# What is uncertainty quantification?

- 217 Downloads

## Abstract

Uncertainty is described by the cumulative distribution function (CDF). Using the CDF one describes all the main cases: the discrete case, the case when an absolutely continuous probability density exists, and the singular case, when it does not, or combinations of the three preceding cases. The reason one does not see any mention of uncertainty quantification in classical books, as Feller’s and Chung’s, is that they found no reason to call a CDF by another name. However, one has to acknowledge that to use a CDF to describe uncertainty is clumsy. The comparison of the CDFs of two random variables to evaluate which one of them is more uncertain is not evident. One feels that there must be a simpler way. Why not to use some small set of statistics to reduce a CDF to a simpler measure, easier to grasp? This seems a great idea, and indeed, one finds it in the literature. Several books are appearing dealing with the problem. We focus the discussion on three main cases: (1) to use mean and standard deviation to construct an envelope with them to make a nice graph; (2) to use mean and coefficient of variation; and (3) to use Shannon entropy, a number, that could allow an ordering for the uncertainties of all CDF that have entropy, a most desirable thing. The reductions consisting of replacing the CDF for a small set of statistics may indeed work in some cases. However, they do not always work, and moreover, the different measures they define may not be compatible. That is, the ordering of uncertainty may vary depending upon what set one chooses. Therefore, the “great idea” does not work so far, but they are happily used in the literature. One of the objectives of this paper is to show, with examples, that the three reductions used to “measure” uncertainties are not compatible. The reason it took so long to find out the mistake is that these reductions methods are applied to very complex problem that hide well the unsuitability of the reductions. Once one tests them with simpler examples, one clearly sees their inadequacy. Therefore, let us safely continue to use the CDF, while a good reduction is not found!

## Keywords

Uncertainty quantification Cumulative distribution function Measures of uncertainty Statistics Entropy Variance Coefficient of variation## Notes

### Acknowledgements

The authors acknowledge the support given by FAPERJ, CNPq, and CAPES.

## References

- 1.de Cursi ES, Sampaio R (2015) Uncertainty quantification and stochastic modeling with matlab. Elsevier, ISTE Press, LondonzbMATHGoogle Scholar
- 2.Smith R (2014) Uncertainty quantification: theory, implementation, and applications. SIAM, PhiladelphiazbMATHGoogle Scholar
- 3.Sulivan T (2015) Introduction to uncertainty quantification. Springer, New YorkCrossRefGoogle Scholar
- 4.Soize C (2017) Uncertainty quantification: an accelerated course with advanced applications in computational engineering. Springer, New YorkCrossRefzbMATHGoogle Scholar
- 5.Soize C (2012) Stochastic models of uncertainties in computational mechanics. American Society of Civil Engineers, New YorkCrossRefzbMATHGoogle Scholar
- 6.Ghanem R, Higdon D, Owhadi H (2017) Handbook of uncertainty quantification. Springer, New YorkCrossRefzbMATHGoogle Scholar
- 7.Feller W (1957) An introduction to probability theory and its applications, vol I and II. Wiley, AmsterdamzbMATHGoogle Scholar
- 8.Chung K (1974) A course in probability theory. Academic Press, CambridgezbMATHGoogle Scholar
- 9.Jaynes E (1957) Information theory and statistical mechanics. Phys Rev 106(4):620–630MathSciNetCrossRefzbMATHGoogle Scholar
- 10.Shannon C (1948) A mathematical theory of communication. Bell Syst Tech 27:379–423 (and 623–659)MathSciNetCrossRefzbMATHGoogle Scholar
- 11.Chen J, Eeden C, Zidek J (2010) Uncertainty and the conditional variance. Stat Probab Lett 80:1764–1770MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Khosravi A, Nahavandi S (2014) An optimized mean variance estimation method for uncertainty quantification of wind power forecasts. Electr Power Energy Syst 61:446–454CrossRefGoogle Scholar
- 13.Nordström J, Wahlsten M (2015) Variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations. J Comput Phys 282:1–22MathSciNetCrossRefzbMATHGoogle Scholar
- 14.Motra H, Hildebrand J, Wuttke F (2016) The Monte Carlo method for evaluating measurement uncertainty: application for determining the properties of materials. Probab Eng Mech 45:220–228CrossRefGoogle Scholar
- 15.Zidek J, Eeden C (2003) Uncertainty, entropy, variance and the effect of partial information, vol 42. Lecture notes-monograph series. Institute of Mathematical Statistics, BeachwoodGoogle Scholar
- 16.Conrad K (2016) Probability distributions and maximum entropy. http://www.math.uconn.edu/~kconrad/blurbs/analysis/entropypost.pdf. Retrieved July 24, 1–27
- 17.Grimmett G, Welsh D (1986) Probability an introduction. Oxford Science Publications, New YorkzbMATHGoogle Scholar
- 18.Sampaio R, Lima R (2012) Modelagem Estocástica e Geração de Amostras de Variáveis e Vetores Aleatórios. Notas de Matemática Aplicada, vol. 70. SBMAC. http://www.sbmac.org.br/arquivos/notas/livro_70.pdf
- 19.Lima R, Sampaio R (2017) Uncertainty quantification and cumulative distribution function: how are they related? In: 37th International workshop on bayesian inference and maximum entropy methods in science and engineering, MaxEnt 2017, Jarinu, SPGoogle Scholar
- 20.Ebrahimi N, Maasoumi E, Soofi E (1999) Ordering univariate distributions by entropy and variance. J Econ 90:317–336MathSciNetCrossRefzbMATHGoogle Scholar
- 21.Khodabin M, Ahmadabadi A (2010) Some properties of generalized gamma distribution. Math Sci 4(1):9–28MathSciNetzbMATHGoogle Scholar