardEventDomain
"Accessor to the mean point in the standard event domain.

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

// ---------------------------------------------------------------------

%feature("docstring") OT::AnalyticalResult::getPhysicalSpaceDesignPoint
"Accessor to the design point in the physical space.

Returns
-------
designPoint : :class:`~openturns.Point`
    Design point in the physical space resulting from the optimization
    algorithm."

// ---------------------------------------------------------------------

%feature("docstring") OT::AnalyticalResult::getStandardSpaceDesignPoint
"Accessor to the design point in the standard space.

Returns
-------
designPoint : :class:`~openturns.Point`
    Design point in the standard space resulting from the optimization
    algorithm."

// ---------------------------------------------------------------------

%feature("docstring") OT::AnalyticalResult::setStandardSpaceDesignPoint
"Accessor to the design point in the standard space.

Parameters
----------
designPoint : sequence of float
    Design point in the standard space resulting from the optimization
    algorithm."

// ---------------------------------------------------------------------

%feature("docstring") OT::AnalyticalResult::getIsStandardPointOriginInFailureSpace
"Accessor to know if the standard point origin is in the failure space.

Returns
-------
isInFailureSpace : bool
    Indicates whether the origin of the standard space is in the failure space."

// ---------------------------------------------------------------------

%feature("docstring") OT::AnalyticalResult::setIsStandardPointOriginInFailureSpace
"Accessor to specify if the standard point origin is in the failure space.

Parameters
----------
isInFailureSpace : bool
    Indicates whether the origin of the standard space is in the failure space."
