Beyond Point Estimates: Risk Assessment Using Interval and Possibilistic Arithmetic

A tutorial workshop
to be held in conjunction with the

Society for Risk Analysis Annual Meeting

1 pm - 5 pm
Sunday, 3 December 2006

Renaissance Innerhabor Hotel
Baltimore, Maryland

This tutorial introduces intervals and possibility distributions and reviews their uses in risk analysis.

Synopsis
Topics descriptions
Presenter
Registration
Venue
More information
Related links

Synopsis

This half-day workshop will introduce the use of interval analysis and possibility theory (fuzzy arithmetic) for propagating uncertainty through calculations in a quantitative risk assessment. These methods can be used even when data are far too sparse for conventional Monte Carlo methods. Interval analysis underlies any reasonable conception of worst case analysis, and, although it is less powerful than other methods when empirical information is abundant, it can be used for analyzing uncertainty of all kinds no matter what its nature or source. We use it as the exemplar calculus for computing with uncertain quantities, and demonstrate many of the commonalities that unite interval analysis and fuzzy arithmetic. Although the basics for uncertainty arithmetic are simple enough for anyone to master, some important details can be very subtle. Inattention to such details is common in risk analysis and it occasionally leads to seriously erroneous conclusions. The methods will be applied to risk assessment problems as examples, including event-tree/fault-tree safety analysis. A workshop booklet containing illustrations used in the presentations will be provided to participants.

Topic descriptions

Introduction to interval and fuzzy arithmetic

The basic arithmetic of intervals for uncertainty propagation is very simple. All of the rules can be derived from answering the question 'what is the largest (smallest) possible answer that could be obtained under this operation?' By generalizing these ideas slightly, an arithmetic for fuzzy numbers can also be supported. Topics include intervals, plus-or-minus ranges, triangular fuzzy numbers, trapezoidal fuzzy numbers, addition, subtraction, multiplication, division and exponentiation.

Worst case analysis requires uncertainty arithmetic

Although worst case analysis has been much criticized, it is clearly useful as a preliminary screening procedure in the regulation of environmental risks. Interval arithmetic provides the necessary foundation for estimating the upper (and lower) bounds required by worst case analysis. The common practice of estimating the 'upper bound' for dose (or any other quantity) by simply combining the upper bounds of all the variables in a deterministic expression for dose does not actually yield the upper bound. Depending on the details of the mathematical expression actually employed, the answer obtained by this method can be off by orders of magnitude. Topics include worst case analyses, theoretical upper bounding estimates (TUBEs), and maximally exposed individual (MEI) estimates.

Example: ground water contamination. A simple published model of well contamination from an octanol spill through ground water flow illustrates a variety of errors great and small that can be made in a risk analysis based on a worst-case argument that is attempted without interval analysis. An approach using fuzzy arithmetic yields results that simultaneously characterize the worst case as well as the best estimate.

The algebra you already know can hurt you

Although often ignored, some computational details can strongly influence the final answer obtained from a risk analysis. The consequences of neglecting them can be quantified in many cases. The algebra of uncertain numbers (whether represented as MEI estimates, TUBEs, intervals, fuzzy numbers or probability distributions) is different from that of real numbers with which most people are familiar. Manipulating expressions using rearrangements that are invalid will lead to incorrect results. However, the algebra for intervals and fuzzy numbers is well known and you can figure out which manipulations are legal by remembering two simple rules about the number of occurrences of variables on either side of the equal sign. Topics include interval-valued and fuzzy-valued functions of intervals (exp, log, etc.), decreasing functions, non-monotonic functions, logical comparisons, multiple occurrence of variables, subdistributivity, subcancellation and enclosures.

Probability theory and interval/fuzzy arithmetic can be combined

In many practical situations, one would like to exploit the conservativism of interval and fuzzy numbers in a problem which is at least partly probabilistic. How can we combine these two traditions of propagating uncertainty?

Example: event-tree safety assessment. The combination of probabilistic and fuzzy methods is illustrated with a model of a safety failure in which both subjective uncertainty and stochasticity are treated simultaneously but in a way that does not confound the two kinds of uncertainty.

References

Alefeld, G. and J. Herzberger 1983 Introduction to Interval Computations. Academic Press, New York.
Cooper, J.A., S. Ferson and L.R. Ginzburg 1996 Hybrid processing of stochastic and subjective uncertainty data. Risk Analysis 16: 785-791.
Dubois, D. and H. Prade 1988 Possibility Theory: An Approach to Computerized Processing of Uncertainty. Plenum Press, New York.
Kaufmann, A. and M.M. Gupta 1985 .Introduction to Fuzzy Arithmetic: Theory and Applications. Van Nostrand Reinhold, New York.
Kulisch, U.W. and W.L. Miranker 1981 Computer Arithmetic in Theory and Practice. Academic Press, New York.
Moore, R.E. 1966 Interval Analysis. Prentice-Hall, Englewood Cliffs, New Jersey.
Neumaier, A. 1990 Interval Methods for Systems of Equations. Cambridge University Press, Cambridge.
Taylor, J.R. 1982 An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Science Books, Mill Valley, California.
Zadeh, L. 1978 Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1: 3-28.

Presenter

J. Arlin Cooper is Distinguished Member of the Airworthiness Assurance Department at Sandia National Laboratories. Cooper works on methodology development and algorithm analysis with particular emphasis on unqiue signal pattern and communication design issues and other nuclear detonation safety assessment strategies. Click here for a description of some of the risk applications of his work in fuzzy arithmetic.

Registration

The registration fee is $175 before 10 November, or $200 on site. You do not need to register for the Annual Meeting to attend the workshop. Registration will be handled by

Secretariat sra@burkinc.com
Society for Risk Analysis www.sra.org
1313 Dolley Madison Boulevard, Suite 402
McLean, Virginia 22101 USA
1-703-790-1745, fax 1-703-790-2672

Venue

The event will be held from noon to 5:00 p.m. on Sunday, 3 December 2006, at

Renaissance Innerharbor Hotel
202 East Pratt Street
Baltimore, Maryland 21202 USA
1-800-535-1201 (toll-free reservations)
1-410-547-1200 (direct to the hotel)
1-410-539-5780 (fax)
http://marriott.com/property/propertypage/bwish?groupCode=srasraa&app=resvlink

Reserve a room at the hotel before 3 November 2006 to obtain the SRA rate of $145 per night (single or double occupancy) plus 12.5% tax. Be sure to mention the Society for Risk Analysis to receive the SRA group rate. This rate is available for stays between 1-9 December 2006, subject to availability. Remember the cut off for this rate is 3 November 2006, or until the SRA room block is sold out. Reserve your room early. Cancellations must be made at least 48 hours in advance. See a description of the hotel at http://marriott.com/property/propertypage/bwish?groupCode=srasraa&app=resvlink. The meeting room for the workshop has not yet been determined; check with the hotel concierge.

More information

More information can be obtained from Arlin Cooper acooper@sandia.gov, telephone 1-505-845-9168.