Editing Sensitivity analysis (section)

=== One-at-a-time (OAT) ===
{{main|One-factor-at-a-time method}}
One of the simplest and most common approaches is that of changing one-factor-at-a-time (OAT), to see what effect this produces on the output.<ref>{{cite journal |first=J. |last=Campbell |year=2008 |title=Photosynthetic Control of Atmospheric Carbonyl Sulfide During the Growing Season |journal=[[Science (journal)|Science]] |volume=322 |issue=5904 |pages=1085–1088 |doi=10.1126/science.1164015 |display-authors=etal |pmid=19008442|bibcode=2008Sci...322.1085C |s2cid=206515456 |url=http://www.escholarship.org/uc/item/82r9s2x3 }}</ref><ref>{{cite journal |first1=R. |last1=Bailis |first2=M. |last2=Ezzati |first3=D. |last3=Kammen |year=2005 |title=Mortality and Greenhouse Gas Impacts of Biomass and Petroleum Energy Futures in Africa |journal=[[Science (journal)|Science]] |volume=308 |issue= 5718|pages=98–103 |doi=10.1126/science.1106881 |pmid=15802601|bibcode=2005Sci...308...98B |s2cid=14404609 }}</ref><ref>{{cite journal |first=J. |last=Murphy |year=2004 |title=Quantification of modelling uncertainties in a large ensemble of climate change simulations |journal=[[Nature (journal)|Nature]] |volume=430 |issue= 7001|pages=768–772 |doi= 10.1038/nature02771|display-authors=etal |pmid=15306806|bibcode=2004Natur.430..768M|s2cid=980153 }}</ref> OAT customarily involves

* moving one input variable, keeping others at their baseline (nominal) values, then,
* returning the variable to its nominal value, then repeating for each of the other inputs in the same way.

Sensitivity may then be measured by monitoring changes in the output, e.g. by [[partial derivatives]] or [[linear regression]]. This appears a logical approach as any change observed in the output will unambiguously be due to the single variable changed. Furthermore, by changing one variable at a time, one can keep all other variables fixed to their central or baseline values. This increases the comparability of the results (all 'effects' are computed with reference to the same central point in space) and minimizes the chances of computer program crashes, more likely when several input factors are changed simultaneously.
OAT is frequently preferred by modelers because of practical reasons. In case of model failure under OAT analysis the modeler immediately knows which is the input factor responsible for the failure.

Despite its simplicity however, this approach does not fully explore the input space, since it does not take into account the simultaneous variation of input variables. This means that the OAT approach cannot detect the presence of [[Interaction (statistics)|interactions]] between input variables and is unsuitable for nonlinear models.<ref>{{cite journal |last=Czitrom|first=Veronica|author-link= Veronica Czitrom |year=1999 |title=One-Factor-at-a-Time Versus Designed Experiments |journal=American Statistician |volume=53 |issue=2 |pages=126–131 |doi=10.2307/2685731|jstor= 2685731}}</ref>

The proportion of input space which remains unexplored with an OAT approach grows superexponentially with the number of inputs. For example, a 3-variable parameter space which is explored one-at-a-time is equivalent to taking points along the x, y, and z axes of a cube centered at the origin. The [[convex hull]] bounding all these points is an [[octahedron]] which has a volume only 1/6th of the total parameter space. More generally, the convex hull of the axes of a hyperrectangle forms a [[hyperoctahedron]] which has a volume fraction of <math>1/n!</math>. With 5 inputs, the explored space already drops to less than 1% of the total parameter space. And even this is an overestimate, since the off-axis volume is not actually being sampled at all. Compare this to random sampling of the space, where the convex hull approaches the entire volume as more points are added.<ref>{{cite journal |last1=Gatzouras |first1=D |last2=Giannopoulos |first2=A |title=Threshold for the volume spanned by random points with independent coordinates |journal=[[Israel Journal of Mathematics]] |date=2009 |volume=169 |issue=1 |pages=125–153 | doi=10.1007/s11856-009-0007-z | doi-access=free}}</ref> While the sparsity of OAT is theoretically not a concern for [[linear model]]s, true linearity is rare in nature.