Uncertainty quantification
From Wikipedia, the free encyclopedia
Uncertainty quantification (UQ) is defined as the quantitative characterization and reduction of uncertainties in applications.
The “mathematics of uncertainties” is used to be mistakenly identified only with 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).
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[edit] 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.
[edit] 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 probably density functions),
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 may be referred to as aleatory (i.e., dependent on chance) uncertainty. Type 2 and 3 are referred to as epistemic 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. 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.
[edit] Goal of uncertainty quantification
The goal of uncertainty quantification is to assign an appropriate mathematical model to real-world situation with respect to objective and subjective uncertainty. The choice of an appropriate uncertainty model primarily depends on the characteristics of the available information. That is, the underlying reality with the sources of the uncertainty dictates the model. In each particular case this information must be analyzed and classified to be eligible for quantification. In general, one of the following three major uncertainty models provides a suitable basis:
1. Random quantification: Data-based information which is characterized by random fluctuations may be described with the aid of a traditional probabilistic model.
2. Fuzziness quantification: The fuzziness uncertainty model lends itself to describing imprecise, subjective, linguistic, and expert-specified information.
3. Fuzziness randomness quantification: The fuzzy randomness uncertainty model is particularly suitable for adequately quantifying uncertainty that comprises only some incomplete, fragmentary objective, data-based, randomly fluctuating information, which can simultaneously be dubious or imprecise and may additionally be amended by subjective, linguistic, expert-specified evaluations.
[edit] References
- [1] Wright, G (1994) Subjective probability, Wiley, Chichester.
- [2] Bernardo, JM, and Smith, AF (1994) Bayesian theory, Wiley, Chichester New York Brisbane Toronto Singapore.
- [3] Mood, AM, Graybill, FA, and Boes, DC (1974) Introduction to the theory of Statistics, McGraw-Hill, New York.
- [4] Bandemer, H (1992) Modelling uncertain Data, Akademie-Verlag, Berlin.
- [5] Zimmermann, H- (1992) Fuzzy set theory and its applications, Kluwer Academic Publishers, Boston London.
- [6] Viertl, R (1996) Statistical Methods for Non-Precise Data, CRC Press, Boca Raton New York London Tokyo.
- [7] Bandemer, H, and Näther, W (1992) Fuzzy Data Analysis, Kluwer Academic Publishers, Dordrecht.
- [8] Kruse, R, and Meyer, KD (1987) Statistics with Vague Data, Reidel, Dordrecht.
