Uncertainty quantification

Uncertainty quantification (UQ) is defined as the quantitative characterization and reduction of uncertainties in applications.

The “mathematics of uncertainties” used to be mistakenly identified with only the fields of finance and economics. However, many problems in the fields of natural sciences and engineering are also rife with sources of uncertainty. Increasing application of computer simulation modeling to study such problems has unfolded a new construct in the form of uncertainty quantification (UQ).

Reasons

The reasons responsible to introduce uncertainty in a model may be:

1. The model structure, i.e., how accurately does a mathematical model describe the true system for a real-life situation,

2. The numerical approximation, i.e., how appropriately a numerical method is used in approximating the operation of the system,

3. The initial / boundary conditions, i.e., how precise are the data / information for initial and / or boundary conditions,

4. The data for input and/or model parameters.

Types of uncertainties

The following three types of uncertainties can be identified:

1. Uncertainty due to variability of input and / or model parameters when the characterization of the variability is available (e.g., with probability density functions, pdf),

2. Uncertainty due to variability of input and/or model parameters when the corresponding variability characterization is not available,

3. Uncertainty due to an unknown process or mechanism.

Type 1 uncertainty, which depends on chance, may be referred to as aleatory or statistical uncertainty. Type 2 and 3 are referred to as epistemic or systematic uncertainties.

It often happens in real life applications that all three types of uncertainties are present in the systems under study. Uncertainty quantification intends to work toward reducing type 2 and 3 uncertainties to type 1. The quantification for the type 1 uncertainty is relatively straightforward to perform. Techniques such as Monte Carlo are frequently used. Pdf can be represented by its moments (in the Gaussian case,the mean and covariance suffice), or more recently, by techniques such as Karhunen-Loève and polynomial chaos expansions. To evaluate type 2 and 3 uncertainties, the efforts are made to gain better knowledge of the system, process or mechanism. Methods such as fuzzy logic or evidence theory are used.

References

Further reading

* Bandemer, H, and Näther, W (1992) Fuzzy Data Analysis, Kluwer Academic Publishers, Dordrecht.
* Bandemer, H (1992) Modelling uncertain Data, Akademie-Verlag, Berlin.
* Bernardo, JM, and Smith, AF (1994) Bayesian theory, Wiley, Chichester New York Brisbane Toronto Singapore.
* Kruse, R, and Meyer, KD (1987) Statistics with Vague Data, Reidel, Dordrecht.
* Mood, AM, Graybill, FA, and Boes, DC (1974) Introduction to the theory of Statistics, McGraw-Hill, New York.
* Viertl, R (1996) Statistical Methods for Non-Precise Data, CRC Press, Boca Raton New York London Tokyo.
* Wright, G (1994) Subjective probability, Wiley, Chichester.
* Zimmermann, H- (1992) Fuzzy set theory and its applications, Kluwer Academic Publishers, Boston London.