Probability bounds analysis: Difference between revisions

Revision as of 03:53, 11 December 2019

Revision as of 04:00, 11 December 2019

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==Sampling-based computation==

Some analysts<ref>Alvarez, D. A., 2006. On the calculation of the bounds of probability of events using infinite random sets. ''International Journal of Approximate Reasoning'' '''43''': 241–267.</ref><ref>Baraldi, P., Popescu, I. C., Zio, E., 2008. Predicting the time to failure of a randomly degrading component by a hybrid Monte Carlo and possibilistic method. ''IEEE Proc. International Conference on Prognostics and Health Management''.</ref><ref>Batarseh, O. G., Wang, Y., 2008. Reliable simulation with input uncertainties using an interval-based approach. ''IEEE Proc. Winter Simulation Conference''.</ref><ref>Roy, Christopher J., and Michael S. Balch (2012). A holistic approach to uncertainty quantification with application to supersonic nozzle thrust. ''International Journal for Uncertainty Quantification'' '''2''' (4): 363–81 {{doi|10.1615/Int.J.UncertaintyQuantification.2012003562}}.</ref><ref>Zhang, H., Mullen, R. L., Muhanna, R. L. (2010). Interval Monte Carlo methods for structural reliability. ''Structural Safety'' '''32''': 183–190.</ref><ref>Zhang, H., Dai, H., Beer, M., Wang, W. (2012). Structural reliability analysis on the basis of small samples: an interval quasi-Monte Carlo method. ''Mechanical Systems and Signal Processing'' '''37''' (1–2): 137–51 {{doi|10.1016/j.ymssp.2012.03.001}}.</ref> use sampling-based approaches to computing probability bounds, including [[Monte Carlo simulation]], [[Latin hypercube]] methods or [[importance sampling]]. These approaches cannot assure mathematical rigor in the result because such simulation methods are approximations, although their performance can generally be improved simply by increasing the number of replications in the simulation. Thus, unlike the analytical theorems or methods based on mathematical programming, sampling-based calculations usually cannot produce [[verified computing|verified computations]]. However, sampling-based methods can be very useful in addressing a variety of problems which are computationally [[NP-hard|difficult]] to solve analytically or even to rigorously bound <ref> Kabir, H. D., Khosravi, A., Hosen, M. A., & Nahavandi, S. (2018). Neural Network-based Uncertainty Quantification: A Survey of Methodologies and Applications. IEEE Access. Vol. 6, Pages 36218 - 36234, {{DOI|10.1109/ACCESS.2018.2836917}} </ref> . One important example is the use of Cauchy-deviate sampling to avoid the [[curse of dimensionality]] in propagating [[Interval (mathematics)|interval]] uncertainty through high-dimensional problems.<ref>Trejo, R., Kreinovich, V. (2001). [http://www.cs.utep.edu/vladik/2000/tr00-17.pdf Error estimations for indirect measurements: randomized vs. deterministic algorithms for ‘black-box’ programs]. ''Handbook on Randomized Computing'', S. Rajasekaran, P. Pardalos, J. Reif, and J. Rolim (eds.), Kluwer, 673–729.</ref>

==Relationship to other uncertainty propagation approaches==