Editing Sensitivity analysis (section)

=== Variogram analysis of response surfaces (''VARS'') ===
One of the major shortcomings of the previous sensitivity analysis methods is that none of them considers the spatially ordered structure of the response surface/output of the model <math>Y=f(X)</math> in the parameter space. By utilizing the concepts of directional [[variogram]]s and covariograms, variogram analysis of response surfaces (VARS) addresses this weakness through recognizing a spatially continuous correlation structure to the values of <math>Y</math>, and hence also to the values of <math> \frac{\partial Y}{\partial x_i} </math>.<ref>{{cite journal|last1=Razavi|first1=Saman|last2=Gupta|first2=Hoshin V.|title=A new framework for comprehensive, robust, and efficient global sensitivity analysis: 1. Theory|journal=Water Resources Research|date=January 2016|volume=52|issue=1|pages=423–439|doi=10.1002/2015WR017558|language=en|issn=1944-7973|bibcode=2016WRR....52..423R|doi-access=free}}</ref><ref>{{cite journal|last1=Razavi|first1=Saman|last2=Gupta|first2=Hoshin V.|title=A new framework for comprehensive, robust, and efficient global sensitivity analysis: 2. Application|journal=Water Resources Research|date=January 2016|volume=52|issue=1|pages=440–455|doi=10.1002/2015WR017559|language=en|issn=1944-7973|bibcode=2016WRR....52..440R|doi-access=free}}</ref>

Basically, the higher the variability the more heterogeneous is the response surface along a particular direction/parameter, at a specific perturbation scale. Accordingly, in the VARS framework, the values of directional [[variogram]]s for a given perturbation scale can be considered as a comprehensive illustration of sensitivity information, through linking variogram analysis to both direction and perturbation scale concepts. As a result, the VARS framework accounts for the fact that sensitivity is a scale-dependent concept, and thus overcomes the scale issue of traditional sensitivity analysis methods.<ref>{{cite journal|last1=Haghnegahdar|first1=Amin|last2=Razavi|first2=Saman|title=Insights into sensitivity analysis of Earth and environmental systems models: On the impact of parameter perturbation scale|journal=Environmental Modelling & Software|date=September 2017|volume=95|pages=115–131|doi=10.1016/j.envsoft.2017.03.031|bibcode=2017EnvMS..95..115H }}</ref> More importantly, VARS is able to provide relatively stable and statistically robust estimates of parameter sensitivity with much lower computational cost than other strategies (about two orders of magnitude more efficient).<ref>{{cite book|last1=Gupta|first1=H|last2=Razavi|first2=S|editor1-last=Petropoulos|editor1-first=George|editor2-last=Srivastava|editor2-first=Prashant|title=Sensitivity Analysis in Earth Observation Modelling|date=2016|isbn=9780128030318|pages=397–415|edition=1st|chapter-url=https://www.elsevier.com/books/sensitivity-analysis-in-earth-observation-modelling/petropoulos/978-0-12-803011-0|language=en|chapter=Challenges and Future Outlook of Sensitivity Analysis|publisher=Elsevier}}</ref> Noteworthy, it has been shown that there is a theoretical link between the VARS framework and the [[Variance-based sensitivity analysis|variance-based]] and derivative-based approaches.