Editing Sensitivity analysis (section)

=== Variance-based methods ===
{{Main|Variance-based sensitivity analysis}}

Variance-based methods<ref>{{cite journal | last1 = Sobol' | first1 = I | year = 1990 | title = Sensitivity estimates for nonlinear mathematical models | journal = Matematicheskoe Modelirovanie | volume = 2 | pages = 112–118 | language = ru }}; translated in English in {{cite journal | last1 = Sobol' | first1 = I | year = 1993 | title = Sensitivity analysis for non-linear mathematical models | journal = Mathematical Modeling & Computational Experiment | volume = 1 | pages = 407–414 }}</ref> are a class of probabilistic approaches which quantify the input and output uncertainties as [[random variable]]s, represented via their [[probability distribution]]s, and decompose the output variance into parts attributable to input variables and combinations of variables. The sensitivity of the output to an input variable is therefore measured by the amount of variance in the output caused by that input.

This amount is quantified and calculated using '''Sobol indices''': they represent the proportion of variance explained by an input or group of inputs. This expression essentially measures the contribution of <math>X_i</math> alone to the uncertainty (variance) in <math>Y</math> (averaged over variations in other variables), and is known as the ''' ''first-order sensitivity index'' ''' or ''' ''main effect index'' ''' or ''' ''main Sobol index'' ''' or ''' ''Sobol main index'' '''.

For an input <math>X_i</math>, Sobol index is defined as following:

<math display="block">S_i=\frac{V(\mathbb{E}[Y\vert X_i])}{V(Y)}</math> where <math>V(\cdot)</math> and <math>\mathbb{E}[\cdot]</math> denote the variance and expected value operators respectively.

Importantly, first-order sensitivity index of <math>X_i</math> does not measure the uncertainty caused by interactions <math>X_i</math> has with other variables. A further measure, known as the ''' ''total effect index'' ''', gives the total variance in <math>Y</math> caused by <math>X_i</math> and its interactions with any of the other input variables. The total effect index is given as following: <math display="block">S_i^T=1-\frac{V(\mathbb{E}[Y\vert X_{\sim i}])}{V(Y)}</math>where <math>X_{\sim i} = (X_1,...,X_{i-1},X_{i+1},...,X_p)</math> denotes the set of all input variables except <math>X_i</math>.

Variance-based methods allow full exploration of the input space, accounting for interactions, and nonlinear responses. For these reasons they are widely used when it is feasible to calculate them. Typically this calculation involves the use of [[Monte Carlo integration|Monte Carlo]] methods, but since this can involve many thousands of model runs, other methods (such as metamodels) can be used to reduce computational expense when necessary.