Editing Sensitivity analysis (section)

=== Derivative-based local methods ===
Local derivative-based methods involve taking the [[partial derivative]] of the output  <math>Y</math> with respect to an input factor <math>X_i</math>:
:<math>
\left| \frac{\partial Y}{\partial X_i} \right |_{\textbf {x}^0 },
</math>
where the subscript '''x'''<sup>0</sup> indicates that the derivative is taken at some fixed point in the space of the input (hence the 'local' in the name of the class). Adjoint modelling<ref>{{cite book |last=Cacuci |first=Dan G. |title=Sensitivity and Uncertainty Analysis: Theory |volume=I |publisher=Chapman & Hall }}</ref><ref>{{cite book |last1=Cacuci |first1=Dan G. |first2=Mihaela |last2=Ionescu-Bujor |first3=Michael |last3=Navon |year=2005 |title=Sensitivity and Uncertainty Analysis: Applications to Large-Scale Systems |volume=II |publisher=Chapman & Hall }}</ref> and Automated Differentiation<ref>{{cite book |last=Griewank |first=A. |year=2000 |title=Evaluating Derivatives, Principles and Techniques of Algorithmic Differentiation |publisher=SIAM }}</ref> are methods which allow to compute all partial derivatives at a cost at most 4-6 times of that for evaluating the original function. Similar to OAT, local methods do not attempt to fully explore the input space, since they examine small perturbations, typically one variable at a time. It is possible to select similar samples from derivative-based sensitivity through Neural Networks and perform uncertainty quantification.

One advantage of the local methods is that it is possible to make a matrix to represent all the sensitivities in a system, thus providing an overview that cannot be achieved with global methods if there is a large number of input and output variables.<ref name="Possible">[https://ieeexplore.ieee.org/abstract/document/9206746 Kabir HD, Khosravi A, Nahavandi D, Nahavandi S. Uncertainty Quantification Neural Network from Similarity and Sensitivity. In2020 International Joint Conference on Neural Networks (IJCNN) 2020 Jul 19 (pp. 1-8). IEEE.]</ref>