Editing Propagation of uncertainty (section)

{{Short description|Effect of variables' uncertainties on the uncertainty of a function based on them}}
{{For|the propagation of uncertainty through time|Chaos theory#Sensitivity to initial conditions}}

In [[statistics]], '''propagation of uncertainty''' (or '''propagation of error''') is the effect of [[Variable (mathematics)|variables]]' [[uncertainty|uncertainties]] (or [[Errors and residuals in statistics|errors]], more specifically [[random error]]s) on the uncertainty of a [[function (mathematics)|function]] based on them. When the variables are the values of experimental measurements they have [[Observational error|uncertainties due to measurement limitations]] (e.g., instrument [[Accuracy and precision|precision]]) which propagate due to the combination of variables in the function.

The uncertainty ''u'' can be expressed in a number of ways.
It may be defined by the [[absolute error]] {{math|Δ''x''}}. Uncertainties can also be defined by the [[relative error]] {{math|(Δ''x'')/''x''}}, which is usually written as a percentage.
Most commonly, the uncertainty on a quantity is quantified in terms of the [[standard deviation]], {{mvar|σ}}, which is the positive square root of the [[variance]]. The value of a quantity and its error are then expressed as an interval {{math|''x'' ± ''u''}}. 
However, the most general way of characterizing uncertainty is by specifying its [[probability distribution]].
If the [[probability distribution]] of the variable is known or can be assumed, in theory it is possible to get any of its statistics. In particular, it is possible to derive [[confidence limits]] to describe the region within which the true value of the variable may be found. For example, the 68% confidence limits for a one-dimensional variable belonging to a [[normal distribution]] are approximately ± one standard deviation {{math|''σ''}} from the central value {{math|''x''}}, which means that the region {{math|''x'' ± ''σ''}} will cover the true value in roughly 68% of cases.

If the uncertainties are [[correlated]] then [[covariance]] must be taken into account. Correlation can arise from two different sources. First, the ''measurement errors'' may be correlated. Second, when the underlying values are correlated across a population, the ''uncertainties in the group averages'' will be correlated.<ref>{{cite web| last1=Kirchner | first1=James | title=Data Analysis Toolkit #5: Uncertainty Analysis and Error Propagation | url=http://seismo.berkeley.edu/~kirchner/eps_120/Toolkits/Toolkit_05.pdf|website=Berkeley Seismology Laboratory|publisher=University of California | access-date=22 April 2016}}</ref>

In a general context where a nonlinear function modifies the uncertain parameters (correlated or not), the standard tools to propagate uncertainty, and infer resulting quantity probability distribution/statistics, are sampling techniques from the [[Monte Carlo method]] family.<ref name="kr11">{{cite book|author-last1=Kroese |author-first1=D. P. |author-last2=Taimre |author-first2=T. |author-last3=Botev |author-first3=Z. I. |title=Handbook of Monte Carlo Methods |year=2011 |publisher=John Wiley & Sons}}</ref> For very large datasets or complex functions, the calculation of the error propagation may be very expensive so that a [[surrogate model]]<ref>{{Cite journal|last1=Ranftl|first1=Sascha|last2=von der Linden|first2=Wolfgang|date=2021-11-13|title=Bayesian Surrogate Analysis and Uncertainty Propagation|journal=Physical Sciences Forum|volume=3|issue=1|pages=6|doi=10.3390/psf2021003006|issn=2673-9984|doi-access=free |arxiv=2101.04038}}</ref> or a [[parallel computing]] strategy<ref>{{cite journal|author-last1=Atanassova |author-first1=E. |author-last2=Gurov |author-first2=T. |author-last3=Karaivanova |author-first3=A. |author-last4=Ivanovska |author-first4=S. |author-last5=Durchova |author-first5=M. |author-last6=Dimitrov |author-first6=D. |year=2016 |title=On the parallelization approaches for Intel MIC architecture |journal=AIP Conference Proceedings |volume=1773 |issue=1 |pages=070001 |doi=10.1063/1.4964983 |bibcode=2016AIPC.1773g0001A}}</ref><ref>{{cite journal|author-last1=Cunha Jr |author-first1=A. |author-last2=Nasser |author-first2=R. |author-last3=Sampaio |author-first3=R. |author-last4=Lopes |author-first4=H. |author-last5=Breitman |author-first5=K. |year=2014 |title=Uncertainty quantification through the Monte Carlo method in a cloud computing setting |journal=Computer Physics Communications |volume=185 |issue=5 |pages=1355–1363 |doi=10.1016/j.cpc.2014.01.006 |arxiv=2105.09512 |bibcode=2014CoPhC.185.1355C |s2cid=32376269}}</ref><ref>{{cite journal|author-last1=Lin |author-first1=Y. |author-last2=Wang |author-first2=F. |author-last3=Liu |author-first3=B. |year=2018 |title=Random number generators for large-scale parallel Monte Carlo simulations on FPGA |journal = Journal of Computational Physics |volume=360 |pages=93–103 |doi=10.1016/j.jcp.2018.01.029 |bibcode=2018JCoPh.360...93L}}</ref> may be necessary.

In some particular cases, the uncertainty propagation calculation can be done through simplistic algebraic procedures. Some of these scenarios are described below.