.. _sensitivity_form: Sensitivity Factors from FORM method ------------------------------------ | Sensitivity Factors are evaluated under the following context: :math:`\vect{X}` denotes a random input vector, representing the sources of uncertainties, :math:`\pdf` its joint density probability, :math:`\vect{d}` a deterministic vector, representing the fixed variables :math:`g(\vect{X}\,,\,\vect{d})` the limit state function of the model, :math:`\cD_f = \{\vect{X} \in \Rset^n \, / \, g(\vect{X}\,,\,\vect{d}) \le 0\}` the event considered here and :math:`{g(\vect{X}\,,\,\vect{d}) = 0}` its boundary (also called limit state surface). | The probability content of the event :math:`\cD_f` is :math:`P_f`: .. math:: :label: PfX11 P_f = \int_{g(\vect{X}\,,\,\vect{d}) \le 0} \pdf\, d\vect{x}. | In this context, the probability :math:`P_f` can often be efficiently estimated by FORM or SORM approximations. | The FORM importance factors offer a way to analyze the sensitivity of the probability the realization of the event with respect to the parameters of the probability distribution of :math:`\vect{X}`. | A sensitivity factor is defined as the derivative of the Hasofer-Lind reliability index with respect to the parameter :math:`\theta`. The parameter :math:`\theta` is a parameter in a distribution of the random vector :math:`\vect{X}`. | If :math:`\vect{\theta}` represents the vector of all the parameters of the distribution of :math:`\vect{X}` which appear in the definition of the isoprobabilistic transformation :math:`T`, and :math:`U_{\vect{\theta}}^{*}` the design point associated to the event considered in the :math:`U`-space, and if the mapping of the limit state function by the :math:`T` is noted :math:`h(\vect{U}\,,\,\vect{\theta}) = g[T^{-1}(\vect{U}\,,\,\vect{\theta}), \vect{d}]`, then the sensitivity factors vector is defined as: .. math:: \nabla_{\vect{\theta}} \beta_{HL} = \displaystyle +\frac{1}{||\nabla_{\vect{\theta}} h(U_{\vect{\theta}}^{*}, \vect{d})||} \nabla_{\vect{u}} G(U_{\vect{\theta}}^{*}, \vect{d}). The sensitivity factors indicate the importance on the Hasofer-Lind reliability index (refer to ) of the value of the parameters used to define the distribution of the random vector :math:`\vect{X}`. Here, the event considered is explicited directly from the limit state function :math:`g(\vect{X}\,,\,\vect{d})`: this is the classical structural reliability formulation. However, if the event is a threshold exceedance, it is useful to explicit the variable of interest :math:`Z=\tilde{g}(\vect{X}\,,\,\vect{d})`, evaluated from the model :math:`\tilde{g}(.)`. In that case, the event considered, associated to the threshold :math:`z_s` has the formulation: :math:`\cD_f = \{ \vect{X} \in \Rset^n \, / \, Z=\tilde{g}(\vect{X}\,,\,\vect{d}) > z_s \}` and the limit state function is : :math:`g(\vect{X}\,,\,\vect{d}) = z_s - Z = z_s - \tilde{g}(\vect{X}\,,\,\vect{d})`. :math:`P_f` is the threshold exceedance probability, defined as: :math:`P_f = P(Z \geq z_s) = \int_{g(\vect{X}\,,\,\vect{d}) \le 0} \pdf\, d\vect{x}`. Thus, the FORM sensitivity factors offer a way to rank the importance of the parameters of the input components with respect to the threshold exceedance by the quantity of interest :math:`Z`. They can be seen as a specific sensitivity analysis technique dedicated to the quantity Z around a particular threshold rather than to its variance. .. topic:: API: - See :class:`~openturns.FORM` - See :class:`~openturns.SORM` .. topic:: Examples: - See :doc:`/auto_reliability_sensitivity/reliability/plot_estimate_probability_form` .. topic:: References: - O. Ditlevsen, H.O. Madsen, 2004, "Structural reliability methods", Department of mechanical engineering technical university of Denmark - Maritime engineering, internet publication.