Intervals and Probability Distributions

Intervals and Probability Distributions

Intervals have long found application in numerical methods for manipulating the distributions of random variables. This page contains links to resources related to this topic. Five ways to think about problems in this area include clouds, copulas, discrete convolutions (i.e. Cartesian products), Monte Carlo simulation, and stochastic dominance.

Scope: this Web site tries to avoid redundancy with the Imprecise Probabilities Project Web site (linked in section 4 below). The reader can find information there as well as here.

1. Request for updates
Please contribute information about relevant work of your own group, in particular individuals, references and download links. Please let us also know of anything we might consider adding to this page. With your input, this page can continue to grow and improve. Send input to berleant@iastate.edu or vladik@cs.utep.edu. Especially - many apologies if we neglected you or your work. Please let us know so we can fix it.

2. People

Please inform us of updates, corrections, and additions.

Daniel Berleant (Ames, Iowa, USA)

Carlos Carreras (Madrid, Spain)

John P. Davis (Bristol, UK)

Scott Ferson (Setauket, New York, USA)

Thomas Fetz (Innsbruck, Austria)

Janos Hajagos (Stony Brook, New York, USA)

Jim Hall (Bristol, UK)

Jon Helton (Albuquerque, USA)

Mac Hyman (Los Alamos, USA)

Igor Kozine (Roskilde, Denmark)

Vladik Kreinovich (El Paso, Texas, USA)

Vladimir Kuznetsov (Russia; deceased, book "Interval-valued statistical models" currently undergoing Russian-English translation)

Weiye Li (Los Alamos, USA)
Weldon Lodwick (Denver, USA)

Johannes Lüthi (Kufstein, Austria)

Charles F. Manski (Evanston, Illinois)

Arnold Neumaier (Wien, Austria)
Michael Oberguggenberger (Innsbruck, Austria)
William Oberkampf (Albuquerque, USA)
John Red-Horse (Albuquerque, USA)
Helen Regan (San Diego, USA)
Gerald Sheblé (Ames, Iowa, USA)
Fulvio Tonon (Salt Lake City, USA)
Lev Utkin (München, Germany)

Ian D. Walker (Clemson, USA)
Steven Wojtkiewicz (Albuquerque, USA)

Marco Zaffalon (IDSIA, Switzerland)
Jianzhong Zhang (Ames, Iowa, USA)

3. Tools

RiskCalc (http://www.ramas.com/riskcalc.htm)

Statool (http://class.ee.iastate.edu/berleant/home/Research/Pdfs/versions/statool/distribution/index.htm)

MatLab is often used as a basis for doing these kinds of calculations

4. Further Links and Resources

- Epistemic Uncertainty Project - has benchmark problems stated using intervals, and links to various papers addressing the problems
- First International Workshop on Soft Methods in Probability and Statistics

- Imprecise Probabilities Project Web site

- 3rd International Symposium on Imprecise Probabilities and Their Applications
- Special Track on Uncertain Reasoning (UR), FLAIRS Conferences: 2003 (http://www.irit.fr/UR-2003/UR-2003.html), 2002 (http://www.stat.uconn.edu/~ims/bulletin/october2001/node32.html)

- Beyond Monte Carlo: Introduction to Imprecise Probabilities (tutorial workshop in conjunction with the World Congress on Risk, 22 June 2003.

- Book by E.T. Jaynes (preprint) (in print)

- SIPTA (http://www.sipta.org/) and its mailing list (http://www.idsia.ch/mailman/listinfo/sipta)

5. References

Books (defined as coherent, non-thesis/dissertation, book-length works)

W. Fellin, H. Lessmann, M. Oberguggenberger, R. Vieider, Analyzing Uncertainty in Civil Engineering, Springer-Verlag, Berlin 2005.

Ferson, S., J. Hajagos, D. Berleant, J. Zhang, W. T. Tucker, L. Ginzburg, and W. Oberkampf, Dependence in Dempster-Shafer Theory and Probability Bounds Analysis, Technical Report SAND2004-XXXX, Sandia National Laboratory, forthcoming. Free. Alternative URL: http://www.ramas.com/d.pdf.
Scott Ferson, Vladik Kreinovich, Lev Ginzburg, David S. Myers, and Kari Sentz, Constructing Probability Boxes and Dempster-Shafer Structures, Sandia National Laboratories, Report SAND2002-4015, January 2003. File in pdf.
Halpern, J.Y., Reasoning About Uncertainty, MIT Press, 2003. ISBN 0-262-08320-5
Helton, J.C. and W.L. Oberkampf, eds., Special Volume on Alternative Representations of Epistemic Uncertainty, Reliability Engineering and System Safety, 85 (1-3) (July-Sept. 2004), pp. 1-369. Collection of papers focusing on the same challenge problems concerning system response under uncertainty. Free if your institution subscribes in paper or electronically. If your institution does not subscribe, maybe you know someone at an institution that does. (You might not want to pay out of pocket: $1116 from Elsevier customer service (not a typo). Maybe they would be willing to negotiate?)
Kuznetsov, Vladimir, Interval Statistical Models (in Russian), Radio i Svyaz, Moscow, 1991. Please send any further info about availability to berleant@iastate.edu.
Manski, C., Partial Identification of Probability Distributions, Springer-Verlag, New York, 2003, ISBN 0-387-00454-8.
Walley, P., Statistical Reasoning with Imprecise Probabilities, Chapman & Hall, N.Y., 1991.

Papers

The following references have been brought to our attention. Please inform us of others.

Jan B. Beck, Vladik Kreinovich, and Berlin Wu, "Interval-Valued and Fuzzy-Valued Random Variables: From Computing Sample Variances to Computing Sample Covariances", In: M. Lopez, M. A. Gil, P. Grzegorzewski, O. Hrynewicz, and J. Lawry (eds.), Soft Methodology and Random Information Systems, Springer-Verlag, 2004, pp. 85-92. Compressed PostScript file; pdf file.

D. Berleant, M.-P. Cheong*, C. Chu, Y. Guan, A. Kamal, G. Sheblé, S. Ferson, and J.F. Peters, “Dependable handling of uncertainty,” Reliable Computing 9(6) (2003), pp. 407-418.

D. Berleant, L. Xie, and J. Zhang, “Statool: a tool for Distribution Envelope Determination (DEnv), an interval-based algorithm for arithmetic on random variables,” Reliable Computing 9 (2) (2003), pp. 91-108.

D. Berleant and J. Zhang, “Using Pearson correlation to improve envelopes around the distributions of functions,” Reliable Computing, 10 (2) (2004), pp. 139-161. (Reviewed. Reliable Computing is the journal of the interval computing community.)

D. Berleant and J. Zhang, "Representation and Problem Solving with the Distribution Envelope Determination (DEnv) Method," Reliability Engineering and System Safety, in press as of 2/03.

D. Berleant and J. Zhang*, “Using Pearson correlation to improve envelopes around the distributions of functions,” Reliable Computing, 10 (2) (2004), pp. 139-161.

Carlos Carreras and Ian D. Walker. "Interval Methods for Improved Robot Reliability Estimation," In the Proceedings of the 2000 IEEE Reliability and Maintainability Symposium, Los Angeles, CA, January 2000, pp. 22-27.

Carlos Carreras and Ian D. Walker. "Interval Methods for Fault-Tree Analysis in Robotics," IEEE Transactions on Reliability, vol. 50, no. 1, March 2001, pp. 3-11.

Martine Ceberio, Vladik Kreinovich, and Lev Ginzburg, "Towards Joint Use of Probabilities and Intervals in Scientific Computing: What is the Best Transition from Linear to Quadratic Approximation?", Proceedings of the Workshop on State-of-the-Art in Scientific Computing PARA'04, Lyngby, Denmark, June 20-23, 2004, Vol. 1, pp. 43-49. Compressed Postcsript file; pdf file

Evgeny Dantsin, Vladik Kreinovich, Alexander Wolpert, and Gang Xiang, "Population Variance under Interval Uncertainty: A New Algorithm", Reliable Computing (to appear). File in pdf and Compressed PostScript

de Cooman, G., Zaffalon, M. (2003). Updating with incomplete observations. In: Kjærulff, U., Meek, C. (Eds), UAI-2003: Proceedings of the 19th Conference on Uncertainty in Artificial Intelligence. Morgan Kaufmann, San Francisco, pp. 142–150.

Vladimir G. Dmitriev, Nina A. Zheludeva, and Vladik Kreinovich. "Applications of interval analysis methods to estimate algorithm errors in measuring systems," Measurement, Control, Automatization, 1985, No.1(53), pp. 31-40 (in Russian).

Diane I. Doser, Kevin D. Crain, Mark R. Baker, Vladik Kreinovich, and Matthew C. Gerstenberger "Estimating uncertainties for geophysical tomography", Reliable Computing, 1998, Vol. 4, No. 3, pp. 241-268.

Didier Dubois, Helene Fargier, and Jerome Fortin, "The empirical variance of a set of fuzzy intervals", Proceedings of the 2005 IEEE International Conference on Fuzzy Systems FUZZ-IEEE'2005, Reno, Nevada, May 22-25, 2005, pp. 885-890.

Didier Dubois and Henri Prade, Random sets and fuzzy interval analysis, Fuzzy Sets and Systems, Volume 42, Issue 1, 5 July 1991, Pages 87-101

Fagiuoli, E., Zaffalon, M. (1998). 2U: an exact interval propagation algorithm for polytrees with binary variables. Artificial Intelligence 106(1), 77–107.

Fagiuoli, E., Zaffalon, M. (1998). A note about redundancy in influence diagrams. International Journal of Approximate Reasoning 19(3–4), 231–246.

Fagiuoli, E., Zaffalon, M. (2000). Tree-augmented naive credal classifiers. In: IPMU 2000: Proceedings of the 8th Information Processing and Management of Uncertainty in Knowledge-Based Systems Conference. Universidad Politécnica de Madrid, Spain, pp. 1320–1327.

Scott Ferson, Lev Ginzburg, and Resit Akcakaya, "Whereof One Cannot Speak: When Input Distributions Are Unknown", Applied Biomathematics Report. pdf file.

Scott Ferson, Lev Ginzburg, Vladik Kreinovich, and Monica Aviles, "Exact Bounds on Sample Variance of Interval Data", Extended Abstracts of the 2002 SIAM Workshop on Validated Computing, Toronto, Canada, May 23-25, 2002, pp. 67-69. Compressed PostScript file; pdf file; detailed version in Compressed PostScript and pdf

Scott Ferson, Lev Ginzburg, Vladik Kreinovich, Luc Longpre, and Monica Aviles, "Exact Bounds on Finite Populations of Interval Data", Reliable Computing, 2005, Vol. 11, No. 3, pp. 207-233. Compressed postscript file; pdf file

Scott Ferson, Lev Ginzburg, Vladik Kreinovich, and Jorge Lopez, "Absolute Bounds on the Mean of Sum, Product, etc.: A Probabilistic Extension of Interval Arithmetic", Extended Abstracts of the 2002 SIAM Workshop on Validated Computing, Toronto, Canada, May 23-25, 2002, pp. 70-72. Compressed PostScript file; pdf file; detailed version in Compressed PostScript and pdf

Scott Ferson, Lev Ginzburg, Vladik Kreinovich, Hung T. Nguyen, and Scott A. Starks, "Uncertainty in Risk Analysis: Towards a General Second-Order Approach Combining Interval, Probabilistic, and Fuzzy Techniques", Proceedings of FUZZ-IEEE'2002, Honolulu, Hawaii, May 12-17, 2002, Vol. 2, pp. 1342-1347. Compressed PostScript file; pdf file.

Scott Ferson, Lev Ginzburg, Vladik Kreinovich, and Harry Schulte, "Interval Computations as a Particular Case of a General Scheme Involving Classes of Probability Distributions", In: Juergen Wolff von Gudenberg and Walter Kraemer (eds.), Scientific Computing, Validated Numerics, Interval Methods, Kluwer, Dordrecht, 2001, pp. 355-366. Compressed PostScript file; pdf file.

Scott Ferson, Lev Ginzburg, Vladik Kreinovich, Luc Longpré, and Monica Aviles, "Computing Variance for Interval Data is NP-Hard", ACM SIGACT News, 2002, Vol. 33, No. 2, pp. 108-118. Compressed PostScript file; pdf file.

Scott Ferson, Vladik Kreinovich, Lev Ginzburg, David S. Myers, and Kari Sentz, "Constructing Probability Boxes and Dempster-Shafer Structures", Sandia National Laboratories, Report SAND2002-4015, January 2003. Abridged version (pdf file); Unabridged version (zipped pdf file).

Emili Garcia-Berthou and Carles Alcaraz, Incongruence between test statistics and P values in medical papers, BMC Medical Research Methodology 2004, 4:13, http://www.biomedcentral.com/content/pdf/1471-2288-4-13.pdf

Laurent Granvilliers, Vladik Kreinovich, and Norbert Mueller, "Novel Approaches to Numerical Software with Result Verification", In: Rene Alt, Andreas Frommer, R. Baker Kearfott, and Wolfram Luther (eds.), Numerical Software with Result Verification, International Dagstuhl Seminar, Dagstuhl Castle, Germany, January 19-24, 2003, Revised Papers, Springer Lectures Notes in Computer Science, 2004, Vol. 2991, pp. 274-305. File in Compressed PostScript" and inn File in pdf"

Hutter, M., Zaffalon, M. (accepted for publication). Distribution of mutual information from complete and incomplete data. Computational Statistics and Data Analysis. Zaffalon, M. (accepted for publication). Reliable classification for environmental problems. Environmental Modelling & Software.

Hutter, M., Zaffalon, M. (2003). Bayesian treatment of incomplete discrete data applied to mutual information and feature selection. In KI-2003: Proceedings of the 26th German Conference on Artificial Intelligence, Lecture Notes in Computer Science, Springer-Verlag.

Cliff Joslyn and Vladik Kreinovich, "Convergence Properties of an Interval Probabilistic Approach to System Reliability Estimation", International Journal of General Systems (to appear). (File in pdf). See also Cliff Joslyn and Vladik Kreinovich, "Convergence Properties of an Interval Probabilistic Approach to System Reliability Estimation", Los Alamos National Laboratory, Technical Report LA-UR-02-6261, Los Alamos, NM, 2002. Compressed PostScript file; pdf file.

W. Kahan. The Improbability of Probabilistic Error Analyses for Numerical Computations. U. California, Berkeley Statistics Colloquium presentation, revised 6/10/98. http://www.cs.berkeley.edu/~wkahan/improber.ps; http://www.cs.berkeley.edu/~wkahan/improber.pdf

Olga Kosheleva and Vladik Kreinovich, "Error estimation for indirect measurements: Interval computation problem is (slightly) harder than a similar probabilistic computational problem", Reliable Computing, 1999, Vol. 5, No. 1, pp. 81-95.

Olga M. Kosheleva and Martine Ceberio, "Processing Educational Data: From Traditional Statistical Techniques to an Appropriate Combination of Probabilistic, Interval, and Fuzzy Approaches", Proceedings of the International Conference on Fuzzy Systems, Neural Networks, and Genetic Algorithms FNG'05, Tijuana, Mexico, October 13-14, 2005, pp. 39-48. pdf file.

Vladik Kreinovich, Daniel J. Berleant, Scott Ferson, and Weldon A. Lodwick, "Combining Interval and Probabilistic Uncertainty: Foundations, Algorithms, Challenges -- An Overview” Proceedings of the International Conference on Fuzzy Systems, Neural Networks, and Genetic Algorithms FNG'05, Tijuana, Mexico, October 13-14, 2005, pp. 1-10. pdf file.

Vladik Kreinovich, "Maximum entropy and interval computations", Reliable Computing, 1996, Vol. 2, No. 1, pp. 63-79.

Vladik Kreinovich, "Random sets unify, explain, and aid known uncertainty methods in expert systems", in John Goutsias, Ronald P. S. Mahler, and Hung T. Nguyen (eds.), Random Sets: Theory and Applications, Springer-Verlag, N.Y., 1997, pp. 321-345. Compressed PostScript file; pdf file.

Vladik Kreinovich, "Probabilities, Intervals, What Next? Optimization Problems Related to Extension of Interval Computations to Situations with Partial Information about Probabilities", Journal of Global Optimization, 2004, Vol. 29, No. 3, pp. 265-280. Compressed PostScript file; pdf file.

V. Kreinovich, J. Beck, C. Ferregut, A. Sanchez, G. R. Keller, M. Averill, and S. A. Starks, "Monte-Carlo-type techniques for processing interval uncertainty, and their engineering applications", Proceedings of the Workshop on Reliable Engineering Computing, Savannah, Georgia, September 15-17, 2004, pp. 139-160. File in Compressed Postscript and in pdf

Vladik Kreinovich, Andrew Bernat, Elsa Villa and Yvonne Mariscal. "Parallel computers estimate errors caused by imprecise data," Interval Computations, 1991, No. 2, pp. 31-46.

Vladik Kreinovich and Scott Ferson, "Computing Best-Possible Bounds for the Distribution of a Sum of Several Variables is NP-Hard", International Journal of Approximate Reasoning (to appear) File in pdf and in Compressed PostScript

Vladik Kreinovich and Scott Ferson, "A New Cauchy-Based Black-Box Technique for Uncertainty in Risk Analysis", Reliability Engineering and Systems Safety, 2004, Vol. 85, No. 1-3, pp. 267-279. Compressed PostScript file; pdf file.

Vladik Kreinovich, Scott Ferson, and Lev Ginzburg, "Exact Upper Bound on the Mean of the Product of Many Random Variables With Known Expectations", Reliable Computing (to appear). Compressed PostScript file; pdf file.

Vladik Kreinovich, Scott Ferson, Lev Ginzburg, Harry Schulte, Matthew R. Barry, and Hung T. Nguyen, "From Interval Methods of Representing Uncertainty To A General Description of Uncertainty", In: Hrushikesha Mohanty and Chitta Baral (eds.), Trends in Information Technology, Proceeedings of the International Conference on Information Technology ICIT'99, Bhubaneswar, India, December 20-22, 1999, Tata McGraw-Hill, New Delhi, 2000, pp. 161-166. Compressed Postscript file.

Vladik Kreinovich and Luc Longpre, "Fast Quantum Algorithms for Handling Probabilistic and Interval Uncertainty'', Mathematical Logic Quarterly, 2004, Vol. 50, No. 4/5, pp. 507-518. http://www.cs.utep.edu/vladik/2003/tr03-22c.ps.gz (CompressedPostCript), http://www.cs.utep.edu/vladik/2003/tr03-22c.pdf (pdf file)

Vladik Kreinovich and Luc Longpré, "Guaranteed Predictions Based on Probabilistic Knowledge: Why and How", International Conference on Interval Methods and their Application in Global Optimization (INTERVAL'98), April 20-23, Nanjing, China, Abstracts, 1998, pp. 61-63.

Vladik Kreinovich, Luc Longpre, Praveen Patangay, Scott Ferson, and Lev Ginzburg, "Outlier Detection Under Interval Uncertainty: Algorithmic Solvability and Computational Complexity", Reliable Computing, 2005, Vol. 11, No. 1, pp. 59-76. File in Compressed PostScript and in pdf.

Earlier versions published as follows:

(1) Vladik Kreinovich, Luc Longpre, Praveen Patangay, Scott Ferson, and Lev Ginzburg, "Outlier Detection Under Interval Uncertainty: Algorithmic Solvability and Computational Complexity", In: Ivan Lirkov, Svetozar Margenov, Jerzy Wasniewski, and Plamen Yalamov (eds.), Large-Scale Scientific Computing, Proceedings of the 4-th International Conference LSSC'2003, Sozopol, Bulgaria, June 4-8, 2003, Springer Lecture Notes in Computer Science, 2004, Vol. 2907, pp. 238-245. Compressed PostScript file; pdf file.
(2) Vladik Kreinovich, Praveen Patangay, Luc Longpré, Scott A. Starks, Cynthia Campos, Scott Ferson, and Lev Ginzburg, "Outlier Detection Under Interval and Fuzzy Uncertainty: Algorithmic Solvability and Computational Complexity", submitted to 22nd International Conference of the North American Fuzzy Information Processing Society NAFIPS'2003, Chicago, Illinois, July 24-26, 2003. Compressed PostScript file; pdf file.

Vladik Kreinovich, Luc Longpre, Scott A. Starks, Gang Xiang, Jan Beck, Raj Kandathi, Asis Nayak, Scott Ferson, and Janos Hajagos, "Interval Versions of Statistical Techniques, with Applications to Environmental Analysis, Bioinformatics, and Privacy in Statistical Databases", Journal of Computational and Applied Mathematics (to appear). Compressed PostScript file; pdf file.

Vladik Kreinovich, Hung T. Nguyen, Scott Ferson, and Lev Ginzburg, "From Computation with Guaranteed Intervals to Computation with Confidence Intervals: A New Application of Fuzzy Techniques", Proceedings of the 21st International Conference of the North American Fuzzy Information Processing Society NAFIPS'2002, New Orleans, Louisiana, June 27-29, 2002, pp. 418-422. Compressed PostScript file; pdf file.

Vladik Kreinovich, Hung T. Nguyen, and Berlin Wu, "On-Line Algorithms for Computing Mean and Variance of Interval Data, and Their Use in Intelligent Systems", Information Sciences (to appear as of 5/04). http://www.cs.utep.edu/vladik/2003/tr03-24c.ps.gz (Compressed PostScript file), http://www.cs.utep.edu/vladik/2003/tr03-24c.pdf (pdf file)

Vladik Kreinovich, Praveen Patangay, Luc Longpré, Scott A. Starks, Cynthia Campos, Scott Ferson, and Lev Ginzburg, "Outlier Detection Under Interval and Fuzzy Uncertainty: Algorithmic Solvability and Computational Complexity", submitted to 22nd International Conference of the North American Fuzzy Information Processing Society NAFIPS'2003, Chicago, Illinois, July 24-26, 2003. Compressed PostScript file; pdf file.

Vladik Kreinovich, Eric J. Pauwels, Scott Ferson, and Lev Ginzburg, "A Feasible Algorithm for Locating Concave and Convex Zones of Interval Data and Its Use in Statistics-Based Clustering", Numerical Algorithms, 2004, Vol. 37, pp. 225-232. Compressed PostScript file; pdf file.

Vladik Kreinovich and Marina I. Pavlovich. "Error estimate of the result of indirect measurements by using a calculational experiment," Izmeritelnaya Tekhnika, 1985, No. 3, pp. 11-13 (in Russian), English translation: Measurement Techniques, 1985, Vol. 28, No. 3, pp. 201-205.

Vladik Kreinovich, Gennady N. Solopchenko, Scott Ferson, Lev Ginzburg, and Richard Alo, "Probabilities, intervals, what next? Extension of interval computations to situations with partial information about probabilities'', Proceedings of the 10th IMEKO TC7 International Symposium on Advances of Measurement Science, St. Petersburg, Russia, June 30-July 2, 2004, Vol. 1, pp. 137-142. Compressed PostScript file; pdf file

Vladik Kreinovich, Gang Xiang, and Scott Ferson, "Computing Mean and Variance Under Dempster-Shafer Uncertainty: Towards Faster Algorithms", International Journal of Approximate Reasoning (to appear). pdf file

Vladik Kreinovich, Gang Xiang, and Scott Ferson, "How the Concept of Information as Average Number of `Yes-No' Questions (Bits) Can Be Extended to Intervals, P-Boxes, and more General Uncertainty", Proceedings of the 24nd International Conference of the North American Fuzzy Information Processing Society NAFIPS'2005}, Ann Arbor, Michigan, June 22-25, 2005 (to appear). Compressed PostScript file; pdf file.

V. Kreinovich, G. Xiang, S. A. Starks, L. Longpre, M. Ceberio, R. Araiza, J. Beck, R. Kandathi, A. Nayak, R. Torres, and J. Hajagos, "Towards combining probabilistic and interval uncertainty in engineering calculations: algorithms for computing statistics under interval uncertainty, and their computational complexity", Reliable Computing (to appear). File in pdf and in compressed postscript

W.A. Lodwick, A. Neumaier and F. Newman, Optimization under uncertainity: methods and applications in radiation therapy, Proc. 10th IEEE Int. Conf. Fuzzy Systems, December 2-5, 2001, Melbourne, Australia, to appear.
abstract and download

Luc Longpré and Vladik Kreinovich, "Interval Computations Related to Privacy in Statistical Databases", working paper Compressed PostScript file; pdf file.

J. Lüthi, Histogram-Based Characterization of Workload Parameters and its Consequences on Model Analysis, chapter, 2005.

Charles F. Manski, Partial Identification of Probability Distributions, Springer-Verlag, New York, 2003, ISBN 0-387-00454-8.

A. S. Moore, Interval risk analysis of real estate investment: a non-Monte-Carlo approach, Freiburger Intervall-Berichte 85/3, Inst. F. Angew. Math., Universitaet Freiburg I. Br., 23-49 (1985).

R. E. Moore, Risk analysis without Monte Carlo methods, Freiburger Intervall-Berichte 84/1, Inst. F. Angew. Math., Universitaet Freiburg I. Br., 1-48 (1984).

A. Neumaier, Clouds, Fuzzy Sets and Probability Intervals, submitted (2003)
abstract and download

A. Neumaier, Fuzzy modeling in terms of surprise, Fuzzy Sets and Systems 135 (2003), 21-38.
abstract and download

A. Neumaier, On the structure of clouds, submitted (2003)
abstract and download

Hung T. Nguyen, Vladik Kreinovich, Vladimir I. Gorodetski, Vyacheslav M. Nesterov, and Alexandre L. Touloupiev, "Applications of Interval-Valued Degrees of Belief: A Survey", In: Alexandre Touloupiev (ed.), Information Technologies and Intellectual Methods, Vol. 3 (IT&IM'3), St. Petersburg Institute for Information and Automation of Russian Academy of Sciences (SPIIRAS), 1999, pp. 6-61 (in Russian).

Hung T. Nguyen, Vladik Kreinovich, and Luc Longpré, "Dirty Pages of Logarithm Tables, Lifetime of the Universe, and Subjective (Fuzzy) Probabilities on Finite and Infinite Intervals", Proceedings of the IEEE International Conference on Fuzzy Systems FUZZ-IEEE'2003, St. Louis, Missouri, May 25-28, 2003 (to appear). Compressed PostScript file; pdf file.

Hung T. Nguyen, Vladik Kreinovich, and Luc Longpré, "Dirty Pages of Logarithm Tables, Lifetime of the Universe, and (Subjective) Probabilities on Finite and Infinite Intervals", Reliable Computing, 2004, Vol. 10, No. 2, pp. 83-106. Compressed PostScript file; pdf file.

Hung T. Nguyen, Vladik Kreinovich, and Gang Xiang, "Foundations of Statistical Processing of Set-Valued Data: Towards Efficient Algorithms", Proceedings of the Fifth International Conference on Intelligent Technologies InTech'04, Houston, Texas, December 2-4, 2004 (to appear). http://www.cs.utep.edu/vladik/2004/tr04-35.ps.gz (Compressed PostScript); http://www.cs.utep.edu/vladik/2004/tr04-35.pdf (pdf file).

Hung T. Nguyen, Vladik Kreinovich, and Qiang Zuo, "Interval-valued degrees of belief: applications of interval computations to expert systems and intelligent control", International Journal of Uncertainty, Fuzziness, and Knowledge-Based Systems (IJUFKS), 1997, Vol. 5, No. 3, pp. 317-358. G.

Hung T. Nguyen, Tonghui Wang, and Vladik Kreinovich, "Towards Foundations of Processing Imprecise Data: From Traditional Statistical Techniques of Processing Crisp Data to Statistical Processing of Fuzzy Data", In: Yingming Liu, Guoqing Chen, Mingsheng Ying and Kai-Yuan Cai (eds.), Proceedings of the International Conference on Fuzzy Information Processing: Theories and Applications FIP'2003, Beijing, China, March 1-4, 2003, Vol. II, pp. 895-900. Compressed PostScript file; pdf file.

H. Regan, S. Ferson and D. Berleant, “Equivalence of methods for uncertainty propagation of real-valued random variables,” International Journal of Approximate Reasoning, in press

S. A. Starks, V. Kreinovich, L. Longpre, M. Ceberio, G. Xiang, R. Araiza, J. Beck, R. Kandathi, A. Nayak, and R. Torres, "Towards combining probabilistic and interval uncertainty in engineering calculations", Proceedings of the Workshop on Reliable Engineering Computing, Savannah, Georgia, September 15-17, 2004, pp. 193-213. http://www.cs.utep.edu/vladik/2004/tr04-20.ps.gz (Compressed Postscript file); http://www.cs.utep.edu/vladik/2004/tr04-20.pdf (pdf file)

Raul Trejo and Vladik Kreinovich, "Error Estimations for Indirect Measurements: Randomized vs. Deterministic Algorithms For `Black-Box' Programs", Sanguthevar Rajasekaran, Panos Pardalos, John Reif, and Jose Rolim (eds.), Handbook on Randomized Computing, Kluwer, 2001, pp. 673-729. Compressed PostScript file; pdf file.

W. Troy Tucker and Scott Ferson, "Probability Bounds Analysis in Environmental Risk Assessments", Applied Biomathematics Report. pdf file.

William Walster and Vladik Kreinovich, "For unknown-but-bounded errors, interval estimates are often better than averaging", ACM SIGNUM Newsletter, 1996, Vol. 31, No. 2, pp. 6-19.

Berlin Wu, Hung T. Nguyen, and Vladik Kreinovich, "Real-Time Algorithms for Statistical Analysis of Interval Data", Proceedings of the International Conference on Information Technology InTech'03}, Chiang Mai, Thailand, December 17-19, 2003, pp. 483-490. Compressed PostScript file; pdf file

G. Xiang, Fast algorithm for computing the upper endpoint of sample variance for interval data: case of sufficiently accurate measurements, Reliable Computing (to appear). pdf file

Gang Xiang, Scott A. Starks, Vladik Kreinovich, and Luc Longpre, "New Algorithms for Statistical Analysis of Interval Data", Proceedings of the Workshop on State-of-the-Art in Scientific Computing PARA'04, Lyngby, Denmark, June 20-23, 2004, Vol. 1, pp. 123-129. Compressed PostScript file; pdf file.

Ronald R. Yager and Vladik Kreinovich, "Decision Making Under Interval Probabilities", International Journal of Approximate Reasoning, 1999, Vol. 22, No. 3, pp. 195-215. Compressed PostScript file; pdf file.

J. Zhang and D. Berleant, Envelopes around cumulative distribution functions from interval parameters of standard continuous distributions, Proceedings, North American Fuzzy Information Processing Society (NAFIPS 2003), Chicago, pp. 407-412.

Zaffalon, M. (2002). Exact credal treatment of missing data. Journal of Statistical Planning and Inference 105(1), 105–122.

Zaffalon, M. (2002). The naive credal classifier. Journal of Statistical Planning and Inference 105(1), 5–21.

Zaffalon, M., Fagiuoli, E. (2003). Tree-based credal networks for classification. Reliable computing 9(6), 487–509.

Zaffalon, M., Hutter, M. (2002). Robust feature selection by mutual information distributions. In: Darwiche, A., Friedman, N. (Eds), UAI-2002: Proceedings of the 18th Conference on Uncertainty in Artificial Intelligence. Morgan Kaufmann, San Francisco, pp. 577–584.

Zaffalon, M., Wesnes, K., Petrini, O. (2001). Credal classification for dementia screening. In: Quaglini, S., Barahona P., Andreassen, S. (Eds), AIME '01: Proceedings of the Eighth European Conference on Artificial Intelligence in Medicine, Lecture Notes in Computer Science, Springer-Verlag, pp. 67–76.

Zaffalon, M., Wesnes, K., Petrini, O. (2003). Reliable diagnoses of dementia by the naive credal classifier inferred from incomplete cognitive data. Artificial Intelligence in Medicine 29(1–2), 61–79.