Presentation on theme: "Fuzzy arithmetic in risk analysis"— Presentation transcript:
1 Fuzzy arithmetic in risk analysis Scott FersonApplied Biomathematics
2 Fuzzy numbers Fuzzy set that’s unimodal and reaches 1 Nested stack of intervals
3 Fuzzy addition1ABA+B0.52468Subtraction, multiplication, division, minimum, maximum, exponentiation, logarithms, etc. are also defined.If distributions are multimodal, possibility theory (rather than just simple fuzzy arithmetic) is required.
4 Kinds of numbersScalars are well known or mathematically defined integers and real numbersIntervals are numbers whose values are not know with certainty but about which bounds can be establishedFuzzy numbers are uncertain numbers for which, in addition to knowing a range of possible values, one can say that some values are more plausible, or ‘more possible’ than others
5 What is possibility? No single definition Depends on your applications Many definitions could be usedSubjective assessmentsSocial consensusMeasurement errorUpper betting rates (Giles)Extra-observational ranges (Gaines)
6 How to get fuzzy inputs Subjective assignments Objective consensus Make them up from highest, lowest and best-guess estimatesObjective consensusStack up consistent interval estimates or bridge inconsistent onesMeasurement errorInfer from measurement protocolsOther special ways
10 When the data are inconsistent Find and emphasize regions of consonanceLet possibility flow to intersectionsDoesn’t work for totally disjoint data setsMay have counterintuitive featuresUse (agglomerative hierarchical) clusteringSingle linkage, complete linkage, UPGMA, etc.Can define ‘similarity’ between intervals in various waysEven works for totally disjoint data sets
12 Betting definitionBy asserting a A, you agree to pay $1 if A is false.If the probability of A is P, then a Bayesian rational agent should agree to assert A for a fee of $(1-P), and should equally well assert not-A for a fee of $P. Although refusing to bet is not irrational, Bayesians don’t allow this.Possibility of A can be measured as the smallest number [0,1], such that, for $, a rational agent will agree to pay $1 if A is found to be false.Possibility is thereby an upper bound on probability.
13 Extra-observational ranges Theoretical ranges are often very wideThe range between the minimum and maximum observed values (where the data is) should be modeled by probability theoryFuzzy/possibility is about the range within the theoretical range but beyond observationstheoreticalminimummaximum1Possibilityminimumobservedmaximumobserved
14 Robustness X de X fe h + g 1 2 3 4 5 6 7 1 1234567XPossibilityTriangular fuzzy numbers are robust characterizations-20-10102030401Xde/(h+g)fePossibilitydeh + gX fed = [0.3, 1.7, 3]e = [ 0.4, 1, 1.5]f = [ 0.8, 6, 10]g = [ 0.2, 2, 5]h = [ 0.6, 3, 6]
15 Distributional results Tails describe possible extremesMore comprehensive than intervalsFull distribution of various magnitudes
16 Comparison Probability theory Possibility theory Axioms 0 P() 1 P(AB) = P(A) + P(B)whenever AB=ConvolutionC(z) = A(x) B(y)Possibility theoryAxioms() = 0() = 1(A) (B)whenever ABConvolutionC(z) = V A(x) B(y)vz=x+yz=x+y
18 Result of convolution12345AB8A+B6If the inputs are fuzzy numbers (unimodal, reach 1), then possibilistic convolution is the same as level-wise interval arithmetic (Kaufmann and Gupta)
19 Probability Possibility X Y X+X Y+Y X+…+X Y+…+Y 1 1 0.5 0.5 2 4 6 8 10 24681024681011X+XY+YProbabilityPossibility0.50.52468102468100.41X+…+X0.20.5Y+…+Y246810246810
20 Computational cost Analysis Operations Deterministic F Interval analysis 4FFuzzy arithmetic MFMonte Carlo NFSecond-order Monte Carlo N2Fwhere M ~ [40,400], and N ~ [1000, ]
21 Data needs Worst case Interval analysis Fuzzy arithmetic Monte Carlo extreme valuesrangesranges or distributionsdistributions and dependencies
22 BackcalculationsDeconvolutions in fuzzy arithmetic are completely straightforward level-wise generalizations of interval deconvolutionsEasy, fastWhen impossible, yields no answer
23 Software FuziCalc Fuzzy Arithmetic C++ Library Cosmet (Phaser) (Windows 3.1) FuziWare,Fuzzy Arithmetic C++ Library(C code) anonymous ftp to mathct.dipmat.unict.it and get \fuzzy\fznum*.*Cosmet (Phaser)(DOS, soon for Windows)Risk Calc(Windows) ;
25 Another exampleConsider a simple example model of octanol contamination of groundwater due to Lobascio (1993 Uncertainty analysis tools for environmental modeling. ENVIRONews 1:6-10). Its assumptions include one-dimensional constant uniform Darcian flow, homogeneous material properties, linear retardation, no dispersion, and the governing equation T = (n + BD foc Koc ) L / (K i).Distance from source to receptor L = [ , , ] mHydraulic gradient i = [0.0003, , ] m m-1Hydraulic conductivity K = [ 300, , ] m yr-1Effective soil porosity n = [ , , ]Soil bulk density BD = [ 1500, , ] kg m-3Fraction of organic carbon in soil foc = [0.0001, , ]Octanol-water partion coefficient Koc = [ , , ] m3 kg-1
27 Reasons to use fuzzy arithmetic Requires little dataApplicable to all kinds of uncertaintyFully comprehensiveFast and easy to computeDoesn’t require information about correlationsConservative, but not hyperconservativeIn between worst case and probabilityBackcalculations easy to solve
28 Reasons not to use it Controversial Are alpha levels comparable for different variables?Not optimal when there're a lot of dataCan’t use knowledge of correlations to tighten answersNot conservative against all possible dependenciesRepeated variables make calculations cumbersome
29 ReferencesDubois, 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.Zadeh, L Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1: 3-28.
30 ApplicationsBardossy, A., I. Bogardi and L. Duckstein 1991 Fuzzy set and probabilistic techniques for health-risk analysis. Applied Mathematics and Computation 45:Duckstein, L., A. Bardossy, T. Barry and I. Bogardi 1990 Health risk assessment under uncertainty: a fuzzy risk methodology. Risk-based Decision Making in Water Resources. Y.Y. Haimes and E.Z. Stakhiv (eds.), American Society of Engineers, New York.Ferson, S Using fuzzy arithmetic in Monte Carlo simulation of fishery populations. Management Strategies for Exploited Fish Populations, T.J. Quinn II (ed.), Alaska Sea Grant College Program, AK-SG-93-02, ppMillstein, J.A Propagation of measurement errors in pesticide application computations. International Journal of Pest Management 40: Simulating extremes in pesticide misapplication from backpack sprayers. 41: