Editing Sensitivity analysis (section)

=== Moment-independent methods ===
Moment-independent methods extend variance-based techniques by considering the probability density or cumulative distribution function of the model output <math>Y</math>. Thus, they do not refer to any particular [[Moment (mathematics)|moment]] of  <math>Y</math>, whence the name.

The moment-independent sensitivity measures of <math>X_i</math>, here denoted by <math>\xi_i</math>, can be defined through an equation similar to variance-based indices replacing the conditional expectation with a distance, as <math>\xi_i=E[d(P_Y,P_{Y|X_i})]</math>, where <math>d(\cdot,\cdot) </math> is a [[statistical distance]] [metric or divergence] between probability measures, <math>P_Y</math> and <math>P_{Y|X_i}</math> are the marginal and [[conditional probability]] measures of <math>Y</math>.<ref name="Borgonovo2014">{{Cite journal |vauthors=Borgonovo E, Tarantola S, Plischke E, Morris MD |date=2014 |title=Transformations and invariance in the sensitivity analysis of computer experiments |journal=Journal of the Royal Statistical Society |series=Series B (Statistical Methodology) |volume=76 |issue=5 |pages=925–947 |doi=10.1111/rssb.12052 |issn=1369-7412}}</ref>
 
If <math>d()\geq 0</math> is a [[Statistical distance|distance]], the moment-independent global sensitivity measure satisfies zero-independence. This is a relevant statistical property also known as Renyi's postulate D.<ref name="Renyi">{{Cite journal |last=Rényi |first=A |date=1 September 1959 |title=On measures of dependence |journal=Acta Mathematica Academiae Scientiarum Hungaricae |volume=10 |issue=3 |pages=441–451 |doi=10.1007/BF02024507 |issn=1588-2632}}</ref>

The class of moment-independent sensitivity measures includes indicators such as the <math>\delta </math> -importance measure,<ref name="Borgonovo2007">{{Cite journal |vauthors=Borgonovo E |date=June 2007 |title=A new uncertainty importance measure |journal=Reliability Engineering & System Safety |volume=92 |issue=6 |pages=771–784 |doi=10.1016/J.RESS.2006.04.015 |issn=0951-8320}}</ref> the new correlation coefficient of Chatterjee,<ref name="Chatterjee">{{Cite journal |vauthors=Chatterjee S |date=2 October 2021 |title=A New Coefficient of Correlation |journal=Journal of the American Statistical Association |volume=116 |issue=536 |pages=2009–2022 |arxiv=1909.10140 |doi=10.1080/01621459.2020.1758115 |issn=0162-1459}}</ref> the Wasserstein correlation of Wiesel <ref name="Wiesel">{{Cite journal |vauthors=Wiesel JC |date=November 2022 |title=Measuring association with Wasserstein distances |journal=Bernoulli |volume=28 |issue=4 |pages=2816–2832 |arxiv=2102.00356 |doi=10.3150/21-BEJ1438 |issn=1350-7265}}</ref> and the kernel-based sensitivity measures of Barr and Rabitz.<ref name="Barr">{{Cite journal |vauthors=Barr J, Rabitz H |date=31 March 2022 |title=A Generalized Kernel Method for Global Sensitivity Analysis |journal=SIAM/ASA Journal on Uncertainty Quantification |publisher=Society for Industrial and Applied Mathematics |volume=10 |issue=1 |pages=27–54 |doi=10.1137/20M1354829}}</ref>

Another measure for global sensitivity analysis, in the category of moment-independent approaches, is the PAWN index.<ref name="PAWN">{{Cite journal |vauthors=Pianosi F, Wagener T |date=2015 |title=A simple and efficient method for global sensitivity analysis based on cumulative distribution functions |journal=Environmental Modelling & Software |volume=67 |pages=1–11 |bibcode=2015EnvMS..67....1P |doi=10.1016/j.envsoft.2015.01.004 |doi-access=free}}</ref> {{citation needed span|It relies on [[cumulative distribution functions|Cumulative Distribution Functions]] (CDFs) to characterize the maximum distance between the unconditional output distribution and conditional output distribution (obtained by varying all input parameters and by setting the <math>i</math>-th input, consequentially). The difference between the unconditional and conditional output distribution is usually calculated using the [[Kolmogorov–Smirnov test]] (KS). The PAWN index for a given input parameter is then obtained by calculating the summary statistics over all KS values.|date=October 2024}}